CHAPMAN & HALL/CRC Research N()tes ill MclthelTIcltic" 398


huangyi Liu
Songmu Zheng


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Sellligroups
associated with
dissipative systeDls

CHAPMAN & HALUCRC
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Zhuangyi Liu
University of Minnesota, Duluth
Songmu Zheng
Fudan University, Shanghai
Semigroups
associated with
dissipative systems
CHAPMAN & HALUCRC
Boca Raton London New York Washington, D.C.

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Contents
Preface ...........................................................................
Chapter 1 Preliminaries ................................"........................ 1
1.1 Some Definitions ...................................................... 1
1.2 Co-semigroup Generated by Dissipative Operator ....................... 2
1.3 Exponential Stability and Analyticity .................................. 4
1.4 The Sobolev Spaces and Elliptic Boundary Value Problems ............. 8
1.4.1 Sobolev Spaces Wm,P(O) ........................................8
1.4.2 The Gagliardo-Nirenberg and Poincae Inequalities ............. 10
1.4.3 Abstract Functions Valued in Banach Spaces ...................12
1.4.4 Linear Elliptic Boundary Value Problems ...................... 12
1.4.5 Interpolation Spaces ...........................................15
1. 5 Notation ............................................................. 1 7
Chapter 2 Linear Thermoelastic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 The Setting of Problems for the One-Dimensional Thermoelastic
System .............................................................. 18
2.2 The Exponential Stability for the Dirichlet Boundary Conditions at Both
Ends ................................................................ 23
2.3 The Exponential Stability for the Stress-Free Boundary Conditions at Both
En ds ................................................................ 28
2.4 The Exponential Stability for the Stress-Free Boundary Conditions at One
En d ................................................................. 37

2.5 The Thermoelastic Kirchhoff Plate Equations ......................... 43
Chapter 3 Linear Viscoelastic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Linear Viscoelastic System ............................................61
3.2 Wave Equation with Locally Distributed Damping .................... 72
3.3 Linear Viscoelastic System with Memory .............................. 77
3.4 The Linear Viscoelastic Kirchhoff Plate with Memory ................. 87
Chapter 4 Linear Thermoviscoelastic Systems ................................. 91
4.1 Linear One-Dimensional Thermoviscoelastic System ................... 91
4.2 Linear Three-Dimensional Thermoviscoelastic System with Memory ... 96
Chapter 5 Elastic Systems with Shear Damping .............................. 112
5.1 Shear Diffusion Equations ........................................... 112
5.2 Laminated Beam wi th Shear Dam ping ............................... 119
Chapter 6 Linear Elastic Systems with Boundary Damping ................... 129
6.1 Second-Order Hyperbolic Equation .................................. 129
6.2 Euler-Bernoulli Beam Equation ..................................... 138
Chapter 7 Uniformly Stable Approximations ................................. 152
7.1 Main Theorem ...................................................... 155
7.2 Approximations of the Thermoelastic System ........................ 163
7.3 Approximation of the Viscoelastic System ............................ 177
Bib 1 i 0 gr a ph y .................................................................. 191
Index ......................................................................... 205

Preface
This book is concerned with exponential stability and analyticity of Co-semigroups as-
sociated with various dissipative systems arising from mechanics. Most of the material
in this book is based on research carried out by the authors and their collaborators in
recent years. It also includes some very recent work by the authors.
One of important motivations of studying exponential stability comes from the
control theory. There are various kinds of damping in a mechanical system, such as
heat conduction, viscosity, friction, etc. It turns out that the corresponding mechanical
system is dissipative. Certainly, there are many good reasons in practice to consider the
corresponding control problem. For instance, controlling the vibration of an antenna
of a satellite moving in outer space is a good example in that respect. There is heat
condution, due to sunlight, in the antenna and the problem can be reduced to a control
problem for a linear one-dimensional thermoelastic system for a rod of length I. If the
exponential stability of the Co-semigroup associated with a dissipative mechanical
system (e.g., the linear one-dimensional thermoelastic system subject to appropriate
boundary and initial conditions) can be shown, then by the standard approach in the
control theory, it implies that the corresponding system with control is stabilizable.
It turns out that the optimal control of a so-called linear quadratic regulator problem
has a feedback form which can be achieved by solving an operator algebraic Riccati
equation. (We refer to Gibson [1] and Gibson, Rao & Tao [1], or Chapter 7 of this
book for more details.)
The study of exponential stability and analyticity is also very important in the
theory of partial differential equations. The property of exponential stability, or, in
other words, exponential decay of a solution to a linear system of partial differen-
tial equations will yield, through a quite standard approach, the global existence and
uniqueness of the corresponding nonlinear system of partial differential equations with

small initial data. (We refer to Zheng [1] and the references cited there in this aspect.)
On the other hand, a dissipative mechanical system can often be described by a cou-
pled system of partial differential equations, part of which is a parabolic equation or
parabolic system. It is well-known that solutions to initial boundary value problems
with a bounded domain, for instance for the heat equation or a large class of linear
parabolic equations, decay exponentially to zero (equilibrium) and are analytic for
t > o. Clearly, it is desirable to know whether or not these two properties will still
preserve for a given dissipative mechanical system. Knowledge of that will certainly
deepen understanding about interaction between "parabolic equations" and "hyper-
bolic equations". The reader of this book will find that the exponential stability is
still preserved for all mechanical systems considered in this book. However, it is also
pointed out in this book that this property is no longer true for linear three-dimensional
thermoelastic systems unless some assumptions on domain and initial data are made.
On the other hand, the reader will find that analyticity is a more sensitive property
and it is not preserved even for some systems considered in this book.
Concerning the method for proving exponential stability and analyticity, what is
presented in this book is a systematic approach developed by the authors and their
collaborators in past years. The spirit of this method for proving exponential stability
is to combine a theorem (see Theorem 1.3.2 in this book) by Gearhart [1] (see also
Wyler [1]) and Huang [1] in the semi group theory with PDE techniques. Combination
of an analogous theorem (see Theorem 1.3.3 in this book) with PDE techniques is the
systematic approach presented in this book for proving analyticity. Certainly, these
approaches are very different from some other methods in the literature, such as the
traditional energy method, the method based on the Datko theorem, and the method
of directly estimating the spectrum. It is our hope that the reader will find that the
methods presented in this book are powerful and simple. We would like to emphasize
that the references cited in this book are not intended to be exhaustive.
In what follows, we briefly describe the main content of this book:

In order that this book be self-contained, in Chapter 1 (mainly a reference chap-
ter), we collect and present some concepts and results in semi group theory, some
results about Sobolev spaces, linear elliptic boundary value problems, and interpola-
,
tion spaces. In particular, Theorem 1.3.2, Theorem 1.3.3, and Theorem 1.2.4 will be
frequently used throughout this book.
In Chapter 2, we are concerned with linear thermoelastic systems, namely, the
linear one-dimensional thermoelastic system and the linear thermoelastic Kirchhoff
plate equations. Results on the exponential stability of Co-semigroups associated
with initial boundary value problems with various boundary conditions for linear one-
dimensional thermoelastic system are presented. It is pointed out that analyticity is
not valid for these semigroups. In the final section of this chapter we consider the
initial boundary value problems for the linear thermoelastic Kirchhoff plate equations.
Results on exponential stability and analyticity or non-analyticity are also presented.
Chapter 3 is devoted to the study of linear viscoelastic systems. In the first section
of this chapter, it is proved that the semi group associated with the initial boundary
value problem for the linear one-dimensional elastic system with viscous damping of
rate type is not only exponentially stable, but is also analytic. In the same section,
an extended model is considered and results on exponential stability and analyticity
are also presented. In the second section, the initial boundary value problem for wave
equation with locally distributed damping is considered. We give a new proof for
the exponential stability of the associated semigroup using the systematic approach
presented in this book. In the third section, a linear viscoelastic system with memory is
investigated in an abstract framework. The exponential stability of the corresponding
semigroup is established. It is also pointed out that analyticity is not valid for such
kind of systems. As an application of the general results in the third section, the final
section of this chapter is devoted to the linear viscoelastic Kirchhoff plate equation
with memory.
Chapter 4 is concerned with linear thermoviscoelastic systems. In other words,

both heat conduction and viscosity coexist in a mechanical system. In the first sec-
tion, the linear one-dimensional thermoviscoelastic system is investigated. It is proved
that if both heat conduction and viscosity are of rate type, then the corresponding
semigroup is not only exponentially stable, but is also analytic. We further consider
the linear three-dimensional thermoviscoelastic system with memory, a model pro-
posed by Navarro [1]-[2]. In contrast to the linear three-dimensional thermoelastic
system, the exponential stability can still be established for this thermoviscoelastic
system.
Chapter 5 is devoted to the study of elastic systems with shear damping, a no-
tion first proposed by D. Russell. Section 5.1 is concerned with the shear diffusion
equations proposed by D. Russell. The exponential stability and analyticity of the
corresponding semigroup is proved by our systematic method. Then, in Section 5.2,
we further consider a model describing laminated beam with shear damping. Again
the exponential stability and analyticity is established.
In Chapter 6, we consider a different kind of damping, i.e., boundary damping due
to friction. In the first section of this chapter, we consider a linear second order hyper-
bolic equation in a bounded domain in IR m with boundary damping. The exponential
stability result was first established by Wyler [1]. But we now use our systematic
method to reprove it. Section 6.2 is devoted to the study of the Euler-Bernoulli beam
equation with boundary damping, which can be considered as a feedback control on
the boundary. The exponential stability is established.
In Chapter 7, the final chapter of this book, we are concerned mainly with uni-
formly stable approximations. In the first section of this chapter, we establish a gen-
eral theorem on the uniformly exponential stability for a sequence of Co-semigroups of
contractions on a sequence of Hilbert spaces. Then, in the next two sections as applica-
tions of this general result, some numerical approximations for the linear thermoelastic
system and viscoelastic system are considered. It is proved that for these particular
numerical schemes, the corresponding semigroups are uniformly exponentially stable.

We would like to take this opprotunity to express our special thanks to Professor
H. Amann for his interest in our research and for acting as the initiator for publication
of this book.
The author Z. Liu would like to record his appreciation of his other collobarators,
including Kangsheng Liu, R. Miller, M. Renardy, S. Trogdon, and Jiongmin Yong for
their fruitful cooperation and stimulating discussions. He also thanks his Ph.D. thesis
advisor, Professor J. A. Burns, at Virginia Polytechnic Institute and State University,
and Professor H. Stech, Head of the Department of Mathematics and Statistics at The
University of Minnesota at Duluth, for their constant support and encouragement.
The author S. Zheng would like to acknowledge the NSF of China and the Ministry
of Education of China for their continuous support. Currently, he is being supported
by the grant No. 19331040 from the NSF of China and the grant No. 96024603 from
the Ministry of Education of China.
Duluth and Shanghai
Professor Z. Liu
Department of Mathematics and Statistics
University of Minnesota at Duluth
Duluth, MN 55812
USA
Email: zliu@d.umn.edu
Professor Songmu Zheng
Institute of Mathematics
Fudan University
Shanghai 200433
P. R. China
Email: szheng@srcap.stc.sh.cn

Chapter 1
Preliminaries
In this chapter we will present some definitions, some results on Co-semigroups, in-
cluding some theorems on exponential stability and analyticity. We will also collect
some results on Sobolev's spaces and elliptic boundary value problems for the sake of
the reader. Some theorems are proved and others are stated without proofs, but the
relevant references are given. The reader may skip this chapter in the first reading,
then return to it for the references of related results.
1.1 Some Definitions
Definition 1.1.1 Let H be a real or complex Hilbert space equipped with the inner
product ( , ) and the induced norm II . II. Let A be a densely defined linear operator on
H, i.e, A: D(A) C H  H. We say that A is dissipative if for any x E D(A),
Re (Ax, x) < o.
(1.1.1)
Definition 1.1.2 A family S(t) (0 < t < 00) of bounded linear operators in a Banach
space H is called a strongly continuous semigroup (in short, a Co-semigroup) if
(i) S(t 1 + t 2 ) = S(t 1 )S(t 2 ), VtI, t 2 > 0,
(ii) S(O) = I,
(iii) For each x E H, S( t)x is continuous in t on [0,00).
For such a semigroup S(t), we define an operator A with domain D(A) consisting
of points x such that the limit
Ax = lim S(h): - x , x E D(A) (1.1.2)
h-O
exists. Then A is called the infinitesimal generator of the semigroup S(t). Given an
operator A, if A coincides with the infinitesimal generator of S(t), then we say that
it generates a strongly continuous semigroup S(t), t > O. Sometimes we also denote
S(t) by eAt.
1

Definition 1.1.3 eAt is said to be exponentially stable if there exist positive constants
a and M > 1 such that
lIeAtll < Me-Oft, V t > o.
(1.1.3)
Hereafter, we also use the notation II . II to denote the norm in £, ( H, H), assuming
that no confusion will occur.
Definition 1.1.4 eAt is said to be analytic if eAt admits an extension T(A) for A E
D,.s = {A E C IlargAI < 8} for some 8 > 0 such that A  T(A) is analytic and
{ lim IIT(A)z - zil = 0,
83>'-+O
T(A + p.) = T(A)T(p.),
V z E H,
(1.1.4)
VA, p. E D,.s,
or equivalently (see Theorem 5.2 in the book Pazy [lJ, p.61-62) there exists a constant
K > 0 such that
IIAeAtl1 < Kt- 1 , V t > O.
(1.1.5)
Let p(A) and u(A) be the resolvent set and the spectrum of A, respectively. We
denote
6.
uo(A) = sup{Re A I A E u(A)}
(1.1.6)
and
(A) 6. 1 . In II eAt II
Wo - 1m .
t-+O t
(1.1.7)
We say that the Co-semigroup eAt has the spectrum determining growth property if
uo(A) = wo(A).
This book is concerned mainly with exponential stability and analyticity of C o -
semigroups of contractions in a Hilbert space generated by dissipative operators arising
from mechanics. In the following two sections we will collect some related results
concerning generation, exponential stability, and analyticity of Co-semigroups.
1.2
Co-sernigroup
Operator
Generated
by
Dissipative
Suppose that the linear operator A generates a Co-semigroup eAt on a Hilbert space
H. Then we have (see pazy [1]):
2

Theorem 1.2.1 (Hille-Yosida) A linear (unbounded) operator A is the infinitesimal
generator of a Co-s emigr oup of contractions S(t), t > 0, if and only if
(i) A is closed and D(A) = H,
(ii) the resolvent set p(A) of A contains IR+ and for every A > 0,
II (AI - A)-III <  .
(1.2.1)
For the sake of the reader, in this section we collect the well-known Lumer-Phillips
Theorem and its corollary concerning generation of the Co-semigroup generated by a
dissipative operator (see Yosida [1] and pazy [1]). All these results are valid in Banach
spaces. However, since this book only uses its version in Hilbert spaces, we restate
these two theorems as follows.
Theorem 1.2.2 Let A be a densely defined linear operator on a Hilbert space H. Then
A generates a Co-semigroup of contractions on H if and only if A is dissipative and
R(I - A) = H.
The following Lumer-Phillips theorem tells us that the condition R(I - A) = H
can be weakened to R(AoI - A) = H for any given AO > O.
Theorem 1.2.3 (Lumer-Phillips) Let A be a linear operator with dense domain
V( A) in a Hilbert space H. If A is dissipative and there is a AO > 0 such that the
range, R(AoI - A), of AoI - A is H, then A is the infinitesimal generator of a C o -
semigroup of contractions on H.
As a collorary of the above theorem, the following result will be frequently used in
this book:
Theorem 1.2.4 Let A be a linear operator with dense domain V(A) in a Hilbert space
H. If A is dissipative and 0 E p( A), the resolvent set of A, then A is the infinitesimal
generator of a Co-semigroup of contractions on H.
Proof By the assumption 0 E p(A), A is invertible and A-I is a bounded linear
operator. By the contraction mapping theorem, it is easy to see that the operator
AI - A = A(AA- I - I) is invertible for 0 < A < IIA-Ili. Therefore, it follows from the
above Lumer-Phillips Theorem that A is the infinitesimal generator of a Co-semigroup
3

of contractions on H. Thus the proof is complete.
o
1.3 Exponential Stability and Analyticity
In this section we collect some results in the literature concerning the necessary and
sufficient conditions for a Co-semigroup being exponentially stable or analytic. The
first result we are going to state is about the necessary and sufficient conditions of
exponential stability of a Co-semigroup on a Hilbert space. The result was obtained
by Gearhart [1] and Huang [1], independently (see also Pruss [1]). The following
statement is due to Huang [1].
Theorem 1.3.1 Let S(t) = eAt be a Co-semigroup on a Hilbert space. Then S(t) is
exponentially stable if and only if
sup{Re A; A E u(A)} < 0
(1.3.1)
and
sup II (AI - A)-III < 00
Re>'O
(1.3.2)
hold.
The following invariant of the result is due to Gearhart [1] (see Wyler [1]).
Theorem 1.3.2 Let S(t) = eAt be a Co-semigroup of contractions on a Hilbert space.
Then S (t) is exponentially stable if and only if
p(A) :> {ij3, j3 E IR} = iIR
(1.3.3)
and
lim II (ij31 - A)-III < 00
1.81--+00
(1.3.4)
hold.
In the following we give the proof of equivalence of these two results under the
condition that S(t) = eAt is a C o - semi group of contractions on a Hilbert space.
It is obvious that (1.3.1)-(1.3.2) imply (1.3.3) and (1.3.4). We now prove that
(1.3.3)-(1.3.4) imply (1.3.1) and (1.3.2) under the condition that lIeAtll < 1. We first
4

notice that by Colollary 3.6 in pazy [1], the resolvent set p(A) of A contains the open
right half-plane, i.e., p(A) :> {A: ReA> OJ, and for such A, II(AI - A)-III < Rh.
This implies that for any given 8 0 < 0, when ReA> 18 0 1, we have
II(U - A)-III < 1;01 " (1.3.5)
Second, we prove that there exists 0'0 < 0 with 10'01 being sufficiently small such that
O'(A) C {A, ReA < O'o}. Indeed, consider for A = J.L+iv,
AI - A = J.LI + ivI - A = (J.L(ivI - A)-I + I)(ivI - A).
(1.3.6)
Then it follows from (1.3.4) that when IJ.L I is sufficiently small, by the contraction
mapping theorem, J.L(ivI - A)-l + I is invertible. Thus, (1.3.4) implies that O'(A) C
{A; ReA < 0'0 < O} with 10'01 small enough. Therefore, O'o(A) < 0'0 < 0 and for
Re A < 18 0 1, II(AI - A)-III < 2M. Combining this with (1.3.5) yields (1.3.2). 0
Concerning analyticity of a Co-semigroup of contractions on a Hilbert space, we
have the following result:
Theorem 1.3.3 Let S(t) = eAt be a Co-semigroup of contractions in a Hilbert space.
Suppose that
p(A) :> {i,81 ,8 E lR} = iIR.
(1.3.7)
Then, S (t) is analytic if and only if
lim 1I,8( i,8I - A)-III < 00
\.8\-00
(1.3.8)
holds.
Proof The statement of Theorem 1.3.3 was first given in the old version of the paper,
Liu & Yong [1], but no proof was given there. We also refer to Liu & Yong [1] for the
statement of a strong version of this theorem. But no proof was given there either.
The proof of the "only if" part of the present theorem is easy and it is actually a direct
consequence of Theorem 5.2 of Pazy [1] (see (5.4) and (5.5) on p. 61 in that book).
The proof of the "if" part consists of the following steps:
(i) Since (1.3.8) holds, there is a positive constant M such that for any given ,8 E IR,
1I,8( i,8I - A)-III < M.
(1.3.9)
5

We now claim that the set PI- Jl < ReA < O} is also contained in p(A). Indeed,
for). = a + i{3 with - < a < 0, we have (a + i(3)I - A = (i{3I - A) + aI =
(if3I - A)(I + a(if3I - A)-I). By (1.3.9) and the contraction mapping theorem,
I + a(if3I - A)-l is invertible. Thus, (a + i(3)I - A is invertible and
1I((a + i(3)I - A)-III <  < II
(1.3.10)
with C = V (4M2 + 1).
(ii) By Corollary 7.5 in pazy [1] (p. 29), for any I > 0 and x E D(A2), we have
.
1 l "+oo At -1
S(t)x = _ 2 . . e (AI - A) XdA,
7r ,,-oo
(1.3.11)
and for every 17 > 0, the integral converges uniformly in t for t E [17,  ].
For any y E H, we have
1 1 ,,+{3
(S(t)x,y) = _ 2 . lim . (eAt(AI - A)-l x ,y)dA.
7r (3-+00 ,,-{3
(1.3.12)
Let 8 1 and 8 2 be two angles such that tan( 8d = - 2h- with  < 8 1 < 7r, 7r < 8 2 <
3f. Consider the closed curve in the complex plane in A: r = r 0 U r 1 U r 2 U r 3 U r 4
with fo = PI ReA = ,,-{3 < 1m). < {3}, f 1 = PI 1m). = {3,-1M < ReA < ,},
f 2 = PI). = pe i01 ,0 < p < fl}, f3 = PI). = pe i02 ,0 < p < fl}, f4 =
PI 1m). = -{3, -1M < ReA < ,}. Since (eAt(>.I - A)-lX, y) is analytic in ). E
PI ReA > O} UP I - JI2).l < ReA < O} C p(A), by the Cauchy theorem in the
theory of analytic functions, we have
l "1'tP (e At ( >.I - A)-lX, y)d)' = j : (e(6+i p )t( (h + i(3)I - A)-l x, y)dh
,,-{3 2M
+  (e(6-i P )t((h _ i(3)I - Aflx, y)dh + e i01 r '!tl (ePteiOl (pe i01 I - A)-lX, y)dp
JJM J o
CI.81 .
+e i02 fa 2M (e Pte 'to 2 (pe i02 I - A)-l x, y)dp
= If + If + Ig + If. (1.3.13)
6

Since S(t) is a Co-semigroup of contractions, by Corollary 3.6 in pazy [1], for A with
ReA> 0, we have
II (AI - A)-III < II"
Combining (1.3.10) with (1.3.14) yields that for t > 0,
IIfl <  ["I...1L e c5t d81lxllllyll -+ 0, as {3 -+ +00.
fJ 2M
(1.3.14)
(1.3.15)
The same argument also yields that IIfl -+ 0, as (3 -+ +00.
By (1.3.10), we have

IIgl < C {2M ePtcos81 dpllxllilyli.
J o p
It follows from (1.3.16) that as (3 -+ +00, Ig converges to
(1.3.16)
. f+oo ie .
13 = e1,O, Jo (e Pte ' (pe1,O, I - A) -1 x, y)dp.
The same argument also yields that I converges to
(1.3.17)
. f+oo ie .
14 = e82 Jo (e Pte 2 (pe82 I - A)-lx, y)dp.
These integrals converge uniformly for t E [1],  ]. Thus we have
(1.3.18)
(S(t)x, y) = 13 + 1 4 , Vt > O.
(1.3.19)
It is easy to see that 13 and 14 are differentiable with respect to t for t > o. Then we
have
. f+oo i .
(S' (t)x, y) = e20, 1, Jo p( e Pte 8, (pe1,O, I - A)-1 X, y)dp
. {+OO i .
+e 202 1, J o p( e pte 82 (pe1,92 I - A)-1 X, y)dp.
(1.3.20)
Therefore,
I( S'( t)x, y) I < C (10+ 00 ePt cos 0, dp + 10+ 00 ePtcos02 dp) IIxllllyll = l llxllllY II (1.3.21)
with
C C
C I = - -
COS(81) cos(8 2 ).
( 1.3.22)
7

Since y is an arbitrary element and D(A 2 ) is dense in H, it follows from (1.3.21) and
the dense argument that
IIS'(t)1I = IIAS(t)11 < 1 , t > O.
(1.3.23)
Thus, from Theorem 5.2 in Pazy [1], it follows that S(t) is analytic.
o
1.4 The Sobolev Spaces and Elliptic Boundary
Value Problellls
In this section we collect some basic results on function spaces and the elliptic partial
differential equations which will be needed in the remainder of the book. Most results
are just recalled without proofs, but the relevant references are given.
1.4.1 Sobolev Spaces Wm,P(O)
Let 0 be a bounded or an unbounded domain of lR n with smooth boundary f. For
m E IN, 1 < p < 00, Wm,P(O) is defined to be the space of functions u in LP(O) whose
distribution derivatives of order up to m are also in LP(O). Then, it is known (see e.g.,
Adams [1] and Lions & Magenes [1]) that Wm,P(O) is a Banach space with the norm
1
Ilullwm,p(o) = ( L IIDOtUllip(o) ) P
IOtIm
(1.4.1)
aOt1 +..-+Otn
where a = {a},...,a n } E JNn, lal = al + ... + an, DOt u = a Ot1 a n . When
Xl ... X n
p = 2, we usually denote Wm,P(O) by Hm(o) and this is a Hilbert space with the
corresponding inner product.
Let Ck(O) (k E IN or k = 00) be the space of k times continuously differentiable
functions on O. Let C(O) be the space of C k functions on 0 with compact support
in O. The closure of CO'(O) in Wm,P(O) is denoted by W,P(O) which is a subspace
of Wm,P(O).
We now recall some important properties of the Sobolev spaces Wm,P(O) (see, e.g.,
Adams [1]).
8

Theorem 1.4.1 (Density Theorem) If 0 is a em domain, m -> 1, 1 < p < 00,
then em(O) is dense in Wm,P(O).
Theorem 1.4.2 (Imbedding and Compactness Theorem) Assume that 0 '/,s a
bounded domain of class em. Then we have
(i) If mp < n, then Wm,p(n) is continuously imbedded in Lq"(n) with i. =  - r;: :
Wm,P(O) '-+ Lq. (0). (1.4.2)
Moreover, the imbedding operator is compact for any q, 1 < q < q*.
(ii) If mp = n, then Wm,P(O) is continuously imbedded in Lq, Vq, 1 < q < 00 :
Wm,P(O) '-+ Lq(O).
(1.4.3)
Moreover, the imbedding operator is compact, Vq,l < q < 00. If p = 1, m = n, then
the above still holds for q = 00.
(iii) If k + 1 > m -  > k, k E IN, then writing m -  = k + Q, k E IN, 0 < Q <
1, Wm,P(O) is continuously imbedded in ek,Q(O) :
Wm,P(O) '-+ ek,Q(O), (1.4.4)
where ek,Q(O) is the space of functions in ek(O) whose derivatives of order k are
Holder continuous with exponent Q. Moreover, if n = m - k - 1, and Q = 1, p = 1,
then (1.4.4) holds for Q = 1, and the imbedding operator is compact from Wm,P(O) to
e k ,I3(O), V 0 < (3 < Q.
Remark 1.4.1 The imbedding properties (i)-(iii) are still valid for smooth unbounded
domains or lR n provided that Lq(O) in (1.4.3) and ek,Q(O) in (1.4.4) are replaced by
Lioc(O) and ek,Q(B) for any bounded domain B c 0, respectively.
Remark 1.4.2 The regularity assumption on 0 can be weakened (e.g., see Adams [1]).
When u E W;,P(O), the above imbedding properties are valid without any regularity
assumptions on o.
Let 0 be a smooth bounded domain of class em and u E Wm,P(O). Then we can
define the trace of u on r which coincides with the value of u on r when u is a smooth
function of em(O).
9

Theorem 1.4.3 (Trace Theorem) Let v = (VI,.. . , v n ) be the unit outward normal
on f and
aju \.J C m ( t=\ ) .
,jU = _ a " , vU E .1G, J = 0,. .. , m - 1.
vJ
r
Then the trace operator, , = {,o, . . . , ,m-I}, can be uniquely extended to a continuous
m-I
. 1
operator from Wm,P(O) to II wm-J-pIP(f) :
j=O
(1.4.5)
m-I
. 1
,: u E Wm,P(O) ,u = {,ou,... "m-IU} E II wm-J-pIP(f).
j=O
( 1.4.6)
Moreover, it is a surjective mapping.
. 1
Notice that Wm-J-pIP(f) are spaces with fractional-order derivatives. Refer to Lions
& Magenes [1] for the definition and more about that.
Let 0 be a bounded domain in lR n with C I boundary f. Then, for any function
U E HI(O), by the trace theorem, we have ulr E Ht(f) C L 2 (f). We now have the
following useful result.
Theorem 1.4.4 Let 0 be a bounded domain in lR n with C I boundary f. Then for
any function u E HI (0), the following estimate holds:
1 1
Il u IIL2(r) < Cll u llk 1 (O) lI u III2(O)
(1.4.7)
with C being a positive constant independent of u.
The proof is actually not difficult. The inequality (1.4.7) clearly holds for u E CJ (IR).
For u E HI(O), we can use the standard technique in PDE: finite covering of 0 and
decomposition of unity to reduce the problem to the corresponding one in lR.
1.4.2 The Gagliardo-Nirenberg and Poincare Inequalities
Throughout this book the following Gagliardo-Nirenberg interpolation inequalities
(see Nirenberg [1] and Friedman [1]) will be frequently used.
First, we introduce some notation. For p > 0, lulp,o = IluIILP(O). For p < 0, set
h = [-  ] , -0: = h +  and define
lulp,O = sup IDhul = L sup ID.Bul, if 0: = 0,
o 1.BI=h 0
(1.4.8)
10

lulp,o = [Dhu]Q,o = L sup[D.Bu]Q
1.BI=h 0
-  ID.Bu(x) - D.Bu(y)1 ; f '" > o.
=  SUp 1 1 ' (. \..(.
1.BI=h x,yeO X - Y Q
If 0 = lR n , we simply write lul p instead of lulp,o.
Theorem 1.4.5 Let j and m be any integers satisfying 0 < j < m, and let 1 < q, r <
00, and p E lR, k < a < 1 such that
1 j 1 m 1
- - - = a(- - -) + (1- a)-. (1.4.10)
p n r n q
(1.4.9)
Then,
(i) For any u E wm,T(lR n ) n Lq(lR n ), there is a positive constant C depending only on
n, m, j, q, r, and a such that the following inequality holds:
IDjul p < CIDmullul-a
(1.4.11)
with the following exception: if 1 < r < 00 and m - j -  is a nonnegative integer,
then (1.4.11) holds only for a satisfying k < a < 1.
(ii) For any u E Wm,T(O) n Lq(O) where 0 is a bounded domain with smooth boundary,
there are two positive constants C 1 , C 2 such that the following inequality holds:
IDjulp,o < CIIDmul,olulft + C 2 1ul q ,o
(1.4.12)
with the same exception as in (i).
In particular, for any u E W:,T(O)nLq(O), the constant C 2 in (1.4.12) can be taken
as zero.
The following two theorems are concerned with the useful Poincare inequalities.
Theorem 1.4.6 Let 0 be a bounded domain in lR n and u E HJ(O). Then there is a
positive constant C depending only on 0 and n such that
Il u IIL2(O) < CII\7uIIL2(Ob VuE HJ(O).
(1.4.13)
Theorem 1.4.7 Let 0 be a bounded domain of C 1 in lR n . There is a positive constant
C depending only on 0, n such that for any u E HI (0),
Il u ll£2(!1) < C (IiVull£2(!1) + I udxl) .
(1.4.14)
11

1.4.3 Abstract Functions Valued in Banach Spaces
For the study of evolution equations it is convenient to introduce abstract functions
valued in Banach spaces.
Let X be a Banach space, 1 < p < 00, -00 < a < b < 00. Then LP( (a, b); X)
denotes the space of LP functions from ( a, b) into X. It is a Banach space with the
norm
1
IIfIlLP((a,b);X) = (t Ilf(t)lI dt) P
(1.4.15)
where the integral is understood in the Bochner sense.
For p = 00, L 00 (( a , b); X) is the space of measurable functions from (a, b) into X
being essentially bounded. It is a Banach space with the norm
IlfIILOO((a,b);X) = sup essllf(t)llx.
te(a,b)
(1.4.16)
Similarly, when -00 < a < b < 00, we can define Banach spaces Ck([a, b]; X) with
the norm
k dif
IIfIlCk([a,b];X) =  m:;] II dt i (t) II X .
(1.4.17)
1.4.4 Linear Elliptic Boundary Value Problems
In this section we introduce some basic results on linear elliptic boundary value prob-
lems (refer to Nirenberg [2], Friedman [1], and Lions & Magenes [1]).
Let n be a domain with smooth boundary r. Any linear partial differential operator
with, for simplicity, Coo coefficients aa( x) in n has the form
P(x, D) = L aa(x)Da.
laltL
(1.4.18)
The operator P is called elliptic in n if the leading homogeneous part of P( x,) does
not vanish for  =1= 0 and x En:
Pp.(x,) = L aa(x )a =1= 0, V x E n,  E lR n \ {O}.
lal=p.
(1.4.19)
12

The Laplace operator  and the biharmonic operator  2 are the most familiar elliptic
operators.
It follows that for an elliptic operator P with real coefficients aa, J.L must be an even
number 2m, m > 0, m E IN. From now on we always assume that P is a uniformly
elliptic operator with real Coo coefficients aa( x) in f2. Usually, one studies boundary
value problems for elliptic operators:
{ Pu = L aa(x)Dau = f, x E 0,
lal2m
B"u l = g o ) "=0...m-1
1 r l' , ,
(1.4.20)
where f and 9j are given functions in nand r, respectively, and Bj are certain partial
differential operators defined on r and the order of each Bj is less than 2m.
Wellposedness
When gj = 0, the problem (1.4.20) is said to be wellposed if
(i) Ker P belongs to Coo and dim(Ker P) = v < 00.
(ii) In suitable function spaces X, Y the operator P: X 1--+ Y is continuous and has
closed range in Y of finite co dimension v*, i.e., P is Fredholm.
The index of operator P is defined as follows:
ind P = v - v*.
(1.4.21)
The boundary operators Bj (j = 0, . . . , m - 1) for which the general boundary prob-
lems (1.4.20) are wellposed have been characterized and are known as the Lopatinsky
boundary conditions. As far as spaces X, Yare concerned, C 2 m+k+a(n), Ck+a(n) (k E
IN, 0 < Q < 1) and W 2 m+k , P (O), Wk,P(O) (1 < p < 00) are two suitable choices of
Sobolev spaces.
Basic Results in ck+a
(i) Fredholm
The mapping
P: {u E C 2 m+k+a(f!)IB j u = 0, on r, (j = 0,. .. , m - I)} 1--+ Ck+a(f!) (1.4.22)
13

is Fredholm and its index is independent of k.
(ii) Regularity
If gj = 0, f E Ck+a(f!) and u is a weak solution of problem (1.4.20) in the distribution
sense, then u E C 2 m+k+a (f!). Thus all functions in Ker P are in Coo.
(iii) A Priori Estimate
There exist two positive constants C1, C 2 independent of u such that for any u E
C 2 m+k+a(f!) satisfying Bjulr = 0, (j = 0,... , m - 1),
IluIIC2m+k+a < C111Pu11Ck+a + C 2 11ullc.
( 1.4.23)
Moreover, if Ker P = 0, i.e., the uniqueness of problem (1.4.20) holds, then there is a
positive constant C 3 independent of u such that
IluIIC2m+k+a < C 3 1IpuIICk+a.
Basic Results in Wk,p(O) (k E IN, 1 < p < 00)
(i) Fredholm
The mapping
P: {u E W 2 m+k,p(O)1 Bju = 0, on f, (j = 0,..., m - I)}  Wk,P(O) (1.4.25)
( 1.4.24 )
is Fredholm and its index is independent of k.
(ii) Regulari ty
If 9j = 0, f E Wk,P(O) and u is a weak solution of problem (1.4.20), then u E
W2m+k,p(o). Thus all functions in ]( er P are in Coo.
(iii) A Priori Estimate
There exist two positive constants C1, C 2 independent of u such that for any u E
W 2 m+k,p(o) satisfying Bjulr = 0, (j = O,...,m - 1) in the trace sense,
IluIIWk+2m.p < C11IPullwk,P + C 2 1IuIILP.
( 1.4.26)
Moreover, if Ker P = 0, i.e., the uniqueness of problem (1.4.20) holds, then there is a
positive constant C 3 independent of u such that
IIUIlWk+2m.p < C 3 11Pullwk,P.
( 1.4.27)
14

In the remainder of this book we will also need the results for the nonhomogeneous
boundary value problem (1.4.20) with gj =I- 0, (j = 0,... , m - 1) being in Sobolev
spaces of fractional order. Let the order of Bj be mj with mj < 2m - 1 and let gj
be in the Sobolev spaces H2m-m) -t of fractional order. Then we have the following
result (see Lions & Magenes [1]).
Theorem 1.4.8 For any u E H2m+k(f!) satisfying Bjulr = gj, (j = 0,. . . , m - 1) in
the trace sense, there exist positive constants C k depending only on k > 0 and f! such
that
IIUIIH2m+k < C k (IIPUIIHk +  IIgjIlH 2m - m ,-! + IIUllu) (1.4.28)
Moreover, if Ker P = 0, i.e., the uniqueness of problem (1.4.20) holds, then there is
a positive constant C k independent of u such that
lIulIH2m+k < C k (IIPUIIHk + 1119jIlH2m_m,_t) .
( 1.4.29)
1.4.5 Interpolation Spaces
We recall a few facts about linear operators associated with a bilinear form and inter-
polation spaces associated with a positive definite operator in Hilbert spaces.
Let V, H be separable Hilbert spaces such that V is dense in H and the injection
V C-....+ H is continuous and compact. Thus, by the Riesz representation theorem we
can write
V C H = H' C V'.
(1.4.30)
The dual product between V and V'is denoted by (, ) and the inner product in H by
(, ).
Let A be a linear continuous operator from V to V'. We can associate it with a
bilinear form a(., .) on V in such a way that
a( u, v) = (Au, v), Vu, v E V.
(1.4.31)
Suppose that a is symmetric:
a(u, v) = a(v, u)
(1.4.32)
15

and also suppose that a is coercive, i.e., there exists a positive constant Q > 0 such
tbat
a(u, u) > Qllull, Vu E V.
(1.4.33)
Let
D(A) = {ulu E V, Au E H}.
(1.4.34)
Then by the Lax-Milgram theorem, a( u, v) can be considered as an equivalent inner
product. Furthermore, A is a strictly positive self-adjoint operator in H and the
spectral theorem allows us to define the powers AS of A for s E JR. Since we assume that
the injection V C-....+ H is compact, there exists (see Yosida [1]) a complete orthonormal
basis {Wj} of H and a sequence {Aj} such that Wj E D(A) and
Aw. - A"W"
1 - 1 l'
) '=12...
" ,
o < Al < A2 < . . . ,
a( Wi, Wj) = Aic5ij,
Aj -+ 00, as ) -+ 00,
(1.4.35)
Vi,j E IN.
Thus, for s > 0, we define
D(A") = {u E H A;SI(u'WjW < 00 },
( 1.4.36)
and
1
lIuIlD(A') = (A;S I(u, WjW) 2
( 1.4.37)
1
For negative s,D(A") is the completion of H for the norm (AJSI(U'WjW) 2. In
particular, we have V = D(At).
Let X and Y be two Banach spaces, with X C Y, X dense in Y, and the injection
X C-....+ Y being continuous. In general, there are several different methods to define the
intermediate spaces between X and Y. The framework described above gives a sort
of definition of intermediate spaces, namely, the interpolation spaces.
Let
1
X = V = D(A2"), Y = H = D(AO),
(1.4.38)
16

and
a(u,v) = (u,v)x.
(1.4.39)
Then interpolation spaces [X, Y]8, (0 < () < 1) are given by
1-8
[X, Y] 8 = D (A "2 ) , V () E [0, 1].
(1.4.40)
The interpolation inequality
Ilull[X,Y]e < C(())llull-8I1ull, Vu E X, () E [0,1]
(1.4.41)
also follows from (1.4.40).
A typical example of the above framework is
v = H(O), H = L 2 (0), A = -
( 1.4.42)
and
D(A) = H 2 (0) n H(O)
(1.4.43)
with 0 being a bounded domain of C 2 .
1.5 Notation
Throughout this book we use the following common notation.
1. In addition to the notation 8 8 t kk ' G 0<1 GO< a O<n ' we also widely use n k to denote
Xl ... X n
the corresponding partial derivatives with respect to x, i.e., n k = t:k . The subscripts
t and X, yare often used to denote the partial derivatives with respect to t and X, y,
.. 8 2 u 8 2 v
respectIvely, I.e., Utt = 8t 2 , v xx = 8x 2 ' etc.
2. We simply denote by (.,.), 11.11 the inner product and norm in L 2 space, respectively.
3. We often use C, Ci(i E IN), K, M to denote a universal positive constant which may
vary in different places.
17

Chapter 2
Linear Thermoelastic Systems
In this chapter we are concerned with linear thermoelastic systems, namely, the linear
one-dimensional thermoelastic system and the thermoelastic Kirchhoff plate equations.
In the first four sections, we are dealing with the exponential stabilit.y of semigroups
associated with the linear one-dimensional thermoelastic system subject to various
boundary conditions. It is also shown that the corresponding semigroups are not ana-
lytic. In the final section of this chapter we consider the thermoelastic Kirchhoff plate
equations. The results on exponential stability and analyticity of the corresponding
semigroups are established.
2.1 The Setting of Problems for the One-
Dimensional Thermoelastic System
In this section we first formulate the initial boundary value problems for the linear
one-dimensional thermoelastic system. Consider a linear homogeneous thermoelastic
bar of length 1 with unit reference density. Let u be the displacement and 0 be the
temperature deviation from the reference temperature. Then u and 0 satisfy the
following linear one-dimensional thermoelastic system (for instance, refer to Dafermos
[1]) :
Utt - au xx + ,Ox = 0, in (0,1) x (0, +00),
CoBt + ,Uxt - kO xx = 0, in (0, 1) x (0, +00).
(2.1.1)
(2.1.2)
In the above, the first equation is the momentum equation and the second equation
is the energy equation. We assume that a, " Co, and k are given constants with
a > 0, Co > 0, k > 0, , 1= o. These constants depend on the material properties.
The above system (2.1.1) and (2.1.2) is subject to the initial conditions
ult=o = uo(x), utlt=o = Ul(X), Blt=o = Oo(x),
(2.1.3)
18

and the boundary conditions which are very often one of the following four pairs at
x = 0 or 1:
{ U = 0
(i) ,
0=0
{ U = 0
(ii) ,
Ox = 0
{ a = 0
(iii) ,
0=0
{ a = 0
(iv)
Ox = 0
(2.1.4)
where a = U x - ,0 is the stress. The mechanical and physical meaning of these
boundary conditions are very clear. For instance, the boundary condition (iv) implies
that at that end of the elastic bar the stress is free and the heat is insulated.
Concerning the initial data, we assume that Uo E HI, UI E L 2 , 0 0 E L 2 satisfying
the compatibility conditions. This means that if conditions (i) or (ii) are considered
at one end of the bar, then Uo is also assumed to satisfy u = 0 in the trace sense at
that end.
In the first two sections of this chapter we always work with the boundary condition
case (i), i.e., at both ends of the bar,
u = 0, 0 = 0, at x = 0, 1.
(2.1.5)
In the third and fourth sections we will treat the other boundary condition cases. We
refer the reader to Liu & Zheng [1] and Burns, Liu & Zheng [1] for the discussion of
these four sections.
Without loss of generality, throughout this chapter we always assume that Co =
a = 1 because for the general case the only thing we have to modify the discussion
throughout this ch apter is to replace, for instance, the usual L 2 norm for 0 by the
equivalent norm j f c o 0 2 dx, etc.
To study the initial boundary value problem (2.1.1)-(2.1.3), and (2.1.5) by the
semigroup theory, we introduce the new variables:
v = Ut.
(2.1.6)
Then the initial boundary value problem (2.1.1)-(2.1.3) and (2.1.5) is reduced to the
19

following abstract initial value problem for a first-order evolution equation:
{ dy
- = Ay Vt > 0
dt ' (2.1.7)
Y\t=o = Yo = (uo, Ut, Oo)T
with
U
y= v (2.1.8)
0
and
0 I 0
A= D 2 0 -,D (2.1.9)
0 -,D kD 2
Here we have used the notation: D i = :I .
Let 0 = (0,1) and
1i = H(O) x L 2 (0) x L 2 (0)
(2.1.10)
equipped with the norm
1
lIy 111i = (II Du 11 2 + IIvl1 2 + 110112) 2 ,
(2.1.11)
where II . II is the L 2 norm in O.
Instead of dealing with (2.1.1)-(2.1.3), and (2.1.5), we will consider (2.1.7) with
the domain of the operator A:
1'(A) = H2nH x H x H2nH.
(2.1.12)
Then it is clear from Chapter 1 that the operator A is a densely defined operator from
1'( A) to 1i. Furthermore, we have the following.
Theorem 2.1.1 The operator A generates a Co-semigroup S(t) = eAt of contractions
on the Hilbert space 1i.
20

Proof We first prove that A is a dissipative operator. Indeed, for any y E V(A), by
the definition (2.1.11) and by integration by parts, we have
(Ay, Y)1{ - 1'" (Dv . Du + D 2 u . v - i DO . v - Dv .0+ kD 2 0 . 0) dx
_ -kIIDOIl 2
< o.
(2.1.13)
This implies that A is a dissipative operator. To prove that A generates a Co-semigroup
of contractions on the Hilbert space 1i, by Theorem 1.2.4 in Chapter 1, it remains to
be proved that 0 E p(A).
For any F = (11,12, I3)T E 1i, consider the equation
Ay = F, (2.1.14)
I.e. ,
v II, (2.1.15)
D 2 u - ,DO - 12, (2.1.16)
-,Dv + kD 2 0 13. (2.1.17)
We plug v = II obtained from (2.1.15) into (2.1.17) to get
kD 2 0 = 13 + ,DII E L 2 .
(2.1.18)
By the standard theory in the linear elliptic equations (see Chapter 1), we have a
unique 0 E H 2 n HJ satisfying (2.1.18). Then we plug 0 just obtained from solving
(2.1.18) into (2.1.16) to get
D 2 u = ,DO + 12 E L 2 .
(2.1.19)
Applying the standard theory in the linear elliptic equations again yields a unique
solvability of u E H2 n HJ for (2.1.19). Thus the unique solvability of (2.1.14) follows.
It is clear from the regularity theory of the linear elliptic equations that lIy liB <
21

KliF/lH with K being a positive constant independent of y. Thus the proof is complete.
D
Since we use the semigroup approach to investigate solvability of problem (2.1.1)-
(2.1.3) and (2.1.5), we have to clearify the relationship between the semigroup solution
given by y(t) = S(t)yo and the previous problem. In other words, we would like to
show in what sense U and 0, the first and third component of y, satisfy problem
(2.1.1 )-(2.1.3) and (2.1.5).
If Yo E V(A), i.e., Uo E H2nHJ, U1 E HJ and 0 0 E H2nHJ, then y(t) = S(t)yo E
C ([0, 00), V( A)) n C1 ([0, 00), 1-£) and (2.1.7) is satisfied in 1-£ for every t > o. It turns
out that U E C([0,00),H 2 nHJ)nC 1 ([0,00),HJ)nC 2 ([0,00),L 2 ), 0 E C([O, 00), H 2
nHJ)nC1([0,00),L2) and they both satisfy equations (2.1.1) and (2.1.2) in L 2 for
every t > o. Initial conditions in (2.1.3) are satisfied in the strong sense and the
boundary conditions (2.1.5) are satisfied in the trace sense. It is also clear that if Yo
is more regular, for instance, Yo E V(A 2 ), then we can obtain classical solution u, 0
from the semigroup solution.
If Yo E 1-£, i.e., Uo E HJ, Ut, 0 0 E L 2 , then y(t) = S(t)yo is only a mild solution to
the first-order evolution equation (2.1.7). Since V(A) is dense in 1-£, there is a sequence
YOn E V( A) converging to Yo in 1-£. Accordingly, we have a sequence Yn (t) = S (t ) YOn
such that Un and On satisfy (2.1.1) and (2.1.2) in L 2 for every t > 0 and for any
T > 0, Un  U in C([O, T], HJ) n C 1 ([0, T], L 2 ), On  C([O, T], L2) n L 2 ([0, T], HJ).
Therefore, for any w, z E HJ, if we multiply equation (2.1.1) and equation (2.1.2) for
Un, On by wand z, respectively, then integrate by parts with respect to x and integrate
with respect to t, and finally pass to the limit, we obtain that
{ (Ut,w) - (UI'W) + af((ux,wx) + ,(Ox,w))dr = 0,
co(O, z) - co(Oo, z) -,(u, zx) + ,(uo, zx) + k f(Ox, zx)dr = o.
(2.1.20)
(2.1.20) is usually called the variational form of problem (2.1.1)-(2.1.3) and (2.1.5).
In other words, when Yo E 1-£, the first and third component U and 0 of the semigroup
solution y is a pair of weak solutions to problem (2.1.1)-(2.1.3) and (2.1.5) in the sense
22

that (2.1.20) is satisfied for any test functions w, z E HJ.
The previous argument also works for other inital boundary value problems for the
system (2.1.1) and (2.1.2).
2.2 The Exponential Stability for the Dirichlet
Boundary Conditions at Both Ends
In this section we will prove that the semigroup S(t), generated by the dissipative
operator A, is exponentially stable.
Theorem 2.2.1 The semigroup S(t), generated by the operator A, defined in (2.1.9)
is exponentially stable, i. e., there are two positive constants M, 0 such that
II S (t) II < Me-ext.
(2.2.1)
Before giving the proof of Theorem 2.2.1, we first recall some related results in the
literature. As far as the linear thermoelastic system is concerned, Dafermos [1] was
probably the first to investigate the asymptotic behavior of solutions to the initial
boundary value problems for the linear thermoelastic system. It was shown in Dafer-
mos [1] that if Uo E HI, UI E L 2 , ()o E L 2 , then the energy function of the system
defined by
E(t) = lIu x l1 2 + II u tll 2 + 11()11 2
(2.2.2)
converges to zero as time goes to infinity. However, no decay rate was given. In 1981,
Slemrod [1] used the energy method to prove that for the system (2.1.1) and (2.1.2) if
u, () satisfy the boundary conditions (ii) or (iii) at both ends (i.e., stress free, constant
temperature or clamped, insulated) and if Uo E H 2 , UI E HI, ()o E H2 satisfy the
compatibility conditions, then there are positive constants M and 0 such that
IIUt(x)1I2 + lIu x (x)112 + lIutt(x)112 + lIu x t(x)112 + Ilu xx (x)1I2
+ II () ( t ) 11 2 + II () t ( t ) 11 2 + II () x ( t ) 11 2 + II () xx ( t ) 11 2
< M(lIuoll2 + IIUIIIl + lI()oll2)e-ext, V t > 0,
(2.2.3)
23

which amounts to the following stimate:
IIS(t)YoIlV(A) < Me-atIlYollv(Ab Vt > o.
(2.2.4)
In i990, Rivera [1] proved that the estimate (2.2.3) still holds if u and () both
satisfy the boundary condition (i) at both ends (i.e., clamped, constant temperature).
Later on, Shibata [1] and Jiang [1] considered the initial boundary value problem with
the boundary conditions (iv) at both ends. Shibata used a spectral analysis method
to obtain the polynomial decay of the solution. His method requires the boundedness
of initial data in more regular spaces. Jiang improved Shibata's results and used the
energy method to obtain the exponential decay of solutions. Again, Jiang's method
requires the boundedness of initial data in more regular spaces. We also refer the
reader to Slemrod [1], Zheng [1], Racke, Shibata & Zheng [1], Zheng & Shen [1],
and Jiang [2] for the results on the global existence and uniqueness for the nonlinear
thermoelastic system with small initial data. However, all of the results mentioned
previously involve estimates of the type (2.2.3). The problem of establishing an energy
estimate of the form
E(t) < Me-atE(O), Vt>O,
(2.2.5)
or equivalently establishing the exponential stability (2.2.1) of the semigroup S( t)
remained open for some time because the estimate (2.2.4) (or (2.2.3)) looks weaker
than (2.2.1) (or (2.2.5)). However, it has now been clear (for instance, see Zheng [1])
that these two statements are equivalent.
When u and () satisfy the boundary conditions (ii) or (iii) at both ends, Hansen [1]
in 1990 succeeded in establishing (2.2.5) using the Fourier series expansion method and
a decoupling technique. We refer to Gibson, Rosen & Too [1] for another approach,
i.e., a combination of semigroup theory and the energy method. When u and () both
satisfy the Dirichlet boundary. conditions, i.e., the boundary conditions (i), Kim [1]
and the authors (Liu & Zheng [1]) independently proved that the estimate (2.2.5) still
holds. The methods in these two papers are quite different. Kim's method is based on
a control theory approach and a uniqueness continuation theorem by J. L. Lions. In
24

Liu & Zheng [1], the authors used a contradiction argument by combining Theorem
1.3.1 with a PDE technique. In a later paper, Burns, Liu & Zheng [1] successively
established (2.2.1) for all the boundary conditions (i)-(iv) using the same technique.
This technique has been developed into a more systematic approach to deal with
other problems, including the higher dimensional problems such as the Kirchhoff plate
equations with thermal or viscous damping, as can be seen in the remainder of this
book. It should be mentioned that for the linear higher dimensional thermoelastic
system, for instance, the linear three-dimensional thermoelastic system, in general, we
can not expect to have the results on exponential stability unless some assumptions
are made on the domain and initial data. We refer the reader to Dafermos [1], Jiang,
Rivera & Racke [1], and Jiang [2] in this direction.
Proof of Theorem 2.2.1
We now use Theorem 1.3.2 to prove Theorem 2.2.1. We first prove (1.3.3). This
consists of the following steps:
(i) It follows from the fact that 0 E p(A) and the contraction mapping theorem that
for any real number /3 with 1/31 < IIA-III-I, the operator i/31 - A = A(i/3A- I - I)
is invertible. Moreover, II (i/31 - A)-III is a continuous function of /3 in the interval
(-IIA-III-I,IIA-III-I).
(ii) If sup{II(i/31 - A)-III I 1/31 < IIA-III-I} = M < 00, then by the contraction
mapping theorem, the operator i/31 - A = (i/301 - A)(I + i(/3 - /30)( i/301 - A)-I) with
1/301 < IIA-III- I is invertible for 1/3 - /301 < it . It turns out that by choosing 1/301 as
close to IIA-III- I as we can, we conclude that {/3 11/31 < IIA-III- I + it } c p(A) and
II (i/31 -A)-III is a continuous function of /3 in the interval (-IIA-III-I-  , IIA-III- 1 +
 ).
(iii) Thus it follows from the argument in (ii) that if (1.3.3) is not true, then there is
w E 1R with IIA-III- I < Iwl < 00 such that {i/3; 1/31 < Iwl} c p(A) and sup{lI(i/3 -
A)-III I 1/31 < Iwl} = 00. It turns out that there exists a sequence /3n E 1R with
/3n  w, l/3nl < Iwl and a sequence of complex vector functions Yn E V(A) with
25

IIYnll1i = (11Du n 1l 2 + IIv n ll 2 + II O n1l 2 )t = 1 such that
II (i,Bn I - A)Ynll1i  0,
(2.2.6)
as n  00, l.e,
i,Bnun - V n  0 in H,
i,BnOn - kD 2 0 n + ,Dv n  0
in L 2 .
(2.2.7)
(2.2.8)
(2.2.9)
i,Bnvn - D 2 u n + ,DOn  0
in L 2
,
Taking the inner product of (i,BnI - A)Yn with Yn in 1i and then taking its real part
yields
Re((i,Bn I - A)Yn, Yn)1i = kli DO nll 2  o.
(2.2.10)
Thus it follows from (2.2.9) and (2.2.10) and the Poincare inequality that
kD 2 0 n -,Dv n  0 in L 2 .
(2.2.11)
Integrating (2.2.11) from 0 to x yields
kDO n - kDOn(O) -,vn(x)  0 in L 2 .
(2.2.12)
Combining (2.2.12) with (2.2.10) yields
kDOn(O) + ,vn(x)  0 in L 2 .
(2.2.13)
By IIYnll1i = 1 and (2.2.7), we get that IIDvnll is uniformly bounded with respect to n.
Thus it follows from (2.2.11) that IID20nil is uniformly bounded. By the Gagliardo-
Nirenberg inequality (1.4.11), we obtain that
IDOn(O)1 < IIDOnllLoo < C 1 1ID 2 0 n lltilDO n lit + C211DOnil  o.
(2.2.14)
Combining (2.2.14) with (2.2.13) yields
V n ( x)  0 in L 2 .
(2.2.15)
26

Taking the inner product of (2.2.8) with Un in L 2 and integrating by parts also yields
II DUn II  0 in L 2 .
(2.2.16)
Thus (2.2.10), (2.2.15) and (2.2.16) contradict IIYnllH = 1, and the proof of (1.3.3) is
complete.
We now prove (1.3.4) by a contradiction argument again. Suppose that (1.3.4) is
not true. Then there exists a sequence /3n with l/3nl  +00 and a sequence of complex
vector functions Yn E V(A) with unit norm in 1i such that (2.2.6) holds. Again we
have (2.2.10). The remaining proof is more delicate than that in (iii) because /3n  00
now. Dividing (2.2.9) by /3n and using the Poincare inequality, we get
kD 2 0 n - ,Dv n 2
 0 in L .
/3n
(2.2.17)
Dividing (2.2.7) by /3n and using (2.2.17), we obtain
k:On _ i'YDun  0 in L 2 . (2.2.18)
Since II DUn II < 1, (2.2.18 ) implies that II k: On II is bounded. Taking the inner product
of (2.2.18) with DUn in L 2 yields
( kD20 n ) . 2
/3n  DUn - z,IIDU n II  o.
(2.2.19)
By integration by parts, we have
( kD 2 0n D ) = kDOnDu n kDOnDu n _ ( kDOn D 2 ) (
!3n ' Un !3n 3;=1 !3n 3;=0 !3n ' Un. 2.2.20)
Dividing (2.2.8) by /3n and using (2.2.10) and the fact that Ilvnll < 1, we deduce that
II n;:n II is bounded. Thus it follows from (2.2.10) and the Cauchy-Schwartz inequality
that
( kDOn D 2 ) 0
/3n' Un  .
By the Gagliardo-Nirenberg inequality (1.4.11), we have
(2.2.21 )
DOn
JI/3J £00
< C1IIDOniit II D 2 0 n II t + C ) DOn II  0,
JI/3J JI/3J
(2.2.22)
27

and
DUn < C1IIDunlit IID 2 u n lit + C2 11Dunii < C
Ji!3:lLoo Ji!3:I Ji!3:I
(2.2.23)
with C being a positive constant independent of n. Then it turns out from (2.2.22)
and (2.2.23) that
kDOnDu n IIDOnllLoo IIDunllLoo
< fIiiI fIiiI -+ o.
f3n Loo - V lf3n I V lf3n I
(2.2.24 )
Combining it with (2.2.19)-(2.2.21) yields
/I DUn II -+ o.
(2.2.25)
Thus by (2.2.1), we get
DV n 0 . L 2
f3n -+ In .
(2.2.26)
Taking the inner product of (2.2.6) with V n in L 2 and dividing the result by f3n, we
obtain that
illv n ll 2 + (Dun, :n ) -+ o.
Therefore, by (2.2.25)-(2.2.27), we obtain that
(2.2.27)
V n -+ 0 in L 2 .
(2.2.28)
Thus (2.2.28), (2.2.25), and (2.2.10) contradict I/Ynll'H = 1. The proof of Theorem
2.2.1 is complete.
o
2.3 The Exponential Stability for the Stress-Free
Boundary Conditions at Both Ends
In this section we consider the one-dimensional thermoelastic system (2.1.1) and (2.1.2)
with the stress-free boundary conditions. The boundary conditions for the temperature
are the following: either it is insulated at both ends, or one end is insulated and
the other is kept at a constant temperature. We can always translate the constant
28

temperature to zero without changing the form of the system. Mathematically, the
boundary conditions considered in this section are the following:
alx=O,l = (U x - ,0) Ix=O,l = 0, Oxlx=O,l = 0,
(2.3.1)
or
alx=O,l = (U x - ,0) Ix=O,l = 0, 0lx=o = 0, Oxlx=l = o.
(2.3.2)
If both ends are kept at a constant temperature, then the boundary conditions are
separated, i.e., u satisfies the Neumann boundary condition and ° satisfies the Dirichlet
boundary condition. This easy case has been considered by Hansen [1] and others (see
Section 2.1).
We first consider the case of boundary conditions (2.3.1). Notice that both
l' Ut dx and l' (() + ,u,,;}dx
(2.3.3)
are conserved all the time. Hence, we make the following substitutions
,C2 (2.3.4)
Yl u -
x 1 + ,2 '
Y2 Ut - Cl, (2.3.5 )
0- C2 (2.3.6)
Y3 1 +,2
where the constants Cl and C2 are given by
1 {l 1 {l
Cl = 1 Jo u1dx, C2 = 1 J o (Oo+,(uo)x)dx.
(2.3.7)
Then Yl, Y2 and Y3 satisfy the boundary conditions
(Yl - ,Y3) Ix=O,l = 0,
Y3x Ix=O,l = 0,
(2.3.8)
as well as the constraints
l' Y2 dx = 0, l' (Y3 + 'Yl)dx = 0
(2.3.9)
29

and the initial conditions
,C2
Yllt=o = YlO = (UO)x - 1 + ,2 ' Y2lt=o = Y20 = Ul - CI,
C2
Y3lt=o = Y30 = (}O - 1 + ,2 '
(2.3.10)
Let Y = (YI, Y2, Y3)T and
1i 1 = {Y E L 2 X L 2 X L 2 subject to (2.3.9)}
equipped with the norm
1
IIYII'HI = (IlY11l 2 + IIY211 2 + IIY311 2 ) 2 .
(2.3.11)
Define the operator Al : V( AI) C 1i 1 --+ 1i 1 by
DY2
Aly = DYI - ,DY3
-,DY2 + kD2Y3
(2.3.12)
with
V(A I ) = {Y E HI X HI X H 2 I subject to (2.3.8) and (2.3.9)}.
(2.3.13)
Then system (2.1.1), (2.1.2) and (2.3.1) can be reduced to the initial value problem
for a first order evolution equation:
{ dy
dt = AIy,
y(O) = (YIO, Y20, Y30)T.
(2.3.14)
In the same way as in the previous section we get that
Re (AIy, Y)'HI = -kllO x ll 2 < 0
(2.3.15)
holds for any Y E V(A I ). Therefore, the operator Al is dissipative. We now use
Theorem 1.2.4 to prove the following theorem.
Theorem 2.3.1 The operator Al generates a Co-semigroup of contractions in HI.
30

Proof By the dense theorem in Chapter 1, V(A 1 ) is dense in 1i 1 . Therefore, it
remains to prove that 0 E p(A 1 ). For any F = (fl, f2, f3)T E 1it, consider
A 1 y = F, (2.3.16)
I.e. ,
DY2 = fl in L 2 (2.3.17)
,
DYI -,DY3 = f2 In L 2 (2.3.18)
,
-,DY2 + kD2Y3 = f3 in L 2 . (2.3.19)
It follows from (2.3.17) and (2.3.9) that Y2 is uniquely given by
_ {X f ( )d f fox fl (s )dsdx
Y2 - Jo 1 S S - 1 .
(2.3.20)
Integrating (2.3.18) with respect to x, we get
YI - 'YY3 = 1'" hdx + C I .
Then, by (2.3.8) and (2.3.9), CI, !o'Yldx and !o'Y 3 dx can be uniquely determined. To
determine Yl and Y3, we multiply (2.3.17) by" then add the result to (2.3.19) to get
(2.3.21 )
kD2Y3 = f3 + ,fl.
(2.3.22)
Then it follows from (2.3.8) that
kDY3 = 1'" (13 + 'Yfl)dx.
(2.3.23)
Since 1 1 Y3dx has been uniquely determined, Y3 and, therefore by (2.3.21), YI can
be uniquely determined. It is clear that lIyllHl < KIIFIIHI with K being a positive
constant independent of F and y. This proves that 0 E p( AI). Thus the proof is
com plete.
Furthermore, we have the following theorem.
o
31

Theorem 2.3.2 The semigroup Sl(t), generated by the operator Al which is defined
in (2.3.12), is exponentially stable, i. e., there are two positive constants M I , al such
that
IISI(t)1I < M l e- cx1t , Vt > o.
(2.3.24 )
Proof We still use Theorem 1.3.2 to prove this theorem. As for Theorem 2.2.2, the
proof consists of the following steps:
(i) Prove that 0 E p(A I ). This has been done in the proof of Theorem 2.3.1.
(ii) If sup{ll(i,81 - AI)-lll I 1,81 < IIAl"IIl-I} = M < 00, then by the contraction
mapping theorem, the operator i,81 - Al = (i,801 - AI)(I + i(,8 - ,80)( i,801 - AI)-l)
with 1,801 < IIAl"IIl-1 is invertible for 1,8 - ,801 < k . It turns out that by choosing 1,801
as close to IIAl"IIl-1 as we can, we conclude that {,8 11,81 < IIAl"IIl-1 + it } C p(A I )
and II (i,81 - AI)-lll is a continuous function of,8 in (-IIAl"IIl-1 - k , IIAl"III-1 + it ).
(iii) Thus it follows from the argument in (ii) that if (1.3.3) is not true, then there is
w E 1R with IIAl"IIl-1 < Iwl < 00 such that {i,8 11,81 < Iwl} C p(A I ) and sup{ll(i,8-
AI)-lll I 1,81 < Iwl} = 00. It turns out that there exists a sequence ,8n E 1R with
,8n --+ w, l,8nl < Iwl and a sequence of complex vector functions y(n) E V(A I ) with
Ily(n)II11 1 = (1Iyn)1I2 + Ilyn)112 + lIyn)1I2) = 1 such that
II (i,8n I - AI)y(n) 11111 --+ 0,
(2.3.25 )
as n --+ 00, I.e.,
i,8nyn) - Dyn) --+ 0 in L 2 ,
(2.3.26)
i,8nyn) - Dyn) + I Dyn) --+ 0 in L 2 ,
(2.3.27)
i,8nyn) - kD2yn) + I Dyn) --+ 0 in L 2 .
(2.3.28)
In the same way as in the previous section, we take the inner product of (i,8nI - AI)y(n)
with y(n) in 1i 1 and then take its real part to get
kIIDyn) 11 2 --+ o.
(2.3.29)
32

Then it follows from (2.3.27) that
i/3nyn) - Dyi n )  0 in L 2 .
(2.3.30)
Multiplying (2.3.26) by" then adding the result to (2.3.28) yields
i/3n( ,yi n ) + yn») - kD2yn)  0 in L 2 .
(2.3.31 )
(n) (n)
Taking the inner product of (2.3.31) with 'YYl (3: Y3 in L2 and integrating by parts,
we get
ilhyn) + yn)1I2 + :n (Dyn), 'YDyn) + Dyn))  o. (2.3.32)
It can be seen from (2.3.30) that D;:n) is uniformly bounded in L 2 with respect to n.
Thus we can deduce from (2.3.32) and (2.3.29) that
,yi n ) + yn)  0 in L 2 .
(2.3.33)
It follows from Ilyn) II < 1 and (2.3.29) that yn) is uniformly bounded in HI. Hence,
by the compactness of the imbedding of HI into L 2 , there is a subsequence of yn), still
denoted by yn), such that yn) is a Cauchy sequence in L 2 , and by (2.3.29), also in
HI. Let Y3 be its liinit. Then we have DY3 = 0 in L 2 , i.e., Y3 = Const. It follows from
(2.3.33) that yi n ) is also a Cauchy sequence in L 2 and its limit YI is also a constant. It
follows from (2.3.26) that Dyn) is a Cauchy sequence in L 2 . Since lIyn) II < 1, by the
same argument as before, we can conclude that yn), if necessary by choosing another
subsequence, is a Cauchy sequence in HI. We denote the limit by Y2. Consequently,
from (2.3.27) we conclude that yi n ) is also a Cauchy sequence in HI. Taking the limit
in (2.3.26) and (2.3.27) yields
iWYI - DY2 = 0
iWY2 = DYI = 0
in L 2
,
in L 2 .
(2.3.34 )
(2.3.35)
Thus, YI = Y2 = 0 and by (2.3.30) and (2.3.33), Y2 = Y3 = o. It contradicts IIYIII 2 +
IIY211 2 + IIY311 2 = 1. Thus the proof for (1.3.3), i.e., ilR C p(A I ), is complete.
33

(iv) We now prove (1.3.4) by a contfadiction argument again. Suppose that (1.3.4)
is not true. Then there is a sequence of complex vector functions y(n) E V(A 1 ) with
lIy(n) II 'HI = (1Iyi n )11 2 + lIyn)1I2 + Ilyn)1I2)t = 1 and a sequence (3n E III such that
l,Bn I ---+ 00 and
II (i,Bn I - A1)y(n) II 'HI ---+ 0,
(2.3.36)
as n ---+ 00, I.e.,
i,Bnyi n ) - Dyn) ---+ 0
in L 2
,
(2.3.37)
i,Bnyn) - Dyi n ) + ,Dyn) ---+ 0
in L 2
,
(2.3.38)
i,Bnyn) - kD2yn) + ,Dyn) ---+ 0
in L 2 .
(2.3.39)
As the same as before, we again have (2.3.29) and (2.3.33). Again there is a subse-
quence of yn), still denoted by yn) such that yn) is a Cauchy sequence in HI with
the limit Y3 = Const. and yi n ) is a Cauchy sequence in L 2 with the limit Yl = Const.
(n) (n)
Since Y;n -+ 0 in L 2 , it follows from (2.3.37) that Y;n is a Cauchy sequence in HI,
whose limit must be zero, and
Yl = O.
(2.3":40 )
It can be seen from (2.3.38) that in order to prove that yn) is a Cauchy sequence in
L 2 . h h 1 .. . ffi h Dy1n) . L 2 W
WIt t e Imlt zero, It su ces to prove t at ,Bn converges to zero In . e can
rewrite (2.3.38) as
D (n) D (n)
. (n) Yl -, Y3 0 I . n L 2 .
Y2 - ,Bn ---+
(2.3.41 )
D (n) D (n)
Taking the inner product of (2.3.41) with YI n' Y3 in L 2 and integrating by
parts yields
D (n) li D (n) D (n) 11 2
. ( Y2 (n) (n» ) Yl -, Y3 0
-  ,Bn ' Yl - 'Y3 - ,B ---+ .
By (2.3.37), we have
(2.3.42)
liD (n) D (n) 11 2
( (n) (n) (n» ) Yl -, Y3 0
Yl , Yl - 'Y3 - ,B ---+ .
(2.3.43)
34

D (n) D (n)
Because yn) --+ 0 and ;: --+ 0 in L 2 , we get that ;: --+ 0 in L 2 . Thus we can
deduce from (2.3.38) that
yn) ---+ 0 in L 2 .
(2.3.44 )
It follows from (2.3.37) and yn) --+ 0 in L 2 that D;:n) --+ 0 in L2. Taking the inner
(n)
product of (2.3.39) with y;n in L 2 , integrating by parts and using (2.3.29), we obtain
that yn) ---+ 0 in L 2 . Thus we again have a contradiction and the proof of the theorem
is complete.
o
We now consider the case of boundary conditions (2.3.2). In this case, only fci Utdx
is conserved. Thus, we make the substitution
v = Ut - CI
(2.3.45 )
where CI is the same constant as defined in (2.3.7). Let Y = (YI, Y2, Y3)T = (u x , V, O)T
and define the operator A 2 the same as Al except that
V(A 2 ) = {y E HI X HI X H 2 I subject to (2.3.2) and l vdx = o} .
Then system (2.1.1)-(2.1".3) and (2.3.2) can be reduced to the initial value problem
(2.3.46)
for a first order evolution equation:
{ dy
dt = A 2 y,
y It=o = (( uO)x, UI - CI, Oo)T.
(2.3.47)
Let
1-£2 = {Y E L 2 X L 2 X L 2 I 1" Y2 dx = 0 }
(2.3.48)
equipped with the usual L 2 x L 2 X L 2 norm. Then it can be proved in the same way
as before, which can be omitted here, that the operator A 2 is dissipative. Moreover,
o E p(A 2 ). Thus from Theorem 1.2.4 in Chapter 1 we can deduce the following.
Theorem 2.3.3 The operator A 2 generates a Co-semigroup of contractions in 1i 2 .
Similarly, we have the next theorem.
35

Theorem 2.3.4 The semigroup S2(t), generated by the operator A 2 which is defined
in (2.3.12), is exponentially stable, i.e., there are two positive constants M 2 , CX2 such
that
IIS2(t)11 < M 2 e- cx2t .
(2.3.49)
Proof The method of the proof here is essentially the same as that used in the proof of
Theorem 2.3.2 except for some slight modifications due to the changes in the boundary
conditions and constraints. Since Olx=o = 0, we can apply the Poincare inequality to 0
so that the proof of this theorem becomes simpler. In order to avoid redundant work,
we will only point out the differences from the proof of Theorem 2.3.2.
(i) We still have 0 E p(A 2 ) as mentioned before.
(ii) If (1.3.3) is not true, then there is w E 1R with IIA2"III-I < Iwl < 00 such that
{i,8 11,81 < Iwl} C p(A 2 ) and sup{lI(i,B - A 2 )-IIIII,B1 < Iwl} = 00. It turns out that
there exists a sequence ,Bn E 1R with ,Bn --+ w, l,Bnl < Iwl and a sequence of complex
vector functions y(n) E V(A 2 ) with Ily(n)II'H2 = (1Iyn)1I2 + Ilyn)112 + Ilyn)112) = 1 such
that
II (i,Bn I - A 2 )y(n) 11'H2 --+ 0,
(2.3.50)
as n --+ 00, I.e.,
i,Bnyn) - Dyn) --+ 0
in L 2
,
in L 2 .
(2.3.51 )
(2.3.52)
(2.3.53)
i,Bnyn) - Dyn) + ,Dyn) --+ 0
i,Bnyn) - kD2yn) + ,Dyn) --+ 0
in L 2
,
Thus we have
kIIDyn) 11 2 --+ 0,
(2.3.54 )
and by the Poincare inequality,
yn) --+ 0 in HI.
(2.3.55 )
In the same manner as before, we have
,yi n ) + yn) --+ 0 in L 2 .
(2.3.56)
36

Therefore,
yi n ) -+ 0 in L 2 .
(2.3.57)
By (2.3.51), we have
D y (n) --+ 0 in L 2
2 ,
(2.3.58)
and by the Poincare inequality,
yn) --+ 0 in L 2 .
(2.3.59)
Thus ilR C p( A 2 ) is proved.
(iii) We now prove (1.3.4) by a contradiction argument again. Suppose that (1.3.4)
is not true. Then there is a sequence of complex vector functions yen) E V(A 2 ) with
lIy(n) 1I'H2 = (lIyn) 11 2 + Ilyn) 11 2 + Ilyn) 112)! = 1 and a sequence /3n E IR such that
l/3nl --+ 00 and (2.3.51)-(2.3.53) hold. Again we have (2.3.55) and (2.3.57). It can be
D (n)
seen from the proof of Theorem 2.3.2 that we still have ; -+ 0 in L 2 and
yn) --+ 0 in L 2 .
(2.3.60 )
Thus, we again have a contradiction and the proof of the theorem is complete. 0
2.4 The Exponential Stability for the Stress-Free
Boundary Conditions at One End
In this section, we turn to the cases that one end of the rod is rigidly clamped, and
the other end is free. This, together with the thermal boundary conditions (insulated
or constant temperature), leads to the following possible boundary conditions:
ulx=o = 0, a/x=l = 0, Oxlx=O,1 = 0,
(2.4.1 )
ulx=o = 0, alx=l = 0, Olx=O,1 = 0,
(2.4.2)
(2.4.3)
ulx=o = 0, alx=l = 0, Olx=o = 0, Oxlx=l = 0,
and
ulx=o = 0, alx=l = 0, Oxlx=o = 0, Olx=l = o.
(2.4.4 )
37

Boundary condition (2.4.4) has been considered by Hansen [1]. He used the method
of combining the Fourier series expansion with decoupling technique. However, his
method failed for boundary conditions (2.4.1), (2.4.2), and (2.4.3) since the system
(2.1.1)-(2.1.3) with these boundary conditions cannot be decoupled.
Let Hl = {f(x) E HI : f(O) = O} and
11.3 = {Y E Hl X L 2 X L 2 Il(Y3 + 'Ylx)dx = o}, 11.4 = 11.5 = Hl X L 2 X L 2 (2.4.5)
equipped with the norm
1
lIylI1i] = (IIDYlIl 2 + IIY211 2 + IIY311 2 ) 2 , j = 3,4,5.
(2.4.6)
In the case of boundary condition (2.4.1), to translate the equilibrium to zero, we make
the following substitution:
,C2
Yl = U - 1 + ,2 X,
C2
Y2 = Ut, Y3 = 8 - 1 + ,2 '
(2.4.7)
where C2 is the constant defined by (2.3.7) in Section 2.2 of this chapter. Then Yl, Y3
satisfy
Yll x=o = 0, (Ylx - ,Y3) IX=I = 0, Y3x 1 x=O,1 = 0, l (Y3 + ,Yl x) dx = o.
(2.4.8)
Define the operators A j : V( A j ) c 1-(,j  1-(,j, j = 3,4,5 by
0
Aj= D2
0
I 0
o -,D
-,D kD 2
(2.4.9)
with
V(A 3 )
V(A 4 )
V(As)
{ (Yt, Y2, Y3)T E H 2 x Hl X H 2 I subject to (2.4.8)} ,
{ (u, Ut, 8)T E H 2 x Hl X H 2 I subject to (2.4.2) } ,
{ (u, Ut, 8)T E H 2 x Hl X H 2 I subject to (2.4.3) } .
(2.4.10)
(2.4.11)
(2.4.12)
38

Thus, the system (2.1.1)-(2.1.3) with the boundary condition (2.4.1), (2.4.2), (2.4.3),
respectively, can be reduced to the initial value problem for a first order evolution
equation:
{  = Ajy, j = 3,4,5,
y(O) = Yo.
(2.4.13)
where y = (Y1, Y2, Y3)T if j = 3; Y = (u, Ut, 8)T if j = 4,5. In the same manner
as before, we can prove that A j , (j = 3,4,5) is the infinitesimal generator of a C o -
semigroup Sj (t) of contractions on the Hilbert space 1-lj. Since the method is exactly
the same, we omit the detail here.
Theorem 2.4.1 The semigroup Sj(t) (j=3,4,S), associated with (2.4.13), is exponen-
tially stable, i. e., there exist constants M j > 0, aj > 0 such that
IISj(t)1I < Mje-cx]t, V t > o.
(2.4.14)
Therefore, the solution Y of (2.4.13) satisfies
lIu x (t)1I2 + Il u t(t)1I2 + 118(t)1I 2
< M j (II (uo)xIl 2 + IIul112 + 11(0112) e-cx]t, if j = 4,5,
(2.4.15)
and
lIu,.( t) - 1 :2 112 + I!Ut(t) 11 2 + 118( t) - 1 :2,2 11 2
< M j (11(uo)xIl 2 + IIUl112 + 11(0112) e-cx]t, if j = 3
(2.4.16)
where M 3 is a constant depending only on M3 and,.
Proof Since the proof is quite similar to Theorem 2.2.1 and Theorem 2.3.2, we will
only point out the major differences:
(i) We still have 0 E p( A j ), j = 3,4, 5.
(ii) For fixed j = 3,4,5, if (1.3.3) is not true, then there is w. E 1R with IIAj111-1 <
Iwl < 00 such that {i,8 11,81 < Iwl} c p(A j ) and sup{ II (i,8 - A j )-11111,81 < Iwl} = 00.
It turns out that there exists a sequence ,8n E 1R with ,8n  w, l,8nl < Iwl and
39

a sequence of complex vector functions yen) E V(Aj) with lIy(n) II'Hj = (IIDyn) 11 2 +
IIY)112 + lIyn)1I2)! = 1 such that
II (i,Bn I - A)y(n) II'H,  0,
(2.4.17)
as n  00, I.e.,
i,Bnyn) - yn)  0 in HI ,
(2.4.18)
i,Bnyn) - D2yn) + ,Dyn)  0
i,Bnyn) - kD2yn) + ,Dyn)  0
in L 2
,
in L 2 .
(2.4.19)
(2.4.20)
Therefore, we still have
kIlDyn) 11 2  o.
(2.4.21 )
For j = 3, we have
,Dyn) + yn)  0 in L 2 .
(2.4.22)
As in the proof of Theorem 2.3.2, there exists a subsequence of yn), still denoted by
yn), such that yn) is a Cauchy sequence in HI with its limit Y3 being a constant.- It
turns out from (2.4.22) that Dyn) is also a Cauchy sequence in L 2 . Since yn) Ix=o = 0,
by the Poincare inequality, yn) is a Cauchy sequence in L 2 , and also in HI. Let YI
be its limit. Then DYI = Const. By (2.4.18), yn) is also Cauchy sequence in HI. It
follows from (2.4.19) that D2yn) is a Cauchy sequence in L 2 . Thus yn) is a Cauchy
sequence in H 2 with D 2 YI = o. Let Y2 be the limit of yn). By passing to the limit in
(2.4.19) and (2.4.18), we get Y2 = 0 and YI = 0, and by ,DYI + Y3 = 0, we deduce that
Y3 = O. Thus, it contradicts IIDYIII 2 + IIY211 2 + IIY31\2 = 1 and the proof of (1.3.3), i.e.,
iIR c p( A 3 ) is complete.
For j = 4, 5, because of the boundary conditions, we can apply the Poincare in-
equality to YI and Y3. It follows from (2.4.21) that yn)  0 in HI. Then we can
deduce from (2.4.20) and (2.4.19) that IID2yn)11 and IID2yn)1I are bounded. Multiply-
ing (2.4.18) by " then adding the result up to (2.4.20) yields
i,Bn,Dyn) - kD2yn)  0 in L 2 .
(2.4.23)
40

Multipiying (2.4.23) by Dyn) and integrating by parts, we get
i,Bn,IIDyn) 11 2 + k(Dyn), D2yn)) - kDyn) Dyn) Ix=l + kDyn) Dyn) Ix=o  O. (2.4.24)
By the Gagliardo-Nirenberg inequality,
IIDyllLoo < C I IID 2 YlltilDYlit + C 2 I1Dyli.
(2.4.25)
Thus, from (2.4.21) we can deduce that two boundary terms in (2.4.24) converge to
zero. Therefore, it follows from (2.4.24) that
Dyn)  0 in L 2 .
(2.4.26)
By the Poincare inequality, we obtain that
yn)  0 in L 2 .
(2.4.27)
Therefore, it follows from (2.4.18) that
yn)  0 in L 2 .
(2.4.28)
A contradiction again. Thus iIR c p( A j ), j = 4,5.
(iii) We now prove (1.3.4) by a contradiction argument. For fixed j = 3,4,5, suppose
that (1.3.4) is not true. Then there is a sequence of complex vector functions y(n) E
V( A j ) with lIy(n) 11ft] = (II Dyn) 11 2 + lIyn) 11 2 + lIyn) 112) t = 1 and a sequence ,Bn E IR
such that l,Bnl  00 and (2.4.18)-(2.4.20) hold. Therefore, we still have (2.4.21).
For j = 3, we have (2.4.22) and a subsequence of yn) and yn), still denoted by yn) and
yn) such that yn)  Y3, yn)  YI in HI with Y3 being a constant and DYI = Const.
It follows from (2.4.18) that
yn)o inL2.
(2.4.29)
Thus, YI = 0, and it turns out that Y3 = 0 also holds. Dividing (2.4.19) by,Bn yields
D 2 (n)
. (n) YI 0
Y2 - 
,Bn
in L 2
(2.4.30)
41

which implies that In D2yn) is uniformly bounded. Taking the inner product of (2.4.30)
with yn) in L 2 and integrating by parts yields
D (n) D (n) D (n)
i ll \n) 11 2 + ( D (n) Y2 ) _ YI _(n) I + YI -(n) I 0
Y2 YI '!3n !3n Y2 x=l !3n Y2 x=O  .
(2.4.31 )
By the Gagliardo-Nirenberg inequality, we have
IIDyn) 11£00 < C IID2yn) II  liD (n) II  + C IIDyn) II
/iAJ - 1 /iAJ YI 2 /iAJ '
(2.4.32)
and
lIyn)liLoo < cI IIDyn)lI lIyn)lI + c2 I1yn)lI .
/iAJ /iAJ /iAJ
(2.4.33)
Thus two boundary terms in (2.4.31) converge to zero. It is clear that the second term
in (2.4.31) also converges to zero. Therefore,
yn)  0 in L 2 .
(2.4.34 )
A contradiction again. Thus the proof for the case j = 3 is complete.
For j = 4,5, we still have yn)  0 in HI and we can deduce from (2.4.20) and
D 2 y(n) D 2 y(n)
(2.4.19) that II !3n 3 II and II !3n I II are uniformly bounded. Now (2.4.24) should be
changed into
D 2 (n) D-(n) I D-(n) I
i'YIIDyn) 11 2 + k(Dyn), I ) - kDyn) ypn x=l + kDyn) ypn x=O -+ O. (2.4.35)
Applying the Gagliardo-Nirenberg inequality, we get (2.4.26) and (2.4.27) again. It
follows from (2.4.19) that
D 2 (n)
. (n) YI 0
Y2 - 
!3n
in L 2 .
(2.4.36)
Then making the same argument as for (2.4.30)-(2.4.34) yields a contradiction again.
Thus the proof for j = 4, 5 is complete. 0
Remark 2.4.1 As proved, for instance, in Rivera & Racke [lJ and Liu & Yong [lJ, the
semigroup associated with linear one-dimensional thermoelastic system is not analytic.
42

2.5 The Therrnoelastic Kirchhoff Plate Equations
In this section we consider the thermoelastic Kirchhoff plate equations. The equations
describing a linear thermoelastic Kirchhoff plate of homogeneous material (see Lagnese
[3]) are the following:
{ Wtt - ,Wtt + 2W + o:() = 0 In n x IR+,
(3()t - TJ() + a() - O:Wt = 0 in n x IR+
(2.5.1 )
where n is a bounded region in]R2 with smooth boundary r r O ur 1 ur 2 and
r 0 U r 1 =/:. 0, f' 0 n f\ n f' 2 = 0; W represents the vertical deflection, and () represents
the relative temperature about the stress free state () = 0; 0:, (3, "I, a > 0" > 0 are
given constants. The boundary conditions considered for system (2.5.1) are often the
following:
(i) the structural boundary conditions
8w
W = - = 0 on ro (clamped edge),
8v
(2.5.2)
W = w + (1 - J.l)Bl W + o:() = 0 on r 1 (simply supported edge), (2.5.3)
{ w + (1 - J.l)B 1 w + o:() = 0,
8w ( ) 8B2W 8Wtt 8fJ 0 on r 2 (free edge), (2.5.4)
8v + 1 - J.l 8T -, 8v + 0: 8v =
where v = (VI, V2) is the unit outward normal vector, and T = (-V2, VI) is a unit
tangent vector to r; J.l is the Poisson ratio with 0 < J.l < ; B 1 and B 2 are the
boundary operators defined by
B 1 w
2 22
VI V 2 W xy - VI W yy - V 2 W xx ,
B 2 w
(V; - vi)w xy + VIV2(Wxx - W yy ),
(2.5.5 )
(2.5.6)
(ii) the temperature boundary condition
8()
AITJ 8v + A2() = 0 on r
with AI, A2 > 0, Al + A2 =/:. O. Al = 0 corresponds to zero temperature; A2 = 0
(2.5.7)
corresponds to zero flux; AI, A2 =/:. 0 corresponds to the Newton's cooling law.
43

In this section we will prove that the corresponding semi group to (2.5.1) with
, > 0 subject to the boundary conditions (2.5.2) and (2.5.3) (i.e., r 2 = 0), (2.5.7) is
exponentially stable. We will also prove that the corresponding semigroup to (2.5.1)
with, = 0 subject to the boundary conditions (2.5.2)-(2.5.4) (i.e., r 2 =I 0 is allowed)
and (2.5.7) is not only exponentially stable, but also analytic.
Let us first recall some related works in the literature. As far as the exponential
stability is concerned, Lagnese in his book (Lagnese [3] published in 1989) proved
that the semi group associated with the thermoelastic Kirchhoff plate equations is
exponetially stable provided that there are additional dissipation terms acting on the
boundary. He thought, as he wrote in his book that for higher dimensional linear
thermoelastic systems, including the system for linear thermoelastic plates, "it seems
unlikely" that the associated semigroup will be expoenetially stable "unless additional
dissipative mechanisms (such as boundary damping) are included" (p. 154). It is
certainly true for the three-dimensional thermoelastic system. However, as far as the
two-dimensional thermoelastic Kirchhoff plate equations are concerned, the situation
is different from what he expected. Actually, the case, = 0 and, > 0 corresponds to
very different dynamics ("parabolic" vs. "hyperbolic" in nature). It is now clear that
when, = 0, the semi group associated with thermoelastic equation (2.5.1) is not only
exponentially stable, but also analytic. However, in the case, > 0, the semigroup is
only exponentially stable, neither compact nor differentiable.
When, = 0, i.e., the rotational force is neglected in the plate equation (2.5.1),
Kim [1] in 1992 considered the problem with the clamped edge and zero temperature
boundary conditions and proved that the corresponding semigroup is exponentially
stable. Rivera & Racke [1] in 1995 considered the problem with boundary conditions
w = w = () = 0, and essentially proved the exponential stability. Liu & Zheng [5],
published in 1997, proved the exponential stability for the problem with the bound-
ary clamped on ro, simply supported on r 1 (with ro u r 1 = r, f\ n f\ = 0), and
the Newton's cooling law temperature boundary condition (2.5.7), essentially using
44

the method described in this book. However, the problem with free edge boundary
condition was left open in their papers. The results in Liu & Zheng [5] were reproved
in Avalos & Lasiecka [1] by a different method. Recently, Lasiecka & Triggiani [2]-[5]
obtained the results on exponential stability as well as analyticity of the correspond-
ing semigroup associated with (2.5.1) subject to all boundary conditions mentioned
above, even including the free edge boundary condition (2.5.4). Their appoarch is very
different from the systematic method described in this book.
As far as analyticity is concerned, we notice that in the present situation once
analyticity is established, exponential stability can be obtained by excluding the pos-
sibility that the generator of the Co-semigroup of contractions has spectrum on the
imaginary axis. In addition to the very recent works by Lasiecka & Triggiani men-
tioned previously, we should also refer to some earlier works in this direction. We
refer to Russell [2] for the problem with the boundary conditions w = w = () = o.
We also refer to Liu & Renardy [1] for the result on the problem with the clamped
edge and zero temperature boundary conditions and Liu & Liu [2] for the problem
with boundary conditions (2.5.2) and (2.5.3) and the Dirichlet boundary condition for
temperature (). In Liu.& Yong [1], the contradiction argument for proving exponential
stability was extended to studying analyticity and other semi group properties.
When, > 0, i.e., the rotational force is also taken into consideration, the first
result on exponential stability was given in Avalos & Lasiecka [1] for the problem with
boundary conditions (2.5.2), (2.5.3) and (2.5.7) (i.e., r 2 = 0). When r 2 = 0 and
the temperature satisfies the Dirichlet boundary condition, the exponential stability
was proved in Liu & Liu [2]. Recently, Avalos & Lasiecka [2] succeeded in proving
the exponential stability for the problem with free edge boundary condition (2.5.4)
(i.e., r 2 =I 0) using a sharp trace theorem given in Tataru [1]. As far as anylycity is
concerned for the case, > 0, as Lasiecka & Triggiani [3], [6] and Chang & Triggaini
[1] recently pointed out, the associated linear semi group is not analytic, even neither
compact nor differentiable.
45

In what follows we first consider the linear thermoelastic Kirchhoff plate equation
(2.5.1) subject to the following initial condition:
Wlt=o = wo(x, y), wtlt=o = Wl(X, y), Blt=o = Bo(x, y)
(2.5.8)
and the boundary conditions (2.5.2), (2.5.3), and (2.5.7) (i.e., r 2 = 0). The presen-
tation here essentially follows the line of argumentation in Liu & Liu [2]. Let us first
convert this problem into a variational evolution equation. For complex functions u, v
defined on 0, we define the following Sobolev spaces and the associated bilinear forms:
Vi
{ 2 au }
U E H (0) I U = 0 on r, a1/ = 0 on ro ,
[uxxvxx+UyyVyy+(UXXVyy+Uyyvxx)
+2(1 - p.)uxyvxy]dxdy, \/ u, v E Vi;
aI(U,v)
HI
HJ(O),
 (rV'uV'v + uv)dxdy, V u, v E HI;
CI ( u, v )
Vi
HI(O),
 (-qV'uV'v + auv)dxdy + !r Auvdf, V u, v E V2i
a2(U,v)
H 2
C2 ( u, v )
L(O),
 (3uvdxdy, V u, v E H 2 o
It is clear that Vi, H i ( i = 1,2) are Hilbert spaces with the corresponding inner products
ai, £;(i = 1,2), and Vi, respectively are continuously and densely imbedded into Hi(i =
1, 2). Furthermore, the imbedding operators are compact. By the results in Section
1.4.5 of Chapter 1, there exist self-adjoint positive definite operators Ai in H i ( i = 1,2),
respectively such that
1 1 1
V(A;) = Vi, ai(u,v) = ci(A;u,Alv), \/u,v E Vj, j = 1,2.
(2.5.9)
46

To motivat the abstract formulation of our problem, suppose that {w, 8} is a
regular solution of our problem (2.5.1) and (2.5.8)-(2.5.11). The pair {w, 8} is then a
solution of the variational equation
CI(Wtt(t), w) + c2(8t(t), 8) + aI(w(t), w) + a2(8(t), 8)
I
+ C2 ( 8 ( t ), B w) - C2 ( B Wt ( t ), 8) = 0, V w E Vi, 8 E V2, t > 0 (2.5.10 )
where the operator B from HI to H 2 is defined as
a vu E V(B) = Vi. (2.5.11)
Bu = -u
(3 ,
Let
Wt = v zn Vi, Vt > 0, (2.5.12)
and take 8 = 0 in (2.5.10). Then we obtain
CI(Vt(t), w) + aI(w(t), w) + c2(8(t), Bw) = 0, Vw E Vi, t > o.
(2.5.13)
Let
1
W = Afw.
(2.5.14)
1
Then w E HI and we can re"write c2(8(t),Bw) as c2(8(t),BA2W). It is clear that
1
the operator BA2 is a linear continuous operator from HI to H 2 . Thus by the
1
definition of adjoint operator in functional analysis, c2(8(t),BA2W) can be rewritten
1 1 l' 1 1
as cI((BA2)*8(t),w) = cI((BA2)*8(t),Afw) = cI(AI(BA2)*8(t),w). Since Vi is
dense in HI, it turns out from (2.5.13) that
1 1
Vt = -AIw - Af(BA2)*8 zn HI, Vt > O.
(2.5.15)
If we take w = 0 in (2.5.10), since V2 is dense in H 2 , we obtain that
8t(t) = Bv - A 2 8 in H 2 , Vt > O.
(2.5.16)
Let z = (w, v, 8)T and 1-(, = Vi X HI X H 2 with the corresponding inner product.
Then combining (2.5.13), (2.5.15) and (2.5.16) yields the following first-order evolution
equation in 1-(,:
dz
-=Az
dt
(2.5.17)
47

with
1 1
V(A) = {z = (W,V,O)T E H I V E Vi,O E V(A 2 ),Alw + (BA2)*O E Vi}, (2.5.18)
and
v
Az=
1 1 1
-AI[Alw + (BA2)*O]
Bv - A 2 0
(2.5.19)
1
where (BA2)* E £(H 2 , HI). Thus this first-order evolution equation can be consid-
ered as the weak formulation of our problem (2.5.1) and (2.5.8)-(2.5.11).
Theorem 2.5.1 A generates a Co-semigroup, S(t) = eAt, of contractions on H.
Moreover, 0 E p(A) with p(A) being the resolvent set of A.
Proof For any F = (f,g, h)T E H, the equation Az = F has an unique solution
z = (w, v, O)T:
v = f,
o = A 2 1 (Bf - h),
1 1
W = -Allg - A2(BA2)*A21(Bf - h).
(2.5.20)
It is easy to verify directly that z E V( A) and
IIZIl1i < KII(f,g,h)TII1i
(2.5.21 )
with some K > 0 independent of F. Therefore, A is closed and 0 E p(A). The
dissipativeness of A can be seen in the following calculation:
1 1
Re(Az,z)1i - Re[al(v,w) - al(w,v) - cI((BA2)*O,Alv) - a2(O,O) + c2(Bv,O)]
- a2 ( 0 , 0) + Re [C2 ( B v, 0) - C2 ( 0, B v ) ]
-a2( 0,0) < O. (2.5.22)
Thus, the conclusion of this theorem immediately follows from Theorem 1.2.4 in Chap-
ter 1.
48
o

Corollary 2.5.1 Let (w(t), v(t), O(t))T = eAtzo. If Zo E V(A), then
{ wE C 2 ([0, 00); HI) n CI([O, 00); Vi),
o E CI([O, 00); H 2 ) n C([O, 00); V(A 2 ))
(2.5.23)
and they satisfy the variational equation (2.5.10).
If Zo = (wo, vo, Oo)T E rt, then
W(.) E CI([O, 00); HI) n C([O, 00); Vi),
0(.) E C([O,00);H 2 ),
v(.) = Wt(.)
(2.5.24 )
and z is a mild solution to the first-order evolution equation (2.5.17). Moreover, using
the dense argument as before, we can easily show that 0 also belongs to L 2 ([0, 00), V2)
and w, 0 satisfy the following weak variational equation: for all w E Vi, 0 E V2 and
t > 0,
CI(Wt(t), w) - CI(Wt, w) + C2(O(t), 0) - C2(OO, 0) - c2(Bw(t), 0) + C2(BwO, 0)
+ l (a!(w(r), w) + a2(O(r), 9) + c2(O(r), Bw))dr = Q. (2.5.25)
Concerning exponential stability, we have the following.
Theorem 2.5.2 The semigroup S(t), generated by the operator A, is exponentially
stable, i. e., there are two positive constants M, a such that
II S (t) II < Me-ext.
(2.5.26)
Proof We still use Theorem 1.3.2 to prove this theorem. The proof consists of several
steps:
(i) To prove that 0 E p( A). This has been done in the previous theorem.
(ii) If sup{lI(iI'I - A)-III I 11'1 < IIA-III- I } = K < 00, then by the contraction
mapping theorem, the operator i(31 - A = (i(3oI - A)(I + i((3 - (3o)(il'oI - A)-I)
with 1(301 < IIA-III- I is invertible for 1(3 - (301 <  . It turns out that by choosing 11'01
as close to IIA-III- I as we can, we conclude that {(3 1 1(31 < IIA-III- I +  } c p(A)
49

and II (iJ3[ - A)-III is a continuous function of 13 in (-IIA-III- I - k , IIA-III- I + k ).
Thus it follows from the above argument that if (1.3.3) is not true, then there is
w E 1R with IIA-III- I < Iwl < 00 such that {iJ3 11131 < Iwl} c p(A) and sup{lI(iJ3-
A)-III I 1131 < Iwl} = 00. It turns out that there exists a sequence J3n E 1R with
J3n --+ w, IJ3nl < Iwl and a sequence of complex vector functions Zn E V(A) with
II Z nll1i = (lIwnll1 + Ilvnllk l + II O nllk 2 )t = 1 such that as n --+ 00,
II (iJ3n - A)znll1i --+ 0, (2.5.27)
I.e. ,
iJ3n w n - V n --+ 0 In Vi, (2.5.28)
I
iJ3n v n + Af Yn --+ 0 In HI, (2.5.29 )
iJ3n O n - BV n + A 2 0 n --+ 0 In H 2 (2.5.30)
III
where Yn = Afw n + (BA2)*On E V(Af). It follows from (2.5.27) that
I
Re((ipn - A)zn, zn)1i = IIAOnllk2 --+ o.
(2.5.31 )
I
Therefore, On converges to zero in H 2 and (BA2)*On converges to zero in HI. We add
up the inner product of (2.5.28) with V n in HI with the inner product of (2.5.29) with
W n in HI to get
I
- 2 2
IIAfwnliH I -lIvnllHI --+ o.
(2.5.32)
This, combinning with the fact that IIZnll1i = 1 and IIOnllH2 --+ 0, leads to
lim IIAf unllk = lim IIvnllk = _ 2 1 .
n-..oo I n-..oo I
(2.5.33)
Furthermore, by (2.5.29) we also have
I 2
lim AfYn 1
n-..oo J3n - 2.
HI"
(2.5.34 )
Dividing (2.5.30) by J3n, from (2.5.31) we obtain
_ BV n + A20n  0
---r In H 2 .
J3n J3n
(2.5.35 )
50

1
Since B E £(Vi, H 2 ) and (BA2)* E £(H 2 , HI), we have
1 1
BA2(BA2)* E £(H 2 ).
(2.5.36)
Thus, it follows that
1 1
BA2 (BA2)*(}n ---+ 0 In H 2 .
(2.5.37)
_1 _1 v n
Inserting the term BAI 2 (BA 1 2 )*{}n into (2.5.35) and using (2.5.28) to replace /3n by
iw n , we obtain that
. BA - [A t (BA - ) *L) ] A2(}n A 2 (}n . BA - O . H
-  I I W n + I un +  =  -  I Yn ---+ In 2.
(2.5.38)
1
N ow we want to prove that A BAll E £( HI, H 2 ). By the definition of Vi and Hi (i =
1,2), it amounts to proving that BAll E £(H I , HI). By the definition of the operator
B, it suffices to prove that V(A I ) E H3(n).
Let W E V(AI),u = Alw. Then we have
al( W, v) = Cl (u, v) = J/ '"fV'uV'v+uv)dxdy = 10 u(l-'"f)vdxdy, \Iv E Vi. (2.5.39)
Therefore, w is a solution of the following elliptic boundary value problem:
2W = (1-/)u E H- I ,
· aw
w=-=O, onf o ,
a1/
w = u + (1 - p,)Blw = 0, on fl.
(2.5.40)
Since f' 0 n f'l = 0, by the regularity results on the elliptic boundary value problems
(see Lions & Magenes [1] or Chapter 1 in this book), we obtain that w E H 3 (n). This
1
proves that ABAl1 E £(H I ,H 2 ).
Thus, it follows from (2.5.31) and (2.5.34) that
( A2(}n BA - ) - (A L) A t BA -I AfYn ) 0
a' I Yn - 2 un, 2 I a ---+ .
n  n 
(2.5.41 )
By (2.5.38) and (2.5.41), we have
1
BA2 Yn ---+ 0 In H 2 ,
(2.5.42)
51

which leads to
BW n --+ 0 in H 2 o
(2.5.43)
Again by the standard estimates in the elliptic PDE theory, as can be seen from
Chapter 1 or from the book by Lions & Magenes [1], we have
IIwilv I < GIIwIlL2 < GIIB w llH2 --+ 0
(2.5.44 )
which contradicts (2.5.34). Thus, (1.3.3) is proved.
(iii) The proof for (1.3.4) is exactly the same as in (ii).
Thus the proof is complete.
o
Remark 2.5.1 As mentioned in Lasiecka & Triggiani [2]-[.4], the semigroup consid-
ered in the above is not analytic.
In what follows we consider the equations (2.5.1) with / = 0 subject to the bound-
ary conditions (2.5.2)-(2.5.4) (i.e., f 2 =I 0 is allowed), (2.5.7) and the initial conditions
(2.5.8).
Let
2 { 2 8w } (
Hro (0) = w Iw E H (0), wlro = 811 1ro = 0 , 2.5.45
Hf l (0) = {w Iw E HI(O), Wirl = OJ, (2.5.46)
[ { 8 2 W 82W 8 2 w 8 2 w 8 2 w 8 2 w 8 2 w 8 2 w 8 2 w 8 2 w }
a(w, w) = 1n ax 2 ax 2 + ay2 ay2 +JL ax 2 ay2 +JL ax 2 ay2 +2(1 - JL) axay axay an
(2.5.47)
and
H = Hfo(O) n Hf l (0) x L 2 (0) x L(O)
(2.5.48)
equipped with the inner product: for any z = (w,v,O)T,z = (w,v,O)T E H,
(z,Z)rt = a(w,w) + (v, v) + (3(0,0).
(2.5.49)
If we denote Wt by v, then the initial boundary value problem (2.5.1 )-(2.5.8) can be
reduced to the initial value problem for a first-order evolution equation:
{ dz(t) = Az ( t )
dt '
z(O) = Zo = (wo, WI, Oo)T
(2.5.50)
52

with
0 I 0
A= _2 0 -a (2.5.51 )
0  (-aI+7J)
(3
and
V(A) = { z E H W E H4(f!) n Hfo(f!) n Hf, (f!), v E Hfo(f!) n Hf, (f!), } .
o E H 2 (0), w, 0 satisfy (2.5.3)-(2.5.8)
(2.5.52)
We recall the following Green's formula (see Lagnese [3]): for any w E H 4 , W E H2,
iY).2w )wdf!
- a(w; w) + £ {[ ovw + (1 - JL) o;w ] w - [w + (1 - JL)BIW]  } dr.
(2.5.53)
Then we have the following theorem.
Theorem 2.5.3 The operator A defined by (2.5.51) and (2.5.52) is the infinitesimal
generator of a Co-semigroup S(t) of contractions on 1i = Hfo (0) n Hf 1 (0) x L 2 (0) x
L 2 (0) .
Proof It is clear that V(A) is dense in ft. For any z = (w, v, O)T E V(A),
Re(Az, z)7-£
- Re {a(v, w) + (_2W - aO, v) + (av - aO + 7JO, O)}
- Re {a(v, w) - a(w, v) - a(O, v) + (av - aO + 7JO, 0)
- [ { [ ow + (1 _ JL) oB 2 w _ a 00 ] v - [w + (1 - JL)BIW _ aO] ov } dr }
Jr a1/ aT a1/ a1/
- -a1lO1l 2 + 7J(O, 0)
- -u1101l 2 -1]IIVOIl 2 - ).1] £ IOl2dr < 0 (2.5.54)
where we have used Green's formula (2.5.53) and the boundary conditions on wand O.
Notice that when Al = 0 or A2 = 0 in the temperature boundary condition (2.5.7), the
boundary integral term in (2.5.54) will not appear; when A > 0, A2 > 0, the boundary
53

integral term in (2.5.54) does appear and A = : . Thus, A is dissipative. It remains
to show that 0 E p(A), i.e, for any F = (f,g, h)T E ft, we want to show that the
equation
-Az=F
(2.5.55 )
has a unique solution z E V(A). This is equivalent to finding w E H 4 and 0 E H 2
satisfying
2W + o:O - 9 E L 2 (0),
aO - TJO - h - o:f E L 2 (0)
( 2.5.56)
(2.5.57)
and the boundary conditions (2.5.2)-(2.5.7). It follows from the standard results on
the linear elliptic PDE theory that the elliptic boundary value problem (2.5.57), (2.5.7)
has a unique solution 0 E H 2 . Furthermore, the elliptic boundary value problem
2W = 9 - o:O E L 2 (0),
aw
w=-=o on fo,
a1/
w = 0, w + (1 - p,)BIw = -0:0 E H 3j 2(f l ) on f l ,
w + (1 - p,)BIw = -0:0 E H3/2(f 2 ) on f 2 ,
aw ( ) aB2W _ ao HI/2 ( f ) f
a1/ + 1 - P, a1/ - -0: a1/ E 2 on 2
(2.5.58)
has a unique solution w E H 4 n Hfo n Hf I .
It is easy to see that the norm of z in H 4 x H 2 X H 2 is bounded by KIIFlhi for
some K > o. Thus, (-A)-I E £(ft). Moreover, A-I is compact in ft. Since A is
dissipative, by Theorem 1.3.4, we can conclude that A generates a Co-semigroup S(t)
of contractions on ft.
o
Furthermore, we have the following.
Theorem 2.5.4 The semigroup S(t), generated by A, is analytic and exponentially
stable.
Proof In the present situation we only need to prove the analyticity, and the expo-
nential stability follows from the analyticity. It amounts to verifying the conditions in
54

Theorem 1.3.3.
(i) We first verify the condition (1.3.7) in Theorem 1.3.3.
Since we have already proved in the last theorem that 0 E p(A) and A-I is compact
in H, it turns out that the spectrum of A consists of its eigenvalues. Thus, if (1.3.7) is
not true, then there is a real number /3 =I 0 such that i/3 is an eigenvalue of A. It turns
out that there is a complex vector function z = (w, V, O)T E V(A) with unit norm in
H such that
(i/31 - A)z = 0, (2.5.59)
I.e.,
i/3w - v = 0 In H 2 (2.5.60)
,
i/3v +  2W + O = 0 In L 2 (2.5.61 )
,
i/30 - Q:v - TJO + <70 = 0 In L 2 . (2.5.62)
Taking the real part of the inner product of (2.5.59) with z in H and applying (2.5.54)
yields
0"118112 + 17I1 V8 11 2 + 17). £ 181 2 df = O.
(2.5.63)
As mentioned before, when Al = 0 or A2 = 0, the boundary integral term in (2.5.63)
does not appear. Anyway, we can deduce from (2.5.63) that
o = 0 in n.
(2.5.64 )
Then it follows from (2.5.62) that
v = 0 In n.
(2.5.65)
It turns out from (2.5.60) that w = 0 in n. Therefore, we also have that 2W = 0 in
n. Owing to (2.5.64), we can deduce from (2.5.61) that v = 0, and by (2.5.60), w = 0
in n, a contradiction. Thus, (1.3.7) is proved.
(ii) Next, we verify (1.3.8). If (1.3.8) is not true, then there exists a sequence /3n E IR:
with /3n ---+ 00 (without loss of generality, we always assume that /3 > 0) and a sequence
55

of complex vector functions Zn = (w n , v n , On)T E V(A) with unit norm in 1i such that
as n ---+ 00,
(if - ;n A)Zn -+ 0 III 1t,
(2.5.66)
I.e. ,
. 1
w - -v ---+ 0
n /3n n
. 1 A2 Q An
Vn + /3n L.1 W n + /3n L.1U n ---+ 0
iO - V - !Lo + O ---+ 0
n /3n n /3n n /3n n
In H 2
,
(2.5.67)
In L 2
,
In L 2 .
(2.5.68)
(2.5.69)
Taking the real part of the inner product of (if - In A)zn with Z'fI. in 1i and applying
(2.5.54) yields
0' II 1 2 Tf 1 2 TfA [ 2
/3n Onl + /3n IIVOnl + /3n J r IOnl df ---+ O.
(2.5.70)
As mentioned before, when Al = 0 or A2 = 0 in the temperature boundary condition
(2.5.7), the boundary term in (2.5.70) will not appear. Anyway, it follows from (2.5.70)
that
1
172110n11Hl ---+ O. (2.5.71)
/3n
In what follows we first show that IIOnll2 ---+ o. Taking the inner product of (2.5.69)
with On in L 2 leads to
illO n l1 2 -  (vn' On) -+ O. (2.5.72)
From (2.5.67) we can deduce that L IIv n llH2 is uniformly bounded with respect to
n. Then by the Gagliardo-Nirenberg interpolation inequality, /2 l1vnllHl is also uni-
/3n
formly bounded. This together with (2.5.71) yields
1
!3n (Y'v n , Y'On) -+ O.
(2.5.73)
Furthermore, by Theorem 1.4.4 in Chapter 1, we have
8v n 1/2 1/2
a < c Ilv n Il H2 (o) IIv n Il H1 (O)'
1/ L2(r)
II O nIlL2(r) < CIIOnll;(o) IIOnllgo)
(2.5.74)
(2.5.75)
56

with C being a positive constant. Then it turns out that
 f aV n On df
!3n Jr av
< ; Vn II On II £2 (r)
fJn V L2 (r)
< C IIvnll;(!1) IIvnll?(!1) IIOnll?(!1) 11 0 I I t
{3/2 {3/4 (3/4 n I
C 1/2
< 174110nIIHl(O)  o.
(3n
Therefore, it follows from (2.5.76) and (2.5.73) that
1 ( ) 1 [aV n - 1
/3n Vn' On = /3n Jr 8v On df - /3n ('Yv n , 'YOn) -+ O.
Combining it with (2.5.72) yields
(2.5.76)
(2.5.77)
II On II  o.
(2.5.78)
In what follows we are going to prove that a( W n , w n ) + IIv n l1 2  0 which will yield the
desired contradiction.
Taking the complex conjugate of the inner product of (2.5.67) with W n in H 2 (n),
then adding it to the inner product of (2.5.68) with V n in L 2 (n) and using Green's
formula (2.5.59), we obtain
i( -a( W n , w n ) + IIv n 11 2 ) +  (On,  V n ) -+ O.
(2.5.79 )
By (2.5.77), the last term in (2.5.79) converges to zero. Hence,
a( W n , W n ) - IIv n ll 2  o.
(2.5.80)
It follows from (2.5.69) and (2.5.78) that
a  2
- /3n Vn - /3n On -+ 0 In L . (2.5.81)
We can deduce from (2.5.67) that ;n lIvnll is uniformly bounded. Then it follows from
(2.5.81) that L IIOnll is bounded. This further implies the uniform boundedness of
57

;n lIil 2 w n ll, due to (2.5.68). Notice that W n satisfies the boundary conditions (2.5.2)-
(2.5.4). Thus, the standard estimates for the elliptic boundary value problems and the
trace theorem (see Lions & Magenes [1] or Chapter 1 of this book) lead to
IIW n IIH4 <
<
<
C (II  2W n II + IIaO n IIH3/2(r))
C (II2Wn II + II O nllH2)
C (II2Wnil + IIOnil + IIOnllHl) .
(2.5.82)
Therefore, ;n IIw n llH4 is uniformly bounded. Since IIw n llH2 < 1, by the Gagliardo-
Nirenberg inequality, /2 I1wnIlH3 is also uniformly bounded. Combining (2.5.67) with
(3n
(2.5.81) yields
iailw n - ;n ilO n - 0 in L 2 .
(2.5.83)
Therefore,
iallilw n ll 2 - ;n (ilO n , ilw n ) - O.
(2.5.84 )
Using integration by parts, the Cauchy-Schwarz inequality and Theorem 1.4.4, we
obtain that
1
(3n (ilO n , ilw n )
< ' ;n II : 11£2(rjllilw n ll£2(r) + ;n II O nllH'IIw n llH3
< c liOn II? ( IIOnll Ilwnll; lIil II  + liOn II? IIW n 11H3 )
(3/ 4 (3/2 (3/ 4 W n (3/ 4 (3/2
< C IIOnll; 0 (2 8 )
1 1/4  . .5. 5
(3n
This, together with (2.5.84), leads to
IIwnil  o.
(2.5.86)
If r 2 = 0, then W n = 0 on r. By the standard estimate for the elliptic boundary value
problems, we have
IIW n llH2 < CIIwnll  0
(2.5.87)
58

which, due to (2.5.80), further yields
V n  0
In L 2
,
(2.5.88 )
a contradiction. Thus, the proof for the case r 2 = 0 is proved.
For the general case that r 2 =1= 0, instead of (2.5.87), we have
IIw n llH2 < C (IIWnll + IIwIIH3/2(r2)) .
(2.5.89)
Therefore, it remains to prove that IIwIIH3/2(r 2 )  o. For this purpose, we first estimate
lIwnIlL2(r2) and II O IIL2(r 2 ). By Theorem 1.4.4, (2.5.86) and (2.5.75), we have
IIWn"L2(r2) < c "Wn"; II  11 1/2  0
1/4 - 1/4 W n ,
(3n (3n
(2.5.90)
and
II On II L2 (r 2 ) 0
1/4 .
(3n
(2.5.91 )
By the trace theorem, we get that
IIw n IIHI(r 2 ) < C IIW n llH2  O.
(3/4 - (3/4
(2.5.92)
If we denote by D".w, D;w the first-order and the second-order tangential derivative
of W on r 2 , respectively, then a straight forward calculation yields that
{ D".w = -V2Wx + VI W y ,
D;w = vwxx - 2VIV2Wxy + V;W yy + (V2 V 2x W x - Vl x V2 W y - VIV2yWx + VI V l Y W y ).
(2.5.93)
Therefore, the first boundary condition on r 2 can be rewritten as
Wn + (1 - J.L) ( -DWn + k O;n ) + aD = 0 on f 2
(2.5.94 )
where k = -( Vlx + V2y). Since the boundary is smooth, k is bounded. It follows from
(2.5.90), (2.5.91) and (2.5.94) that
IID;w n IlL2(r2)  0
1/4 .
(3n
(2.5.95)
59

Notice that H 2 (r 2 ) is a Sobolev space on the one-dimensional manifold r 2 . Therefore,
a straight forward calculation shows that IIw n llk 2 (r2) is equivalent to IIDwnlli2(r2)
+ Ilw n llk 1 (r 2 ). Then we deduce from (2.5.95) and (2.5.92) that
Ilw n llH2(r2)  0
1/4 .
(3n
(2.5.96)
By the interpolation inequality for the Sobolev spaces on the manifold (for instance,
see Hormander [1]) and Theorem 1.4.4, we have
1 1
IIw n llH3/2(r 2 ) < Cllw n ll12(r2) IIw n ll11(r 2 )
( " wn IIH2(r 2 » ) 1/2 1 1 1
< C 1 p/4 IIw n ll1211PJ w n lll, -7 0,
due to (2.5.96) and the boundedness of W n in H2(0,) and (3/2Wn in H1(0,). Thus, the
(2.5.97)
proof is complete.
o
60

Chapter 3
Linear Viscoelastic Systems
In this chapter we are concerned with exponential stability of semigroups associated
with linear viscoelastic systems. We begin in the first section of this chapter by
considering a very simple linear viscoelastic equation for (x, t) E (0,1) X 1R+:
Utt - au xx - ,Uxxt = 0
(3.0.1)
with a" being given positive constants and subject to certain initial and boundary
conditions. The exponential stability and analyticity of the corresponding semigroup
are proved in the first section. Then we further extend this model to an abstract
setting in operator forms, using the systematic method described in this book. The
second section is devoted to the wave equation with locally distributed damping. In
the third section, we consider the linear viscoelastic system with memory and prove
the exponential stability under certain assumptions on the kernel. We also point out
that the analyticity is not valid for such kind of semigroups. In the final section,
we consider the Kirchhoff plate equation with memory and give the results on the
exponential stability of the corresponding semigroup, as an application of the general
result established in the third section.
3.1 Linear Viscoelastic SystelD
Let us first consider the motion of an elastic rod in the x direction with the reference
configeration of length 1. Suppose that the stress is of rate type, i.e.,
(j = au x + ,Uxt.
(3.1.1)
Then, the momentum equation is written as (3.0.1). To fix our idea, let us assume
that the rod is clamped at both ends, x = 0 and x = 1. Then we have the following
boundary conditions:
ulx=o = 0, Ulx=l = O.
(3.1.2)
61

and also the following initial conditions:
ult=o = uo(x), Utlt=o = UI(X), Vt > O.
(3.1.3)
In order to convert the above problem into the first-order evolution equation, we
introduce
v = Ut
(3.1.4)
and let 1-l = HJ(0,1) x L 2 (0,1) with HJ = {ulu E HI, u(O) = u(1) = O} equipped
with the norm lIullH' = ( [I alDul 2 dx)L As in the previous chapters, we always denote
o J o
D i = a fY . Then the initial boundary value problem (3.1.1 )-(3.1.3) is reduced to the
X
following initial value problem for a first-order evolution equation in 1-l:
{ dy
dt = Ay, Vt > 0
ylt=o = (uo, UI)T
(3.1.5)
with
y=(:),
Ay - ( D(aDu V + i Dv ) )
(3.1.6)
(3.1.7)
and
V(A) = {y E 1-llv E HJ, (aDu+,Dv) E HI}.
(3.1.8)
It is clear, for instance from Chapter 1, that V(A) is dense in H. We now have the
following.
Theorem 3.1.1 The operator A generates a Co-semigroup of contractions in 1-l
Proof For y E V(A), by integration by parts we have
( Ay, Y )1-£
l (aDuDv + D(aDu + iDv)v)dx
-lilDvl2dx < o.
(3.1.9)
62

This shows that A is dissipative. We now prove that 0 E p(A). For any F = (f, g)T E
H, consider the equations Ay = F, i.e.,
v = f E H,
(3.1.10)
D(aDu + ,Dv) = 9 E L 2 .
(3.1.11)
We plug v = f, obtained from (3.1.10), into (3.1.11) to get
aD(Du) = 9 _,D 2 f E H- I
(3.1.12)
where H- I is the dual space of HJ. Thus by the standard result in the elliptic equations
(see Chapter 1, for instance), we can conclude that (3.1.12) admits a unique solution
u E HJ. It turns out that we obtain a unique (u, v)T E H such that (u, v)T E V( A)
and satisfies (3.1.10) and (3.1.11). It is clear that II (u, v)TII'H < KIIFII'H with K being a
positive constant. Thus 0 E p(A) is proved. By Theorem 1.2.4, the proof is complete.
D
Furthermore, we have the following.
Theorem 3.1.2 The semigroup S(t), generated by A, is exponentially stable, z.e.,
there exist positive constants M, a: such that
II S (t) II < Me-at.
(3.1.13)
Proof As in the previous chapter, we still use Theorem 1.3.2 to prove this theorem.
We first prove (1.3.3) which consists of the following steps:
(i) Prove 0 E p(A) which has been done in Theorem 3.1.1.
(ii) It follows from Remark 2.1.1 and the contraction mapping theorem that for any
real number (3 with 1(31 < IIA-III- I , the operator i(31 -A = A(i(3A- I -l) is invertible.
Moreover, II (i(31 - A)-I II is a continuous function of (3 in (- IIA -1 II -1, IIA -1 II -1).
(iii) Thus, if (1.3.3) is not true, then there is w E 1R with IIA- I II-I < Iwl < 00 such
that {i(3II(31 < Iwl} C p(A) and sup{II(i(3-A)-IIIII(31 < Iwl} = 00. It turns out that
there exists a sequence (3n E 1R with (3n --+ w, l(3nl < Iwl and a sequence of complex
63

vector functions Yn E V(A) with unit norm in 1{ such that
II (i,Bn I - A)Ynll'H  0,
(3.1.14)
as n  00, I.e.,
i,Bnun - V n  0 In HJ,
(3.1.15)
i,Bnvn - D(aDu n + ,Dv n )  0 In L 2 .
(3.1.16)
Taking the inner product of (i,BnI - A)Yn with Yn in 1{ and then taking its real part
yields
Re((i,Bn I - A)Yn, Yn) = ,IIDv n ll 2  o.
(3.1.17)
By the Poincare inequality, we get that
V n  0 In L 2 .
(3.1.18)
Then it immediately follows from (3.1.15) that
O . H I
un  In 0 .
(3.1.19)
This contradicts IIYnN'H = 1. Thus (1.3.3) is proved.
We now prove (1.3.4) by a contradiction argument. Suppose that (1.3.4) is not
true. Then there exists a sequence ,Bn with l,Bnl  +00 and a sequence of complex
vector functions Yn E V(A) with unit norm in 1{ such that (3.1.14) holds. Again we
have (3.1.17). Dividing (3.1.15) by,Bn yields (3.1.9). Thus the proof is complete. 0
We would like to emphasize that the viscoelastic system which we are now con-
sidering is quite different from the linear thermoelastic system in the sense that the
corresponding semigroup is not only exponentially stable, but is also analytic, as shown
in the following theorem.
Theorem 3.1.3 The semigroup S(t), generated by A, is analytic.
Proof On the basis of the proof of the above theorem, by Theorem 1.3.3, we only
need to prove (1.3.8). We still use a contradiction argument. Suppose that (1.3.8) is
64

not true. Then there exists a sequence !3n with l!3nl  +00 and a sequence of complex
vetor functions Yn E V(A) with unit norm in 1i such that
II (if - n A)Ynll1t -+ 0,
(3.1.20 )
as n --+ 00, I.e.,
· 1 0 H I
un - !3n v n --+ In 0'
iV n - n D(aDu n + ,Dv n ) -+ 0 III L 2 .
In the same manner as before, we get
(3.1.21)
(3.1.22)
1
(3n IIDV n 11 2 -+ o.
(3.1.23)
Now we take the inner product of (3.1.21) with Un in HJ and apply the Cauchy-
Schwartz inequality to obtain that
1 1 1
IIDu n ll 2 < (3n I(Dv n , Dun)1 + 0(1) < 2 II DUn 11 2 + 2(3 IIDv n ll 2 + 0(1)
(3.1.24)
where 0(1) --+ 0, as n --+ +00. Thus it follows from (3.1.23) and (3.1.24) that
O . H I
un --+ In 0 .
(3.1.25)
Taking the inner product of (3.1.22) with V n in L 2 , and then integrating by parts and
applying the Cauchy-Schwartz inequality, we obtain that
/ a
II V n 11 2 < (3n II DV n 11 2 + (3n I ( DUn, Dv n ) I + o( 1 )
(3.1.26)
Thus, V n --+ 0 in L 2 follows from (3.1.25)-(3.1.26). A contradiction again. The proof
is complete.
o
Remark 3.1.1 As in the previous chapter, we can also deal with other boundary con-
ditions.
In what follows, we consider an extended model to (3.0.1) or (3.1.5)-(3.1.7). Sup-
pose that A is a self-adjoint positive definite operator with the domain V( A) being
dense in a Hilbert space H. As known from Chapter 1, we can define the fractional
65

power AQ with Q E JR. Consider the following the initial value problem for a second-
order evolution equation in the operator form:
{ Utt + Au + AQut = 0,
ult=o = Uo, utlt=o = Ul,
,
(3.1.27)
where 0 < a: < 1. When Q = 1, the equation (3.0.1) subject to the corresponding
boundary conditions (3.1.2) becomes a special case for the above abstract setting.
When a: = 0, the following equation with appropriate initial and boundary conditions
Utt - U xx - Ut = 0,
(3.1.28)
which is the momentum equation of a rod with the damping due to friction, is a special
case for the above abstract setting. However, for 0 < a: < 1, the mechanical meaning of
the equation in (3.1.27) does not seem very clear. We now use the semigroup approach
to investigate solvability of problem (3.1.27).
Let
1
1-l = V(A2) X H.
(3.1.29)
If we introduce v = Ut, then (3.1.27) can be reduced to the following initial value
problem for a first-order evolution equation in 1-l:
{ dy
dt = AQy,
ylt=o = Yo = (uo, Ul)T
(3.1.30)
with
y = ( : ) , (3.1.31)
V(A Q ) = {y E 1-l I (u, v)T E V(A) x V(A), Au + AQ-v E V(A)}, (3.1.32)
and
AQy =
( V )
1 1 1.
-A2(A2U + AQ- 2 v)
(3.1.33)
66

It is easy to verify that for y E V(Ao),
(AQy,y)'H = (Atv,Atu)H - (At(At u + AQ-tV),V)H = -IIAvll < O. (3.1.34)
Thus A Q is a dissipative operator. We now have the next theorem.
Theorem 3.1.4 The operator A Q generates a Co-semigroup SQ(t) of contractions in
H.
Proof It follows from V(A) x V(A) C V(A Q ) that V(A Q ) is dense in 1-(,. By Theorem
1.2.4, it suffices to prove that 0 E p(A Q ). For any F = (f, g)T E V(A Q ), consider the
following equation:
AQy = F,
(3.1.35)
l.e,
1
V = f E V(A2"),
(3.1.36)
and
1 1 1
- A2"(A2"u + A Q -2"v) = g E H.
(3.1.37)
We plug v = f obtained from (3.1.36) into (3.1.37) to get
1 1 1
- A2"(A2"u + AQ-2" f) = 9 E H.
(3.1.38)
Since At is invertible, (3.1.38) has a unique solution:
-1 -1 1
U = -AT(ATg + A Q -2"f).
(3.1.39)
It turns out that (u,v)T belongs to V(A Q ) and satisfies (3.1.36) and (3.1.37). We also
1 1 1 1
notice that for 0 < a: < 2' it follows from v E V(A2") that AQ-2"v E V(A2"). Thus
u E V(A) and A(Au + AQ-v) = Au + AQv. It is clear that lI(u, v)TII'H < KIIFII'H
with K being a positive constant. Thus, the proof is complete.
Furthermore, we have the following.
Theorem 3.1.5 Let SQ(t) be the Co-semigroup of contractions generated by A Q . Then
for t < a: < 1, SQ(t) is exponentially stable and analytic; for 0 < a: < !' SQ(t) is
o
exponentially stable.
67

Remark 3.1.2 If we use the results by Taylor [ll concerning the characterization of
Gevrey's class instead of (1.3.4), then we can also prove that for 0 < a < , the
semigroup SQ(t) of contractions generated by A Q is in Gevrey's class.
Proof of Theorem 3.1.5. First for 0 < a < 1 we want to prove that the semigroup
SQ(t) is exponentially stable. The basic strategy is still to use Theorem 1.3.2.
(i) 0 E p(A Q ). This has been proved in the previous theorem.
(ii) We now prove (1.3.3). Suppose that it is not true. Then there is wEIR with
IIAlll-l < Iwl < 00 such that {i,8 11,81 < Iwl} C p(A Q ) and sup{lI{i,8-A Q )-lllll,81 <
Iwl} = 00. It turns out that there exists a sequence ,8n E IR with ,8n -+ w, l,8nl < Iwl
and a sequence of complex vector functions Yn E V{A Q ) with IIYnll1l = (IiAtunll k +
IIvnllk)t = 1 such that as n -+ 00,
i,8ny(n) - AQYn -+ 0 in 1i,
(3.1.40)
I.e. ,
i,8nun - V n -+ 0 In V{A),
(3.1.41)
1 1 1
i,8nvn + A2(A 2 u n + A Q -2V n ) -+ 0 In H.
(3.1.42)
It follows from (3.1.40) that
Re(i,8nYn - AQYn, Ynht = -IIAvnll -+ o.
(3.1.43)
Thus,
Q
IlvnllH < GIIA2V n llH -+ o.
(3.1.44)
By (3.1.42), we get that
1 1 1
A2'(A 2 u n + A Q -2V n ) -+ 0 In H.
(3.1.45)
Taking the inner product of (3.1.45) with Un in H yields
1 2 Q Q
IIA 2 u n li H + (A2V n , A 2 u n )H -+ o.
(3.1.46)
68

Since IIA UnIiH is bounded, it follows from (3.1.43) and (3.1.46) that
IIAtunll k  o.
(3.1.47)
(3.1.44) and (3.1.47) contradict IIYnllrt = (IiAtunll k + IIvnllk)t = 1. Thus (1.3.3) is
proved.
(iii) We now prove (1.3.4) by a contradiction argument. Suppose that (1.3.4) is not
true, then there exists a sequence !3n with l!3nl  +00 and a sequence of complex
vector functions Yn E V(A Q ) with unit norm in 1-l such that (3.1.40) holds. As in the
above, we still have (3.1.44). Dividing (3.1.41) by !3n, then taking the inner product
with Un in V( At), we obtain that
1 1
. IIA 1 11 2 (A2Vn, A2U n )H 0
 2 Un H + !3n  .
(3.1.48)
Dividing (3.1.42) by !3n, then taking the inner product with V n in H yields
(Aun' AVn)H + IIAvnllk -+ o.
!3n
(3.1.49)
Thus combining (3.1.49) with (3.1.48) and (3.1.44) yields
IIA!Unll  0,
(3.1.50 )
which together with (3.1.44) contradicts the unit norm of Yn. Thus the proof for
exponential stability is complete.
We now continue to prove that for! < a < 1, the semigroup is analytic. By
Theorem 1.3.3, it suffices to prove that (1.3.8) holds. Suppose that (1.3.8) is not true.
Then there exists a sequence !3n with l!3nl  +00 and a sequence of complex vector
functions Yn E V(A Q ) with unit norm in 1-l such that as n  00,
. A Q Y n 0 . '1J
Yn - !3n  In rL,
(3.1.51)
I.e. ,
iU n - ;: -+ 0 In V(At),
(3.1.52)
69

111
" A2'(A2U n + AO'-2'v n ) 0
Vn + (3n -+ In H.
Taking the inner product of (3.1.51) with Yn in 1-l, we obtain that
IIAvnll 0
(3n -+.
(3.1.53)
(3.1.54)
Taking the inner product of (3.1.52) with Un and A-O'v n in V(At), respectively, yields
1 1
" 'I A ! 11 2 (A2Vn, A2U n )H 0
 2 Un H - (3n -+ ,
(3.1.55)
and
1=2. 2
" (A !=.g. A !=.g. ) IIA 2 VnllH
 2 Un, 2 V n H - (3n -+ o.
(3.1.56)
Since t < a < 1,
I-a a
IIA "T"VnIlH < GIIA2"v n IlH.
(3.1.57)
Then it follows from (3.1.54), (3.1.56), and (3.1.57) that
I-a I-a
(A"T"u n , A"T"Vn)H -+ O.
(3.1.58)
We further take the inner product of (3.1.53) with A1-O'u n in H to obtain that
" (A l-a A 1=2. ) IIAl-unllk (At vn , AtUn)H 0
 2 V n , 2 Un H + /3n + /3n -+.
(3.1.59)
Adding (3.1.59) up with (3.1.55), then taking the imaginary part yields
1 2 I-a I-a
IIA2U n liH + (A"T"v n , A"T"Un)H -+ O.
(3.1.60)
It follows from (3.1.58) and (3.1.60) that
IIAtunll -+ O.
(3.1.61)
Taking the inner product of (3.1.53) with V n in H and using (3.1.54) and (3.1.61), we
immediately get
IIVnll -+ O.
(3.1.62)
Thus, we have contradiction again and the proof of this theorem is finally complete.
D
70

We now investigate the relationship between the semigroup solution and the so-
lution to problem (3.1.27). In other words, we want to explain in what sense the
semigr9up solution satisfies problem (3.1.27).
If Yo E V(A Q ), then y E C([O,oo),V(AQ))nCl([O,oo), 1i) satisfies (3.1.30) point-
wise. It turns out that U belongs to C 2 ([0, 00), H) n C1([0, 00), V(A)) and satisfies
1 1 1
Utt + A2"(A2'u + AQ-2'Ut) = 0
(3.1.63)
in H for every t > 0 and the initial conditions of (3.1.27) in the strong sense. Notice
that when 0 < a < t, as mentioned before, we have A(Au + AQ-ut) = Au + AQut,
i.e, (3.1.27) is satisfied in H pointwise for every t > o.
If Yo E 1i, then y E C([O, 00), 1i) (i.e., U E C([O, 00), V(A)), v E C([O, 00), H)) is
a mild solution to problem (3.1.30). Since V(A Q ) is dense in 1i, we have a sequence
YOn E V(A Q ) converging to Yo in 1i. Accordingly, we have a sequence Yn = (un, vn)T
satisfying (3.1.63). Moreover, for any T > 0, Un -+ U in C([O, T], V(At)), V n = Unt -+ V
in C([O, T], H). Let W be an arbitrary element in V(At). Taking the inner product
of (3.1.63) for Un with W in H, then integrating with respect to t and passing to the
limit, we obtain that
1 1 1 1 r t 1 1
(Ut,W)H - (Ul,W)H + (A Q -2'u,A2'w)H - (A Q -2' u o,A2'w)H + 10 (A2'u,A2'w)HdT = 0
(3.1.64)
or equivalently,
d 1 1 1 1
dt ((Ut,W)H + (A Q -2"u,A2"w)H) + (A2"u,A2"w)H = 0, Vt > O.
(3.1.65)
(3.1.64) or (3.1.65) is usually called the variational form of (3.1.27). The above discus-
sion shows that for Yo E 1i, u, the first component of y = S(t)yo, is a weak solution of
problem (3.1.27) in the sense that the variational form (3.1.64) or (3.1.65) is satisfied
1
for any test element W E V(A2').
Remark 3.1.3 The extended model considered above seems to be initiated in Chen &
Russell [1]. They considered the damped elastic system in the following form:
Utt + BUt + Au = 0
(3.1.66)
71

with A being a positive definite self-adjoint operator densely defined in a Hilbert space
Hand B being -another positive self-adjoint operator. Let v = Ut. Then equation
(3.1.66) can be reduced to a first-order evolution equation:
dy
dt = ABy (3.1.67)
with
ABy = ( )
-Au v_ Bv .
(3.1.68)
We refer the reader to Chen & Russell [lJ for the conjectures on when AB will generate
an analytic semigroup, and Huang [2J-[3J for the complete proofs of these conjectures.
When B is an operator comparable to A'\ 1/2 < a < 1, we refer to Huang [4J and
Chen & Triggiani [1 J for the results and discussions about generation of an analytic
semigroup by AB . We also refer to Chen & Triggiani [2J for the results of generation
of a semigroup in Gevrey's class when 0 < a < 1/2. The method used in Chen &
Triggiani [lJ-[2J is to directly estimate the spectrum of A B . We refer to Liu & Liu
[3J for two counter examples showing the possible illposedness for some operators B.
The same conclusion on the exponential stability and analyticity for 0 < a < 1 was
also obtained in Liu & Liu [3J, . using a different factorization of Au + AO'v from what
presented here.
3.2 Wave Equation with Locally Distributed
Damping
Notice that equation (3.1.28) indicates that the friction damping is distributed in a
whole rod. However, it is desirable in practice to consider the problem with locally
distributed damping. More precisely, in this section we will consider the following
wave equation with locally distributed damping:
Utt - U xx + a(x)ut = 0, x E (0,71"), Vt > 0
(3.2.1)
where a(x) is a given function such that a E Wl,oo, a(x) > 0 in [0,71"] and
ao == a(x)dx > Q.
(3.2.2)
72

This means that a(x) can vanish at some points of the interval [0,71"], but the measure
of its support is positive. We consider the initial boundary value problem for equation
(3.2.1) subject to the following boundary conditions and initial consitions:
ulx=Q = ulx=1f' = 0,
(3.2.3)
and
ult=Q = uQ(x), utlt=Q = Ul(X), x E [0,71"].
(3.2.4 )
If we introduce v = Ut and the following Hilbert space
1-l = HJ X L 2
(3.2.5)
equipped with the inner product: for Yl = (Ul, Vl)T, Y2 = (U2, V2)T,
(Yl, Y2)1i = 10''' (UI x U2x + VI v2)dx,
(3.2.6)
then problem (3.2.1), (3.2.3), and (3.2.4) can be reduced to the following initial value
problem for a first-order evolution equation on the Hilbert space 1-l:
{ dy
dt = Ay, t > 0,
ylt=Q = (uo, Ul)T
(3.2.7)
with y = (u,v)T,
Ay = ( )
U xx _Va(x)v '
(3.2.8)
and
V(A) = {y = (u,v)T E 1-l I v E HJ, U E H2nHJ}.
(3.2.9)
We should mention that this initial boundary value problem (3.2.1), (3.2.3), and
(3.2.4) has been studied by many people (for instance, see Chen, Fulling, Narcowich &
Sun [1] and the references cited there). What we would like to do now is to give a new
proof for the exponential stability of the associated semi group using the systematic
method presented in this book. A new feature presented here is the combination of the
73

contradiction argument with the frequency domain multiplier technique. This tech-
nique has also been applied to certain elastic systems with damping locally distributed
in the domain (see Liu & Liu [4]). We first prove the following.
Theorem 3.2.1 The operator A defined in {3.2.8} generates a Co-semigroup S(t) of
contractions on the Hilbert 1-l.
Proof It is clear from Chapter 1 that V(A) is dense in 1-l. Therefore, by Theorem
1.2.4, it suffices to prove that 0 E p( A) and A is dissipative. For any y = (u, v)T E
V( A), a straight forward calculation shows that
Re (Ay, y)7t
Re 10" (vxu x + (u xx - a(x)v )v)dx
- 10" a(x)lvI 2 dx < 0,
(3.2.10)
i.e., A is dissipative. Therefore, it remains to prove that 0 E p(A). For any F =
(f, g)T E 1-l, consider the equation
Ay = F,
(3.2.11)
l.e. ,
v = f E HJ,
u xx - av = gEL 2 .
(3.2.12)
(3.2.13)
We plug v = f obtained from (3.2.12) into (3.2.13) to get
u xx = a f + gEL 2 .
(3.2.14)
It easily follows from (3.2.14) or the standard result on the linear elliptic equations
that (3.2.14) has a unique solution u E H 2 n HJ. Therefore, 0 E p(A). Moreover, it
can be easily seen that A -1 is compact in 1-l. The proof is complete. 0
The main result in this section is the following.
Theorem 3.2.2 The semigroup S(t), generated by A, is exponentially stable, z.e.,
there exist two positive constants a, M such that
IIS(t)1I < Me-Qt, Vt > o.
(3.2.15)
74

Proof By Theorem 1.3.2, it suffices to verify (1.3.3) and (1.3.4). We use the contra-
diction argument again.
(i) If (1.3.3) is not true, then there must be a (3 E IR such that (3 =1= 0, i(3 is in the
spectrum of A. Since A-I is compact, i(3 must be an eigenvalue of A. It turns out
that there is a vector function y = (u, v)T E V( A), II Y 111t = 1 such that
i(3y - Ay = 0,
(3.2.16)
I.e. ,
{ i(3u - v = 0,
(3.2.17)
i(3v - U xx + a(x)v = o.
Taking the inner product of (3.2.16) with y in 1-(" then taking its real part yields
Re (i(Jy - Ay, y) = fa" a(x )lvl 2 dx = O.
(3.2.18)
It follows from the following estimate
lIavll 2 = fa" a 2 1vl 2 dx < lI a liLoo fa" alvl 2 dx = 0
(3.2.19)
that
a(x)v = 0, \Ix E [0,71"].
(3.2.20)
Combining it with (3.2.17) yields that
- (32u - U xx = o.
(3.2.21 )
Since u satisfies the Dirichlet boundary condition, it follows from (3.2.21) that there
must be an integer n E IN such that
a- n
fJ - ,
(3.2.22)
and
u = Csinnx
(3.2.23)
with C =1= 0 being a constant. Therefore, it follows from the first equation in (3.2.17)
that
o = a(x)v = iCna(x) sin nx, \Ix E [0,71"]
(3.2.24 )
75

which contradicts the assumptions on a( x) that a > 0 and the measure of its support
is positive. Thus (1.3.3) is proved.
(ii) Suppose that (1.3.4) is not true. There then exists a sequence {3n with {3n -+ 00
and a sequence vector functions Yn = (un, vn)T E V( A) with unit norm in 1-l such that
as n -+ 00,
i{3Yn - AYn -+ 0, In 1-l,
(3.2.25 )
I.e. ,
!n=i{3nun-vn-+O In HJ,
(3.2.26)
(3.2.27)
9n = i {3n V n - U nxx + a ( x ) V n -+ 0 In L 2 .
Taking the inner product of (3.2.25) with Yn in 1-l and using the estimate (3.2.19)
yields that
aV n -+ 0 In L 2 .
(3.2.28)
On the other hand, we can easily deduce from (3.2.27) that
- (3un - U nxx = 9n + i{3nfn - av n .
(3.2.29)
Let q( x) be a real function in Cl which will be chosen later. Taking the inner product
of (3.2.29) with q( x )u nx in L 2 , integrating by parts, we obtain that
fal' qx( IUn:c1 2 +.BlunI2)dx-q(1) IUn:c(l W +q(O) lun:c(OW = 2(gn, qun:c) -2( i.Bn(fnq )x, un).
(3.2.30)
Since U nx , {3nun are uniformly bounded in L 2 , the terms on the right hand side of
(3.2.30) converge to zero. Taking q = x, 1 - x, respectively, we deduce from (3.2.30)
and IIYnlirt = 1 that
1 U nx ( 1 ) 1 2 -+ 1, I U nx ( 0) 1 2 -+ 1.
We now take q( x) = fox a( s )ds in (3.2.30) to obtain that
(3.2.31 )
(a.Bnun, .Bnun) + (au nx , u nx ) -+ ao = fol' a( x )dx > O.
(3.2.32)
76

In what follows we prove that this IS a contradiction. Indeed, it follows from the
first equation of (3.2.27) and (3.2.28) that the first term on the left hand side of (3.2.32)
converges to zero. Therefore, it remains to be proved that the second term on the left
hand side of (3.2.32) also converges to zero. Taking the inner product of the second
equation of (3.2.27) with au n , then integrating by parts yields
i( av n , !3nun) + (au nx , u nx ) + (ax Un, u nx ) -+ o.
(3.2.33)
We can easily deduce from the first equation of (3.2.27) and (3.2.28) that the first
term on the left hand side of (3.2.33) converges to zero. Dividing the first equation of
(3.2.27) by!3n yields that Un -+ 0 in L 2 . Therefore, the third term on the left hand side
of (3.2.33) also converges to zero. Then it turns out that the second term on the left
hand side of (3.2.33) converges to zero, a contradiction. Thus, the proof is complete.
D
3.3 Linear Viscoelastic System with Memory
In this section we consider the following linear viscoelastic system
Utt(t) + A [g(O)u(t) + 10+ 00 g'(s)u(t - s)ds] = 0
(3.3.1)
where A is a linear positive definite, self-adjoint unbounded operator on a Hilbert
space Hand g( s) is a "history kernel" which satisfies the following conditions:
(gl) g(s) E C 2 (0, +00) n C[O, +00), g' E L 1 (0, +00);
(g2) g(s) > 0, g'(s) < 0, g"(s) > 0 on (0, +00);
(g3) g( +00) > 0, which is the equilibrium elastic modulus in linear viscoelasticity;
without loss of generality, in the sequel we always assume that g( +00) = 1;
(g4) gl/(s) + 8g'(s) > 0 on (0,00) for some constant 8 > 0 , and there exist positive
constants sl,K such that for s > Sl,g"(S) < Klg'(s)l.
77

Notice that condition (g4) implies that "k g" < -g'(s) < -g'(sl)e- c5 (s-Sl) for s > Sl > o.
Condition (gl) also allows g' to be singular at s = o. It is easy to see that the weakly
singular kernel of the form
-C3 S
g' ( s) = -Cl e SC2 ' 0 < C2 < 1, Cb C3 > 0
(3.3.2)
satisfies the above conditions. This is a fractional derivative model modified by an
exponential decay factor. Fractional derivative models have been successfully used to
fit experimental complex modulus data for some real materials.
Equation (3.1.1) can be written as an abstract first order evolution equation
dz = Az
dt
(3.3.3)
in a suitable Hilbert space 1-l. Let us first esta.blish an abstract setting. Assume that
V and H are a pair of Hilbert spaces with V c H, a continuous dense injection. Let
V* be the dual space of V. We identify H with its dual so that V c H = H* c V*.
Consider a symmetric sesquilinear form a on V such that
la( u;v) I < Gllullv IIvllv for u, v E V
a( u, u) > wllull
(3.3.4)
(3.3.5)
where w > o. Let A E £(V, V*) be defined by
a( u, v) = (Au, v)v*,v for u, v E V.
(3.3.6)
Then, the restriction of A on H defines a positive definite and self-adjoint operator,
which is exactly the operator we work with in the above, with
V(A) = {u E V I Au E H}.
(3.3.7)
1
Furthermore, we have V(A2') = V, and
1 1
a( u, v) = (A2'u, A2'V)H for u, v E V.
(3.3.8)
78

Thus, V can be equipped with a scalar inner product (u,v)v = a(u,v). Let W =
L;,(O, +00; V) be the Hilbert space of all V-valued, square integrable functions defined
on the meure space ((0, +00), V, Ig'lds) equipped with the norm
( f+oo ) 1/2
Ilwllw = Jo Ig'(s) Illw(s) lIds .
(3.3.9)
Define
v = Ut, w(t, s) = u(t) - u(t - s),
(3.3.10)
for t > 0, s E (0, +00). Then equation (3.3.1) can be written as an abstract first order
evolution equation (3.3.3) on the Hilbert space 1i = V x H x W equipped with the
norm
( ) 1/2
IIzll1i = lIull + Ilvll + IIwll .
(3.3.11)
Here z = (u,v,w)T and
v
Az = -A (u - 1+ 00 g'(s)w(s)ds)
v - Dsw
(3.3.12)
with
V(A) = {z E 1i I U -1+ 00 g'(s)w(s)ds E V(A)j v E Vj Dsw E W; w(O) = O}
(3.3.13)
and Zo = (u(+O),v(+O),u(+O) - u(+O - s)), s E (0,+00) being the initial history.
The energy associated with the viscoelastic system (3.3.3) is defined by
1
E(t) = 211zll
(3.3.14)
in which three parts are potential, kinetic, and memory energy.
Now let us recall some related works in the literature. When A = -Ll subject to
the Dirichlet boundary conditions, Dafermos first proved (see Dafermos [2], published
in 1970) that E (t) tends to zero asymptotically under the assumptions that 9 satisfies
conditions (g2), (g3) and
79

(g 1') g' E C 1 [0, 00 ), g', g" EL I ( 0, 00 ) ,
and that g' is convex. Later on, a similar result without the assumption of convexity
on 9 was obtained (see Dafermos [3]). The first result giving an explicit rate at which
E(t) decays to zero was provided by Day in 1980 (see Day [2]):
Let
a(t) = 1'00 g(s)ds+a(oo). (3.3.15)
Assume that g(8) satisfies (gl)-(g3), and a(t) - a(oo) E L 1 (IR+) and is log convex.
Then E(t) = o(t- 1 ) as t -+ 00 if the initial data (uo)x and (U1)X are uniformly bounded.
It was shown by Desch & Miller [1]-[2] that in the case where the kernel g( 8) satisfies
conditions (gl'), (g2) and (93), and decays exponentially, the solution u and Ut will
also decay to zero exponentially at a rate no better than the kernel. Note that the
initial history was taken to be zero in that paper. The assumption (gl') also excludes
the weakly singular kernel (3.3.2) for the interesting fractional derivative model in
linear viscoelastic solids. We also refer to Hannsgen & Wheeler [1]-[2] for the re-
sults with exponentially decay kernels. Their results also imply the exponential decay
rate of the energy function, however, with the additional smoothness assumption on
the kernel 9(8) and with zero initial history. We also refer the reader to the works
by Ito & Fabiano [1]-[2], but their results do not imply the exponential stability of
the energy function for the singular kernels. In 1991, Fabrizio & Lazzari [1] investi-
gated the three-dimensional viscoelastic system with memory and obtained the result
on the exponential stability of the energy function E( t) with almost the same as-
sumptions on the kernel 9 as stated here, except that there was no assumption like
8 > 81, g" (8) < K Ig' (8) I in (g4). This assumption seems to be needed for the conver-
gence of 1+ 00 g"(s)lIw(s)lIds. The method they used is the Laplace transform and
the Datko theorem. What we present in this section mainly follows the early paper
by the authors (see Liu & Zheng [4]), using the systematic method described in this
book. We also refer to the paper Liu & Liu [1] for the spectrum determined growth
rate property of this model.
80

We first prove the following.
Theorem 3.3.1 The linear operator A, defined by (3.3.12) and (3.3.13), generates a
Co-semigroup S(t) of contractions on 1-l.
This theorem has been proved under a weaker condition on 9 in Fabiano & Ito [1].
We will give a proof here for the sake of the reader. Before giving the proof of this
theorem, we first prove some lemmas.
Lemma 3.3.1 Suppose that 9 satisfies the conditions (gl) -(g3). Then for any w E W
with w(O) = 0 and Dsw E W, we have
Is g' ( s ) 1 --+ 0, as s --+ 0,
(3.3.16)
and
1 g' ( s ) 1 " w ( s ) II  --+ 0, as s --+ O.
(3.3.17)
Proof By condition (g2), g' is monotone increasing and Ig'l = -g' is monotone
decreasing. By condition (gl), if (3.3.16) is not true, then there is a sequence Sn >
0, Sn --+ 0 and a constant TJ > 0 such that Sn Ig'( sn) I > TJ for all n. By the monotonicity
of g', we have snlg'(s)1 > TJ for s E (O,Sn). Integrating with respect to S yields that
fSn
o < TJ < Jo Ig' (s ) 1 ds --+ 0 as Sn tends to zero, a contradiction. Thus the proof of
(3.3.16) is complete.
The proof of (3.3.17) is very similar. Suppose that (3.3.17) is not true. Then there
is a sequence Sn > 0 , Sn --+ 0 and a constant TJ > 0 such that Ig'(sn)llIw(sn)lI > TJ. It
follows from
W(Sn) = In w'(r)dr
and the Cauchy-Schwartz inequality that
(3.3.18)
IIw(sn)lI < Sn In IIw'(r)lIdr.
(3.3.19)
Since -g'(s) = Ig'(s)1 is monotone decreasing, we have
17 < Ig'(sn)llIw(sn)lI < Sn In Ig'(r)llIw'(r)lIdr -+ 0 as Sn -+ 0,
(3.3.20)
81

a contradiction. Thus the proof of (3.3.17) is complete.
Furthermore, we have the following.
Lemma 3.3.2 Suppose that 9 satisfies the conditions (gl)-(g3). Then for any w E W
with w(O) = 0 and Dsw E W, we have
o
10'00 g"(s )lIw(s)lIds < 00,
(3.3.21 )
and
Ig' ( s ) III w( s ) II t  0, as s  +00.
(3.3.22)
Proof By condition (g2), for any a > 0, [0 g"(s)lIw(s)lIds < 00. Let f(t) =
g'(s)llw(s)II. Then by the assumptions on w, f E Ll(a, 00). By the Cauchy-Schwartz
inequality, f'(s) = g"(s)lIw(s)lI + 2g'(s)(w,w s )v also belongs to Ll(a,oo). Then
by the well-known result in analysis, we can conclude that (3.3.22) holds. To prove
(3.3.21), for any small", > 0 and large M > 0, we use integration by parts to obtain
that
i M g"(s)lIw(s)lIds
g'(M)lIw(M)II - g'(7])lIw(7])II - 2 i M g'(s)(w(s), ws(s))v ds . (3.3.23)
By the assumptions on w, the third term on the right hand side of (3.3.23) converges
as M  00 and ",  o. The first two terms converge to zero, due to (3.3.22) and
(3.3.17). Thus (3.3.21) is proved.
o
Proof of Theorem 3.3.1 First we prove that the operator A is dissipative. Indeed,
for z E V(A), we have
(Az, z)'H
(v, u}v + (-A(u - {O g'(s)w(s)ds), V}H
+(v - Dsw, w)w
-(Dsw,w)w
- 1+ 00 g"(s)lIw(s)lIds < o.
(3.3.24 )
82

Thus, the operator A is dissipative. Next we prove that 0 E p(A). For any F =
(11, 12,13)T E 1i, consider the unique solvability of the equation
Az=F, (3.3.25)
I.e. ,
v = 11, (3.3.26)
- A(u -1'00 g'(s)w(s)ds) = 12, (3.3.27)
v - Dsw = 13. (3.3.28)
We can get a unique v E V from (3.3.26), and then from (3.3.28) we can get
w = f (v - f3(r))dr = sv - f f3(r)dr.
(3.3.29)
It is clear that w(O) = 0, Dsw E W. To prove that w E W, for any T > 0, € > 0, by
(g4) and the Cauchy-Schwartz inequality, we have
J.T Ig' (s) IlIw( s) lIds
<  J. T g" ( S ) II w ( s ) II  ds
 g'(T)llw(T)II -  g'( f)lIw( f)ll -  J.T g'(s )(w, D.w}v ds
1 1 J. T
< - 8 g'(€)lIw(€)II + 2  Ig'(s)llIw(s)lIds
+ :2 J.T Ig'(s)IIID.wllds. (3.3.30)
Thus,
J. T 2 4 J. T
f Ig'( s) IlIw( s) lIds < -8"9'( f) IIw( f) II + 8 2 f Ig'( s) IIID.wllds.
(3.3.31 )
As can be seen from the proof of Lemma 3.2.1, the following holds:
-  g' ( f) II w ( f ) II   0, as f  O.
(3.3.32)
It turns out from (3.3.31) by letting T --+ +00 and € --+ 0 that w E Wand
4 f+oo
Ilwll < 8 2 Jo Ig'(s)IIIDswllds.
(3.3.33)
83

Finally we can get a unique u E V, u - 1 00 g'(s)w(s)ds E D(A) by solving (3.3.27).
Thus the unique solvability of (3.3.25) with Z = (u, V, w)T E V(A) is proved. Moreover,
it can be easily seen from (3.3.26), (3.3.28), (3.3.31), and (3.3.27) that there is a positive
constant K independent of z = (u, v, w)T such that IIZIl1-l < KIIFII1-l. This implies that
I
o E p(A) and IIA- 1 11 < K. Since A is dissipative, as we have already proved before,
by Theorem 1.2.4, A generates a Co-semi group of contractions in 'H. 0
Furthermore, we have the following theorem.
Theorem 3.3.2 The semigroup S(t) generated by A is exponentially stable, i.e., there
exist two positive constants M, a such that
II S (t) II < Me-at, \It > O.
(3.3.34 )
Proof We still use Theorem 1.3.2 to prove this theorem. The proof consists of the
following steps:
(i) Since it has been proved in Theorem 3.2.1 that 0 E p(A), in the same manner as
before, we can show that if (1.3.3) is not true, then there is w E 1R with II A -111- 1 <
Iwl < 00 such that {iP IIPI < I£:VI} C p(A) and sup{lI(ip - A)-II1IIPI < Iwl} = 00. It
turns out that there exists a sequence Pn E 1R with Pn --+ w, IPnl < Iwl and a sequence
of complex vector functions Zn = (un, V n , wn)T E V(A) with unit norm in 1-l such that
as n --+ 00,
ipnzn - AZ n --+ 0 In 1-l, (3.3.35 )
I.e. ,
iPnun - V n --+ 0 In V, (3.3.36)
i{3nvn + A(u -1 00 g'(s)wn(s)ds) -+ 0 In H, (3.3.37)
iPnwn - V n + Dsw n --+ 0 In W (3.3.38)
In the same manner as before, taking the inner product of (3.3.35) with Zn in 1-l, then
taking its real part, we obtain
Re(Az n , Zn),i = - 1+ 00 g"(s)lIwn(s)lIds -+ o.
(3.3.39)
84

Then it follows from condition (g4) and (3.3.39) that W n -+ 0 in W which implies that
lIunll + IIvnll -+ 1.
(3.3.40)
Taking the inner product of (3.3.36) with V n in H and taking the inner product of
(3.3.37) with Un in H, respectively, yields
i{3n( Un, Vn)H - IIvnll;, -+ 0,
(3.3.41 )
and
f+OO
i{3n( V n , Un)H + lIunll - Jo g'( s)( W n , vn)v ds -+ O.
(3.3.42)
By the Cauchy-Schwartz inequality and by the fact that W n -+ 0 in W, the last term
in (3.3.42) converges to zero. Adding (3.3.41) and (3.3.42) together, then taking its
real part, we get
lIunll - IIvnll;, -+ O.
(3.3.43)
Thus combining (3.3.43) \vith (3.3.40) yields
lIunll -+ , IIv n II -+ .
(3.3.44 )
In what follows we want to show that this is a contradiction.
It is clear from the conditions imposed on g(s) that both s2g'(S) and s2g"(S) belong
to L 1 (0, +00). We can also easily verify that ;: E W. Dividing (3.3.38) by {3n, then
taking the inner product with s;: in Wand using the fact that W n -+ 0 in W, we
obtain that
II {3 v n II f+oo slg'lds - {3 1 f+oo sg'(s )(Dsw n , V n )vds -+ o.
n Jo n Jo Pn
(3.3.45)
We now prove that the second term of (3.3.45) converges to zero. First, it follows from
the Cauchy-Schwartz inequality, Lemma 3.2.1 and Lemma 3.2.2 that
Isg'(s )(wn(s), ;: }vl
< s Ig' (s) IlIw n ( s) II vII ;: IIv
< Slgis)I (lIwn(S)II+II ;: II)-+O, as s-+O
(3.3.46)
85

and
s g' ( s ) ( W n ( S ), ;: } v
< slg'( s) I IIw n ( s) IIvil ;: II v
< 1g'(s)IIIwn(s)II + S2Ig(s)1 II ;: II  0, as s  +00. (3.3.47)
Then we use integration by parts, the Cauchy-Schwartz inequality, (3.3.39) and the
fact that W n --+ 0 in W to obtain
f+OO v
- Jo sg'(s )(Dsw n , (3: }v ds
1+ 00 sgl/(s )(w n , ;: }v ds + 1+ 00 g'(s )(w n , ;: }v ds
<  (1+ 00 gl/(s )IIWnllds ) t +  (1+ 00 sV'(s )ds) t II ;: IIv
1
+IIwnllw (1+ 00 Ig'(s)lds) 2 11 ;: IIv  o.
(3.3.48)
Therefore, we can deduce from (3.3.45) that
II ;: II   o.
(3.3.49)
It follows from (3.3.36) divided by /3n that
Un --+ 0 In V.
(3.3.50)
A contradiction. Thus the proof of (1.3.3) is proved.
(ii) Since in the above proof in (i) we only use the fact that l/3n I is bounded below
from zero, the proof of (1.3.4) is exactly the same as that in (i). So we can omit the
detail here.
o
Remark 3.3.1 It is known that for the system with history memory, in general we
cannot expect that the corresponding semigroup is analytic. (see, for instance, Liu &
Liu [lJ).
86

3.4 The Linear Viscoelastic Kirchhoff Plate with
Memory
In this section, we turn to the study of the semi group associated with the linear
viscoelastic Kirchhoff plate with memory. Suppose that a thin plate of the Kirchhoff
type occupies a bounded region n E IR? with smooth boundary f = f o Uf I Uf 2 . When
the viscoelastic damping is considered, the vertical deflection of the plate satisfies
Wtt(t) - 'YWtt(t) + 2g(0)w(t) + 21000 g'(s)w(t - s)ds = 0
(3.4.1)
with I > 0 (see Lagnese [3]). The boundary conditions considered in this section are
the following:
ow
w= ov =0 onf o , t>O,
w = 8 1 (w(t) + 10 00 g'(s)w(t - s)ds) = 0 on rl, t > 0,
(3.4.2)
(3.4.3)
and
{ 81(W(t) + 10 00 g'(s)w(t - s)ds) = 0
00 OWtt
82(W(t) + 10 g'(s)w(t - s)ds) - 'Y 01/ = 0
on f 2, t > 0
(3.4.4 )
where
otlw oB 2 w
B 1 W = tl w + (1 - J.L) B 1 W, B 2 W = OV + (1 - J.L) OT
with the operators B I , B 2 being the same as in Chapter 2:
02w 02w 02w
B 1 W = 21/11/2 oxoy - 1/i oy2 - 1/ ox 2 '
02w 02w 02w
B 2 w = (1/i - 1/n oxoy + 1/11/2( oy2 - ox 2 )
and v = (VI, V2) being the unit outward normal to f, and T = {-V2, VI}. In (3.4.5),
J.L (! > J.L > 0) is the Poisson ratio. In the sequel, we always assume that fo U f l =F 0
and fonf\ nf\ = 0.
(3.4.5 )
(3.4.6)
We still make the same assumptions (91)-(94) on the relaxation function 9 as in
the previous section. The initial state of the plate is
w(O+) = wo, w'(O+) = WI, w(-s) = Wh(S), for 0 < S < 00
(3.4.7)
87

with W O , WI, wh being given functions.
In order to convert the problem to a first-order evolution equation, as in Chapter 2
for the Linear Kirchhoff plate with thermal damping, we use the variational approach.
Let
H = Hfourl = {u I u E HI(O),ulrourl = O},
(3.4.8)
and
v = Hfo nHf, = {v I v E H2(!1),vlro = : Iro = vir, = O}.
(3.4.9)
For u, v E V, as in Chapter 2, we define
a(u, v) = 10 [uxxv xx + UyyV yy + p.(uxxv yy + UyyV xx )
+2(1 - p.)uxyvxy]dxdy.
(3.4.10)
We simply denote a( u, u) by a( u). H and V are equipped with the following norms,
respecti vely:
1
IIwllH = (lIwll2 + I'll V'wll 2) 2 ,
(3.4.11)
and
1
Ilwllv = a(W)2.
(3.4.12)
Since fo Uf l # 0, the norm defined in (3.4.12) is an equivalent norm in H 2 . It is clear
from Chapter 1 that V is dense and continuously imbedded in H. Therefore, there is
a self-adjoint and positive definite operator A with V(A) = {ulu E V, Au E H} such
1 1
that (A 2 u, A 2 v)H = (u, v)v.
The total energy corresponding to (3.4.1 )-(3.4.6) is defined by
E(t) =  1o (a(w(t)) + [Wt(tW + I'[ VW t(tW) d!1-  10 00 1og'(s)a(w(t)-w(t-s))d!1ds.
(3.4.13)
Suppose now that w is a regular solution to the problem (3.4.1)-(3.4.6), and (3.4.7).
We introduce
v = Wt, h = w( t) - w( t - s).
(3.4.14)
88

Then we multiply (3.4.1) by W E V and take integration by parts to get
(Vt, W)H + a( w, w) + a(fo+oo -g'(s )h(s )ds, w) = O.
Let W = L 2 (0, 00; Ig'(.)I; V) equipped with the norm
(3.4.15)
1
II hll w = (fo+oo Ig'( s )111 h( s) lIds ) 2
(3.4.16)
and let 1i = V x H x W. Since V is dense in H, it follows from (3.4.15) that
[+00
Vt = -A(w - J o g'(s)h(s)ds).
(3.4.17)
By the definition of h, it is clear that
ht = v - Dsh.
(3.4.18)
Let z = (w, v, h)T. Then (3.4.14), (3.4.17), and (3.4.18) are reduced to an abstract
first-order evolution equation
dz = Az
dt
(3.4.19)
with
v
Az = -A (w - fo+oo g'(s)h(s)ds)
v - Dsh
(3.4.20)
V(A) = {z E 1i I w - fo+oo g'(s)h(s)ds E V(A); v E Vj D.h E Wj h(O) = O},
(3.4.21 )
and
zlt=o = Zo = (WO, WI, WO - Wh( +0 - s))T, S E (0, +00).
(3.4.22)
l,From the above reduction, the initial value problem (3.4.19)-(3.4.22) in 1i can be
considered as the weak formulation for the problem (3.4.1)-(3.4.6), and (3.4.7). Notice
that the setting here exactly falls into the general framework discussed in the previous
section. Therefore, from Theorem 3.2.1 and Theorem 3.2.2, we immediately have the
following.
89

Theorem 3.4.1 The operator A defined by (3.4.20) generates a Co-semigroup S(t)
of contractions in 11,. Moreover, the semigroup S(t) is exponentially stable, i.e., there
exist two positive constants M, a such that
IIS(t)1I < Me-at, Vt > o.
(3.4.23)
Remark 3.4.1 As the same as in the previous section, in general, the semigroup S(t)
is not analytic.

Chapter 4
Linear Thermoviscoelastic Systems
In this chapter we will discuss the semigroup associated with the linear thermovis-
coelastic system. In other words, the dissipative mechanisms is not solely due to heat
conduction or viscous damping. Instead, we will consider the problems with both
dissipative mechanisms. In the first section, we are concerned with motion of a rod
of length 1 with the dissipative mechanism of heat conduction and viscous damping
of rate type. We will prove that the semi group associated with that system is not
only exponentially stable, but also is analytic. As we have seen from previous t¥lO
chapters, heat conduction is not strong enough to make the semigroup analytic, while
the viscous damping of rate type itself is strong enough to do so. The results shown
in the first section of this chapter, therefore, is not surprising because we now have
both dissipative mechanism. In the second section of this chapter we will discuss the
linear three-dimensional thermoviscoelastic system in which both heat conduction and
viscous damping have memory. Under certain conditions on the relaxation functions
we will prove that the corresponding semigroup is exponentially stable. As remarked
in Chapter 3, with this kind of memory, in general, we cannot expect that the corre-
sponding semigroup is analytic.
4.1 Linear One-Dimensional Thermoviscoelastic
System
In this section we consider the following linear system of partial differential equations
in (0,1) x (0,00) :
Ult - aU2x = 0,
U2t - aUl x + {3u3x - j.L U 2xx = 0,
U3t + {3U2x - kU3xx = 0
(4.1.1)
91

with constants a > 0, {3 # 0, p. > 0, k > o. Here the subscripts t and x denote the
partial derivatives with respect to t and x, rspectively. This system is a model of linear
one-dimensional thermoviscoelasticity with Ul being the scaled deformation gradient,
U2 the velocity, and U3 the deviation of temperature from a given temperature. (4.1.1)
is also a linearized system for motion of compressible viscous and heat-conductive
fluids in Lagrange coordinates. System (4.1.1) is supplemented by initial conditions:
Ullt=o = u(x), U2lt=o = u(x), u3lt=o = u(x)
(4.1.2)
and boundary conditions. As in Chapter 2, various kinds of boundary conditions can
be considered. To fix our idea, without loss of generality, let us consider the following
boundary conditions for system (4.1.1):
u2lx=o = u2lx=1 = u3lx=o = u3lx=1 = O.
(4.1.3)
The mechanical meaning of the boundary conditions is clear: the rod of length 1 is
clamped at both ends and the deviation of temperature is given (to be zero). It
follows from the first equation of (4.1.1) and the boundary conditions u2Ix=O,1 = 0 that
10 1 Ultdx = 0, i.e., 10 1 Ul dx is conserved for t > O. Without loss of generality, we assume
that l Ul dx = O. Otherwise, we can make the substitution Ul = Ul -  10 1 U( X )dx
which does not change the system (4.1.1). We can now rewrite problem (4.1.1)-(4.1.3)
as a first-order evolution system on the Hilbert space 11, = {zl Ul E L2, U2 E L 2 , U3 E
L 2 , luldx = O} :
{ dz
-=Az
dt '
Zlt=o = zo( x) = (u, ug, ug)T
(4.1.4)
with z = (u, V, w)T = (Ul, U2, U3)T,
aDU2
Az = D(aul + P.DU2) - {3 Du 3
-{3 DU 2 + kD2U3
(4.1.5)
92

and
'D(A) = {z E 1t aUI + P. DU 2 E HI, l uIdx = 0, U2 E H ,u3 E H2nH }.
(4.1.6)
Now we have the next theorem.
Theorem 4.1.1 The operator A generates a Co-semigroup S(t) of contractions in 11,.
Moreover, the semigroup is exponentially stable, i.e., there exist two positive constants
M, a such that
IIS(t)1I < M e-Ott, Vt > O.
(4.1.7)
Proof This theorem has been proved by one of the authors, S. Zheng, in his mono-
graph (Zheng [1]) by the energy method. Now we use the systematic method described
in this book to give a new proof.
For z E V(A), we have
( Az, z) 1-£
= l (auI Du 2 + U2 D ( aUI + P. DU 2) - !3U2 Du 3 - !3u3 Du 2 + kU3D2U3)dx
= -JLIIDu2112 - kllDu3112 < O. (4.1.8)
Thus, the operator A is dissipative. We now further prove that 0 E p( A). For any
F = (II, f2, f3)T E 11" we consider
Az=F.
(4.1.9)
It turns out from the first equation of the vector form (4.1.9) that U2 is uniquely given
by
1 1 3:
U2 = - II dx E HJ.
a 0
(4.1.10)
Substituting this expression of U2 into the third equation of (4.1.9), we get
kD2U3 = f3 + (3 DU 2 E L 2 .
(4.1.11)
It follows from the standard result in the linear elliptic equations that (4.1.11) has a
unique solution U3 E H 2 n HJ. Once U2, U3 have been obtained, it turns out from the
93

second equation of (4.1.9) that UI is uniquely given by
1 L x
UI = -( f2() d + C - P. DU 2 + /3U3)
a 0
(4.1.12)
with
c = - (.8 fa' U3 dx + fa'fo': h d dX) . (4.1.13)
Thus A is invertible with A-I being a bounded operator from 11, to 11,. By Theorem
1.2.4, we can conclude that A generates a Co-semigroup of contractions in 11,. We now
want to apply Theorem 1.3.2 to prove that the semigroup S(t) is exponentially stable.
The proof consists of the following steps:
(i) Since 0 E p(A), it follows from Remark 2.1.1 and the contraction mapping the-
orem that for any real number /3 with 1/31 < IIA-III- I , the operator i/31 - A =
A(i/3A- I - I) is invertible. Moreover, lI(i/31 - A)-III is a continuous function of
/3 in (-II A-Ill-I, IIA-III-I). Thus, if (1.3.3) is not true, then there is w E 1R with
IIA-III- I < Iwl < 00 such that {i/3 11/31 < Iwl} c p(A) and sup{ll(i/3 - A)-III 11/31 <
Iwl} = 00. It turns out that there exists a sequence /3n E 1R with /3n  w, l/3nl < Iwl
and a sequence of complex vector functions Zn = (un, V n , wn)T E V(A) with unit norm
in 11, such that
II (i/3n I - A)zn 111i  0,
(4.1.14)
as n  00, I.e.,
i/3nun - aDv n  0
In L 2
,
i/3nvn - D(au n + p.Dv n ) + /3Dw n  0
In L 2
,
(4.1.15)
(4.1.16)
(4.1.17)
i/3nwn + /3Dv n - kD 2 w n  0 in L 2 .
Taking the inner product of (i/3nI - A)zn with Zn in 11, and then taking its real part
yields
Re((i/3nI - A)zn, zn)1i = -p.IIDv n Il 2 - kllDw n ll 2  o.
(4.1.18)
Thus it follows from the Poincare inequality that
VnO, wnO in L 2 .
(4.1.19)
94

Dividing (4.1.15) by /3n and using (4.1.18), we obtain that
Un -+ 0 in L 2 .
( 4.1.20)
Thus, we get a contradiction. The proof of (1.3.3) is complete.
(ii) To prove (1.3.4), we use a contradiction argument again. If (1.3.4) is not true, then
there exists a sequence /3n E 1R with l/3nl -+ +00 and a sequence of complex vector
functions Zn = (un, V n , wn)T E V(A) with unit norm in 11, such that (4.1.14) holds.
Following exactly the same argument as in (i) yields a contradiction again. Thus, the
proof is complete.
Furthermore, we have the following theorem.
Theorem 4.1.2 The semigroup S(t), generated by A, is analytic.
Proof This theorem has been proved in Liu & Yong [1] as an example of their general
theorem. Now we give a direct proof. By Theorem 1.3.3, it suffices to prove that (1.3.8)
holds. We use a contradiction argument again. Suppose that (1.3.8) is not true. Then
there exists a sequence /3n E 1R with l/3nl -+ +00 and a sequence of complex vector
functions Zn = (un, V n , wn)T E V(A) with unit norm in 11, such that
o
iZ n - ;n AZ n -+ 0 in 1t,
(4.1.21)
as n -+ 00, I.e.,
. Q D O . L 2
ZU n - /3n V n -+ In ,
(4.1.22)
iV n - ;n (D(au n + p.Dv n ) - fJDw n ) -+ 0 In L2, (4.1.23)
iW n + ;n (fJDv n - kD 2 w n ) -+ 0 in L 2 . (4.1.24)
Taking the inner product of (if - In A)zn with Zn in 11, and then taking its real part
yields
. 1 1 2 2 )
Re((tI - fJn A)zn, Znhi = - fJn (p.IIDvnll + kllDwnll -+ O.
Then it follows from (4.1.25), (4.1.23) and (4.1.24) that
( 4.1.25)
iV n - ;n D(au n + p.Dv n ) -+ 0
In L 2
,
(4.1.26)
95

and
. k n 2 2
ZW - - W -+ 0 In L .
n f3n n
Taking the inner product of (4.1.27) with W n in L 2 , then integrating by parts ad
,
using (4.1.25), we get W n -+ 0 in L 2 . Taking the inner product of (4.1.26) with V n in
L 2 , then integrating by parts, we get
( 4.1.27)
illv n ll 2 + ;n ((aun,Dv n ) + JLIIDv n Il 2 ) -+ o.
( 4.1.28)
Using (4.1.25) and the fact that lIunll < 1, we deduce from (4.1.28) that
IIv n l1 2 -+ o.
(4.1.29)
It immediately follows from (4.1.25) and (4.1.22) that Un -+ 0 in L 2 . Thus, we get a
contradiction again. The proof is complete.
o
4.2 Linear Three-Dimensional Thermoviscoelastic
System with Memory
In this section we are concerned with the linear three-dimensional thermoviscoelastic
system with fading memory.
Suppose that a body occupies a bounded domain n c 1R? with smooth boundary
r. We assume that the reference configuration is a natural state in which stress is zero
and base temperature Bo is a strictly positive constant. Let x E n be the position of a
material point at time t and let u(x, t) be the displacement, which is a vector function
valued in 1R?, and let B(x, t) be the temperature difference from Bo, which is a scalar
function. If we assume that the Cauchy stress T and the specific entropy difference TJ
are given by functionals depending on both displacement and temperature difference
history in the following form: (see Marsden & Hugens [1], Navarro [1], and Coleman
& Mizel [1]):
T(x, t) g(x, O)V'u(x, t) - B(x, t)l(x, 0)
+ 1''''' (g'(x, s )Vu(x, t - s) -l'(x, s )9(x, t - s ))ds,
(4.2.1)
96

p( x )71( x, t) l( x, 0) · VUt(-x, t) + p( x )c( x, 0) 9(; t)
+ looo [1'(X, s) . VUt(X, t - s) + p(x )c'(x, s) 9(X,;0- S) ] ds (4.2.2)
where p( x) is the mass density in the natural state and the prime denotes the derivative
with respect to s. The material functions g(x,s),l(x,s) and c(x,s) are the relaxation
tensors of fourth, second and zero order, respectively. Their values at s = 0 are
called the instantaneous elastic modulus, instantaneous stress-temperature tensor, and
instantaneous specific heat, respectively.
We assume Fourier's law for the heat flux vector q(x, t):
q(x, t) = -(x)V'8(x, t)
(4.2.3)
where (x) is the thermal conductivity in the reference configuration and it is a tensor
of second order. Let v denote velocity. Then the equations for balance of momentum
and energy read as follows:
p(x)Vt(x, t) = div T(x, t),
(4.2.4)
(4.2.5 )
8oP(x)7Jt(x, t) + div q(x, t) = 0
where the subscript t denotes the partial derivative with respect to t.
Substituting the expressions given by (4.2.1) and (4.2.3) into (4.2.4) and (4.2.5), we
obtain the following equations for a body in JR3 composed of a non-homogeneous
anisotropic linear thermoviscoelastic material:
p(x)Utt(x, t) div [g(x, O)V'u(x, t) - 8(x, t)l(x, 0)
+ looo (g'(x, s )Vu(x, t - s) -l'(x, s )9(x, t - s »ds] , (4.2.6)
p(x)c(x, 0)8t(x, t) div[(x)V'8(x, t)] - 8 o l(x, 0) . V'Ut(x, t)
looo [9 0 1' (x, s) . VUt( x, t .:.... s) + p( x )c'( x, s )9t( x, t - s)] ds
(4.2.7)
for (x, t) E n x 1R+, n c 1R 3 . The boundary conditions considered here are
U ( x, t) = 0, 8 ( x, t) = 0 on r x 1R+
(4.2.8)
97

while the prescribed initial histories for the displacement and temperature difference
are given by
u(x, -s) = wo(x, s),
8(x, -s) = yo(x, s),
- +
x E f!, s E IR .
(4.2.9)
We refer the reader to Navarro [1] for the derivation of the equations and for the re-
sults on the existence, uniqueness, and asymptotic stability of the generalized solutions
as well as the semigroup approach. However, the issue of the exponential stability of
the semi group was not discussed in that paper. Our main purpose in this section is to
prove that under certain reasonable assumptions on material properties, the semigroup
associated with the prior mentioned system is exponentially stable. In this direction,
we refer to the paper Liu & Zheng [4] by the authors on the exponential stability of
the semigroup associated with the linear one-dimensional thermoviscoelastic system
with fading memory for the displacement. We also refer to the recent work Rivera &
Barreto [1] for the results on the exponential stability for the linear three-dimensional
thermoviscoelastic system. In that paper the assumptions on material properties are
qui te special and it can be considered as a special case for our problem considered
here. Besides, the relaxation functions are not allowed to have singularity at s = 0 in
Rivera & Barreto [1]. The method used in that paper is the energy method. It turns
out that more regularity requirements are needed for the relaxation functions.
The main assumptions made on the material properties are the following:
(HI) g, I, c, K, p are independent of the space variable x. This assumption IS not
essential and is only for the simplicity of exposition.
p > O.
(H 2 ) g(s) = gT(s), i.e, 9ijkl(S) = 9klij(S) (i,j, k, 1 = 1,2,3), for s > 0;
g(s) E C[O,oo),g'(s) E LI(0,oo)nC1(0,oo), i.e, 9ijkl(S) E C[O,oo), 9jkl(S)
E LI(O,oo)nCI(O,oo); there is a positive constant 6 > 0 such that for any
ij E lR, i,j = 1,2,3,
9fjklijkl > 6lj
( 4.2.10)
98

where 9i J o kl = lim 9ijkl (s) and hereafter the summation convention is used.
8-00
There exists a positive, monotone decreasing scalar function 91 (s) E L1 (0,00) n
C1 (0, 00) and constants k 1 > 1, k 2 , Sl > 0 such that
k 1 91 (s )lj > -9jkl( s )ijkl > 91 (s )lj' \I s > 0
(4.2.11)
and
k 2 91 (s )lj > 9jkl (s )ijkl, \I s > SI,
( 4.2.12)
for all ij E lR, i, j = 1,2,3.
(H3) K = K T and there is a constant 6 > 0 such that for all i E lR, i = 1,2,3,
K,ijij > 6l.
( 4.2.13)
(H4) e(s) E C[O,+oo), e'(s) E L1(0,00)nC 1 (0,00) and there are positive constants
co, Sl , k3 such that
e(O) > Co > 0,
k3 e' ( s) > - e" ( s ) ,
\Is > Sl.
(4.2.14)
(4.2.15)
(4.2.16)
e' ( s) > 0, e" ( s) < 0, \I s > 0,
(Hs) l(s) = IT(s), i.e., lij(s) = lji(s) for s > 0; l(s) E C[O,+oo), l'(s) E L 1 (0,00)
n C 1 (0, 00). Furthermore, there is a constant Q E (0,1) such that
1
111'(s)1I = (l:j(s)z:j(s))t < a(g}(s))t (  c'(s)) 2", Vs > 0, (4.2.17)
111"(s)1I = (l:j(s)Z:j(s))t < a(g}(s))t( - :0 c"(s))t, Vs > O. (4.2.18)
These assumptions on material properties are quite comparable with those made in
Navarro [1]. Notice that we now allow g'(s),l'(s) and e'(s) to have singularity at s = 0
(see Chapter 3 in this aspect). We will use the semigroup approach to prove that under
these assumptions, the initial boundary value problem for the linear three-dimensional
thermoviscoelastic system with memory (4.2.6)-(4.2.9) defines a Co-semigroup in an
99

appropriate Hilbert space. We will also prove the exponential stability of that semi-
group under the following additional assumptions on 9 ( s) and e( s ):
(Hs)
9jkl(S)ijkl > 9t(S);j' Vs > 0, Vij E 1R
(4.2.19)
where 9t also satisfies the following further assumption:
- k49 ( s) > 9t ( s ) ,
Vs> 0
( 4.2.20)
with k4 being a given positive constant.
( H 7) There is a posi ti ve constant ks such that
- ks e" ( s) > e' ( s ) ,
Vs > o.
( 4.2.21)
Let
W(x, t, s) = u(x, t) - u(x, t - s), y(x, t, s) = (}(x, t - s).
( 4.2.22)
Define the Hilbert space
1-l - H(O) x L 2 (0) x L 2 (0) X Lg(O, +00; H(O)) x L(O, +00; L 2 (0))
U X V x e x W x Y (4.2.23)
equipped with the following inner product:
for Zt = (Ut, Vt, (}I, WI, Yt)T, Z2 = (U2, V2, (}2, W2, Y2)T E 1-l,
(Zt, Z2)1t
(Ut,U2)U + (Vt,V2)V + ((}t,(}2)e + ({Wt,Yl}, {W2,Y2})WxY
 [900'VUl . 'VU2 + PVl . V2 +  c(O)6 1 6 2 ] dn-
10 00  '(S)'VWl . 'V W2 + Yl(l'(S) . 'V W 2) + (I'(s) . 'VwIHh -  C'(S)YIY dnds.
( 4.2.24)
100

By assumptions (H 2 ), (H4)-(Hs),
10''''' fo ( -g'(s)'Vw. 'V w - yl'(s). 'V w - yl'(s). 'Vw +  c'(s)lyI2) dnds
> 10''''' (9 1 (S) fo l'Vwl 2 dn +  c'(s) fo lyl2 dn) ds
-Q 10 00 (91(S) fo l'Vwl 2 dn +  c'(s) fo lyl2 dn) ds
- (1 - Q) 10 00 (91(S) fo l'Vwl 2 dn +  c'(s) fo lyl2 dn) ds
> o. (4.2.25)
It is obvious that the equal sign in (4.2.25) holds if and only if w = 0, y = o. This,
together with assumption (H 2 ), shows that (4.2.24) is indeed an inner product on 1-£.
Let z = (u, v, 8, w, y)T. Then the initial boundary value problem (4.2.6)-(4.2.9) can
be reduced into the initial value problem for a first-order evolution equation in 1-£:
{ dz
-=Az
dt '
z(O) = Zo
( 4.2.26)
where
v
Az=
1 10 00
-div{gOOyu -1(0)8 - (g'Vw + l'y) ds}
p 0
1 1 00
( ) {-8 0 1(0) . 'Vv - (8 0 1' . 'Vw' + pc'y') ds + div(K'V8)}
pc 0 0
( 4.2.27)
v -w'
-y'
with
v E H(f2), w' E L(O, +00; H(f2)), w(O) = 0,
V(A) = z E 1-£ div {gOOVu -1(0)8 - foOO(g'Vw + l'y) ds} E L 2 (f2), . (4.2.28)
y' E L(O, 00; L 2 (f2)), y(O) = 8
Theorem 4.2.1 Under the hypotheses (Hl)-(Hs) , the operator A defined in (4.2.27)-
(4.2.28) generates a Co-semigroup S(t) = eAt of contractions on 1-£.
101

Proof We want to use Theorem 1.2.4 in Chapter 1 to prove the present theorem. It is
easy to see that V(A) is dense in 1i. To prove the dissipativeness of A, by integration
by parts, we have
Re(Az, z)'H I
Re {(1I, u )u + G dill(goo"Vu -l(O)O - looo (g'(s)"Vw + l'(s)y) ds), 11) v
+ CcO) ( -Ool(O) . "V1I - looo (Ool' . "Vw' + pc' ( s )y') ds + dill( K, "VO)) , B) e
looo k (g'(s)"V(lI - w') . "V w + yl'(s) . "V(1I - w') - y'l'(s) . "V w
+  c'( s )Y'Y) dnds}
looo k [ :((:: _ :o( :' s) ] ( yw: , ) . ( y"Vw B ) dnds
1 1 -
- -KV8. v8do.
n ()o
<-o}Jf-+oo  k [::: _ :o( ;(S) ] ( y"Vw o ) . ( y"Vw B ) dn s=<
1 i N 1 [ g"(s)
--Re lim
2 €-o+ ,N-oo € n 1"( s)
1 1 -
- -KV8. V8 dO..
n ()o
s=N
l"(s) ] ( v w ) ( V w )
-  d'(s) y-O. y-O
do.ds
(4.2.29 )
Define
h(s)  - k [::; _ :,( ::(S) ] ( y"Vw o ) . ( y"Vw o ) dn.
By assumptions (H 2 ), (H 4 ), and (Hs) and w' E Lg(O, +00; H6), y' E L(O, +00; L 2 ),
we can easily deduce that h( s), h'( s) E L 1 (S1, 00). Hence
( 4.2.30)
lim h(N) = o.
N-oo
(4.2.31 )
In what follows, we prove by a contradiction argument that h( c)  0, as c  o.
Suppose that it is not true. Then there is a constant 6 > 0 and a sequence Sn, Sn  0+
102

such that h( sn) > 6 > 0 for all n. By (4.2.11) and (4.2.17), we have
h(s) < (1 - (1) (91 (s) !r/ilwI2 df! +  c'(s)  Iy - 81 2 df!) .
We now prove that 91(Sn) In lV'w(sn)1 2 df2  o. Indeed, as Sn  0+,
( 4.2.32)
91(Sn)  'Vw(Sn) . 'V w (Sn) df!
- 91(Sn)lIV'w(sn)1I2
- 91(sn)11 fn 'Vw'(7") d7"112
< Sn91 (sn) fn lI'Vw' (7") 11 2 d7"
< Sn fn 91 ( 7" ) II 'V w' ( 7" ) 11 2 d7" -+ 0, as Sn -+ 0+.
( 4.2.33)
Similarly, we can also get
c'( Sn) lIy( Sn) - 811 2  0, as c  0+.
( 4.2.34)
Thus combining (4.2.32) with (4.2.33) and (4.2.34) yields a contradiction. This proves
that h( s)  0, as S  o. Then it follows from (4.2.29) that
1 00 [ g"(s) l"(s) ] ( V' W ) ( V' w )
[ [ p " . _ df!ds
- "2 io in l" (s) - 8 0 c (s) y - 8 ii - 8
( 4.2.35)
is convergent and
Re(Az, z)
1 00 [ g"(s)
= -"2 fa  l"(s)
1 1 -
-- KV'8. V'8df2.
(}o n
l"(s) ] ( V' W ) ( V' w )
p " . _ df2ds
- (}o c (s) y - 8 Y - 8
( 4.2.36)
By assumptions (H 2 ), (H3), and (Hs), each term on the right hand side of (4.2.36) is
non-positive. Thus the dissipativeness of A follows.
Now it remains to prove that 0 E p(A). Let F = (/1,/2,/3,/4, fs)T E 11,. We
consider the equation
-Az=F
( 4.2.37)
103

I.e. ,
-v = 11 E U, (4.2.38)
-  div {gooVU -1(0)0 - fooo [g'(s)Vw + l'(s)y] ds} = 12 E V, (4.2.39)
PCO) {0 0 1(0) . Vv + fooo [Ool'(s) . Vw' + pc'(s)yj ds - div(KVO)} = 13 E e,
( 4.2.40)
( -V; w' ) = ( : ) E W x Y.
From equations (4.2.38) and (4.2.41) we deduce that
(4.2.41 )
v = - /1 E U,
( 4.2.42)
( W ) = ( 1o"[/4(:)-/l]dr ) .
y 0 + fo Is(r)dr
Since {w',y'} = {/4(s) - 11'!s(s)} E W x Y, as in the proof of Theorem 3.2.1, we
( 4.2.43)
can conclude that w E Wand y E Y. Moreover,
( -v + w' )
y' E W x Y.
(4.2.44 )
Substituting (4.2.42) and (4.2.43) into equation (4.2.39) and (4.2.40), we obtain that
1 --
--div(gOOVu -IOOB) = /2'
P
( 4.2.45)
and
-1. -
pc(O) d'lV(KVO) = 13
( 4.2.46)
where 1 00 = lim l( s) and
8-00
12 - 12 -  div {fooo fo8 [g'(s)(V/4(r) - VII) + 1'(s)/s(r)] drds} E H-\
1
13 - 13- pc(O) Ool(O).V/l
+ pcO) {fooo 10" [Ool'(s) . (V 14( r) - V II) + pc'(s )/s( r)] drds} E L 2 .
104

It follows from the standard results in the elliptic boundary value problems (see
Chapter 1 or Lions & Magenes [1]) that equation (4.2.46) has a unique solution
8 E HJ(f2)nH 2 (f2). Moreover,
6 0 l(0) . 'Vv + 1o'X> [6 0 l'(s) . 'Vw' + pc'(s)y'] ds - div(K.'V6) E 0.
(4.2.47)
Define the following bilinear form on U = H6(f2):
b( u, u) = p  goo'Vu . 'Vu dr!.
( 4.2.48)
Clearly it is bounded and coercive. Then by the Lax-Milgram theorem, there is a
unique u E U = H6(f2) such that
b(u,u) = P(fl'U) + (IOO8, V'u)v, Vu E H6
( 4.2.49)
where (, ) denotes the dual product between U = H6 and U' = H- I . Furthermore,
it follows from (4.2.45) that
div {goo'Vu -l(0)6 - looo [g'(s)'Vw + l'(s)y] ds} E V
( 4.2.50)
Finally, it follows from (4.2.43) that
w(O) = 0, y(O) = 8.
(4.2.51 )
Thus, we have proved that for any F E 1i, the equation -Az = F has a unique
solution z = {u, v, 8, w, y} E V(A). It is easy to verify that IIzll'H < KIIFII'H for some
constant K > o. Thus 0 E p(A) and by Theorem 1.2.4 in Chapter 1 we can conclude
that A generates a Co-semigroup of contractions on 1-£. 0
Under further assumptions (Hs) and (H7), we have the following result on the
exponential stability.
Theorem 4.2.2 Under the hypotheses (HI )-(H7), the Co-semigroup S(t), generated
by A, is exponentially stable.
Proof We still use Theorem 1.3.2 in Chapter 1 to prove the present theorem.
First we prove (1.3.3) by a contradiciton argument. The proof consists of the following
105

steps:
(i) Since 0 E p(A), i!3 - A is invertible for !3 E 1R and 1!31 < IIA-Ill- I .
(ii) If (1.3.3) is not true, then there is w E 1R with IIA-I\I-I < Iwl < 00 and a sequence
of complex vector functions Zn E V(A) with IIZnll'H = 1 and a sequence of !3n with
l!3nl < Iwl, !3n  w such that as n  +00,
lim II (i!3n I - A)znll'H = 0,
n-+oo
( 4.2.52)
I.e. ,
i!3nun - V n  0 in U, (4.2.53)
i!3n v n - !.div{gooV'u - 1(0)8 - [00 [g'( S )V'w n + 1'( s )Yn] ds}  0 in V, (4.2.54)
p h
if3n B n + PCO) {Bol(O) . 'Vv n + 10''''' [Bol' (s) . 'Vw' + pc'( s )y'] ds + div (K, 'VBn) }
 0 in 8, (4.2.55)
if3n ( :: ) _ ( V n _y  )  (  )
in W x Y.
( 4.2.56)
It follows from (4.2.52) that
Re( -Az n , zn)'H  o.
( 4.2.57)
Thus, we obtain from (4.2.36) that
00 [ g"(s) l"(s) ] ( V'wn ) ( V' wn )
!o  l"(s) -  c"(s) Yn _ Bn . fin _ Bn dnds  0,
( 4.2.58)
and
118nIlHJ(O)  o.
( 4.2.59)
By assumptions (Hs)-(H7) and (4.2.58), we have
10''''' g"(S)'VWn.'V Wn dnds+ {'"  - :/ '(s)IYn-BnI2dnds0.
106
( 4.2.60)

Since both terms in (4.2.60) are non-negative, we get
{ 10'>0 hp"( s )'Vw n . 'V w n dnds -+ 0,
10 00 10 -  CIl(S)IYn-6nI2dnds -+ 0,
(4.2.61 )
which further implies, due to assumption (Hs)-(H7), that
{ 1000 91(S) 10 l'Vw n l 2 dnds -+ 0,
10 00  c'(s) 10 IYn - 6nl 2 dnds -+ O.
( 4.2.62)
Hence,
( Yn wn 6 n) Xy
- - 10 00 10 [:(: _ ;,( ;S) ] ( Y W n ) . ( Y W ;n ) dnds
< 210 00 10 -g'(s)'Vwn.'V wn dnds+  c'(s)IYn-6nI2dnds
< 2k 1 10 00 91(S) 10 l'Vw n l 2 dnds + 2 10 00  c'(s) 10 IYn - 6nl 2 dnds
 O. ( 4.2.63)
By (4.2.59), we obtain that
(:n) 2
WxY
-  10 00 10 c'(s)16nI2dnds
-  (COO - c(0))1I 6 nIl 2 -+ O.
(4.2.64 )
Combining (4.2.64) with (4.2.63) yields
( :: )
 0,
( 4.2.65)
WxY
I.e. ,
W n  0 in W, Yn  0 in Y.
( 4.2.66)
107

Since IIZnll1i = 1, it follows from (4.2.59) and (4.2.65) that
IIU n lib + IIV n II  1.
( 4.2.67)
We now take the inner product of (4.2.53) with ;: in U and (4.2.54) with ;: in V to
get
i II Un lit, - ;n (V n , Un)U --+ 0,
illVnll + ;n (Un, Vn)u - ;n 10 1(0) . 'V vn O n dO.
+ ;n 10 00 10 (g'(S)'VW n + 1'(S)Yn)' 'V V n dnds --+ O.
( 4.2.68)
( 4.2.69)
In what follows, we prove that the last two terms of (4.2.69) converge to zero. Indeed,
by integration by parts, we have
;n 10 1(0) . 'V vn O n dO. < KlIvnllvll'VOnll --+ 0,
(4.2.70)
and
;n 10 00 10 (g'(S)'VW n + 1'(S)Yn) . 'V V n dnds
( wn ) 1 ( Vn )
- ( Yn ' (3n 0 )WxY
< (::) WXY' ;n ( V On ) WxY
< K ( w n ) II ;: II U --+ 0
Yn WxY
(4.2.71)
where we have used (4.2.53), (4.2.65) and the fact that Ilu n Ilu < 1.
Adding (4.2.68) up to the complex conjugate of (4.2.69) yields
Ilunllb - Ilvnll  o.
(4.2.72)
Hence,
2 1 2 1
Ilunllu  2' Ilvnllv  2.
(4.2.73)
108

Dividing (4.2.56) by (3n and using (4.2.65), we have
1 ( Vn ) 1 ( w )
(3n 0 - (3n Y -+ 0
In WxY.
(4.2.74)
Since
 ( SVn ) 2
(3n 0
WxY
lo OO 2 1 VVn V V n
< Kl S 9I(S)ds -. -df2
o (2 (3n (3n
< Kllunll& < K,
- lo OO 1 2 ' ( ) VVn V V n d {") d
- -s 9 s -. - .1G S
o (2 (3n (3n
(4.2.75)
we can take the inner product of (4.2.74) with n ( Sn ) in W x Y to get
n ( s:n ) 2 _  \ ( :' ) , ( Sn ) )
WxY n WxY
- (I) + (I I) -+ O.
(4.2.76)
In what follows, we prove that (I I) converges to zero.
By integration by parts with respect to t and assumptions (H 2 )-(H 7 ), we get that
IIII - ; 110 00 10 sg'(s)"Vw n . "V vn df!ds + 10 00 10 sl'(s). "V Vn Ydf!dsl
<  110 00 10 sg"( s )"Vw n . "V v n df!dsl +  110 00 10 g'(s )"Vw n . "V v n df!dsl
+  110 00 10 (sl"( s) . "V vn )Yn df!dsl +  110 00 10 (I'(s) . "V vn )Yn df!dsl.
(4.2.77)
By assumptions (H 2 )-(H 7 ), the terms on the right hand side of (4.2.77) can be esti-
mated as follows:
; 110 00 10 sg"( s )"Vw n . "V v n df!dsl
1 ) 1
1 00 2" 00 Vv V v 2
< (3n (10 10 g"(s )"VW n . "V W n df!ds) (10 10 s2g"(S) (3n n . (3n n df!ds
109

< ;n (loo 1ogl/(s)Vw n . V Wn dndS)t (10"' 10 S2gl/(S) :n . :n dnds
1
+00 V 2 ) '2
+ l k 2 s 2 91 (s)ds 10 /3: n dn
<  (loo 1ogl/(S)VW n . V Wn dnds)t --+ 0, (4.2.78)
; Iloo 10 -g'(s)Vw n . V Vn dndsl
1
< ;n (loo 10 -g'(S)VWn.V Wn dnds)t (lOOk 191 (S)dS 10 :n 2 dn)'
1
<  (loo 91(S) 10 IVWnl2dndsr --+ 0, (4.2.79)
and
; Iloo 10 sll/(s) . V Vn Yn dndsl
1
< i (lOO 10 S291(S) :n 2 dnds) , (loo 10 -  cl/(S)IYnI 2 dnds) t
K ( [00 P [ ) t
< /3n 10 0 0 c'(s) 10 IYnl 2 dnds --+ 0,
( 4.2.80)
; Iloo 10 sl'(s). V Vn YndndSI
1
<  (lOO 10 S291(S) :n 2 dndS) , (loo 10  c'(s)IYnI 2 dnds) t
1
<  (loo 10 d(S)IYnI2 dnds r --+ O. (4.2.81)
Therefore, (I I) converges to zero. It turns out from (4.2.76) that (I) also converges
to zero. Finally, since
) 1 00 ( ) 1 VVn V V n I VVn 1 2
(I > S91 S ds -. -df2 > vi-I u
o (2!3n!3n !3n
with v being a positive constant, we obtain that
VV n -+ 0
!3n u .
( 4.2.82)
( 4.2.83)
110

By (4.2.53), we conclude that
Ilunll -+ 0
(4.2.84 )
which contradicts (4.2.73). Thus (1.3.3) is proved.
(iii). The proof of (1.3.4) is exactly the same as in (ii) because in the proof of (1.3.3)
we only use the fact that /3n converges to a limit which is different from zero. 0

Chapter 5
Elastic Systems with Shear Damping
In this chapter, we consider some linear elastic systems with shear damping. Section
5.1 is devoted to the so-called shear diffusion equations proposed in Russell [2]. Section
5.2 deals with the laminated beam equations with shear damping derived in Liu,
Trogdon & Yong [1]. Our goal is to obtain the exponential stability and analyticity of
the semigroups associated with these equations.
5.1 Shear Diffusion Equations
Let us start from the following Timoshenko beam equations (see Timoshenko, Young
& Weaver [1]):
8 2 w ( 84W 8 3 r ) ( 84W fJ3r )
p 8t 2 - Ip 8x 4 + 8t 2 8x + EI 8x 4 + 8x 3 = 0, in [0, L] x 1R+ (5.1.1)
I p ( a;x + :: ) + aT - EI ( : + : ) = 0, in [0, L] x lR+ (5.1.2)
where w is the lateral deflection and r is the shear angle, L is the length of the beam,
and p, a, Ip, E, I are positive constants.
When the viscous force which affects the evolution of r is taken into account, a
term 20" with 0" > 0 should be added to equation (5.1.2). If we further assume
that the rotatory inertia Ip is relatively small so that the corresponding terms can
be neglected, then equations (5.1.1) and (5.1.2) become the shear diffusion model
proposed in Russell [2] for the Euler-Bernoulli beam equations:
8 2 w ( 8 4 w 83r )
p 8t 2 + EI 8x 4 + 8x 3 0,
8r ( 8 3 W 82r )
20" 8t + ar - EI 8x 3 + 8x 2 = o.
The boundary conditions and initial conditions considered here are
(5.1.3)
(5.1.4)
8w
w = 8x + r = 0, at x = 0,
(5.1.5)
112

a 2 w ar fJ3w a 2 r
aX 2 + ax = aX3 + aX2 = 0, at x = L,
w(x,O) = wo(x), Wt(x,O) = WI(X), r(x,O) = rO(x).
(5.1.6)
(5.1.7)
The energy associated with this model is defined by
[ ( ) 2 ( 2 ) 2 ]
1 L aw a w ar
E(t) = 2" 10 p at + ar 2 + EI ax2 + ax dx.
(5.1.8)
Let z = (w, v, r)T and
H= {z
w E H 1 (0,L),wl x =o = O,v,r E L2(0,L), }
(Dw + r) E HI(O, L), (Dw + r)lx=o = O.
(5.1.9)
equipped with the norm
II Z Il1i = (EIIID(Dw + r)1I2 + pllvl1 2 + allrl1 2 )t
(5.1.10)
and the corresponding inner product. As usual, in the above we denote D; = ::J for
j = 1,2,... and we denote by II. II the L2(0, L) norm. Hereafter we also denote by (.)
the inner product in L 2 (0, L). It is clear that 1i is a Hilbert space. Let v = Wt. Then
the system (5.1.3)-(5.1.7) can be reduced to the following initial value problem for a
first-order evolution equation on 1i:
{ dz ( t )
dt = Az(t), Vt > 0
z(O) = Zo = (wo, WI, ro)T
(5.1.11)
with
v
Az=
EI 3
--D (Dw + r)
p
EI 2 a
-D(Dw+r)--r
20" 20"
(5.1.12)
and
v E HI(O, L), vlx=o = 0, Dw + r E H 3 (0, L),
V(A) = z E 1i D(Dw + r)lx=L = D2(Dw + r)lx=L = 0,
Dv - 2: r E HI(O,L), (Dv + : D2(Dw + r) - 2: r)lx=o = 0
(5.1.13)
113

Theorem 5.1.1 The operator A defined in {5.1.12} and {5.1.13} generates a C o -
semigroup S(t) = eAt of contractions on the Hilbert space H.
Proof Clearly, V(A) is dense in H. By integration by parts, we have
(Az, z)1i
{ EI a
EI(D(Dv + 20" D 2 (Dw + r) - 20" r), D(Dw + r)) - EI(D 3 (Dw + r), v)
EI a }
+ a( 20" D 2 (Dw + r) - 20" r, r)
1
- 20" IIEI D 2 (Dw + r) - a r l1 2 < O. (5.1.14)
Thus, A is dissipative. To apply Theorem 1.2.4 in Chapter 1, it remains to show that
o E p(A). For any F = (f, g, h)T E H, consider the following equation:
Az = F.
(5.1.15)
Equation (5.1.15) gives
v
f E H 1 (0, L)
pg E L 2 (0, L)
20"h E L 2 (0, L).
(5.1.16)
(5.1.17)
(5.1.18)
-EI D 3 (Dw + r)
EI D 2 (Dw + r) - ar
To show that it yields a unique solution z E V(A), we introduce another inner product
space
{ wE HI(O, L), wlx=o = 0, r E L2(0, L), }
HI = {w,r}
(Dw + r) E Hl(O, L), (Dw + r)lx=o = O.
(5.1.19)
equipped with the inner product
({ w, r}, {iV, r} )1i 1 = EI(D(Dw + r), D(Dw + r)) + a(r, T).
(5.1.20)
It is clear that HI is also a Hilbert space. Thus, by the Riesz representation theorem,
there is a unique {w, r} E HI such that
({ w, r}, {iV, r} )1i 1 = -(pg, iV) - (20"h, T)
(5.1.21)
114

for all {w, r} E Ht.
In what follows, we use the standard variational approach to discuss regularity
property of a pair of solutions {w, r} and what boundary conditions they satisfy. For
this purpose, first by taking w = 0 and r E CO'(O, L) in (5.1.21), we have
EI(D(Dw + r), Dr) + a(r, r) = -(2uh, r).
( 5 .1. 22 )
It follows from (5.1.22) that Dw + r is a weak solution of
EI D 2 (Dw + r) = 2uh + ar E L 2 (0, L)
( 5.1.23)
in the distribution sense. Thus, it follows that
Dw + r E H 2 (0, L).
(5.1.24 )
Similarly, we take r = 0 in (5.1.21) to get that
EI(D(Dw + r), D 2 w) = -(pg, w)
( 5.1.25)
holds for all w E CO'(O, L). Thus, we also obtain that Dw + r is a weak solution of
- EID 3 (Dw + r) = pg
( 5 .1. 26)
in the distribution sense which implies that Dw + r E H3(0, L). Moreover, since
(5.1.25) holds for any w with {w, O} E Ht, by integration by parts, we can easily
deduce that D(Dw + r)lx=L = D2(Dw + r)lx=L = O. Other boundary conditions in
the definition of V(A) can be easily verified by the assumption, F E H. Therefore,
a unique solution z = (w, v, r)T E V(A) to (5.1.15) is obtained. Moreover, it is clear
from (5.1.16)-(5.1.18) that IIZII'H < KIIFII'H with K being a positive constant. Thus,
o E p(A). By Theorem 1.2.4, the proof of this lemma is complete. 0
Theorem 5.1.2 The Co-semigroup S(t), generated by A, is exponentially stable.
Proof We will use the systematic method described in this book, i.e., apply Theorem
1.3.2, and use the contradiction argument to prove the present theorem.
(i) Suppose that condition (1.3.3) is false. Since 0 E p(A), then there is an w E 1R with
115

IIA-III- I < Iwl < 00 such that {i,8II,81 < Iwl} C p(A) and sup{lI(i,8 - A)-III I 1,81 <
Iwl} = 00. Thus, there exists a sequence,8n -+ w, l,8nl < Iwl, and a sequence of complex
vectors Zn = (w n , V n , Tn)T E V(A) with IIZnll1i = 1 such that as n -+ 00,
II (i,8n I - A)zn 111i -+ 0,
(5.1.27)
I.e. ,
. ,8 ) EI 2 Q
Z nD(Dwn + Tn - D(Dv n + _ 2 D (Dw n + Tn) - -Tn) -+ 0 In
u 2u
i{3n v n + EI D 3 (Dw n + Tn) -+ 0 in £2(0,£),
p
. EI 2 ( Q 2
z,8n T n- 2u D DWn+Tn)+ 2u Tn-+0 in L (O,L).
L 2 (0, L),
(5.1.28)
(5.1.29)
(5.1.30)
Since
IRe(Az n , zn)1i1 < II (i,8n I - A)zn 111i,
it follows from (5.1.14), (5.1.27), and (5.1.31) that
IIEI D 2 (Dw n + Tn) - QTnl1 -+ 0,
(5.1.31)
(5.1.32)
which further implies II,8nTnll -+ 0, due to (5.1.30). This yields
IITnl1 -+ 0
(5.1.33)
because l,8nl is bounded below from zero. It follows from (5.1.32) that
IID2(Dw n + Tn)11 -+ 0,
( 5 .1. 34 )
which further leads to
IID(Dw n + Tn)11 -+ 0,
(5.1.35)
due to the Poincare inequality.
In what follows, we prove that V n converges to zero in L 2 (0, L). Combining this
with (5.1.33), (5.1.35), and Ilznll1i = 1 will yield a contradiction. To do so, we first
take the inner product of (5.1.29) with 7f:. in L2 and integrate by parts to get
ipllv n ll 2 - EI(D 2 (Dw n + Tn), :n ) -+ O. (5.1.36)
116

On the other hand, owing to (5.1.35), we can rewrite (5.1.28) as
D( DVn EI D 2 (D ) Q ) 0 l . n L 2 . ( 37)
/3n + 20"/3n W n + Tn - 20"/3n Tn -+ 5.1.
Next, we take the inner product of (5.1.37) with D(Dw n + Tn) in L 2 and integrate by
parts to obtain
( DVn 2 ( ) ) E I I 2 ) 2
----r.;-, D DW n + Tn + _ /3 I D (Dw n + Tn II
n 20" n
Q 2
- 2u/3n (r n , D (Dw n + r n )) -+ O. (5.1.38)
It follows from (5.1.34) that the second and third terms in (5.1.38) converge to zero.
Thus, combining (5.1.38) with (5.1.36) yields
IIV n II -+ 0,
(5.1.39)
i.e., the desired contradiction. Thus (1.3.3) is verified.
(ii) Suppose that (1.3.4) is not true. Then we can repeat exactly the same argument
as in part (i) to get the same contradiction because in the proof of part (i) we only
use the fact that /3n is bounded below from zero. Thus, the proof is complete. 0
Furthermore, we have the subsequent theorem.
Theorem 5.1.3 The semigToup S(t), geneTated by A, is analytic.
Proof Since we have already proved that ilR c p(A) in the previous theorem, we
need only to verify condition (1.3.8) of Theorem 1.3.3. Suppose it is false. Then there
exists a sequence /3n -+ 00, (/3n > 0 without loss of generality) and a sequenece of
complex vectors Zn = (w n , V n , Tn)T E V(A) with IIZnll1i = 1 such that as ri -+ 00,
(if - ;n A)zn 1{ -+ 0,
(5.1.40)
I.e. ,
iD(Dw n +rn)-D( ;n DV n + 2:;n D 2 (Dw n +r n )- 2:/3n r n ) -+ a
iv n + E /3 f D 3 (Dw n +r n )-+ 0 III L 2 (0,L),
p n
. EI 2 Q 2 ( )
ZT n - 20"/3n D (Dw n + Tn) + 20"/3n Tn -+ 0 In L 0, L .
In L 2 ( 0, L ) ,
(5.1.41)
(5.1.42)
(5.1.43)
117

Since
;n Re(Az n , Zn),i < (iI - ;n A)zn 7{
it follows from (5.1.40) and (5.1.14) that
( 5 .1.44 )
1 22
,an IIEID (Dw n + r n ) - arnll - 0,
(5.1.45 )
which implies
IIrnli  0,
( 5.1.46)
due to (5.1.43). Now (5.1.45) leads to
-';'IID 2 (Dw n + rn)1I  o.
(3J
In view of (5.1.42), kllD3(Dwn + rn)1I is uniformly bounded in n. Then, it follows
from (5.1.41) that kllD(Dvn - rn)11 is uniformly bounded in n. On the other hand,
since EIIID(Dw n + r n )1I 2 < 1, by the Poincare inequality, IIDw n + rnll is uniformly
(5.1.47)
bounded in n, which further leads to the uniform boundedness of IIDwn11 because
IIrnll  o. It follows from
;n IID(DV n + ;:' Dw n ) II < ;n GID(DV n - 2: rn )11 + 2: IID (Dw + rn)ll) (5.1.48)
that ;n IID(Dv n + 2':. Dw n )11 is uniformly bounded in n. By the Gagliardo-Nirenberg
inequality, we deduce that ;! IIDv n + 2 Dwnll is also uniformly bounded. Therefore,
1
IIDvnll < K, \In
(3J
(5.1.49)
for some constant K > O.
We now take the inner product of (5.1.42) with V n in L 2 (0, L) and integrate by parts
to get
ipllv n l1 2 - ;n EI(D 2 (Dw n + r n ), Dv n ) - O.
Therefore, by (5.1.49) and (5.1.47) we have
(5.1.50)
IIVnll  o.
(5.1.51)
118

Similarly, taking the inner product of (5.1.41) with D(Dwn+r n ) in L 2 (0, L), integrating
by parts and applying (5.1.46), (5.1.47), and (5.1.49), we also obtain
IID(DW n + r n ) II  O.
(5.1.52)
In summary, we have proved IIZnll'H  0 which contradicts IIZnll1-l = 1. Thus, the proof
is complete. 0
Remark 5.1.1 For the simply supported boundary conditions, i.e.,
W = D 2 w + Dr = 0, at x = 0, L,
(5.1.53)
the exponential stability of the associated semigroup was proved in Liu & Zheng [3J.
The analyticity of this semigroup was proved in Russell [2J as an example of his general
results. A quite strong assumption he imposed is that all differential operators appeareq
in A in his general setting are mutually commutative. This assumption rules out most
of other boundary conditions including the one considered here. A recent result on
analyticity of the Co-semigroup associated with a class of general linear c0'!lpled systems
without the "commutative" restriction has been obtained in Liu & Yong [1 J which also
includes an application to the shear diffusion equations. But only the example with
simply supported boundary conditions was given.
5.2 LaIIlinated BeaIIl with Shear DaIIlping
In this section, we discuss the exponential stability and analyticity of the Co-semigroup
associated with a three-layer laminated beam model given in Liu, Trogdon & Yong [1].
Consider a three-layer laminated beam of unit width and length L with 0 < x < L.
The beam is composed of a top and a bottom face-plate of thicknesses h (1) and h (3) ,
and a core layer of thickness h (2). The top and bottom layers are made of linear elastic
material. The core layer is assumed to be a material of Kelvin-Voigt type with a shear
stress-strain relation
T = C , + C1/t.
(5.2.1)
119

Under some assumptions, we are led to the following coupled system of partial diffe:G-
ential equations (we refer to the reader to the paper Liu, Trogdon & Yong [1] for the
detail of derivation):
04W Q 0 ( 03W ( 2 / )
phWtt = -Eele ox 4 - k ox Q ox 3 - 2h(2) ox2 '
1 ( 03w ( 2 / )
Cl1t = -C 1 - - Q- - 2h(2)-
k OX 3 OX 2
(5.2.2)
where p, h, Ee, Ie, Q, k, h(2), Cl, C are given positive constants depending on the mate-
rial properties and reference configuration of the beam; wand 1 are the transversal
displacemnt and shear strain of the core layer, respectively. When Cl = 0, one can
eliminate 1 in equation (5.2.2) to obtain a sixth-order partial differential equation
ck ph ph 02Wtt ck ( Q2 ) 04W 06W
2h(2) EeIe Wtt - EeIe ox2 + 2h(2) 1 + EeIe k ox 4 - ox 6 = o.
(5.2.3)
By scaling in time, equation (5.2.3) coincides with the three-layer laminated beam
equation derived in Mead & Markus [1] and D.K. Rao [1].
This system is slightly different from the shear diffusion model discussed in the
previous section. However, their physical interpretation has essential differences. For
the three-layer laminated beam model, shear damping in the core layer is passed
on to the other layers through the interfaces to cause the energy dissipation. From
mathematical point of view it significantly differs from the classical thermoelastic beam
model in some aspects. Equations (5.2.2) are coupled via a spatial derivative of the
beam displacemant, while the classical thermoelastic beam equations are coupled via
the spatial derivative of beam velocity.
The boundary conditions and the initial conditions considered in this section are
the following:
ow
W = - = 1 = 0, at x = 0,
ox
0 1 02W
oX = ox 2 = 0, at x = L,
03W Q ( 03w ( 2 / )
EeIe ox 3 + k Q OX3 - 2h(2) OX 2 = 0,
(5.2.4)
(5.2.5)
at x = L,
120

W ( X, 0) = Wo ( x ) , Wt ( X, 0) = WI ( X ) , l' ( X, 0) = 1'0 ( X ) .
(5.2.6)
The total energy of the system (5.2.2)-(5.2.5) is defined by
E(t) =.!. fL [ Ee1e ( 02W ) 2 +.!. ( a: 02W _2h(2) 01' ) 2 +2h(2)cl +PhW 2 ] dx (5.2.7)
2 J o ox 2 k ox 2 Ox t
where the first three terms in (5.2.7) represent the energy due to bending, stretching
and shearing, and the last term is the kinetic energy.
Let Hi(O, L) be a closed subspace of the usual Sobolev space Hm(o, L) defined as
follows: Hi(O, L) = {u E Hm(o, L)I ulx=o = . . . = Dm-1ulx=0 = O}. Let
1-(, = Hl (0, L) x L 2 (0, L) x Hl (0, L )
( 5.2.8)
equipped with the norm
1
II (w, v, i ?1I1i = (EelellD2wll2 + phllvl1 2 +  lIaD 2 w - 2M2) Dill 2 + 2h(2)cIl711 2 ) 2"
(5.2.9)
and the corresponding inner product. Then it is easy to see that 1i is also a Hilbert
space. To study the initial boundary value problem (5.2.2)-(5.2.6) by the semigroup
theory, we denote by v = Wt and z = (w, v, 1')T and rewrite problem (5.2.2)-(5.2.6) as
the initial value problem for a first-order evolution equation in 1i:
{ dz ( t )
 = Az(t), \It > 0
z(O) = Zo = (wo, WI, 1'o)T
(5.2.10)
with
W
A v
l'
and
v
1 ( 4 a: 3 ( 2 ) 2 ))
ph -Ee1e D w - k D(aD w - 2h D i
 ( -Cf - .!.( aD 3 w - 2h(2) D2i ) )
Cl k
(5.2.11)
W E H 4 , V E Hr,1' E H 3 , D1'lx=L = D 2 wlx=L = 0,
V(A) = (w, v, r)T E 1i (a:D 3 w - 2h(2)D 2 1')lx=0 = 0,
[ Ee 1 e D3w + a: (a:D 3 w - 2h(2) D 2 1' ) ] I = 0
k x=L
(5.2.12)
121

Now we have. the foUowing.
Theorem 5.2.1 The operator A defined in (5.2.11) and (5.2.12) generates a C o -
semigroup S( t) = eAt of contractions on the Hilbert space ft.
Proof It is clear that V(A) is dense in ft. We first prove that A is dissipative. For
any (w,v,!,)T E V(A), by integration by parts we have
Re(A(w, v, !,)T, (w, V, !,)T)1-£
- Re [Ee1e(D2v,D2w) + (-Ee 1 e D4w -  D(a:D3W - 2M2)D 2 "{), v)
+(a:D2V - 2M2) ( -cD!'- D(a:D3W - 2h(2) D 2 !') ) , a:D 2 w - 2h(2) D!')
k  k
+2h(2)C( :1 (-cy -  (a:D 3 w - 2h(2) D2"{)) , "()]
2h (2) 1
= --lIc!' + - ( a:D 3 w - 2h(2) D 2 r ) 112 < O. (5.2.13)
Cl k
Thus, A is dissipative.
We then show that 0 E p(A). In doing so, for any F = (f,g,p) E ft, we consider
the equation A(w, v, r)T = F, i.e.,
v = f E H1,
Ee1e D 4 w + D ( a:D3W - 2h(2)D 2 ", ) = - g E L 2 , (5 214)
ph kph I . .
C 1 ( 3 ( 2 ) 2 ) 1
-!' + - a:D w - 2h D!, = -p E HI .
Cl k Cl
In order to obtain unique solvability of a pair of solutions {w,!'} to the second and
third equations of (5.2.14), we introduce a bilinear form on H1(0, L) x Hl(O, L) as
follows:
b( {w, r}, {w, r:}) = Ee 1 e(D 2 w, D 2 w)
+  (a:D 2 w - 2h(2) D"{, a:D 2 w - 2h(2) D7) + 2h(2)C("{, 7).
(5.2.15)
It is easy to see from the Poincare inequality that this bilinear form is bounded and
coercive on H1(0, L) x Hl(O, L). Thus by the Lax-Milgram theorem, there is a unique
{w, r} E H1(0, L) x Hl(O, L) such that
b( {w, r}, {w, r:}) = -ph(g, w) - 2h(2)Cl (p, 7)
(5.2.16)
122

for" all {w, 7} E Hl(O, L) x Hl(O, L). In what follows, we use the standard variational
approach and the bootstrap argument to prove that (w,V,1')T E V(A) and it is the
unique solution to (5.2.14).
Indeed, by taking w = 0 , we obtain that
1
k (a:D 2 w - 2h (2) D1', - 2h (2) D7) + 2h (2) c( 1', 7) = - 2h (2) Cl (p, 7)
(5.2.17)
holds for any r E cg:> (0, L ). This means that w, l' satisfy the following eq au tion in the
distri bu tion sense:
-=-, + k 1 D (aD 2 w - 2h(2) D 1 /) = -p E Hl(O, L).
Cl Cl
(5.2.18)
Since 1', P E Hl, we can conclude that
D(a:D 2 w - 2h(2)D1') E Hl.
(5.2.19)
Now, we take 7 = 0 in (5.2.16) to get that
Ee 1 e(D 2 w, D 2 w) +  (aD 2 w - 2h(2) D/, aD 2 w) = -ph(g, w)
( 5.2.20)
holds for any w E Cg:>(O, L). Thus, w, l' satisfy the following equation in the distribu-
tion sense:
;:e D4W+ k;h D2(aD2w-2h(2)D1/) =-gEL 2 (0,L). (5.2.21)
It follows from (5.2.19) and (5.2.21) that w E H 4 (0, L), l' E H 3 (0, L). Since w, l' satisfy
(5.2.18) and (5.2.17) holds for any 7 E Hl, by integration by parts, we deduce that
(a:D 2 w - 2h(2) D1')lx=L = o.
(5.2.22)
Similarly, since w, l' satisfy (5.2.21), and (5.2.20) holds for any w E Hl, we obtain that
(Ee 1 e D3w +  (aD 3 w - 2h(2) D 2 /))lx=L = 0,
( 5.2.23)
and
(Ee1eD2w +  (aD2w - 2h(2)D/))lx=L = O.
(5.2.24 )
123

Combining (5.2.22) with (5.2.24) yields
D 2 wlx=L = Dllx=L = O.
(5.2.25)
It follows from (5.2.23), (5.2.25), and F E fi that A( w, V, I)T = F has a unique
solution (w, V, I)T E V(A). Thus by Theorem 1.2.4 in Chapter 1, A generates a
Co-semigroup of contractions in fi. The proof is complete.
Furthermore, we have the subsequent theorem.
Theorem 5.2.2 The semigroup S(t) = eAt, generated by A, is analytic and exponen-
o
tially stable.
Proof We first prove the analyticity of S(t) by verifying two conditions in Theorem
1.3.3 in Chapter 1.
(i) Suppose that condition (1.3.7) is false. Since we already have proved in the previous
theorem that 0 E p(A), it turns out that there exists an w E 1R with IIA-11I- 1 < Iwl <
00, a sequence of !3n E 1R with l!3nl < Iwl, !3n  wand a sequence of complex vectors
Zn = (w n , V n , In)T E V(A) with IIznll'H = 1 such that as n  00,
II (i!3n I - A)znll'H  0,
( 5.2.26)
I.e. ,
i!3nwn-vno In H 2 ,
i !3 v + Eele D 4 w + D ( a:D 3 w - 2h(2) D 2 ", )  0
n n ph kph n In
i!3nln + ln + k 1 (aD 3 w n - 2h(2) D2in) - 0 in HI,
Cl Cl
In L 2
,
( 5.2.27)
( 5.2.28)
(5.2.29)
Since
IRe(Az n , zn)'H1 < II (i!3n I - A)znll'H,
(5.2.30)
by (5.2.13) we have
1
IIc 1n + k (a:D 3 w n - 2h(2) D 2 1n ) II  0,
(5.2.31 )
which further implies
IIlnll  0
( 5.2.32)
124

due to (5.2.29). Combining (5.2.31) with (5.2.32) yields
IIa:D 3 w n - 2h(2) D 2 1n II  o.
( 5.2.33)
By the Poincare inequality, we have
IIa:D 2 w n - 2h(2) DIn II  o.
(5.2.34 )
On the other hand, if we take the inner product of (5.2.27) with V n in L 2 , add it to
the complex conjugate of the inner product of (5.2.28) with W n , integrate by parts and
use (5.2.34), then we obtain
EelellD2wnll2 - phllv n ll 2  o.
(5.2.35 )
Note that equation (5.2.28) implies that
;n II Ee 1 e D4w n +  D(aD 3 w n - 2M2) D 2 / n ) II < K
( 5.2.36)
for some positive constant K independent of n. Applying the Poincare inequality
and using (5.2.33), we also get the uniform boundedness of ;n IID 2 / n ll and ;n IID 3 wll.
Taking the inner product of (5.2.29) with ; D 2 / in L 2 , and integarting by parts, we
get that
c 1 i1lD/ n 11 2 - ;n (Cfn +  (aD3Wn - 2M2)D 2 /), D 2 / n } - O.
Combining it with (5.2.31) yields
( 5.2.37)
II DIn II  0,
( 5.2.38)
which further leads to
IID 2 w n II  0, IIv n \l2  0,
(5.2.39)
due to (5.2.38), (5.2.34), and (5.2.35). In summary, we have obtained IIZnll1-l  o.
This is a contradiction.
(ii) We now prove (1.3.8), i.e.,
lim 1,8II1(i,8 - A)-III < 00.
1.81-+00
(5.2.40)
125

Suppose it is false. Then there exists a sequence !3n > 0 (without loss of generality),
!3n  00, and a sequence of complex vectors Zn = (w n , V n , l'n)T E V(A) with IIZnll1-l = 1
such that as n  00,
II (iI - ;n A)zn II'H --+ 0,
(5.2.41 )
I.e. ,
iW n - ;n V n --+ 0 in H 2 , (5.2.42)
. Eel e 4 Q 3 (2) 2 2
ZV n + ph/3n D W n + phk/3n D(aD W n - 2h Din) --+ 0 In L , (5.2.43)
. C 1 3 (2) 2 1
Zin + ----r:/in + k /3 (aD W n - 2h Din) --+ 0 in H . (5.2.44)
ClfJn Cl n
Since
IRe( ;n Az n , zn}'H I < II( iI - ;n A)znll'H'
( 5.2.45)
by (5.2.13) we have
1 1
/3n IICfn + k (aD 3 w n - 2h(2) D2in)1I2 --+ 0
( 5.2.46)
which implies
lIl'n II  0,
(5.2.47)
and
;n II (aD 3 w n - 2h(2) D2in) 11 2 --+ 0, (5.2.48)
due to (5.2.44). Taking the inner product of (5.2.44) with l'n in Hl and using the fact
that !3n  00, we have
iliDinll 2 + k 1/3 (D( aD 3 w n - 2h(2) D2in), Din} --+ O.
Cl n
(5.2.49)
It is easy to see from (5.2.43) and (5.2.44) that both ;n IID(aD 3 w n - 2h 2 D2in)11 and
;n IID 4 w n ii are uniformly bounded. Thus, ;n II D3in II is also uniformly bounded. By
the Gagliardo-Nirenberg inequality, /2 1ID2inll is also uniformly bounded. Owing to
!3n
126

(5.2.48), by integration by parts we can deduce that
1
(3n (D( aD 3 w n - 2h(2) D2'Yn), D'Yn)
1 3 (2) 2 1 2
-- (l12(aD W n - 2h DIn), l12 D In)  o.
{3n {3n
(5.2.50)
Therefore, it follows from (5.2.49) that
IIDlnll  o.
(5.2.51 )
By the Gagliardo-Nirenberg inequality again, we have
1 2 1 3 1 1
l12I1D Inll < (- /3 liD In 11)2" II DIn 112"  o.
{3n n
( 5.2.52)
Combining (5.2.52) with (5.2.48), we obtain
1 3
l12 I1D Wnll  o.
{3n
(5.2.53)
By IIZnll1i = 1 and (5.2.42), ;,., II D 2 v n II and IIvnll are unifomly bounded. Then by the
Gagliardo-Nirenberg inequality, /2 1IDVnll is also uniformly bounded. We now take
(3n
the inner product of (5.2.43) with V n in L 2 and integrate by parts to get
. 11 11 2 Eele 1 3 1 )
 v n - h(l12 D W n , l12 DVn
P {3n (3n
a 1 3 (2) 2 1
- hk (l12(aD w n -2h DIn), l12Dvn) o.
P {3n {3n
(5.2.54 )
The last two terms in (5.2.54) converge to zero because of (5.2.48) and (5.2.53). Hence,
IIV n II  o.
(5.2.55)
Furthermore, we take the inner product of (5.2.42) with W n in Hl, then add it to the
complex conjugate of the inner product of (5.2.43) with V n in L 2 . After integration by
parts we arrive at
Ee1e II D 2 W n 11 2 - phllv n ll 2  o.
( 5.2.56)
127

Combining (5.2.56) with (5.2.55) yields
IID 2 wll  o.
( 5.2.57)
In summary, (5.2.47), (5.2.51), (5.2.55), and (5.2.57) imply
IIZnll1i  0,
( 5.2.58)
which contradicts IIZnll1i = 1. The proof for analyticity of S(t) is complete.
The exponential stability of S(t) is just a direct consequence of analyticity since
iIR c p(A) and S(t) is a Co-semigroup of contractions in the Hilbert space 'H. 0
128

Chapter 6
Linear Elastic Systems with Boundary
Damping
In the previous chapters, we have discussed the semigroup properties of some linear
elastic systems with various damping. These dampings are all distributed on the do-
main, either OIl the whole domain or on a subset of the domain whose measure is
positive. We now turn our attention to some linear elastic systems with boundary
damping. Exponential stability of the semigroup associated with a second-order hy-
perbolic equation with boundary damping, and a nonhomogeneous Euler-Bernoulli
beam equation with boundary damping will be proved in section 6.1 and 6.2, respec-
tively. We will still use the contradiction argument as in the previous chapters. As in
Section 3.2, a combination of the contradiction argument with the frequency domain
multiplier technique is used.
6.1 Second-Order Hyperbolic Equation
Consider the following linear second-order hyperbolic equation with boundary damping
02W m 0 ow
p(x)- -  -(aij(x)-) + q(x)w = 0
ot 2 . .- 1 OXi OX j .
,j-
in n x (0,00),
w=o
m ow ow
 aij-Vi + b(x)- = 0
. . I OX j . ot
,j=
w(x,O) = wo(x), Wt(x,O) = vo(x)
on fox [0, 00 ) ,
on fIx [0,00),
(6.1.1)
where n is a bounded domain in JRm(m > 2) with a C 2 boundary f = fo U f l and
fo n f l = 0; V = (VI,..., v m ) is the unit outward normal vector on f, b(x) E CI( f l )
and b( x) > b o > 0 on fl. Boundary f 0 is called an energy reflecting surface, and
f I, an energy absorbing surface. It is well-known that vertical vibration of a thin
elastic membrane of shape n E JR2 with one edge fo being clamped and another edge
f I subject to friction which is proportional to velocity, can be reduced to problem
129

(6.1.1).
We assume that the real-valued coefficient functions in (6.1.1) satisfy the following
conditions: p(x) E C 2 ( f2) , q(x) E LOO(f2) and aij(x), i,j = 1,..., m are in C 1 ( f2) and
p(x) > Po > 0, q(x) > 0 in f2; A(x) = (aij(x)) is an m x m symmetric and positive
defini te matrix.
Let Ht o (f2) = {w E H 1 (f2) I wlro = O} and we introduce the following Hilbert
space
H = Hi-o (f2) x L;(f2)
(6.1.2)
equipped with the inner product: for Zl = (WI, Vl)T, Z2 = (W2, V2)T E H,
(ZI, Z2}1i = 10 ((A(X)VWI) . VW2 + q(X)WIW2 + P(X)VIV2) dx.
(6.1.3)
When wand v are complex-valued functions, we can define the inner product, accord-
ingly. The energy of system (6.1.1) is defined by
E(t) =  10 (pw + (AVw) . Vw + qw 2 ) dx.
(6.1.4)
In this section, we want to show that under some conditions on the coefficient functions
p( x), aij (x) and on the geometry of f2, E (t) decays exponentially to zero, i.e., there
exist constants M, a > 0 such that
E( t) < Me-at E(O), t > 0,
(6.1.5)
as long as E(O) < +00.
Let us first recall the related works in the literature. Quinn & Russell [1] is a
pioneering work in this direction. In the case that aij (x) = 8 ij and p( x) = 1 (the equa-
tion in (6.1.1) then becomes the wave equation), Chen [1], published in 1979, was the
first one to succeed in showing (6.1.5) on (star-complemented)-(strongly star-shaped)
domain (f2, r 0, r 1) (see Chen [1] for the precise definition of such kind of domains).
In 1981, Chen [2] obtained (6.1.5) again under the following relaxed conditions: there
exists a vector field l(x) = (ll(x),...,lm(x))T E C 4 (IR m ,IR m ) with compact support
such that
130

(i) I". v < 0 on r 0;
(ii) I. v > II > 0 on r 1 for some constant II;
( 1 . 1 . 1 . ) 01i ( ) 1 c. . I .. d fi .
ox. x - 20ij IS strIct y posItIve e nIte;
J
(iv) there exists constant 12 > 0 small enough such that
0 3 1.
 < (')
 2 - 12, on [,;
UXjUXi
0 21 i
< 12 on r 1 .
ox jOXi -
(6.1.6)
Later on in 1983, Lagnese [1]-[2] improved Chen's result by changing condition (iii)
and (iv) into
(iii)' the matrix [li,j + lj,i] is possitive definite on n where li,j = gi.
J
Lagnese's result has subsequently been reproved by Lasiecka & Triggiani [1] and
Triggiani [1] using a different method. In all of the above papers, the estimate (6.1.5)
was obtained by the energy method and by employing a result of Datko [1] from the
estimate on 10 00 E(t) dt.
For the general second-order linear hyperbolic equation (6.1.1), in 1994 Wyler [1]
obtained (6.1.5) under conditions (i), (ii) and
(1 . 1 . 1 . ) " h . 1 I v p . I t"7 I . . £ I .. d fi .
t e matrIx Pij = aik j,k + ajk i,k + p aij - v aij. IS uni orm y posItIve e nite
on fl .
Hereafter the summation convention is used. Notice that condition (iii)' is just a special
case of (iii)" when the equation in (6.1.1) is the wave equation. Wyler's method is based
on Gearhart's theorem (see Theorem 1.3.2 in Chapter 1). He used a contradiction
argument to prove (1.3.3). However, for the proof of (1.3.4), he used a direct estimate
on the resolvent.
In what follows, we will reprove Wyler's result by a contradiction argument both
for the proof of (1.3.3) and (1.3.4). To convert the system (6.1.1) into an abstract
131

first:or<;ler evolution equation, we define the unbounded operator A and its domain
V(A) as follows:
{ T w E H 2 , v E Hfo (0), }
V( A) = z = (w, v ) E 1{ ,
v. (A(x)Vw) + b(x)vlrl = 0
A( :) = ( ptX) [VO(A(X;VW)-q(X)W]) 0
Then (6.1.1) can be reduced to the initial value problem for an abstract first-order
(6.1.7)
(6.1.8)
evolution equation
{ z = Az, "It > 0
ztO) = Zo
where z = (w, v)T, V = Wt, Zo = (wo, vo)T.
Theorem 6.1.1 The operator A defined in (6.1.7)-(6.1.8) generates a Co-semigroup
S(t) = eAt of contractions on the Hilbert space 1{.
(6.1.9)
Proof We use Theorem 1.2.4 in Chapter 1 to prove the present theorem.
Clearly, V( A) is dense in 1{. In what follows, we first prove the dissipativeness of A.
Let Z = (w,v)T E V(A). Since
(Az, z)1i
-  ((A(x)Vw) . Vv + q(x)vw)dx +  (V 0 (A(x)Vw) - q(x)w)vdx
1r v 0 (A(x)Vw)vdu
- [ b( x ) 1 V 1 2 dO" < 0, ( 6 .1.1 0 )
Jr 1
A is dissipative. Therefore, it suffices to show that 0 E p(A). For any (f, g)T E 'H,
consider the following equation
A ( : ) = ( ; ), (:) E V(A),
(6.1.11)
l.e. ,
{ V = f E Hfo
\7. (A(x)Vw) - q(x)w = p(x)g E L 2 (0).
(6.1.12)
132

To solve the second equation in (6.1.12) subject to the following boundary conditions
{ wlro = 0,
· v. (A(x)Vw)lr, = -b(x)flr, E H!(f 1 ),
(6.1.13)
we introduce the following bilinear form on Hfo,
a(w,u) = k(A(x)VwoVu+q(x)wu)dx,
(6.1.14)
which is clearly coercive and bounded. Then, problems (6.1.12) and (6.1.13) can be
converted into the following weak formulation:
to find w E Hfo such that
a(w, u) + [ b(x)fudu = - [p(x)gudx, Vu E Hf .
irl in 0
(6.1.15)
By the well known Lax-Milgram theorem, there is a unique weak solution w E Hfo. By
the standard regularity result o the elliptic boundary value problems (see Chapter 1),
we can conclude that w E V(A). Moreover, it easily follows from v = f and (6.1.15)
that lI(w,v)Tllrt < KII(f,g)Tllrt with K being a positive constant. Thus, 0 E p(A),
and the proof is complete.
N ow we have the following main result in this section.
Theorem 6.1.2 Suppose that there is a vector field l(x) = (ll(X),..., lm(x)) of class
C 1 ( fl ) satisfying {i}, {ii}, (iii}". Then the Co-semigroup S(t) on 1{ generated by A is
D.
exponentially stable, i.e., there exist positive constants M, a such that
II S ( t ) II < M e - at, t > O.
(6.1.16)
Remark 6.1.1 Inequality (6.1.5) immediately follows from (6.1.16) andz(t) = S(t)zo.
Remark 6.1.2 In Wyler [1 J, the author remarked that the assumption (ii) is equiva-
lent to
(ii) , I . v > 0 on r 1.
133

Proof of Theroem 6.1.2 It suffices to verify conditions (1.3.3) and (1.3.4) in The-
orem 1.3.2. The whole proof consists of the following steps:
(i) If (1.3.3) is false, then there is i(3 with (3 i= 0 such that i(3 E u(A). It is easy to see
from the proof of the previous theorem that A-I is compact from H to H. Therefore,
i(3 must be an eigenvalue. It turns out that there is a z = (w, v)T E V(A), z i= 0 such
that
Az = i(3z,
(6.1.17)
I.e. ,
V. (A(x)Vw) - q(x)w = i(3pv.
(6.1.18)
(6.1.19)
v = i(3w,
Therefore, w E H 2 with w i= 0 satisfies
V. (A(x)\7w) - q(x)w = _(32pw
w=o
m ow
L aij- lI i + i(3b(x)w = 0
" " l OX ) "
,)=
In fl,
on fo,
on fl.
(6.1.20)
By integration by parts, we have
o = Re( _(32 pw - \7 . (A(x )Vw) + qw, i(3w) £2(0.)
= [ b(x )(32IwI 2 du.
Jr 1
(6.1.21)
It turns out that w = 0 on f 1. We can deduce from the boundary condition on f 1
in (6.1.20) that .f aij ::. /Ii = 0 on fl. By Lemma 2.5 in Wyler [1], which is an
,) = I )
easy consequence of a 'Unique Continuation Theorem' of L. Hormander for the elliptic
operator, we have w = 0, hence by (6.1.18), v = 0 in fl, a contradiction. Thus, (1.3.3)
is proved.
(ii) Suppose that (1.3.4) is false. There then exists a sequence (3n, (3n --+ 00, and a
sequence of complex vectors Zn = (w n , vn)T E V(A) with unit norm in H such that as
n --+ 00,
II (i(3nI - A)znll1-l --+ 0,
(6.1.22)
134

i(3n w n - V n = in  0 in Hfo (0),
i13n v n- ptx) [\7.(A(x)\7w)-q(X)W]=gnO in L;(!1).
( 6.1.23)
( 6 .1. 24 )
Since
II (i(3n I - A)znll1i > IRe((i(3n I - A)zn, zn)1i1 = IRe(Az n , zn)1iI,
(6.1.25)
by (6.1.22) and (6.1.10), we obtain
f b( x) Iv n l 2 du  0,
Jr 1
( 6 .1. 26)
I.e. ,
IIv n llL2(rd  o.
(6.1.27)
Then it follows from (6.1.23) and the trace theorem that
II(3nwnIlL2(rd  o.
(6.1.28)
Next, we solve V n from equation (6.1.23) and substitute it into (6.1.24) to get
- 13wn - p() [\7 . (A(x)\7w n ) - q(x)w n ] = gn + i13nfn  0 in L;(!1). (6.1.29)
Taking the inner product of (6.1.29) with W n in L;(O) and integrating by parts yields
- 10 p(x)l13n w nI 2dx + 1o[(A(x)\7w n ). \7 w n + q(x)lw n I 2 ]dx
- f v. (A(x)Vwn) wn du = (gn + i(3nin, W n )L2.
Jr 1 P
(6.1.30)
It can be seen from (6.1.23) that (3nwn is bounded in L;(O), which also implies that
W n converges to zero in L 2 because (3n  00. This together with the fact that 9n, in
converge to zero in L;(O) implies that the inner product on the right-hand side of
(6.1.30) also converge to zero. Since IIwnllHl < 1, by the trace theorem and the
Cauchy-Schwartz inequality, we have
Ii, v . (A(x)\7w n ) w n dO" I Ii, b(x)vn w ndO" I
1
< KllwnllHf o (£. b(x)lv n I2 dO"):2  O. (6.1.31)
135

Therefore, it follows from (6.1.30) that
- 10 p(x) l.8nwn 1 2 dx + 10 [(A(x)Vw n ) . V W n + q(x)lw n I 2 Jdx - O.
(6.1.32)
Combining this with (6.1.23) yields
[[(A(x)\7w n ) . \7 w n + q(x)lw n I 2 )dx -lIv n lli2  o.
in p
(6.1.33)
Recall that II (w n , vn)Tllrt = 1. We then obtain that
1o[(A(x)VW n ) . V W n + q(x)lw n I 2 Jdx - , IIvnlli - .
( 6 .1. 34 )
In what follows, we will show that (6.1.34) is a contradiction. In doing so, we first
take the inner product of (6.1.29) with l(x) . \7w n in L;(O) to get
(-(3;w n - _ ( 1 ) [\7. (A(x)\7w n ) - q(x)wn],l(x). \7W n )L2 = (gn + i(3nfn,l(x). \7W n )L2.
p x p p
(6.1.35)
By integration by parts, (6.1.28) and the trace theorem, we have
I ((3nfn, I . \7w n ) L I
1£, p(x)(l . v)fn(.8n w n )du - 10 .8n w n (p(x)lj,dn + p(x)ldn,j + V p .lfn)dxl
< Kllfn IIHo (lI.8n w nIlL2(r,) + lI.8n w n II L2(O) ) - O. (6.1.36)
By the Cauchy-Schwartz inequality, we have
I (gn, I . \7w n ) L I < Klign IIL IIw n llHl  o.
(6.1.37)
Therefore, the inner product on the right-hand side of (6.1.35) converges to zero. In
what follows, we investigate three terms on the left hand side of (6.1.35). By integration
by parts, we get
Re(-(3wn, I. \7Wn)L
- £, p(x )(1 . v)l.8n w nI 2du +  10 (lj,jp(x) + V p .1)I.8n w nI 2dx . (6.1.38)
136

By (6.1.28), the boundary integral on the right hand side of (6.1.38) converges to zero.
Since II V W n II is uniformly bounded and W n  0 in L 2 , for the last term on the left
hand side of (6.1.35), by the Cauchy-Schwartz inequality we have
1
Re (- ( ) q(X)Wn, I. VW n )L2  o.
p x p
(6.1.39)
For the second term on the left hand side of (6.1.35), by integration by parts, we obtain
that
1
Re (-- ( ) V. (A(x)\7w n ), (I. VW n ))L2
p x P
- -Re fr v. ( A ( x ) Vw )( I.V w ) du +  1 w . ( a" k Z. k+ a" k Z. k -Va"..I ) w .dx
n n 2 n," 3, 3', '3 n,3
r n
+i £(1. v)(A(x)Vw n ) . V wn du - i 10 Ij,j(A(x)Vw n ) . V wn dx. (6.1.40)
We then substitute (6.1.36)-(6.1.40) into (6.1.35) to get
 f w . ( a" k Z. k+ a" k Z" k -Va...l ) w .dx
2 J n n,' '3, 3', '3 n,3
+ i 10 Ij,j [p(x) l.8nwnl2 - (A(x)Vw n ) . V wn ]dx
+ i 10 (V p . I) l.8nwn1 2 dx - Re £ v. (A(x)Vwn)(1 . V wn )du
+ i £(l. v)(A(x)Vw n ) . V wn du - O. (6.1.41)
Since W n = 0 on fo, the real and imaginary parts of VW n are in the same direction of
v. By the definition of vector inner product, we have the following equality:
Re (v. A(x)Vwn)(I. V w n ) = (I. v)(A(x)\7w n ) . V w n .
(6.1.42)
Thus, the boundary integrals in (6.1.41) can be rewritten as follows:
I = - £0 (I. v)(A(x)Vw n ). V wn du +  £. (I. v)(A(x)Vw n ) . V wn du
+Re f b(x )vn(l . \7 w n )du. (6.1.43)
J r1
By condition (ii), the third term in (6.1.43) can be bounded as follows:
IRe [ b(x)vn(l. V wn )dul < !. [ (I. v)(A(x)Vw n ) . V wn du + Kllv n ll12(rd (6.1.44)
Jr 1 4 Jr 1
137

with K being a positive constant. Thus, it follows from (6.1.26) and (6.1.44) that
'1 1
I > -- 2 f (I. v)(A(x )Vw n ). V wn da + - f (I. v)(A(x )Vw n ). V wn da + 0(1) (6.1.45)
Jro 4 J r1
with 0(1)  0, as n  00. Now it is easy to see that by conditions (i) and (ii), two
terms of boundary integrals in (6.1.45) are positive, definite quadratic forms.
On the other hand, if we take the inner product of (6.1.29) with (ljp),jw n in L 2 (0),
then by the same argument as before, we have
f 2 Vp.l
Jo.[(p(x)lj,j + V' p .1)I.8n w nl - (lj,j + p )A(x)V'w n . V' wn ]dx - O.
( 6 .1. 46 )
Thus, the terms on the left hand side of (6.1.41) can be rewritten as the sum of I
and 11 with
I i [ Vp.l ]
11= - w" a' k l' k +a' k l" k+ a""-Va...l w .dx
2 n,z Z J, J, J J n,J
o p
(6.1.47)
which is also a positive definite quadratic form by condition (iii)". Now it follows
from (6.1.41) and (6.1.45) that W n must converge to zero in H 1 (0). This contradicts
(6.1.34). Thus, the proof is complete. 0
6.2 Euler-Bernoulli Beam Equation
In this section, we consider the following initial boundary value problem for the Euler-
Bernoulli equation of a non-homogeneous beam with certain boundary damping:
p(x)Wtt + (p(x)w")" = 0, (x, t) E (0,1) x (0,00),
w(O, t) = w'(O, t) = 0, t > 0,
-p(l)w"(I, t) = m(t), -(pw")'(I, t) = h(t), t > 0,
w(x,O) = wo(x), Wt(x,O) = vo(x)
(6.2.1)
where w is the vertical displacement of the beam and the prime represents the deriva-
tive with respect to the spacial variable x. In (6.2.1), we have normalized the length
of the beam to one by scaling of the variable x and we have taken p( x) = {J( Lx),
p(x) = L- 4 EI(Lx) with L, {J(x) > 0, El(x) > 0 being the length, density, and
138

Young's modulus of the beam. The mechanical meaning of boundary conditions at
one end of beam x = 1 in (6.2.1) is that the bending moment and shear force ap-
plied to that end is the given function m(t) and h(t), respectively. In this section, we
will consider the case ,that functions m(t), h(t) are given in the feedback form. More
precisely, let
( m(t) ) = diag(l, -l)F ( w(l, t) ) (6.2.2)
h(t) wt(l, t)
where F is a given 2 x 2 matrix of complex numbers and it will be specified later. In
the control theory, this means that one measures Wt, w at x = 1, then feeds them back
to the bending moment and shear force at x = 1 as a control. If the resulting system
(a close-loop system) is exponentially stable, then we say that the original system is
exponentially stabilizable. Throughout this section, we always assume that p( x) E
C 1 [0, 1], p(x) E H2[0, 1] and p(x), p(x) > O. We are now interested in the exponential
stabilization problem: find conditions on F such that the semigroup associated with
system (6.2.1) is exponentially stable. The work in this section essentially follows the
paper Liu & Liu [5].
The energy of the beam at time t is defined by
1 fl
E(t) = 2 Jo [p(x)lw"(x, tW + p(x)lwt(x, tW] dx.
(6.2.3)
We will see later that the exponential stability of the associated Co-semigroup is equiv-
alent to the exponential decay of the energy, i.e.,
E(t) < Me-atE(O),
t > 0
(6.2.4)
for some constants a > 0, M > 1. In this section we will consider the following three
cases of feedback scheme:
(i) F > ko!, ko > 0 (i.e., both bending moment and shear force are given simulta-
neously in a feedback form);
(ii) F = diag (0, k 2 ), k 2 > 0 (i.e., shear force is given in a feedback form and bending
moment is free);
139

(iii) F = diag (k 1 , 0), k 1 > 0 (bending moment is given in a feedback form and shear
force is free).
In what follows, we reduce (6.2.1) and (6.2.2) to an abstract first-order evolution
equation in a Hilbert space.
Let
1
V = {w E H 2 (O, 1)1 w(O) = w'(O) = O}, IIwllv = (l p(x)lw"1 2 dx r
(6.2.5)
and
H = L;(O, 1),
1
IIvllH = (l p(x)lv I2dx r.
(6.2.6)
Then both V and H are Hilbert spaces and so is 1i = V x H equipped with the norm
1
lI(w, vflbi = [l (p(x)lw"1 2 + p(x)lvI2) dx] 2" .
(6.2.7)
If we introduce v = Wt and define in 1i
V(A) = {z = (w,vf
W, v E V, p(x)w" E H 2 (0, 1), } ,
(-p(l)w"(l), (pw")'(l))T = F(v'(l), v(l))T
(6.2.8)
and
AZ= ( )
- p) (p(: )W")"f '
then the closed-loop system (6.2.1) and (6.2.2) can be reduced to the initial value
(6.2.9)
problem for an abstract first-order evolution equation in 1i:
{ dz
dt = Az, Vt > 0
z(O) = Zo = (wo, vo)T.
(6.2.10)
We first prove the following
Theorem 6.2.1 Suppose that F is symmetric and non-negative definite, i.e., (FTJ, TJ) >
o for any TJ E m? Then A generates a Co-semigroup S(t) = eAt of contractions on 1i.
Moreover, A-I E .c(1i) is a compact operator.
140

Proof For any (w,v)T E V(A), by integration by parts we have
(A( w, v f, (w, V f),i - 10 1 (pvIW" - (pw")"v)dx
- -(pw"),vlx=l + pw"v'lx=1
- -(V'IX=I' VIX=I)T . F(v'lx=I' VIX=I)T < 0 (6.2.11)
where the dot "." denotes the inner product in m? Thus, A is dissipative.
In order to apply Theorem 1.2.4 in Chapter 1, it suffices to verify that 0 E p(A).
For any (f, g)T E 1i, consider the equation
A(W,V)T = (f,g)T,
(w, v)T E V(A).
( 6.2.12)
By the definition of A, we can easily deduce from (6.2.12) that
v = f,
w(x) = l' xp)s (-1, s - If . F(f'(I), f(I)f ds
fX x - Y fY fS
- Jo p(y) J1 A p(r)g(r)drdsdy.
(6.2.13)
It is easy to see that there exists a constant M > 0 independent of (f, g)T such that
II(w,V)Tllrt < MII(f,g)Tllrt.
(6.2.14)
Therefore, A-I E .c(1i), 0 E p(A). By Theorem 1.2.4, A generates a Co-semigroup
S(t) of contractions on 1i. Moreover, it easily follows from (6.2.13) that IIVIlIP <
KII(f,g)Tllrt and IIwllH4 < KII(f,g)Tllrt with K being a positive constant. By the
compactness of imbedding operators H2(0, 1) '---+ L 2 (0, 1) and H 4 (0, 1) '---+ H2(O, 1), we
can conclude that A-I is also a compact operator. 0
It is clear from the definition of 1i that the exponential decay property (6.2.4)
holds if and only if S (t) is exponentially stable. Now we are in position to state and
prove our main result in this section.
Theorem 6.2.2 Suppose that F, p(x) and p(x) satisfy one of the following conditions:
(i) F > koI, ko > 0, p" is piecewisely continuous on [0,1]
141

(ii) F = diag(O, k 2 ), k 2 > 0, p + xp' > 0 and 3p - xp' > 0 on [0, 1]
(iii) F = diag(k 1 , 0), k 1 > 0 and there is a constant c > 0 such that
, ( {X dS ) p
p c + Jo p( s) < p '
_p' (c+ fox p:) ) < 3 on [0,1].
(6.2.15)
Then the Co-semigroup S(t), generated by A, is exponentially stable.
Before giving the proof of this theorem, we first recall the related works in the
literature. When p and p are constants, the energy decay property (6.2.4) was estab-
lished for cases (i) and (ii) in Chen, Delfour, Krall & Payre [1] by means of energy
method. We also refer to Lagnese [3] and Chen, Coleman & Liu [1] for the plate and
shell counterparts of cases (i) and (ii). The decay property (6.2.4) for case (iii) with
constant p, p was established by Chen, Krantz, Ma & Wayne [1] by means of the fre-
quency domain method. The proof in that paper uses the representation of solution
to a fourth-order ordinary differential equation. The feedback scheme (iii) is simple
and attractive. To our knowledge, so far there has been no result on the boundary
stabilization of the plate equation by using the feedback scheme like case (iii). For
the problem considered in this section with variable coefficients, auxiliary functions
or so-called frequency domain multiplier techniques were used in Liu & Liu [5]. We
would like to point out that if p and p are constants, then the conditions in Theorem
6.2.2 are clearly satisfied.
Proof of Theorem 6.2.2
We divide the proof into several steps which consist of the following five lemmas.
Lemma 6.2.1 Suppose F satisfies one of the previous three conditions and suppose
that ilR c p(A). Then S(t) is exponentially stable.
Proof We use a contradiction argument to prove this lemma. By Theorem 1.3.2, the
exponential stability is equivalent to (1.3.3) and (1.3.4). Therefore, if the conclusion
is not true, then there exists a sequence of real numbers /3n ---+ 00 and a sequence of
complex-valued vectors Zn = (wn,vn)T E V(A) with IIZnll'H = 1 such that
II (i/3n I - A)zn II'H ---+ 0, as n ---+ 00,
(6.2.16)
142

I.e. ,
i/3n w n - V n
= In  0
zn V,
(6.2.17)
(6.2.18)
. /3 1 ( ( ) /1 ) /1
 nVn+ p(x) P X W n
= 9n  0
zn H.
In view of (6.2.11) and (6.2.8), we have
d n = -( v(l), v n (l))T . F( v(l), v n (l))T = Re (Az n , Zn)'H  o.
(6.2.19)
It follows from (6.2.17) and (6.2.18) that
i/3nllwnll - (v n , Wn)V  0,
i/3nll v nllir + (w n , Vn)V  0,
(6.2.20 )
(6.2.21)
which further imply
IIwnll - Ilvnllk  o.
(6.2.22)
Therefore, by IIZnll'H = 1 and (6.2.17), we obtain that
lim IIwnll = lim IIvnllir = lim II/3nwnllir = .
n-+oo n-+oo n-+oo 2
(6.2.23)
Now we claim that
W n E C 3 [O, 1],
lim /3nwn(l) = lim w(l) = lim (pw)'(l) = o.
n-+oo n-+oo n-+oo
(6.2.24 )
This is a key step in our proof. We post pone the proof of this claim to the end of the
proof of this lemma. Now we solve V n from equation (6.2.17) and substitute it into
(6.2.18) to get
- /3:w n + ptX) (P(x)w]" = gn + i/3nfn. (6.2.25)
Let q(x) be a function in C 2 ([0, 1]) with q(O) = o. Taking the inner product of (6.2.25)
with qw in H yields that
(-/3:W n + ptx) (P(x )w]" , qW)H = (gn + i/3nfn, qW)H.
( 6.2.26)
143

We now prove that the inner product on the right-hand side of (6.2.26) converges to
zero.
Dividing (6.2.17) by /3n, we get IIwnliH  o. It is easy to see by the Poincare
inequality that the norm in V is equivalent to the norm in H 2 . Thus, by the Gagliardo-
Nirenberg inequality we have
1 1
,--
IIwnliH < CllwnlllIwnll1r.
( 6.2.27)
Combining it with (6.2.23) yields that IlwIIH also converges to zero. Therefore,
1{9n, qW)HI < CI19nllHllw n llv  o.
( 6.2.28)
By the claim (6.2.24) and integration by parts, we also obtain that
I {/3nfn, qW)HI
< C(IIfnllvll II/3n w nllH + Ifn(l )/3n w n(l) I)
< Cllfnllv(II/3n w nllH + l/3nwn(l)1)  O.
( 6.2.29)
Concerning the inner product on the left-hand side of (6.2.26), by integartion by parts
we have
1 1 fl
Re(-/3wn, qW)H = -2 P (1)q(1)I/3n w n(1)1 2 + 2 Jo (pq),l/3n w nI 2dx
(6.2.30)
and
Re( p( ) (P( x )w]" , qw) H
1
- Re([q(pw)' - pq'w](l)w(l)) - 2P(1)q(1)lw(1)12
fl 1 2
+ J o [2(3 pq ' - p' q)lw1 + Re(pq"ww)]dx.
(6.2.31 )
All the boundary terms in (6.2.30) and (6.2.31) converge to zero due to (6.2.24) and
the boundedness of W n in V. Since IIwIIH  0, it follows that
l pq"wwdx -+ o.
( 6.2.32)
144

We then substitute (6.2.30) and (6.2.31) into (6.2.26) to get
1 fl 1 fl
2 J o (pq' + p'q)l,Bn w nI 2dx + 2 J o [3pq' - p'q]lw12dx  o.
( 6.2.33)
One can always choose the function q( x) such that
, , 0
pq + p q > ,
3pq'-p'q>0 on [0,1].
(6.2.34 )
For example, we can choose q( x) = e ax - 1 with a satisfying
{ PI PI }
max - , _ 3 < a < 00
po Po
(6.2.35 )
where
po = min p(x), Po = min p(x),
xe[O,I] xe[O,I]
(6.2.36)
and
PI = max Ip'(x )1, PI = max Ip'(x )1.
xe[O,I] xe[O,I]
Therefore, it follows from (6.2.33) that
( 6.2.37)
Ilwnllv  0,
(6.2.38)
which contradicts (6.2.23).
We now finish the proof of Lemma 6.2.1 by proving (6.2.24). The conclusion
W n E C 3 [0,1] follows from pw E H2(0,1) C CI[O,l] and 0 < P E C I [O,l]. We now
give the proof of (6.2.24) one by one for the three cases.
Case (i) (F > ko!, ko > 0).
By (6.2.19) we have
lim v n (l) = lim v(l) = o.
n-+oo n-+oo
(6.2.39 )
Hence,
II([pw](l), [pw]'(l))TII < IIdiag( -1, l)FIIII(v(l), v n (l))TII  O.
(6.2.40)
In view of (6.2.17) and the Sobolev imbedding theorem, we also obtain that lim ,Bnwn(l)
n-+oo
= o. Thus, (6.2.24) follows.
145

Case (ii) (F =diag(O, k 2 ), k 2 > 0).
We now have
p(l)w(I) = 0,
[pw]'(I) = k 2 v n (I).
(6.2.41)
Moreover, (6.2.17) and (6.2.19) yield that lim ,8nwn(l) = lim v n (l) = o. Thus, (6.2.24)
n-+oo n-+oo
also follows.
Case (iii) (F =diag(k 1 , 0), k 1 > 0).
In this case, we have
p(1 )w (1) = -k 1 v(I),
[pw]'(I) = O.
(6.2.42)
Thus, from (6.2.17), (6.2.19), and the Sobolev imbedding theorem we obtain that
lim w/(I) = lim w(I) = lim ,8nw(I) = lim v(I) = o.
n-+oo n-+oo n-+oo n-+oo
(6.2.43)
Now it remains to prove that lim ,8nwn(l) = o. To do so, we introduce another
n-+oo
mul ti plier.
Let (x) E C 2 ([0, 1]) satisfy
 ( 0) > 0,  ( 1) = 0, ' ( x) <  1 < O.
(6.2.44 )
The precise form of this function will be given later. Denote 4>n = JTB:\. We then
take the inner product of (6.2.25) with te-cPn in L 2 (0, 1) to get
( A-..3 -cPn ) + (( /1 ) /1 1 -cPn ) - ( + . ,8 f p -cPn )
- '+' n W n , pe L2 pw n , 4>n e L2 - 9n  n n, 4>n e L2 .
(6.2.45 )
The right-hand side of (6.2.45) converges to zero because
I(gn, :n e- tPnt }£21 < ClIgnliH -+ 0,
( 6.2.46)
and
1(4)nfn, pe- M }£21 = Il ; fnde-tPnt(x) I
- ;V n(l) -l ( ; fn)' e-Mdx < Cllfnllv -+ o.
(6.2.47)
146

For the second term on the left-hand side of (6.2.45), we have
(( /1 ) /1 1 -cPn )
pw n , 4>n e L2
[ (pw)' :n e -4>nt + pw e ( x )e -4>n{ J:
+ (pw, 4>n(/)2e-cPn - /le-cPn)L2.
(6.2.48)
Dividing (6.2.25) by /3 and using the fact that W n  0 in HI, we get that
A... -4 [p /I ] /I O . L 2
'+'n W n  In .
(6.2.49 )
Thus, integrating (6.2.49) over [0,1] and using (6.2.43), we obtain that
lim 4>4(pW)'(0) = lim 4>4(pW)(0) = O.
n-oo n-oo
(6.2.50)
Combining this with (6.2.43) and (6.2.44) yields that the boundary terms in (6.2.48)
converge to zero. Moreover, it easily follows from (6.2.44) and /3n  00 that
lIe-cPn"L2  0
(6.2.51)
which implies that
I(pw , /le-cPn)1 < C"wn"vlle-cPn"  o.
( 6.2.52)
Thus, by (6.2.48) we have
((pw)", :n e-4> n t)£2 = (pw , 4>n(02 e -4>n t )£2 + 0(1)
(6.2.53)
where 0(1)  0, as n  00. Integrating by parts again and using (6.2.43) and
w(O) = 0, we have
(pw , 4>n(/)2e-cPn)L2 = -(4)nw , (p(/)2)'e-cPn)L2 + (4)w , p(/)3e-cPn)L2 + 0(1).
(6.2.54 )
By (6.2.23) and the Gagliardo-Nirenberg inequality, we can deduce that "4>nwIIL2 is
bounded. Therefore,
1(4)nw , (p(/)2)'e-cPn)L21 < CII4>nwIIL2 "e-cPnIIL2  o.
(6.2.55)
147

Then it follows that
(pW , <Pn(/)2e-4>n) = (<pw , p(/)3e-4>n) + 0(1).
( 6.2.56)
Now we use integration by parts once more to obtain that
(<p w , p( /)3 e-4>n) L2 l,Bn Iw n (l )p(l)( ' (1))3 - (l,Bn Iw n , (p( /)3)' e-4>n) L2
+ (4)wn' p(/)4e-4>n). (6.2.57)
By the uniform boundedness of lI,BnwnllL2 and (6.2.51), we have
1(,Bnwn, (p(/)3)'e-4>n)L21 < CII,BnwnIlL21Ie-4>nIIL2  o.
( 6.2.58)
Then,
(<pW , p(/)3e-4>n)L2 = l,Bnlwn(1)p(1)(/(1))3 + (<pwn , p(/)4e-4>n)L2 + 0(1).
( 6.2.59)
Combining this with (6.2.53)-(6.2.59) and (6.2.45), we obtain that
l,BnIWn(1)p(1)(/(1))3 + (<PWn , [P(/)4 - p]e-4>n) = 0(1).
( 6.2.60)
Now it is clear that in order to have ,Bnwn(l)  0, it suffices to choose (x) to satisfy
the following equation
p(X)(/(X))4 - p(x) = o.
(6.2.61 )
Clearly, the following choice of function 
1
((x) = [ ( :;D · ds
( 6.2.62)
satisfies (6.2.44). With this special choice of , (6.2.60) gives us the desired result.
Thus, the proof of Lemma 6.2.1 is complete. 0
Now we turn our attention to the conditions on p,p under which one can conclude
that
iIR c p(A).
( 6.2.63)
148

Lemma 6.2.2 If for each {3 E 1R, the following boundary value problem
{ (pw")" - {32 pw = 0, w E V, pw" E H 2 (0, 1)
w" ( 1) = w'" ( 1) = 0, F ( w' ( 1 ), w ( 1 ) ) T = (0, 0) T
(6.2.64 )
has unique solution w = 0, then ilR C p(A) for A defined in (6.2.8).
Proof We have already proved the compactness of A-I in Theorem 6.2.1. Thus,
0'( A), the the spectrum of A only consists of eigenvelues of A. Therefore, the claim
ilR C p(A) is equivalent to that the equation
( i {3 I - A) ( w, v) T = (0, 0) T , {3 E 1R, (w, v) T E V( A)
(6.2.65)
has only a trival solution. For {3 = 0, multiplying the differential equation in (6.2.64) by
w, then integrating by parts twice, we conclude that pw" = o. Then by the boundary
conditions at x = 0, we immediately get w = o. This is nothing else, but what we
have proved in Theorem 6.2.1: 0 E p(A).
For {3 E 1R, {3 =1= 0, equation (6.2.65) can be equivalently reduced to (6.2.64). In
fact, we can get v = i{3w from the first equation in (6.2.65). Then we can eliminate v
from the second equation in (6.2.65) to get (pw")" - {32pw = o. Since (w, v)T E V(A),
we have w E V, pw" E H 2 (0, 1) and
(-p(i)w"(l), (pw")'(l))T = F(v'(l), v(l))T.
( 6.2.66)
Taking the inner product of (6.2.65) with (w,v)T in 1-l and using (6.2.11), we get
(v'(l),v(l))T. F(v'(l),v(l))T = 0 which, due to the assumption that F is symmetric
and non-negative definite, is equivalent to
F(v'(l),v(l))T = o.
( 6.2.67)
Therefore, due to (6.2.67), w"(l) = w'''(l) = 0 is equivalent to (6.2.66). Since v = i{3w
and {3 =1= 0, (6.2.67) is equivalent to F(w'(l),w(l))T = (0,0). Thus the proof is
completed by the derived equivalence between (6.2.65) and (6.2.64). 0
In what follows, we prove that if one of three conditions in the statement of Theorem
6.2.2 is satisfied, then ilR C p( A).
149

Lemma 6.2.3 If condition (i) in the statement of Theorem 6.2.2 is satisfied, then
ilR C p().
Proof In this case, since F is positive definite, we deduce from F(w'(I),w(I))T =
(O,O)T that w(l) = w'(I) = O. By Lemma 6.2.2, we only need to verify that the
following problem
{ (pw")" - (32 pw = 0,
w(l) = w'(I) = w"(I) = w"'(I) = 0
w E V, pw" E H 2 (0, 1),
( 6.2.68)
has only trivial solution. This follows from uniqueness of solutions to the above initial
value problem for ordinary differential equation.
o
Lemma 6.2.4 If condition (ii) in the statement of Theorem 6.2.2 is satisfied, then
ilR C p(A).
Proof In this case, F(w'(I), w(I))T = (O,O)T implies w(l) = O. By Lemma 6.2.2 we
only need to verify that the boundary value problem
{ (pw")" - (32 pw = 0,
w(O) = w'(O) = w(l) = w"(I) = w"'(I) = 0
(6.2.69)
has only a trivial solution. Multiplying the ordinary differential equation in (6.2.69)
by xw' and integrating by parts, noticing the boundary conditions in (6.2.69), we have
l [(p + xp)'I,8wI 2 + (3p - xp')lw"1 2 dx = O. (6.2.70)
By condition (ii), we deduce from (6.2.70) that w = O. Therefore, ilR C p(A). 0
Lemma 6.2.5 If condition (iii) in the statement of Theorem 6.2.2 is satisfied, then
ilR C p(A).
Proof In this case, F( w'(I), w(I))T = (O,O)T implies w'(I) = o. By Lemma 6.2.2 and
Theorem 6.2.1, we only need to verify that for each (3 E 1R, (3 =1= 0, the boundary value
problem
{ (pw")" - (32 pw = 0,
w(O) = w'(O) = w'(I) = w"(I) = w"'(I) = 0
(6.2.71)
150

has only a trivial-solution.
Let
1 ( [X ds )
q( x) = p C + J o p( s)
(6.2.72) ,
where c E IR is the constant appearing in condition (iii) in the statement of Theorem
6.2.2. Taking the inner product of the ordinary differential equation in (6.2.71) with
q(pw")' in L 2 (0, 1) and integrating by parts, we obtain that
q(O)I(pw")'(OW + l[q'l(pw")'1 2 + (3 + pqp')I.8w'1 2 ]dx = O.
(6.2.73)
By condition (iii) in the statement in Theorem 6.2.2, we have
q' = ( p _ ( c + [X  ) pI )  > 0,
p J 0 p( s ) p2
3 + pqp' = 3 + (c + fox p:) ) p' > 0,
c
q(O) = - > O.
p
(6.2.74)
Therefore, {3w = 0 in HJ (0, 1) which implies w = 0 for {3=1=O. Thus, the proof of Lemma
6.2.5 is complete. 0
Finally, we return to the proof of Theorem 6.2.2. Now it is clear that the proof of
Theorem 6.2.2 is completed by combining Lemmas 6.2.1-6.2.5.
o
151

Chapter 7
Uniformly Stable Approximations
All the systems discussed in previous chapters are treated as an abstract first-order
evolution equation
{ dt) = Az(t) + j(t),
z(O) = Zo
Vt > 0
(7.0.1)
on a Hilbert space H, and A is the infinitesimal generator of a Co-semigroup T(t) = eAt
on H. In order to compute the solution z(t), one often has to use various numerical
approximation schemes. The most common approach for the approximation of (7.0.1)
is to consider a sequence of finite-dimensional systems
{ dZ;t(t) = AN zN(t) + jN(t), \It > 0
zN(O) = zf
(7.0.2)
on a sequence of finte-dimensional subspaces H N of H, where AN is the infinitesimal
generator of a Co-semigroup TN(t) = e ANt on H N . Generally, (7.0.2) is derived from
(7.0.1) using various discretization techniques.
Definition 7.0.1 A sequence of Co-semigroups SN(t) is said to be uniformly expo-
nentially stable in N if there exist positive constants M and a independent of N such
that
IISN(t)11 < Me-at, t > O.
(7.0.3)
Our main interest in this chapter is to find the conditions under which (7.0.3) holds.
Our motivation arises from the study of the linear-quadratic regulator problem (in
short, LQR problem) in the control theory. In what follows, we briefly recall the
related results in the control theory (refer to Gibson [1] and Gibson, Rosen & Tao [1]
for the detailed discussion in this aspect).
152

For f(t) = Bu(t) in (7.0.1) with B = B(IR m , H), i.e., B is a linear bounded operator
from IR m to H, u(t) E lR m , we want to choose u to minimize the following functional
J(u) = lXJ [(Qz(t), z(t))1{ + uT(t)Ru(t)] dt
(7.0.4)
where Q E B(H, H) is a given non-negative self-adjoint linear bounded operator and
R E IR mxm is a given symmetric positive definite matrix. This problem is usually
called the LQR problem in the control theory. We say that the operator pair (A, B) is
stabilizable if there exists a bounded linear operator K such that A - B K generates
an exponentially stable Co-semigroup S(t) = e(A-BK)t. It is known from the control
theory that if (A, B) is stabilizable, then there is a unique non-negative self-adjoint
operator solution II E B(H, H) to the following operator algebraic Riccati equation
A*II + IIA - IIBR- 1 B*II + Q = O.
(7.0.5)
It turns out that the optimal control of this LQR problem mentioned previously has
the feedback form
u(t) = -K z(t), t > 0
(7.0.6)
where
K = R- 1 B*II E B(H, IR m )
(7.0.7)
which is called the gain operator. A sufficient condition to guarantee the stabilizability
of (A, B) is the exponential stability of T( t) = eAt because we can take K = O. That
is one of the important reasons why we would like to study the exponential stability
of a Co-semigroup in a Hilbert space in this book.
The finite-dimensional approximation of the LQR problem is:
For fN(t) = BNu(t) in (7.0.2) with B N E B(IR m , H N ) and u(t) E IR m , we want to
choose u(t) to minimize the following approximate functional
IN (u) = 10"'" [(QN zN (t), zN (t))1{N + u T (t)Ru(t)] dt
(7.0.8)
where QN E B(H N , H N ) is self-adjoint and non-negative. Accordingly, we say that
the operator pair (AN, B N ) is uniformly stabilizable, if there exists a bounded linear
153

operator K N such that AN - B N K N generates a Co-semigroup SN (t) = e(.AN -BNKN)t
satisfying (7.0.3) for all N. It is also known from the control theory that if (AN, B N )
is uniformly stabilizable, then for every positive integer N, there is a unique non-
negative, self-adjoint solution n N E B(H N , H N ) to the following finite-dimensional
algebraic Riccati equation
(AN)*n N + n N AN - rr N B N R- 1 (B N )*n N + QN = o.
(7.0.9)
The corresponding optimal control has the following feedback form
uN(t) = _K N zN(t), t > 0
(7.0.10)
with
K N = R- 1 (B N )*n N E B(HN,IR m ).
(7.0.11)
A natural question is: What is the relation between uN and u, and will uN converge to
u as N goes to infinity? To answer this question, let us assume that the approximation
scheme we work with is convergent in the sense that for each N, there exists a linear
mapping (projection) pN from 1i onto H N such that
lim pNz=z, lim QNpNz=Qz lim BNu=Bu, VuEIR m , VzEH,
N-oo N-oo N-oo
(7.0.12)
lim TN(t)pN z = T(t)z,
N-oo
Vz E H, t > o.
(7.0.13)
lim (TN (t))* pN z = (T(t) )*z,
N-oo
It is known from the control theory that if
sup IlnN11 < 00
N
(7.0.14)
and SN(t) = e(AN-BNKN)t is uniformly exponentially stable in N, then
lim rr N pN z = rrz, z E 1i,
N-oo
lim SN (t)pN z = S(t)z, z E H
N-oo
(7.0.15)
(7.0.16)
154

with the convergence being uniform in any bounded intervals of t. Moreover, it follows
from (7.0.15) that
lim III{N pN - KIIB ( 1i lRm ) = o.
Noo '
(7.0.17)
This leads to the strong convergence of optimal control uN (t) of the finite-dimensional
LQR problem to optimal control u( t) of the original infinite-dimensioanl LQR problem,
I.e. ,
lim II uN (t) - u(t)lllR m = 0
Noo
(7.0.18)
uniformly in any bounded intervals of t. As mentioned in Gibson, Rosen & Tao [1], the
easiest way to guarantee (7.0.14) and the exponential stability of SN (t) = e(AN _BN KN)t
uniformly in N is to show that TN (t) = e AN (t) itself possesses the exponential stability
uniformly in N.
Then, it is clear from the previous discussion that when we construct an approx-
imation scheme for the infinite-dimensional LQR problem, we need to know how to
verify the uniformly exponential stability in N for a sequence of Co-semigroups SN (t)
on the Hilbert spaces H N .
In Section 7.1, we will characterize the property (7.0.3) similar to Theorem 1.3.2.
Then in Sections 7.2 and 7.3 we will apply the result to the approximations of the
linear thermoelastic and viscoelastic systems.
7.1 Main Theorem
In this section we will give necessary and sufficient conditions to characterize the
uniform exponential stability of Tn(t), a sequence of Co-semigroups on the Hilbert
spaces Hn.
For a single Co-semigroup T( t) on a Hilbert space, the characteristic conditions
of exponential stability, as mentioned in Chapter 1, were given by Gearhart [1] and
Huang [1]. In what follows, we extend their results to a sequence of Co-semigroups
Tn(t).
Theorem 7.1.1 Let Tn(t) (n = 1,. . .) be a sequence of Co-semigroups of contractions
155

on the Hilbert spaces Hn and An be the corresponding infinitesimal generators. Then
Tn( t) are uniformly exponentially stable if and only if the following three conditions
hold:
sup {ReA I A E O"(An)} = 0"0 < 0,
nEN
(7.1.1)
there exists 0" E (0"0,0) such that
sup {II (AI -An)-lll} = Mo < 00,
Re)..q, nEN
(7.1.2)
and there exists a constant M 1 > 0 independent of n such that
IITn(t)1I < M 1 < 00, Vt > 0, Vn E N.
(7.1.3)
Proof If Tn(t) are uniformly exponentially stable, i.e., there exist M, a > 0 such that
IITn(t)11 < M e- CLt , V t > 0, Vn E N,
(7.1.4)
then
wo(An) de! lim In II Tn(t) II < -a.
t-++oo t
(7.1.5)
Thus, (7.1.1) follows from the following property:
O"o(An) de! SUp {Re A; I A E O"(An)} < wo(An) < -a.
(7.1.6)
Let 0" = -. Then 0"0 < 0" < O. For Re A > 0", we have
II(>.! - An)- IX II 111+ 00 e-AtTn(t)Xdtll
< M IIx 1l 1+ 00 e-Re>.te-cxtdt
M IIxll
a+ReA
< 2M IIxll
a
(7.1.7)
This implies that (7.1.2) holds. Furthermore, (7.1.3) immediately follows from (7.1.4).
Thus, the proof of the "only if" part is complete.
156

Now suppose (7.1.1)-(7.1.3) hold. Let
- 0'
An = A n -"2 I .
(7.1.8)
Then
{ - } 0' 0'
sup Re A; I A E O'(An) < -- + 0'0 < - < 0,
nEN 2 2
(7.1.9)
and
sp { II (u - Anfl ll } = Mo < 00.
Re '\21 nEN
(7.1.10)
In what follows, we prove that (7.1.9), (7.1.10), and (7.1.3) imply that there exists
a positive constant M > 0 independent of n such that the corresponding semigroups
Tn(t) = Tn(t)e-t to the infinitesimal generators An satisfy
IITn(t)11 < M,
(7.1.11)
which results in (7.1.4) with a = - > o.
To prove (7.1.11), we use the same technique as in Huang [1]. First, by (7.1.3) we
have
IITn(t)11 < Mle-t.
(7.1.12)
Therefore,
- 0'
wo(An) < - 2 .
Our next step is to prove the following estimate:
cM 2
I (Tn(t )x, y) I < 2''/ Ilx IIlIyll, for t > 1, x, Y E ?-In, \In,
(7.1.13)
(7.1.14)
with constant c > O. For this purpose we first prove the following two lemmas.
Lemma 7.1.1 For any x E Hn,T > -, as a function ofw E 1R,
II ((T + iw)I - Anfl xii E L 2 (lR), II ((T + iw)I - Anfl xll-+ 0, as Iwl -+ 00.
Moreover,
1: 00 11((T+iw)I -Anfl xl1 2 dJ.J < 7r1;1I2 ,
1: 00 II((T - iw)I - Afl xl1 2 dJ.J < 7r1;1I2 .
(7.1.15)
(7.1.16)
157

Proof By Hille- Yosida's theorem (see pazy [1]) we have
11((7 + iw)l - Anfl xf
- (100 e-(T+iw)tTn(t)xdt, 100 e-(T+iw)STn(s)xds)
- 100 100 e -T(t+s)e-iw(t-s) (Tn(t)x, Tn( s)x) dtds
- 100 I e-T(u+2s)e-iwu (Tn(U + s)x, Tn(s)x) duds
_ 100 e- iwu (100 e- T (u+2s) (Tn(u +s)x,Tn(s)x) ds) du
+ 1: e- iwu (1: e- T (u+2s) (Tn(u + s)x,Tn(s)x) ds) du
del 1: 00 f(u)e-iWUdu
(7.1.17)
with
{ rOO e -T(U+2S) (Tn ( U + s )x, Tn( s)x) ds,
f(u) = looo __ __
Iu e- T (u+2s) (Tn(u + s)x, Tn(s)x) ds,
Therefore, by (7.1.12) we have for u > 0
u > 0,
(7.1.18)
u < o.
If( u)1 < 100 e- T (u+2s) IITn( u + s )xll.IITn(s )xll ds
< M 2 11xll 2 [00 e-T(u+2s)e-(u+s)e-sds
1 lo
M{ IIxll 2 -(T+)U
- 2(1 + ) e ,
and for u < 0,
If(u)1 < Mi IIx 11 2 1: e- T (u+2s)e- f (U+S)e- fS ds
_ M{lIxll 2 e(T+)U
2( 1 + ) .
It turns out that f( u) E L 1 (lR) n LOO(lR),
Ilfll£oo < M{lI x I1 2 . ( 7.1.19 )
- 2(/+)
By the result in Hewitt & Stromberg [1], we conclude that 11((1 + iw)l - An)-lxIl 2 E
L1(lR) and
1: 00 11((7 + iw)l - An)-lxIl 2 dw < 211"lIfll£<>o.
(7.1.20)
158

Moreover, from the Riemann-Lebesque theorem, 11(( T + iw)I - An)-l xII  0 as Iwl 
00. Combining (7.1.20) with (7.1.19) yields (7.1.15). Inequality (7.1.16) can be proved
the same way.
Lemma 7.1.2 For any x E Hn, wEIR we have
o
II(iwI - An)- lX II < 2 m 11(( -(7 + iw)I - An)- lX II
(7.1.21)
with m being an integer such that
m - 1 < -2M o (7 < m.
(7.1.22)
Proof Let
1
T m = -(7, tl T = 2M 0 ' Ti = T m - (m - i) tl T, (i = m - 1, . . . , 0).
(7.1.23)
Then TO < O. Since
II [I - (Ti - T)((Ti + iw)I - An)-l] xii
> Ilxll- (Ti - T)II((Ti + iw)I - An)-lXII
> IIxll- tlT . Mo .!lxll
>  IIx II, for 7 E (7;-1, 7i],
we have
II [1 - (7i - 7)((7i + iw)1 - An)-lr 1 11 < 2.
( 7.1.24 )
For any fixed i, (i = m, . . . , 1) and any T E [Ti-l, Ti], we obtain that
II((T + iw)I - An)- lX II
- II [1 - (7i - 7)((7; + iw)1 - An)-lr 1 ((7; + iw)1 - An)- IX II
< II [1 - (7i - 7)((7i + iw)1 - An)-lr 1 11.11((7 i + iw)1 - An)- IX II
< 211((Ti + iw)I - An)-lXII. (7.1.25)
This leads to (7.1.21) by induction. 0
159

Lemma 7.1.2 and 7.1.3 imply that
1: 00 II(iwI - An)- IX I1 2 dMJ < cM;lIxl1 2
and
( 7 .1. 26 )
1: 00 II( -iwI - A)-IXI12 dMJ < cM;lIxll 2
(7.1.27)
with a positive constant c depending only on (7.
We are now in position to prove (7.1.14). Let 12 > 11 > - > o. Then for
x E V(A), y E 1i n , by the inverse formula (see Pazy [1], p. 29, Corollary 7.5) we
have
(Tn(t)x, y)
1 1 71 +w ( ,.., )
_ 2 . lim . eAt (AI - An)-IX, y dA
7r w-++oo 71-W
[ At .
1 e ,.., 71 + w
_ 2 . lim - ( (AI - An)-IX, Y )I .
7r w-++oo t 71-W
1 71 +iw eAt ,.., ]
+ . -((AI-An)-2x, y)dA .
71-W t
( 7 .1. 28 )
It turns out from Lemma 7.1.1 that
,.., 1 1 7 1 +iw eAt ,..,
(Tn(t)x, y) = lim -: . -((AI-An)-2 x , y)dA.
w-++oo 27r 71-W t
Since sup {Re Aj I A E u(A n )} <  < 0, for t > 0, e;t ((AI - An)-2X, y) is analytic
nEN
in the domain: {A I Re A E (, 12)}. Let f w be the curve composed of fo = {Re A =
11, -w < 1m A < w}, f 1 ,2 = {O < ReA < 11, 1m A = :f:w} and f3 = {ReA =
0, -w < 1m A < w}. From
At
{ ((AI-An)-2X, y)dA=O
Jr w t
and, due to Lemma 7.1.1,
(7.1.29 )
(7.1.30)
fr eAt ,..,
lim - ( (AI - An)-2x, Y ) dA = 0
w-++oo r 1 ,2 t
it follows that
(Tn(t)x, y)
160
(7.1.31)
1 j iw eAt ,..,
-: lim . - ( (AI - An)-2X, Y ) dA
27r w-++oo -w t
.
1 j +oo ewt ,..,
- - ((iwI - An)-2x, y) dw.
27r -00 t

Therefore,
I (Tn(t)x, y)1 < 2 1: 00  II(iwI - A n )- I X II.II(-iwI - A)-lyll d;,,;
< 2t (1: 00 II(iwI - An)- IX I1 2 d;,,;)  (1: 00 II( -iwI - A)-lyr d;,,; ) . (7.1.32)
Combining it with (7.1.26), (7.1.27) yields (7.1.14) for x E 1J(.A), y E ?-In. Since
V(.A) is dense in ?-In, (7.1.14) also holds for any x, y E ?-In. By taking y = Tn(t)x, we
conclude that for t > 1
cM 2
IITn(t) II < .
(7.1.33)
For 0 < t < 1, by (7.1.3) we have
IITn(t)1I = IITn(t)e-tll < Mle-.
(7.1.34 )
Therefore,
IITn(t)11 < max ( C{ , Ml e - f ) de! M, Vt > 0
(7.1.35)
which results in
IITn(t)1I = IITn(t)etll < Met, (j < 0, Vt > O.
(7.1.36)
Thus, the proof of Theorem 7.1.1 is complete.
o
Theorem 7.1.2 Let Tn(t) (n = 1,...) be a sequence of Co-semi groups on the Hilbert
spaces ?-In and An be the corresponding infinitesimal generators. Then Tn (t) are uni-
formly exponentially stable if and only if {7.1.1}, {7.1.3}, and
sup { II ( A I - An) -III} < 00
Re XO,nEN
(7.1.37)
hold.
Proof We only need to prove that (7.1.1), (7.1.3), and (7.1.37) imply (7.1.1)-(7.1.3).
Let
M = SUp { II (AI - An) -III} < 00,
Re AO,nEN
(7.1.38)
161

and A = (I + iw), 1 E [-,O]. Then, we have
II (I + r(iwI - An)-I) xii > Ilxll- 2 II(iwI - An)- IX II
1 1
> Ilxll - 211xll = 2 11xll .
This implies that
II (I + r(iwI - An)-lflll < 2.
(7.1.39)
By
AI - An = (I + l(iwI - An)-l) (iwI - An),
(7.1.40 )
we conclude that AI - An is invertible and
II(AI - An)-111 I! (iwI - An)-1 (I + r(iwI - An)-lflll
< 211(iWI-An)-111 < 2M.
(7.1.41)
Let
( 1 0'0 )
0' = max - 2M ' 2 .
(7.1.42)
Then, we have
era < er < 0, er E [- 2 ' 0).
Therefore, (7.1.2) follows from (7.1.41) and (7.1.43).
As a corollary of Theorem 7.1.2, we have the following theorem.
Theorem 7.1.3 Let Tn (t) be a sequence of semigroups of contractions on the Hilbert
spaces 1i n and An be the corresponding infinitesimal generators. Then Tn (t) are uni-
(7.1.43 )
o
formly exponentially stable if and only if
p(An) :) {i,8l,8 E lR}, Vn E IN
( 7.1.44 )
and
sup II (i,8I - An)-ll1 < 00.
{3elR, nelN
(7.1.45 )
162

Proof Since Tn(t) is a Co-semigroup of contractions, we only need to show the equiva-
lence of (7.1.1) and (7.1.37) with (7.1.44) and (7.1.45). Obviously, (7.1.1) and (7.1.37)
imply (7.1.44) and (7.1.45). We now prove that (7.1.44) and (7.1.45) also imply
(7.1.1) and (7.1.37). By Corollary 3.6 in pazy [1], the resolvent set p(An) of An
contains the open right half-plane, i.e., p(An) :) {A; Re A > O}, for all n. Furthermore,
II (AI - An)-III <  uniformly in n when ReA> o. Thus, for any 6 0 > 0 we have
1
sup II(AI - An)-III < 6.
Re A>oo,nEIN °
(7.1.46 )
Note that (7.1.44) and (7.1.45) imply
sup {II ( i,8 I - An) -111 I ,8 E 1R} = M < 00.
nEIN
(7.1.47)
Let A = a + i,8. Then
AI - An = aI + i,81 - An = (a(i,8I - An)-I + I)(i,8I - An).
(7.1.48 )
For lal < , by the contraction mapping theorem, a(i,8I - An)-I + I is invertible
for all n. Thus, we have proved (7.1.1). Moreover, we also get
sup {II(AI - An)-111 I I Re.X I < 2 } < 2M.
nEIN
(7.1.49)
Combining (7.1.46) and (7.1.49) yields (7.1.37).
o
7.2 Approximations of the Thermoelastic System
In this section, as an application of Theorem 7.1.3, we first present a general approxi-
mation scheme for the linear thermoelastic system
{ Utt - U1'1' + "{81' _ 0,
()t + ,Uxt - ()xx - 0
(7.2.1)
on (0, 7r) X (0,00) with the boundary conditions
U Ix=o,1(" = () Ix=o,1(" = o.
(7.2.2)
163

Then we will use Theorem 7.1.3 to show the uniformly exponential stability of the
corresponding Co-semigroups associated with a particular approximation scheme whick
is often refered to as a modal method. We also provide a convergence proof of this
scheme.
Recall that the corresponding abstract evolution equation for (7.2.1) and (7.2.2) is
d
dt
v
u 0 I 0
=A v - D 2 0 -i D
() 0 -,D n 2
u
u
v
(7.2.3)
()
()
in the Hilbert space rl = HJ(o') x L 2 (O,) x L 2 (O,), where 0, = (0,7r) and V(A)
H 2 n HJ x HJ X H 2 n HJ.
In Section 2.2, we have proved that the Co-semigroup S(t) = eAt is exponentially
stable. Let H(o'), H(o') and H;(o') be the n-dimensional subspace of HJ(o'), L 2 (O,)
and L 2 (0,) with basis {<PI,..., <Pn}, { 1/;1, . . . , 1/;n} and {1'..., n}' respectively. Since
H 2 n HJ is dense in L 2 , we c"an choose <Pi E H 2 n HJ, 1/;i E HJ, i E H 2 n HJ. Let
rl n = H(o') x H(o') x H;(o') with a basis (j = 1,. .. , n)
<Pj 0 0
Ej = 0 , E n + j = 1/;j , E 2n + j = 0 (7.2.4)
0 0 j
The inner product on rl n is that induced by the inner product of rl which has been
given in Section 2.2. We consider the approximation to the solution of (7.2.3) in the
form
3n
Zn = LZj(t)Ej(x)
j=1
(7.2.5)
which is required to satisfy the following variational system:
( dn , E j ) 1t = (Az n , E j )1t, j = 1, . . . ,3n.
(7.2.6)
164

Then we have
M(l) .:...(1 ) 0 --T 0 ......(1 )
n zn Dn zn
MnYn = M(2) .:...(2) -Dn 0 -,Fn ......(2) = AnYn
n Zn Zn
M(3) .:...(3) 0 ,F; -G n ......(3)
n Zn zn
(7.2.7)
with
(MA 1 ))ij = (D<Pi, D<pj)L2, (MJ2))ij = ("pi, "pj)L2, (MA 3 ))ij = (i, j)L2,
( Dn )ij = (D<Pi, D"pj)L2, (Fn)ij = (Di, "pj)L2, (Gn)ij = (Di, Dj)L2
and
(7.2.8)
......( i) - ( ...... . . .. ......" ) T . - 1 2 3
Zn - Z(t-1)n+I, , Ztn , Z - , , .
(7.2.9)
By construction, the matrix MJi) is symmetric and positive definite.
Therefore,
there exists a lower triangle matrix L i such that MJi) = (Li)(Li)T. Let
Ln = diag(L 1 ,L 2 ,L 3 ) and we denote LYn by Yn = (z1),Z2),z3))T. Then to ob-
tain approximate solution Zn, we are led to solving the following ordinary differential
equations
dYn _
dt = AnYn
(7.2.10)
with
0 L- 1 DT (LT)-l 0
1 n 2
An = -L"2 1 Dn (Lf)-l 0 -,L"2 1 Fn(LI)-l
0 ,L -1 PT(LT)-l _L 3 1 G n (Lr)-1
3 n 2 ,
(7.2.11)
It is easy to see that
R (A - - ) - (G (L T ) -1-(3) (L T ) -1-(3) ) < 0
e nYn, Yn C3n - - n 3 Zn, 3 Zn C n -
(7.2.12)
because G n is semi-positive definite.
The modal approximation scheme is to choose the eigenvectors of the system as
basis vectors. Here we will use the eigenvectors of the uncoupled thermoelastic system,
i.e., , = 0 in (7.2.3), and we still call it the modal approximation. Let
A..  1 .. 10/'  .. t  .. . 1
o/j = --:- SlnJX, o/j = - SlnJX, j = - SlnJX, J = ,..., n.
'TrJ 'Tr 'Tr
(7.2.13)
165

A straight forward calculation following (7.2.8) and (7.2.11) yields that M n = I and
0 Dn 0
An= -Dn 0 -,Fn
0 , F'[ , -D 2
n
(7.2.14)
with
1
n
F;j = {
4 ij
- - "2 .2'
7rZ -J
0,
Ii - jl = odd
(7.2.15)
Dn =
otherwise.
Notice that with the previous choice of basis in ?-in, (7.2.5) defines an isomorphism
between 1i n and IR 3n which is equipped with the usual inner product. Let
An = PnAPn
(7.2.16)
which is an operator from 1i n to itself. Then (7.2.6) can be considered as an evolution
equation in 1i n :
dz
dt = Anz.
(7.2.17)
Notice that for Zn E 1i n , we have (Anz n , zn)1t n = (AnYn, Yn)IEtn.
Theorem 7.2.1 An generates a C o - semigroup of contractions on ?-in.
Proof To apply Theorem 1.2.4, we have to show that An is dissipative and 0 E p(An).
The dissipativeness of An comes from the straight forward calculation
(Anz n , zn)1tn = (AnYn, Yn)IIfn = -IIDn3)1I2 < o.
(7.2.18)
Hereafter we also denote by II . II the [2 norm in IR n or cn when no confusion occurs.
We now prove that 0 E p(A n ). For any F E ?-in, by (7.2.5), we have fn = (fIn, f2n, f3n)T
E IR 3n , accordingly. It follows from the definition of An that Anzn = F is equivalent
to
AnYn = fn,
(7.2.19)
166

I.e. ,
D Z(2)
n n
-D Z(I) -.-y Z(3)
n n Inn
.-y FT Z(2) _ D 2 Z(3)
Inn n n
fIn,
f2n
f3n.
(7.2.20)
Notice that Dn is invertible. Then it is easy to get
-(1)
Zn
-(2)
Zn
-(3)
Zn
-D:;;I(f2n -iFnD:;;2(f3n -iF; D:;;l fIn)),
D;,l fIn,
-D:;;2(f3n - iF; D:;;l fIn).
(7.2.21 )
Thus, by Theorem 1.2.4, the proof is complete.
o
Theorem 7.2.2 The semigroups Sn(t) = e Ant , generated by An which is defined in
(7.2.16), are uniformly exponentially stable, i. e., there exist positive constants M and
a, independent of n, such that
II S n(t)IIL:(1tn l 1tn) < Me-at.
(7.2.22)
Proof By Theorem 7.1.3 we need only to prove that (7.1.44) and (7.1.45) hold.
(i) We first prove (7.1.44) by a contradiction argument. Since An is an operator
from finite-dimensional space 1-l n into itself, An is compact. Therefore, every point
in the spectrum of An must be an eigenvalue. If (7.1.44) is not true, then there
must exist at least an m E IN and (3m E 1R such that i{3m is an eigenvalue of Am.
Since 0 E p(A n ), Vn E IN, as proved before, (3m is not zero. Let Zm 'E V(Am) with
IIZmll1tm = 1 be the corresponding eigenvector. Then we have
(i{3mI - Am)zm = O.
(7.2.23)
Let Ym = (u m , V m , 8m) E C 3m be the corresponding coordinate vector to Zm. Taking
the real part of the inner product of (7.2.23) with Zm in 1-l n , we obtain that
IIDm8mll2 = O.
( 7 . 2.24 )
Since Dm is invertible, 8m = O. Notice that (7.2.23) is equivalent to
(i{3mI - Am)Ym = O.
(7.2.25)
167

Taking the first 2m rows of (7.2.25) into consideration, we have
(i{3m I - Am») ( :: ) = 0
(7.2.26)
with
Am) = [ _ m m] 0
(7.2.25) implies that i13m is an eigenvalue of Am). It is clear that Am) has eigenvalues
(7.2.27)
ij, j = 1,...,m,
(7.2.28)
and the corresponding eigenvectors
1 ( e. )
£.__ J
J - yI2 iej ,
1 ( ej )
£_j = - , j = 1,. . . , m,
yI2 -iej
(7.2.29)
with ej being the standard jth unit vector in lR m . It turns out that there must be
some integer jm, (-m < jm < m) such that 13m = jm and (um,Vm)T = £jm, i.e.,
1 · 1 1
U m = yl2 e 1j m l and V m = yl2 e 1j m l or V m = - yl2 eljmlo Now taking the last m rows of
(7.2.25) into consideration yields
F::' V m = o.
(7.2.30)
Then a contradiction easily follows from the expression of FJ:. and v m . Thus (7.1.44)
is proved.
(ii) We now prove (7.1.45) by a contradiction argument. If (7.1.45) is false, then there
must exist a subsequence of An, still denoted by An' a sequence of 13n E lR, and a
sequence of Zn E ?-In with II Zn lI'Hn = 1 such that as n -+ +00,
(i13nI - An)zn -+ 0 in ?-In.
(7.2.31 )
Let Yn = (un, V n , On) E C 3n be the corresponding coordinate vector to Zn. Then (7.2.31)
is equivalent to
II (i13n I - An )Yn II C 3n -+ o.
(7.2.32)
168

I t follows from
Re (( i/3n I - An)Yn, Yn)C 3n -+ 0
(7.2.33)
that
IIDnOnll2 -+ o.
(7.2.34 )
Therefore, we have
IIOnll < IIDnOnll -+ o.
(7.2.35)
Taking the first 2n rows of (7.2.32) into consideration yields
(i/3nI-A) ( Un ) + ( 0 ) -+0
V n ,FnOn
C 2n
(7.2.36)
where the matrix
n _ [ 0 Dn ]
Ao -
-Dn 0
(7.2.37)
has eigenvalues
 ij,
) "=1... n
, "
(7.2.38)
and the corresponding eigenvectors
1 ( ej )
£j = M . '
y2 ze"
)
1 ( ej )
£_j = - , j = 1, . . . , n
V2 -iej
(7.2.39)
with ej being the standard jth unit vector in lR n .
We now claim that
II FnOn II -+ o.
(7.2.40)
In fact,
(FnOn)i  ( DSinix, iJOn)jSinjx )
)=1 £2
- -  ( sin ix, t,j(On)j C OSjX )
)=1 £2
(7.2.41)
169

By Parsaval's inequality, we have
2
2' n
" F n On 11 2 < - L, j ( On) j COS j X
7r . 1
J=
n
< L,j2\(On)j\2 = II DnOn 1\2 .
£2 j = 1
(7.2.42)
Thus, (7.2.40) follows from (7.2.35). Moreover, we deduce from (7.2.36) that
( i I3n I - A) ( :n ) -+ O.
n C 2n
(7.2.43)
Since the eigenvectors {£;1;j} form a normalized basis in C 2 n, we have
( Un ) n
= L, a:nj£j.
V n j=-n,j#O
(7.2.44 )
It follows from IIYnllc3n = 1 and IIOnil -+ 0 that
( :: ) 2
C 2n
n
L, Ia:nj 1 2 -+ 1.
j=-n,j#O
(7.2.45 )
Substituting (7.2.44) into (7.2.43), we obtain
2
( i I3nI - A) ( :: )
C 2n
2
n
L, (il3n - ij)a:nj£j
j=-n,j#O C 2n
n
L, Il3n - j 121a: n j 1 2 -+ 0, as n -+ +00.
j=-n,j#O
(7.2.46)
n
If for n large enough, Il3n - j I > {; > 0 for all j, then L, Ia:n,j 1 2 -+ 0, a contradiction
j=-n,j#O
with (7.2.45). Thus, we derive from (7.2.45) and (7.2.46) that there exists j(n) E
{1, 2,. . . , n} such that as n -+ +00,
I3n - j ( n ) -+ 0,
n
L, Ia:n,j 1 2 -+ 0, (7.2.47)
j=-n,j#O,j(n)
Ia:n,j( n) 1 -+ 1,
170

and
( :: ) - Qn,j(n)£j(n) -+ o.
C 2n
(7.2.48)
Taking the last n rows of (7.2.32) into consideration, we obtain that
Ilif'nOn + DOn - ,F; vnll-+ o.
(7.2.49)
By (7.2.47), (7.2.49), and j(n) =I- 0, if we denote j() On by x n , then
II II de! · . ( ) D 2 . ,Qn,j(n) F T 0
9n = 1, J n X n + n X n - 1, m. I n e /j (n) I -+ .
v 2 1J(n)
(7.2.50)
Taking the real part of the inner product of 9n with DYn yields
R ( D 2 ) IID 2 11 2 R ( . ,Qn,j(n) F T D 2 )
e 9n, nXn = nXn - e 1, V2lj(n)1 n e/j(n)" n Xn
(7.2.51)
We now estimate the last term on the right hand side of (7.2.51). Indeed,
( IT 2 )
Re Jj(n)( n elj(n)" Dn xn -
. ) Re t r -(D sin ix) sinj(n)xdx . i 2 (x n );
1rJ n i=l J o
- Re ti l 1r sinixcosj(n)xdx. i(x n );
1r i=l 0
- -Re t l 1r cosixsinj(n)xdx 0 i(On);
1r i=l 0
-  Re t ( cos i x cos j ( n ) x) 1r 0 -:-- ( 1 ) i ( On) i . (7.2.52)
1r i=l 0 J n
Therefore,
I -
<
-
R ( . ,Qn,j(n) F T D2 )
e 1, V2lj(n)1 n elj(n)" n Xn
'YIQn)1 (lisinj(n)xllv IIDnOnl1 + 2l1Dh n ll Loo )
'YIQn)1 G IIDnOnll + 2 II Dh n IILoo )
(7.2.53)
where
h n = {2 t(x n );sinixo
V ;: i=l
(7.2.54 )
171

By the well-known Gagliardo-Nirenberg inequality, we have
1 1
IIDhnll Loo < CIIID2hnll2I1DhnIlI2 + C 2 11 Dh nllL2
1
- C 1 IIDXnI12 IIDnxnll  + C 2 11 D n X nli
with C 1 , C 2 being positive constants independent of h n .
Combining (7.2.53) with (7.2.55) and applying Young's inequality yields
(7.2.55)
I < '"Ylan)1 G IIDn1q + 2C2l1Dnxnll) +  IIDxnI12 + C 3 11 D n x nll t . (7.2.56)
On the other hand, we have
IRe (gn, Dxn)1 < lIgnll2 + IIDxnIl2.
Thus, combining (7.2.56) and (7.2.57) with (7.2.51) and (7.2.35) yields
IIDxnll -+ o.
Let W n be the unique solution to the equation
. O ( ) D 2 . ,Qn,j(n) F T 0
1,) n W n + n W n - 1, to . n elj(n)1 = .
v 2 1J(n)1
Then
{ -sign(j (n) )i2J2,Qn,j(n) i
(Wn)i = 7r(ij(n) + i 2 )(i 2 - j2(n)) ,
0,
Ii - j(n)1 = odd,
otherwise.
From (7.2.50) and (7.2.59), we obtain that
II gn ll de! Ilij(n)(x n - W n ) + D(xn - wn)ll-+ o.
Taking the real part of the inner product of gn with D(xn - w n ) yields
IID(xn - wn)1I -+ o.
It immediately follows from (7.2.58) and (7.2.62) that
IIDwnll -+ o.
172
(7.2.57)
(7.2.58)
(7.2.59)
(7.2.60)
(7.2.61 )
(7.2.62)
(7.2.63)

Now we are ready to show a contradiction. Since lan,j(n) I -+ 1, as n -+ 00, we obtain
for n large enough that
II 11 2  8,2l a n,j(n) 1 2 i 6
Dwn = L..J 2( "2() "4)("2 "2( ))2
i=l,li-j(n)l=odd 7r J n + z z - J n
2,2lan,j(n) 12( Ij (n) I + 1 )6
> 7r2((ljn)1 + 1); + j2(n2)(2Ij(n)1 + 1)2'
2, lan,j(n) I (n - 1)
7r 2 ((n - 1)4 + n 2 )(2n - 1)2'
> <5 > 0
Ij(n)1 < n
Ij(n)1 = n
(7.2.64)
with <5 being a constant independent of n,j(n). Thus, we have a contradiction and the
proof of Theorem 7.2.2 is complete.
o
For the cases of the Dirichlet-Neumann and Neumann-Dirichlet boundary con-
ditions, we can show that the corresponding Co-semigroups obtained by the modal
method are also uniformly exponentially stable. For exam pIe, if u satisfies the Dirichlet
boundary condition and 8 satisfies the Neumann boundary condition, then fo1r 8(x, t)dx
= fo1r 8o( x )dx which is derived by integrating the second equation of (7.2.1) with re-
spect to x and t, where Oo( x) is the initial temperature distribution of the rod.
Having changed to the new dependent variable
1 1 1r
0=0-- Oo(x)dx,
7r 0
(7.2.65)
we can choose the state space 1i = { (Yb Y2, Y3) E Ho x £2 x £21 fo1r Y3dx }, and choose
<Pj, 'ljJj as before and j to be /7r cosjx. In this case, the matrix Dn is same as
(7.2.15). Moreover, the Matrix Fn = -Dn. Therefore, the proof of IIDYnll -+ 0 and
II D W n II -+ 0 can be even more easily carried out. Accordingly, we have
( w ) . = -sign(j(n)),an,j(n)
n  yI2(i 2 + ij(n))
(7.2.66)
and
IIDWnI12
n 2 1 1 2'4
"' , an,j(n) Z
 2(i 4 + j2(n))
173

> ,2!a n ,j(n)1 2 !4(n) > {; > o.
- 2(J4(n) + J2(n)) -
(7.2.67)
Remark 7.2.1 In Theorem 2.2.1, we have used a simpler contradiction argument than
that used here to show the exponential stability of the Co-semigroup associated with the
thermoelastic system (7.2.1) with the Dirichlet-Dirichlet boundary conditions (7.2.2).
However, it seems difficult to apply the same argument to the finite-dimensional ap-
proximation of this problem. Although we still use a contradiction argument in the
proof of Theorem 7.2.2, the proof becomes much more involved. If we have the equiv-
alence between FJ'v n and Dn V n , i. e.,
C 2 11 D n v nli < IIFJ'vnll < C1lIDnvnll,
(7.2.68)
then the argument in the proof of Theorem 2.2.1 can also work here. In the case of the
Direchlet-Neumann boundary conditions, FJ' = -Dn. Thus, (7.2.68) holds and we can
use the same argument as in Section 2.2 to obtain the uniform exponential stability for
a sequence of Co-semigroups associated with the modal approximation. However, in
the case of the Dirichlet-Dirichlet boundary conditions, we are only able to get the sec-
ond inequality in (7.2.68). This is not surprising because for different approximation
schemes, we will have different structures of the matrix An. Even for some approxima-
tion schemes, the corresponding Co-semigroups are not uniformly exponentially stable
(refer to Liu & Zheng [2J for the detailed discussion of this aspect).
We now turn the discussion onto the convergence of our approximation scheme.
Let Pn be the orthogonal projection from 'H to 'Hn. Then, as mentioned before, the
matrix An in (7.2.14) is the matrix representatoin of the operator An = PnAPn where
A is defined in (7.2.3). Let
v = V(A) n (H 4 x H 3 X H 4 ).
(7.2.69)
It is easy to see that V is dense in 'H. Since (I - A)V(A) = 'H, we also know
that (I - A)V is dense in H. To show the strong convergence of the approximation
semi groups Tn(t) to T(t), by the dissipativeness of A and An and the Trotter-Kato
174

theorem (see pazy [1], p.88, Theorem 4.5), we only need to show Anz  Az in H
for all z E V.
Theorem 7.2.3 Tn(t), (Tn(t))*  T(t), (T(t))* in H, respectively. Moreover, the
convergence is uniform in any bounded intervals of t.
Proof Let z E V. Then
1;. sin j x 0 0
00 J
z=E a. 0 + b j SIn J x + Cj 0
J
j=l
0 0 SIn J x
(7.2.70)
. h { .3 b .3 "4 } oo b . 1 2 F h h
WIt ajJ, jJ , CjJ 1 elng sequences. urt ermore, we ave
00
E b i sin ix
i=l
00 00
Az = E( -aii -, E Cjj!ij) sin ix
i=l j=l
00 00
E( -, E bjj fij - Cii2) sin ix
i=l j=I
(7.2.71)
and
n
E b i sin ix
i=l
n n
Anz = E( -aii -, E Cjj !ij ).sin ix
i=l j=l
n n
E( -, E bjj fij - Cii2) sin ix
i=l j=l
(7.2.72)
where Ii; = I! (cosjx, sinix)L2o Now Az - A,.z can be written as
00
E b i sin ix
i=n+ 1
00 00
E (-aii - , E Cjj fij) sin ix
i=n+l j=l
00 00
E (-, E bjj fij - c i i 2 ) sin ix
i=n+l j=l
o
n 00
,E( E Cjjfij)sinix
i=l j=n+l
n 00
,E( E bjj fij) sin ix
i=l j=n+l
= I + I I.
(7.2.73)
It follows from Az E H that IIIIIH -+ 0, as n -+ 00. The second entry of I I can be
175

estimated as follows
t ( . f: Cjj j;j ) sin ix 2
=1 J=n+l
_ ; f ( . f: Cjj j;j ) 2
=1 J=n+l
< ; f ( .f ICjl2aj2.f I Cj!2-2alfijI2 )
=1 J=n+l J=n+l
< ; ( .f !Cj!2ap ) ( .f !CjI2-2af II; j I 2 )
J=n+l J=n+l =1
; ( .f ICjl2ap ) ( .f ICjI2-2a )
J=n+l J=n+l
(7.2.74)
00 00 00
Since E ICjl2j6 is a convergent series, then Elcjl2aj2 and E ICj12-2a are also
j=n+l j=l j=n+l
convergent as long as  > Q > . Therefore, by (7.2.74), we obtain that
t ( . f: Cjj Ii j ) sin ix 2 ---+ O.
=1 J=n+l
(7.2.75)
Similarly, we can get
t ( . f: bjj Iij ) sin ix ----+ o.
=1 J=n+l
(7.2.76)
Thus, we have proved that
lim IIAz - AnzllH = 0, Vz E V.
n-oo
(7.2.77)
The convergence of approximate adjoint semigroups can be verified in a similar way
since A and A * only differ from the sign in front of the coupling coefficient I (see
Hansen [1]).
o
Remark 7.2.2 For the cases of the Dirichlet-Neumann and Neumann-Dirichlet bound-
ary conditions, the convergence ofTn(t) and T:(t) is obvious from above analysis since
there is no need to expand cosjx in terms of sin ix in (7.2.71) and (7.2.72).
176

Remark 7.2..3 There are some works in the literature on the uniform exponential
stability for the approximation of certain elastic systems. We refer to Banks, [to &
Wang [lJ for the result on the wave equation with boundary damping. We also refer to
Fabiano [1 J and [2J for the "equivalent norm" method to construct such approximations.
7.3 Approximation of the Viscoelastic System
In this section, we study the uniformly exponentially stable approximation of the
following linear viscoelastic system with memory on a finite history interval:
Utt(t) + A [g(O)u(t) + f g'(s)u(t - s)ds] = 0
(7.3.1)
where A is a positive definite, self-adjoint unbounded operator on a separable Hilbert
space H, and g( t) satisfies the following conditions:
(g 1) 9 ( s) E C 2 ( 0, r] n C [0, r], g' ( s) EL I ( 0, r ) ;
(g2) 9 ( s) > 0, g' ( s) < 0, g" ( s) > 0 on (0, r);
(g3) g"( s) + 8g'( s) > 0 on (0, r) for some constant 8 > o.
Without loss of generality" we can assume g(r) = 1. Notice that equation (7.3.1)
is slightly different from equation (3.2.1) where the history interval is (0,00). By
taking a finite history interval (0, r), we can employ some approximation schemes
proposed by Fabiano & Ito [1]. Under conditions (gl) and g(2), the convergence
of these approximation schemes in the sense of (7.0.13) was proved in their paper
mentioned previously. To our knowledge, such a convergent approximation scheme
is not available in the literature when r = 00. Our purpose in this section is to
show the uniform exponential stability of the Co-semigroups associated with these
approximation schemes if condition (g3) is also satisfied.
The semi group setting for equation (7.3.1) is exactly the same as in Section 3.2.
Let V = V(At) with inner product
All
(u, v)v = 0"( u, v) = (A2"u, A2"V)H, Vu, V E V
(7.3.2)
177

and W = L;,(O, r; V) with the inner product
(WI, W2}W = f Ig'(s)l(wI, W2}V ds.
(7.3.3)
Define
v = Ut, W ( t, s) = U ( t) - U (t - s),
(7.3.4)
for t > 0, s E (0, r). Let z = (u, v, w)T and the Hilbert space 1-£ = V x H x W equipped
with the norm
1
Ilzll1t = (lIulI + Ilvll + IIwll) 7: .
(7.3.5)
Then equation (7.3.1) can be reduced to the following abstract first-order evolution
equation on 1-£
dz(t) = Az(t)
dt
(7.3.6)
where
v
Az = -A(u - f g'(s)w(s) ds)
v - Dsw
(7.3.7)
with
V(A) = { z E 1i u - f g'(s)w(s)ds E,V(A), } . (7.3.8)
v E V,Dsw E W,w(O) = 0
It is known (refer to Fabiano & Ito [1]) that under conditions (gl) and (g2) A generates
a Co-semigroup S(t) of contractions on 1-£. Moreover, S(t) is exponentially stable if ,
in addition, condition (g3) is also satisfied (refer to Liu & Zheng [4]).
We now present the approximation scheme of equation (7.3.1) introduced in Fabi-
ano & Ito [1]. Let V N be a given sequence of finite-dimensional subspaces of V.
Assume that for any u E V, there exists a sequence uN E V N such that
IIUN - ullv -+ 0, as n -+ 00.
(7.3.9)
Define a continuous linear operator AN : v N -+ V N by
(A N U, v) H = (j ( u, v), for u., v E V N .
(7.3.10)
178

Thus, the spaces V N complete the discretization of the spatial variable in the sense
that they provide a sequence of finite-dimensional subspaces of V and H, and (7.3.10)
gives an approximation of the operator A. This can often be realized by choosing a
standard finite-element scheme. The two approximation schemes in the time variable
for the resulting equation considered here are the averaging scheme (see Banks & Burns
[1]) and a spline-based scheme (see Ito & Kappel [1]). They all involve discretization
of the history interval [0, r) and approximation of the differential operator D s.
Let M be a given integer. Then we obtain the discretization of [0, r) with mesh:
sf1 = ih, i = 0, . . . , M and h = lvt . Let Bf1, i = 1,. . . , M be the corresponding linear
spline elements:
Bf1 ( s) =
1 M M M
i (s - S;_I), for S;_1 < S < S; ,
h (Sftl - s), for sf1 < s < sftu
0, elsew here,
i=I,...,M-l,
(7.3.11)
BkJ(s) = {  (s - S_I)' for S-1 < S < s,
0, elsew here.
We also denote the characteristic function on [Sl' sf1) by Ett, z -
each positive integer N, M, define the subspaces of W by
(7.3.12)
1, . . . , M. For
{WEW M W E V N } ,
W N,M - W = E bJ:J B?J (7.3.13)
t t'
i=l
{WEW M a E V N } .
WN,M W = E aJ:J E?J (7.3.14)
t t'
i=l
In order to approximate the operator Ds, we consider a sesquilinear form aN,M on
W N,M x WN,M:
aN,M(wN,M, hN,M) = (DswN,M, hN,M)w
(7.3.15)
for wN,M E W N,M and hN,M E WN,M. Since aN,M is continuous, there exists a linear
operator DN ,M : W N,M -+ WN,M such that
aN,M(wN,M, hN,M) = ( DN ,MwN,M, hN,M)w.
(7.3.16)
179

Notice that Ds W N,M = WN,M. Hence, a simple calculation shows that DN ,M is given
by
DN ,M wN,M = .!.  ( b¥ - b¥ ) E?J
h  t t-I t
t=1
(7.3.17)
M
for wN,M = E bf1 Bf1, where br E V N , b = O. Finally, we introduce the following
i=1
isomorphisms from W N,M to WN,M:
if,M wN,M
M
- Ebf1Ett,
i=1
(7.3.18)
i,M wN,M
M b M +b M
- E i-I i Ett.
2
(7.3.19)
i=1
Thus, we obtain the approximations of Ds by defining operators D:,M
WN,M by
WN,M -+
D N,M _ D N ,M ( .N,M ) -I
k - 'lk ,
k = 1,2.
(7.3.20)
M
For wN,M = """ a¥ E?J we have
L...J t t'
i=1
D N,M N,M
2 W
1 M M M
- h (ai - ai_I)E i ,
t=1
1  M M ) M
- h L...J( b i - b i - I E i ,
i=1
(7.3.21)
Df,M wN,M
(7.3.22)
b¥ + b¥
where t 2 t-I = af1, i = 1, 2, . . . , M and b = O. Construction of two approxima-
tion schemes is now completed.
Let 1i N ,M = V N X H N X WN,M equipped with the norm induced from the norm
in 1i. We now have two approximation schemes of equation (7.3.6):
d
dt ZN,M(t) = Af,M zN,M(t), k = 1,2
where zN,M(t) = (uN(t),vN(t),wN,M(t))T and
(7.3.23)
A N,M N,M _
k z -
v N
-AN (uN - f9'(s)w N . M (s)ds)
v N _ D:,M wN,M
(7.3.24 )
180

Lemma 7.3.1 Af,M (k = 1,2) is dissipative in 1f,N,M and generates a Co-semigroup
of contractions on 1f,N,M. Moreover, for zN,M = (uN, v N , wN,M)T E 1f,N,M with wN,M =
M
Eaf4 Ef1,
i=1
NM NM NM 1 ( M E -II M M M 2 1 M M 2
Re ( A I ' z ' , z ' ) 'U (g 9 )ll a II + 9 Ii a II
n. - 2" i=1 h i - i+1 i V h M M V
+   gfdlla - alll) , (7.3.25)
and
R (A N,M N,M N,M ) 1 ( I 1 ( M M )llb M l1 2 1 M llb M l1 2 )
e 2 z , Z 1{ = 2  h g; - g;+1 ; v + h gM M V
(7.3.26)
where
l S ¥
af1 = (bf4I + b)/2, gf'/ = & g'(s)ds, i = 1,..., M, a = b = o.
- M
Si_l
(7.3.27)
Proof We first prove that Af,M (k = 1, 2) is dissipative in 1f,N,M. By the conditions
on g( s), we have
gf'! < gff-I < 0, i = 1,. . . , M - 1.
(7.3.28)
Thus,
Re (A,M zN,M, zN,M)1-£
Re {a(vN,uN)-a(u N - fa' g'(s)wN,M(s)ds, v N )
- fa' g'(s)(v N - D,MwN.M, wN,M}v ds }
Re [ g'(s)(  f(a - al)Efd,wN,M}vds
o t=1
1  R ( M M M ) M
h L..J e ai - ai-I' ai Vgi
i=1
2 1 h fgfd(lIalI -lIalll + lIa - all1n
t=1
1 M-I 11 M
2h  (gfd - gl)lIalI + 2h glIalI + 2h Egfdlla - alll
t=1 t=1
< o. (7.3.29)
181

Similarly, we have
Rs (A,M ZN,M, ZN,M)1i
- Re {a(vN,u N ) - a(u N - 10' g'(S)WN,M(S)ds, V N )
fT N M }
- J o g'(S)(V N - D 2 ' WN,M, WN,M)vds
- Re 19'(s)(  I=(W -bf!.l)EfI,W N ,M)vds
o t=1
- 2 1 h I= Re(W - bf!.l, W + bf!.l)Vgfl
t=1
- 2 1 h I=gfl (IIWII - IIbf!.lll)
t=l
1 1 ( M M )11 M I 2 1 M I M 1 2
- 2h  9i - 9i+1 b i Iv + 2h 9M IbM I v
t=1
< o.
(7.3.30)
Thus, to prove that A:,M (k = 1,2) generates a Co-semigroup, Sf,M, of contractions
on 1i N ,M, by Theorem 1.2.4 in Chapter 1, it suffices to prove that 0 E p(A:,M) (k =
1,2). For any F = (!1,!2,!3)T E 1i N ,M, consider the equatons A:,MzN,M = F, Le.,
V N - f
- 1,
(7.3.31)
(7.3.32)
(7.3.33)
-AN(U N - 10' g'(s)wN,M(s)ds) = 12,
N D N,M N,M - f
v - k W - 3.
It follows from (7.3.31) and (7.3.33) that
D N,M W N,M - f - f
k - 1 3.
(7.3.34 )
By induction, we can obtain afl, (i = 1, . . . , M), thus a unique wN,M in case k = 1. In
case k = 2, by induction we can obtain b, (i = 1,..., M), thus afl, (i = 1,., M) and
a unique wN,M. Substituting wN,M just obtained into (7.3.32), we can obtain a unique
uN by the assumption that A is a positive definite and self-adjoint operator and AN
is its restriction to V N . Thus, it is clear from the above that 0 E p(A:,M) (k = 1,2)
and the proof is complete.
182
o

In what follows we- will show that Sr:,M (t) are uniformly exponentially stable.
Theorem 7.3.1 lfthe kernel g(s) satisfies conditions (gl)-(g3), then the Co-semigroup
Sr:,M (t) generated by the operator A:,M defined in (7.3.20), is uniformly exponentially
stable, for k = 1, 2.
Proof It suffices to verify conditions (7.1.44) and (7.1.45) of Theorem 7.1.3 for the
cases of k = 1, 2 , respectively.
Case 1 (k = 1 Averaging Scheme) :
(i) Suppose (7.1.44) is not true. Then there is an A,M and an w E 1R such that iw
is an eigenvalue of A,M. This implies that there exist a zN,M = (uN, v N , wN,M)T E
V(Af,M) with IIZN,MII1tN,M = 1 such that
(iwI - A,M) zN,M = 0 In 1i,
(7.3.35)
I.e. ,
iwu N - v N = 0 In V,
iwv N + AN (uN - 10' 9'(s)w N ,M(s)ds) = 0 III H,
iwwN,M(s) - v N + D,MwN,M(s) = 0 in W.
(7.3.36)
(7.3.37)
(7.3.38)
Taking the real part of the inner product of (7.3.35) with zN,M yields
Re (( i{3N,M I - Af,M )zN,M, zN,M)1t
_ -Re (Af,M zN,M, zN,M)1t
1 ( l 1 M M M 2 1 M M 2  1 M M M 2 )
- -2 6. h (9i - 9iH)lI a i IIv + h 9MllaMliv + 6. h 9i Ila i - ai-Illv
- o. (7.3.39)
All terms in (7.3.39) are non-negative. Thus,
M-l1 (7.3.40)
E h (9f':-1 - 9i'f)llalI - 0,
i=l
1 M M 2 0, (7.3.41)
h gM lIaMliv -
f: 1 Mil M M 11 2 o. (7.3.42)
. h gi ai - ai-l v -
t=l
183

Then, by the condition on 9, we immediately get
M
IIwN,MII = - E 9f1l1af1l1 = o.
i=l
(7.3.43)
It follows from (7.3.38) that v N = O. Since 0 E p(Af,M), we deduce from (7.3.36) that
uN = 0, a contradiction.
(ii) Suppose (7.1.45) is not true. Then there exist a subsequence of A,M, still denoted
by Af,M, a sequence of {3N,M E 1R and a sequence of zN,M = (uN, v N , wN,M) E
V(Af,M) with Ilz N ,MII1t = 1 such that
lim II (i,BN,MI - Af,M)zN,MII1t = 0,
N,Moo
( 7 . 3.44 )
I.e. ,
i{3N,M u N - v N --+ 0 In V, ( 7 . 3.45 )
ij3N,Mv N + AN (uN - 10' g'(s)wN,M(s)ds) --+ 0 In H, (7.3.46)
i{3N,M wN,M (s) - v N + Df,M wN,M (s) --+ 0 In w. (7.3.47)
It follows from Lemma 7.3.1 and (7.3.44) that
Re (( i{3N,M I - Af,M)zN,M, zN,M)1t
_ -Re (Af,M zN,M, zN,M)1t
1 ( l 1 M M M 2 1 M M 2  1 M il M M 11 2 )
- -2  h (gi - giH)lIai IIv + h gMllaMllv + (:: h gi ai - ai-l v
--+ O. (7.3.48)
All terms in (7.3.48) are non-negative. Thus, we have
M- 1 1 (7.3.49)
E h (91 - 9f1)lIaf1l1 --+ 0,
i=l
1 M M 2 0, (7.3.50)
h 9M lIaMliv --+
f: 1 Mil M M 11 2 --+ O. (7.3.51 )
h 9i ai - ai-l v
i=l
184

Next, we show that (7.3.49) and (7.3.50) imply that
M
IlwN,MII = - E gf1l1af1l1 --+ o.
i=l
(7.3.52)
Since
(gftl - gfl)
h
1. S [1.1 g'(S) - g'(S - h) ds
S¥ h
&
1.ttl g"(Ods (s - h <  < s)
&
> - 1.ttl 6g'(Ods (by (g3))
&
> C 1. s [1.1 ' ( )d c M
-0 9 s s = -ogi+l'
S¥
&
(7.3.53)
and
M M-l M-l
- E gf1l1arll = E (gf:-l - gf1)lIarll - E glllarll - glIaII, (7.3.54)
i=l i=l i=l
we can see that (7.3.52)' is an immediate result of (7.3.49) and (7.3.50). Therefore, it
follows from II zN,M 111-£ = 1 that
IIUNII + IIvNII --+ 1.
(7.3.55)
By (7.3.45) and the continuous injection of V into H, i{3N,Mu N - v N also converges
to zero in H. Thus, we have
i{3N,M (uN, VN)H - IIvNII --+ o.
(7.3.56)
Taking the inner product of (7.3.46) with uN in H, we have
i(3N,M (v N , uN) H + u( uN - 10' g' (s )wN,M (s )ds, uN)
= i(3N,M(v N ,U N }H + IluNII - 10' g'(s)(wN,M(s),uN}vds -+ O.
By (7.3.52) and the following estimate
110' -g'(s )(wN,M (s), uN}vdsl < IluNllv 10' -g'(s )lIwN,M (s )llvds
1
< lIuNliv (10' -g'(s)ds) 2 IlwN,Mllw, (7.3.58)
185
(7.3.57)

the third term in (7.3.57) converges to zero. Adding the complex conjugate of (7.3.57)
to (7.3.56) yields
IIUNII - IIvNII --+ o.
(7.3.59)'
Therefore, it follows from (7.3.56) abd (7.3.59) that
1
lIuNII --+ 2'
1
IIvNII --+ -
2.
(7.3.60)
Now we claim that It'N,MI is bounded below by a positive constant for N, M large
enough. Otherwise, there is a subsequence of t'N,M, still denoted by t'N,M, such that
lim t'N,M = O. Combining this with (7.3.45) implies that IlvNIIH --+ o. Thus, we
N,M-oo
have a contradiction to (7.3.60).
Therefore, we can divide equation (7.3.45) by t'N,M to obtain
v N 2 1
N --+ -. (7.3.61)
(3 ,M v 2
The rest of the proof is to show that (7.3.61) is a contradiction. We rewrite (7.3.47)
as
v N 1 N,M N,M. ( )
- j3N,M + j3N, M D l w (s) -+ 0 tn W, 7.3.62
N
then take the inner product with ;,M E W in W to obtain that
j3v:.M : { sg'(s)ds - j3;'M { sg'(s)(Df",MwN,M(s), j3v:.M )vdS -+ O. (7.3.63)
In what follows, we will show that the second term in (7.3.63) converges to zero.
Therefore, the first term in (7.3.63) must converge to zero, which gives the desired
contradiction. In doing so, we first get the following estimate:
IT v N
10 sg' (s )(Df",M wN,M, j3N,M )vds
M-1 ( ) M ( ) M N ( ) M N
" 89 i+1 - 89 i ( J:J v ) _ 89 M ( aM v )
 h at , r.lN,M V h M, {3 N,M V
1.=1 fJ
< j3v:.M V (l (Sg)itl  (sg)f"I IlalIv - (st: II at: II v ) , (7.3.64)
186

{s[J
where (sg)f4' = JSl sg'(s)ds for i = 1,..., M. A simple calculation also leads to
Ml ( ) M ( ) M
?= Sg;+1; sg; Ilaf4'llv
t=l
l l1>f.l s g'(s) - {(s - h) ds + f.1>f.l g'(s - h)ds lIarliv
< [s g'(s) - '(s - h) 1: 1 1IarllvEf1ds + 1: 1 Igf1ll1 a rllv
h i=l i=l
1 1
< ([ s2 g'(S) - '(s - h) dS) 2 (l  (gt!l _ gr)lIarll) 2 + IIwN,Mllw
1
< K G \9t!1 - gf1)llarll ) 2 + IlwN,Mllw -4 0 (7.3.65)
for some constant K > O. Here we have used (7.3.49), (7.3.52), and the fact that
8 2 g"(8) E Ll(O,r). Using the Cauchy-Schwarz inequality and (7.3.50), we also have
(sM IlaMliv
1 1
< (f.h -sy(s) dS) 2 G gMllaMII) 2
< K G gMllaMII ) t -4 O. (7.3.66)
Finally, we substitute (7.3.65) and (7.3.66) into (7.3.64) to obtain the desired result.
Thus the proof for Case 1 (k = 1) is complete.
Case 2 (k = 2, Spline- based scheme) :
The proof is very similar to that for Case 1. Therefore, we will only point out the
main differences and omit the redundant argument.
(i) Suppose (7.1.44) is false. Then a similar proof to part (i) of Case 1 also works here.
Therefore, we can focus our attention to the proof of part (ii).
(ii) Suppose (7.1.45) is false. Then there exist a subsequence of A,M, still denoted by
A,M, a sequence of {3N,M E 1R and a sequence of zN,M = (uN, v N , wN,M) E V( A,M)
with IIZ N ,MII7-l = 1 such that
lim II (i{3N,Mj - A,M)zN,MII7-l = 0,
N,M-oo
(7.3.67)
187

i{3N,Mu N _ V N  0 in V, (7.3.68)
i(3N,Mv N + AN(u N - 10' g'(s)wN,M(s)ds)  0 in H, (7.3.69)
i{3N,M wN,M (s) - v N + nf,M wN,M (s)  0 in W. (7.3.70)
It follows from Lemma 7.3.1 and (7.3.67) that
Re (( i{3N,M I - A,M )zN,M, zN,M)1-l
_ - Re (Af,M zN,M, zN,M)1-l
1 ( I 1 M M M 2 1 M M 2 )
- 2 6. h (gi - gi+1 )lIb i IIv + h gM II b M IIv
 o.
(7.3.71)
Therefore,
M- I 1
 h (9f:-I - 9f1) II bf1l1  0,
t=I
1 M M 1 2
h 9MllbMI V  o.
(7.3.72)
(7.3.73)
b M + b M 1 sM
. M i-I i M M' I .
SInce ai = 2 ' b o = 0, and 9i = M 9 (s) ds, we obtaIn
S,_1
M
IIwN,MII - - L9f1llarll
i=I
< - fgfl (lIblll + IIWII)
t=I
1 I M M 2 1 I M M 2 1 M il M I1 2
- -2:  9i+Illb i IIv - 2:  9i IIbi IIv - 2 9M b M V
=I t=I
M-I 1 M-I 1
- - L gftlllWII + 2 L (gftl - gfl)IIWII - 2glIblI
t=I =I
1 M-I 1 M-I
< 6h L (gftl - gfl)lIbflll + 2  (gftl - gfl)IIWII
t=I t=I
1 M M 1 2
-29M IIbMI v,
(7.3.74)
188

where we have used (7.3.53). Combining (7.3.72)-(7.3.74) yields
"WN,MII  0,
(7.3.75)
and
M-l
E gf1llbf1"  o.
i=1
(7.3.76)
We now repeat the argument from (7.3.55)-(7.3.60) to get
1
"uNII  2'
1
IIvNllk  2.
(7.3.77)
In the same manner as before, I,BN,MI is bounded from below by a positive constant
for N, M large enough. Therefore, we can divide equation (7.3.68) by ,BN,M to obtain
that
v N 2 1
(3N.M V -+ 2". (7.3.78)
The rest of the proof is to show that (7.3.78) leads to a contradiction. We first rewrite
(7.3.70) as
v N 1 N,M N,M .
- (3N.M + (3N.M D 2 w (s) -+ 0 m W.
N
Then, we take the inner product of (7.3.79) with ;.M E W in W to obtain
V N 2 lo r 1 lo r V N
, ) ' ( ) ( N,M N,M ( ) )
(3N.M V 0 sg (s ds - (3N.M 0 sg S D 2 W S, (3N.M vds -+ O.
To obtain a contradiction to (7.3.78), it suffices to prove that the second term in
(7.3.79)
(7.3.80)
(7.3.80) converges to zero. This can be proved as follows:
A simple calculation yields
<
{r v N
- Jo sg'(s)(Df.MwN.M, (3N.M ) V ds
lo r 1 M N
, M M M V
- 0 Sg(s)( h (bi -bi-I)E i , (3N.M )vds
M-l ( ) M ( ) M N ( ) M N
 8g i+l - 8g i ( b V ) _ 8g M ( b M V )
 h t , ,B N,M V h M, ,B N,M V
t=1
(3V;M V (l (Sg)f1-1  (sg)ff 1IW"lIv - (s IIblIv ) ,
(7.3.81)
189

where (sg)!'l = l: M sg'(s)ds for i = 1,...,M.
8'_1
By (7.3.72) and (7.3.76), we have
M-1 ( ) M ( ) M
L sg i+1; sg i IIWllv
i=l
_ 1 11 s g'(s) - r(s - h) ds + 11 g'(s - h)ds IIbflllv
< 1: 1 [ s g'(s) - '(s - h) IIbflllvElds + 1: 1 lgfllllWllv
=1 2.=1
1 1
< ([ s2 g'(S) - '(s - h) dS) 2 (1  (gl _ gfl)IIWII) 2 - gflIlWII
< K G \91 - gfl)IIWII) t - gflllbflll -+ 0 (7.3.82)
for some constant K > O.
On the other hand, by (7.3.73), we get
(s IIblIv
1 1
< (lh - s2g(s) dS) 2 G glIbII) 2
1
< K G 9lIbll r -+ 0
(7.3.83)
for some constant K > o. Therefore, we have a contradiction to (7.3.78). The proof
is complete.
o
190

Bibliography
R.A. Adams
[1] Sobolev Spaces, Academic Press, New York, 1975.
G.A valos and I. Lasiecka
[1] Exponential stability of thermoelastic system without mechanical dissipation,
Rend. Istit. Mat. Univ. Trieste Suppl., Vol. XXVIII(1997), 1-28.
[2] Exponential stability of a thermoelastic system with free boundary conditions
without mechanical dissipation SIAM J. Math. Anal., Vol. 29, No.1 (1998),
155-182.
A. V. Balakrishnan
[1] Applied Functional Analysis, Second Edition, Springer-Verlag, New York, 1981.
H.T. Banks and J.A. Burns
[1] Hereditary control problems: numerical methods based on averaging approxima-
tions, SIAM J. Control and Optimization, Vol. 16, No. 2(1978), 169-208.
H.T. Banks, K. Ito, and C. Wang
[1] Exponentially stable approximations of weakly damped wave equations, "Es-
timation and Control of Distributed Parameter Systems," Proceedings of an
International Conference on Control and Estimation of Distributed Parameter
Systems, Vorau, July 8-14, 1990, W. Desch, F. Kappel & K. Kunisch, eds.,
Birkhauser, 1991.
H.T. Banks and K. Kunisch
[1] The linear regulator problem for parabolic systems, SIAM J. Control and Opti-
mization, Vol. 22(1984), 684-698.
H.T. Banks and C. Wang
[1] Optimal feedback control of infinite dimensional parabolic evolution systems:
Approximation techniques, SIAM J. Control and Optimization, Vol. 27, No.
5(1989), 1182-1219.
191

J.A. Burns and R.H. Fabiano
[1] Feedback control of a hyperblic partial-differential equation with viscoelastic
damping, Control-Theory and Advanced Technology, Vol. 5, No. 2(1989), 157-
188.
J .A. Burns, Z. Liu, and R.E. Miller
[1] Approximations of thermoelastic and viscoelastic control systems, Numerical
Functional Analysis and Optimization, Vol. 12(1991), 79-136.
J .A. Burns, Z. Liu, and S. Zheng
[1] On the energy decay of a linear thermoelastic bar, J. of Math. Anal. Appl., Vol.
179, No. 2(1993), 574-591.
S.K. Chang and R. Triggiani
[1] Spectral analysis of thermo-elastic plates with rotational forces, Optimal Con-
trol: Theory, Algorithms, and Applications, pp. 84-115, W.W. Hager and P.M.
Pardalos, eds., Kluwer Academic Publishers B.V., 1998.
G. Chen
[1] Energy decay estimates and exact boundary value controllability for wave equa-
tion in a bounded domain, J. Math. Pures. Appl., 58(1979), 249-273.
[2] A note on the boundary stabilization of the wave equation, SIAM J.. Contr. and
Opt., 19(1981), 106-113.
G. Chen, M.P. Coleman, and K. Liu
[1] Boundary stabilization of Donnell's shallow circular cylindrical shell, to appear
in J. Sound & Vibration.
G. Chen, M.C. Delfour, A.M. Krall, and G. Payre
[1] Modeling, stabilization and control of serially connected beams, SIAM J. Cont.
and Opt., 25(1987), 526-546.
192

G. Chen, S.A. Fulling, F.J. Narcowich, and S. Sun
[1] Exponential decay of evolution equations with locally distributed damping, SIAM
J. Appl. Math., Vol. 51, No. 1 (1991), 266-301.
G. Chen, S.G. Krantz, D.W. Ma, and C.E. Wayne
[1] The Euler-Bernoulli beam equation with boundary energy dissipation, in Oper-
ator Methods for Optimal Control Problems, S.J. Lee ed., Lecture Notes in Pure
and Appl. Math. , pp. 67-96, Marcel Dekker, Inc., New York, 1987.
G. Chen and D. L. Russell
[1] A mathematical model for linear elastic systems with structural damping, Quart.
Appl. Math., 39(1981/1982), 433-454.
S. Chen and R. Triggiani
[1] Proof of extensions of two conjectures on structural damping for elastic system,
the case  < a < 1, Pacific J. Math., 39(1989), 15-55.
[2] Gevery class semigroups arising from elastic systems with gentle dissipation: the
case 0 < a < , Proc. AMS, Vol. 110, No. 2(1990), 401-415.
B.D. Coleman and V.J...Mizel
[1] Norms and semigroups in the theory of fading memory, Arch. Rat. Mech. Anal.,
Vol. 23 (1966), 87-123.
R.F. Curtain and A.J. Pritchard
[1] Infinite Dimensional Linear System Theory, Springer-Verlag, New York, 1978.
C.M. Dafermos
[1] On the existence and the asymptotic stability of solution to the equations of
linear thermoelasticity, Arch. Rat. Mech. Anal., Vol.29(1968), 241-271.
[2] Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal., Vol. 37(1970),
297-308.
[3] An abstract Volterra equation with applications to linear viscoelasticity, J. Dif-
ferential Equations, Vol. 7(1970), 554-569.
193

R. Datko
[1] Extending of a theorem of Liapunov to Hilbert Space, J. Math. Anal. AppL,
32(1970), 610-616.
A. Day
[1] Heat Conduction with Linear Thermoelasticity, Springer-Verlag, New York, 1985.
[2] The decay of energy in a viscoelastic body, Mathematika, Vol. 27(1980), 268-
286.
W. Desch and R.K. Miller
[1] Exponential stabilization of Volterra integrodifferential equations in Hilbert space,
J. Differential Equations, Vol. 70(1987), 366-389.
[2] Exponential stabilization of Volterra integral equations with singular kernels, J.
Integral Equations Applications, Vol. 1(1988), 397-433.
R.H. Fabiano
[1] Stability and approximation for a linear viscoelastic model, Journal of Mathe
matical Analysis and Applications, Vol. 204 (1996), 206-220.
[2] Stability preserving Galerkin approximations for linear distributed parameter
systems, preprint.
R.H. Fabiano and K. Ito
[1] Semigroup theory and numerical approximation for equations arising in linear
viscoelasticity, SIAM J. Math. Anal., Vol. 21, No. 2(1990), 374-393.
[2] An approximation framework for equations in linear viscoelasticity with strongly
singular kernels, to appear in Quarterly of Applied Mathematics.
M. Fabrizio and B. Lazzari
[1] On the existence and asymptotic stability of solutions for linearly viscoelatic
solids, Arch. Rat. Mech. Anal., Vol. 116(1991), 139-152.
A. Friedman
[1] Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
194

J .S. Gibson
[1] The Riccati integral equations for optimal control problems on Hilbert spaces,
SIAM J. Control and Optimization, Vol. 17(1979), 537-565.
J.S. Gibson and A. Adamian
[1] Approximation theory for LQG optimal control of flexible structures, SIAM J.
on Control and Optimization, Vol. 29, No.1(1991), 1-37.
J.S. Gibson and LG. Rosen
[1] Numerical approximation for the infinite-dimensional discrete-time optimal Lin-
ear Quadratic Regulator problem, SIAM J. on Control and Optimization, Vol.
26(1988), 428-451.
J.S. Gibson, LG. Rosen, and G. Tao
[1] Approximation in control of thermoelastic systems, SIAM J. on Control and
Optimization, Vol. 30, No. 5(1992), 1163-1189.
C. Giorgi, V. Pata, and M.G. Naso
[1] Exponential stability in linear heat conduction with memory: a serrngroup ap-
proach, preprint, 1998.
K.B. Hannsgen, Y. Renardy, and R.L. Wheeler
[1] Effectiveness and robustness with respect to time delays of boundary feedback
stabilization in one-dimensional viscoelasticity, SIAM J. Control and Optimiza-
tion, Vol. 26, No. 5(1988), 1200-1233.
K.B. Hannsgen and R.L. Wheeler
[1] Viscoelastic and boundary feedback damping: Precise energy decay rates when
creep modes are dominant. J. Integral Equations and Applications, Vol. 2(1990),
495-527.
[2] Moment conditions for a Volterra integral equation in a Banach space, Proceed-
ings of Delay Differential Equations and Dynamical Systems, Claremont, Lecture
Notes in Mathematics, Vol. 1475, pp. 204-209, Springer-Verlag, 1991.
S.W. Hansen
[1] Exponential energy decay in a linear thermoelastic rod, J. of Math. Anal. Appl.,
Vol. 167(1992), 429-442.
195

S. Hewitt and K. Stromberg
[1] Real and Abstract Analysis, Springer-Verlag, New York, 1965.
L. Hormander
[1] The Analysis of Linear Partial Differential Operators, Springer Grundlehren,
Heidelberg, 1985.
F.L. Huang
[1] Characteristic condition for exponential stability of linear dynamical systems in
Hilbert spaces, Ann. of Diff. Eqs, 1(1) (1985), 43-56.
[2] On the holomorphic property of the semi group associated with linear elastic
systems with structural damping, Acta Math. Sci., (in Chinese), 55(1985), 271-
277.
[3] A problem for linear elastic systems with structural damping, Acta Math. Sci.
Sinica, 6(1986), 107-113.
[4] On the mathematical model for linear elastic systems with analytic damping,
SIAM J. Control and Optimization, Vol. 26( 1988), 714-724.
K. Ito and F. Kappel
[1] On variational formulations of the Trotter-Kato theorem, CAMS Report, # 91-7,
Univ. of Southern California, April, 1991.
S. Jiang
[1] Global solution of the Neumann problem in one-dimensional thermoelasticity,
Nonlinear Analysis, 19(1992), 107-121.
[2] Exponential decay and global existence of spherically symmetric solutions in
thermoelasticity, to appear in Chin. Ann. Math., series A (in Chinese).
S. Jiang, J .E.M. Rivira, and R. Racke
[1] Asymptotic stability and global existence in thermoelasticity with symmetry, to
appear in Quart. Appl. Math.
196

M. Katsoulakis and S. Koike
[1] Viscosity solutions of monotone systems for Dirichlet problems, Differential and
Integral Equations, Vol. 7, No.2(1994), 367-382.
J.U. Kim
[1] On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math.
Anal., 23(1992), 889-899.
J. Lagnese
[1] Decay of solutions of wave equations in a bounded region with boundary dissi-
pation, J. Diff. Eqs, 50(1983), 163-182.
[2] Boundary stabilization of linear elastodynamics, SIAM J. on Control and Opti-
mization, Vol. 21, No. 6(1983), 968-983.
[3] Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics,
Vol. 10, Society for Industrial and Applied Mathematics, Philadelphia, 1989.
I. Lasiecka
[1 ] Controllability of a viscoelastic Kirchhoff plate, N umer. Math., 91 (1989), 237-
247.
I. Lasiecka and R. Triggiani
[1] Uniform exponential energy decay of the wave equation in a bounded region
with L 2 (0, 00; L2(r))-feedback control in the Dirichlet boundary condition, J.
Differential Euqations, 66(1987), 340-390.
[2] Two direct proofs on the analyticity of the S.C. semigroup arising in abstract
thermo-elastic quation, to appear in Advances in Differential Equation.
[3] Analyticity, and lack thereof, of thermo-elastic semigroups, to appear in the
Proceedings of the Conference on PDE Control.
[4] Analyticity of thermo-elastic semigroups with coupled Hinged/Neumann B.C.,
to appear in Abstract and Applied Analysis.
[5] Analyticity of thermo-elastic semigroups with free B. C. submitted to Anneli
Scoulls Normale Superiors di Pisa.
[6] Structual decomposition of thermo-elastic semigroups with rotational forces, to
appear in Semigroup Forum.
197

G. Leugering
[1] On boundary feedback stabilization of a viscoelastic membrane, Dynamics and
Stability of Systems, Vol. 4, No. 1(1989), 71-79.
[2] On boundary feedback stabilizability of a viscoelastic beam, Proc. Roy. Soc.
Edinburgh, Sec. A, Vol. 114, No. 1(1990), 57-69.
J. L. Lions and E. Magenes
[1] Nonhomogeneous Boundary Value Problems and Applications, Springer, Heidel-
berg, 1972.
Z. Liu
[1] Approximation and control of a thermoviscoelastic system, Ph.D. dissertation,
Virginia Polytechnic Institute and State University, Department of Mathematics,
August 1989.
K. Liu and Z. Liu
[1] On the type of Co-semigroup associated with the abstract linear viscoelastic
system, ZAMP, 47(1996), 1-15.
[2] Exponential stability and analyticity of abstract linear thermoelastic systems,
ZAMP, 48(1997) 885-904.
[3] Analyticity and differentiablity of semigroups associated with elastic systems
with damping and gyroscopic forces, Journal of Differential Equations, Vol. 141,
No. 2(1997), 340-354.
[4] Exponential decay of energy of the Euler-Bernoulli beam with locally distributed
Kelvin- Voigt damping, SIAM J. Control and Optimization, Vol. 36, No. 3(1998),
1086-1098.
[5] Boundary stabilization of a nonhomogeneous Euler-Bernoulli beam by the fre-
quency domain multiplier method, preprint, 1997.
Z. Liu and M. Renardy
[1] A note on the equation of a thermoelastic plate, Appl. Math. Lett., Vol. 8, No.
3(1995), 1-6.
198

z. Liu, S.A. Trogdon, and J. Yong
[1] Modeling and analysis of a laminated beam, to appear in Mathematical and
Computer Modeling.
Z. Liu and J. Yong
[1] Qualitative properties of certain Co semigroups arising in elastic systems with
various dampings, to appear in Advances in Differential Equations.
Z. Liu and S. Zheng
[1] Exponential stability of semi group associated with thermoelastic system, Quar-
terly Appl. Math., Vol. LI, No. 3(1993), 535-545.
[2] Uniform exponential stability and approximation in control of thermoelastic sys-
tem, SIAM. J. Control and Optimization, Vol. 32, No. 5(1994), 1226-1246.
[3] Exponential energy decay of the Euler-Bernoulli beam with shear diffusion or
thermal dissipation, Journal of Mathematical Analysis and Applications, Vol.
196(1995),467-478.
[4] On the exponential stability of linear viscoelasticity and thermoviscoelasticity,
Quarterly Appl. Math., Vol. LIV, No.1(1996), 21-31.
[5] Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping,
Quarterly of Applied Mathematics, No. 3(1997), 551-564.
[6] Uniform exponential stability of approximation in linear viscoelasticity, Journal
of Mathematical System, Estimation and Control, Vol.8, No. 2(1998), 177-180.
J.E. Marsden and T.J.R. Hughes
[1] Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, N J,
1983.
D.J. Mead and S. Markus
[1] The forced vibration of a three-layer, damped sandwich beam with arbitrary
boundary conditions, J. Sound Vibration, 10(1969), 163-175.
199

C.B. Navarro
[1] Asymptotic stability in linear thermoviscoelaticity, J. Math. Anal. Appl., Vol.
65(1978), 399-431.
L. Nirenberg
[1] On elliptic partial differential equations, Annali della Scoula Norm. Sup. Pisa,
13 (1959), 115-162.
[2] Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sci-
ences, New York, 1974.
A. Pazy
[1] Semigroups of Linear Operators and Applications to Partial Differential Equa-
tions, Springer, New York, 1983.
J. Pruss
[1] On the spectrum of Co-semigroups, Trans. AMS, 284 (1984), 847-857.
J.P. Quinn and D.L. Russell
[1] Asymptotic stability and energy decay rates for solutions of hyperbolic equations
with boundary damping, Proc. Roy. Soc. of Edinburgh, 77 A(1977), 97-127.
R. Racke
[1] Lectures on Nonlinear Evolution Equations, Aspects of Mathematics Vol. 19,
VIEWEG, Bonn, 1992.
[2] On the time-asymptotic behaviour of solutions in thermoelasticity, Proc. Roy.
Soc. Edinburgh, 107 A(1987), 289-298.
[3] On the Cauchy problem in nonlinear 3-d-thermoelasticity, Math. Z., 203 (1990),
649-682.
[4] Blow-up in nonlinear three-dimensional thermoelasticity, Math. Meth. Appl.
Sci., 12 (1990), 267-273.
R. Racke and Y. Shibata
[1] Global smooth solutions and asymptotic stability in one-dimensional nonlinear
thermoelasticity, Arch. Rat. Mech. Anal., Vol. 116(1992), 1-34.
200

R. Racke, Y. Shibata, and S. Zheng
[1] Global solvability and exponential stability in one-dimensional nonlinear her-
moelasticity, Quarterly of Applied Mathematics, No. 4(1993), 751-763.
B. Rao
[1] A compact perturbation method for the boundary stabilization of the Raleigh
beam equation, to appear in App1. Math. Optim.
D.K. Rao
[1] Frequency and loss factors of sandwich beams under various boundary condi-
tions, J. Mechanical Engineering Sciences, 20(1978),271-282.
R. Rebarber
[1] Exponential stability of coupled beams with dissipative joints: a frequency do-
main approach, SIAM J. Control and Optimization, 33(1995),1-28.
K. Rektorys
[1] Variational Methods in Mathematics, Science and Engineering, D. Reidel Pub-
lishing Company, Holland, 1977.
J .E.M. Rivera
[1] Energy decay rate in linear thermoelasticity, Funkcial Ekvac., Vo1. 35 (1992),
19-30.
[2] Asymptotic behaviour in linear viscoelasticity, Quarterly of Applied Mathemat-
ics, Vo1.LII, N 0.4( 1994), 629-648.
J .E.M. Rivera and R.K. Barreto
[1] Uniform rates of decay in anisotropic thermo-viscoelasticity, preprint in Labora-
torio Nacional de Computacao Cientifica (LNCC).
J.E.M. Rivera and L.H. Fatori
[1] Smoothing effect and propagations of singularities for viscoelastic plates, Journal
of Mathematical Analysis and Applications, 206(1997), 397-427.
201

J .E.M. Rivera, E.C. Lapa, and R. Barreto
[1] Decay rates for viscoelastic plates with memory, Journal of Elasticity, 44(1996),
61-87.
J.E.M. Rivera and R. Racke
[1] Smoothing properties, decay and global existence of solutions to nonlinear cou-
pled systems of thermoelastic type, SIAM J. Math. Anal., Vol. 26 (1995),
1547-1563.
[2] Large solutions and smoothing properties for nonlinear thermoelastic systems,
to appear in J. Differential Equations.
D.L. Russell
[1] A comparison of certain elastic dissipation mechanisms via decoupling and pro-
jection techniques, Quarterly Appl. Math., Vol. XLIX, No. 2(1991),373-396.
[2] A general framework for the study of indirect damping mechanisms in elastic
systems, J. of Math. Anal. Appl., Vol. 173, No. 2(1993), 339-358.
Y. Shibata
[1] Neumann problem for one-dimentional nonlinear thermoelasticity, preprint, 1991.
M. Slemrod
[1] Global existence, uniqueness, and asymptotic stability of classical smooth solu-
tions in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal.,
76 (1981), 97-133.
D. Tataru
[1] On the regularity of boundary traces for the wave equation, preprint.
S. Taylor
[1] Gevrey Semigroups, Ph.D. Thesis, Chapter 5, University of Minnesota, 1989.
202

R. Temam
[1] Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied
Mathematical Sciences 68, Springer-Verlag, New York, 1988.
S. Timoshenko, D.H. Young, and W. Weaver, Jr
[1] Vibration problems in engineering, John Wiley & Sons, New York, 1974.
R. Triggiani,
[1] Wave equation on a bounded domain with boundary dissipation: an operator
approach, J. of Math. Anal. and Appl., 137(1989), 438-461.
A. Wyler
[1] Stability of wave equations with dissipative boundary conditions in a bounded
domain, Differential and Integral Equations, Vol. 7, No.2(1994), 345-366.
Songmu Zheng
[1] Nonlinear Parabolic Equations and Hyerbolic-Parabolic Coupled Systems, Pit-
man series Monographs and Survey in Pure and Applied Mathematics, Vol. 76,
Longman, 1995.
Songmu Zheng (Zheng, Songmu) and Weixi Shen (Shen, Weixi)
[1] Global solutions to the Cauchy problem of a class of quasilinear hyperbolic-
parabolic coupled systems, Scientia Sinica, 4A (1987), 357-372.
[2] Global solutions to the Cauchy problem of the equations of one-dimensional
thermoviscoelasticity, Journal of Partial Differential Equations, Vol. 2, No. 2
(1989), 26-38.
203

Index
Analyticity, see also analytic semigroup,
4, 5, 8, 18, 43, 44, 45, 46, 52, 54, 61,
64, 67, 72, 86, 90, 91, 95, 112, 117,
119, 124, 128
Approximation
approximation of the thermoelastic
system, 163
approximation of the viscoelastic sys-
tem, 1 77
.finite-dimensional approximation,
153
modal approximation, 165, 174
uniformly stable approximation,
152
Beam
Euler- Bernoulli beam equations,
112, 129, 138
laminated beam, 119
Timoshenko beam equations, 112
thermoelastic beam, 120
Damping
boundary damping, 129, 138
friction damping, 66, 72
locally distributed damping, 72
shear damping, 112, 119, 120
thermal damping, 25, 88
viscoelastic damping, 87
viscous damping, 25, 91
Dissipative operator, 1, 3, 21, 23, 30, 35,
48, 63, 67, 74, 82, 83, 84, 91, 102,
103, 114, 122, 132, 141
Elliptic boundary value problem, 8, 12
well posedness, 13
regularity, 14, 21, 51, 133
Exponential stability, exponentially
stable, 2, 4, 18, 23, 24, 32, 36, 39,
44, 45, 46, 49, 50, 54, 61, 63, 64, 67,
68, 69, 72, 73, 74, 80, 84, 90, 91, 93,
94,98, 100, 105, 112, 115, 119, 124,
128, 129, 130, 133, 139, 142, 153,
155, 164, 174, 178
205

uniformly exponential stability, 152,
154, 155, 156, 161, 162, 164,
16'7,173,174,177,183
Hille- Yosida Theorem, 3, 158
Inequality
Gargiliardo- Nirenberg inequality,
10, 11, 26, 42, 43, 58, 118, 126,
127, 147, 171
Poincare inequality, 10, 11, 26, 27,
36, 40, 41, 94, 118, 122, 125,
144
Interpolation spaces, 15
Kirchhoff plate, 25, 43, 44, 45, 46, 88
with memory, 87
Linear thermoelastic system, 18, 42
Linear thermoviscoelastic system,
91, 92, 96, 98, 99
with memory, 96
Linear viscoelastic system, 61
with memory, 61, 77
Lumer-Phillips Theorem, 3
Semigroup
206
Co-semigroup
analyticity of, see Analyticity
definition of, 1
definition of infinitesimal gen-
erator of, 1
exponential stability of, see
Exponential stability
analytic semigroup, 2, 4, 5, 8, 18,
43, 44, 45, 46, 52, 54, 61, 64,
67, 72, 86, 90, 91, 95, 112, 117,
119, 124, 128
Shear diffusion, 112, 120
Sobolev space
compactness theorem, 9
definition, 8
density theorem, 9
imbedding theorem, 9, 145, 146
trace theorem, 10, 58, 59
Solution
classical solution, 22
weak solution, 22, 71
Stabilizable 153
uniformly stabilizable, 153, 154

AB()Vl' l'HIS V()I UME
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