\documentclass[twoside]{article}
\usepackage{amssymb} % used for R in Real numbers
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\markboth{\hfil Exponential stability of a Von Karman model \hfil EJDE--1998/07}%
{EJDE--1998/07\hfil Assia Benabdallah \& Djamel Teniou \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~07, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
   Exponential stability of a Von Karman model with thermal effects
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35M10, 73B30.
\hfil\break\indent
{\em Key words and phrases:} Thermoelastic systems, Von Karman, asymptotic
behaviour, \hfil\break\indent exponential stability, semigroup, Lyapunov function.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted December 29, 1997. Published February 27, 1998.
\hfil\break\indent Research partially supported by the French-Algerian
agreement No. 96 MDU 378.} }
\date{}
\author{Assia Benabdallah \& Djamel Teniou}

\maketitle

\begin{abstract}
A one-dimensional Von Karman model with thermal effects is studied. 
We derive the equations that constitute the mathematical model, and 
prove existence and uniqueness of a global solution. Then using 
Lyapunov functions, we show that solutions decay exponentially.
\end{abstract}

\newcommand{\grad}{\mathop{\rm grad}}
\newcommand{\diver}{\mathop{\rm div}}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}

In the last few years, the asymptotic behaviour of the coupling between elastic
and heat phenomena has been studied by several authors.
Most of their results concern the linear case, see for example  
\cite{Da, Ra, G.R.T, A-B, Liu} and references therein.
Analysis of these articles shows that the linear
thermoelastic plate models (coupling of plate and heat) and the standard
linear thermoelastic system (coupling between the wave and heat equations)
have different properties. The first model is always exponentially
stable (namely the energy approaches zero exponentially when time approaches 
infinity), while the second model has this property only in certain domains. 
The second model consists of the system 
\begin{eqnarray*}
&\partial _{tt}u-\Delta u-\beta \grad(\diver  u)+m\grad\theta 
=0\quad\mbox{in }\Omega&  \\
&\partial _{t}\theta -k\Delta \theta +m\diver \partial _{t}u
=0\quad\mbox{in }\Omega & \\
&u = \theta =0\quad\mbox{on }\partial \Omega \,,&
\end{eqnarray*}
where $\beta ,m,k$ are positive constants, $u$ is the displacement and $\theta$
the temperature. 
For this model, D.\ B. Henry, A. Perissinitto and O. Lopes \cite{H-P-L} proved that the exponential stability is equivalent to that of the
decoupled system
\begin{eqnarray*}
&\partial _{tt}u-\Delta u-\beta \grad(\diver u)+(m^2/k)\grad \Delta
^{-1}\diver \partial _{t}u =0\quad\mbox{in }\Omega& \\
&\partial _{t}\theta -k\Delta \theta +m\diver \partial _{t}u
=0\quad\mbox{in }\Omega&  \\
&u =\theta =0\quad\mbox{on }\partial \Omega\,.& 
\end{eqnarray*}
Here the operator $\grad \Delta ^{-1}\diver $ is a projection  whose 
range is the irrotational part of the velocity field. The question is whether
the control of this part of the velocity field is sufficient to ensure
uniform stability. In the one-dimensional model, in higher dimensions 
in the presence of symmetry properties, and in very special domains
(excluding convex domains) the answer is positive.
See  \cite{ H-P-L,L-Z} for the
one-dimensional case, \cite{B-S, Ra} for the presence
of symmetry, and \cite{K, Le-Zu} for special domains. 

Results  for various nonlinear models have been obtained in 
\cite{B-V-M-Z, P-Z, Ra} and their references. 
In particular, \cite{B-V-M-Z, P-Z} concern Von Karman models with thermal 
effects. In \cite{A-B}, the authors construct simple Lyapunov functions for 
a different thermoelastic plate model.
In this paper, we use these functions to prove stability results for a Von 
Karman model with thermal effects.

The plan of this paper is to derive the equations, then
prove existence and uniqueness of a global weak solution, and
finally demonstrate exponential stability of the model.

Our proof of existence and uniqueness of weak solutions is directly
inspired by the techniques used in \cite{La-Le}, where uniform stabilization
of a nonlinear beam by a nonlinear boundary feedback is obtained.

We restrict our work to the one-dimensional problem, for the following 
two reasons. The first one is the difficulty in obtaining uniqueness
for the multi-dimensional Von Karman models in the energy space we consider.
To our knowledge, there exist only partial results in this case,
\cite{Pu-Tu, Ta-Tu}. 
In \cite{Pu-Tu},  existence and uniqueness of a global strong solution 
in two dimensional bounded domains is proven, but without uniqueness for 
finite energy solutions.
In \cite{Ta-Tu}, the authors prove existence and uniqueness for finite energy 
solutions in ${\Bbb R}^2$, in rectangular domains, and outside a convex obstacle.
These difficulties also appear in the thermal case. In fact, for  (\ref{eq1})
with $\gamma >0$, it is known that the linear part has no 
regularization property. 
The second reason is the presence of planar strain in the coupling 
(see the first and third equation in (\ref{eq1})).
Recall that exponential stability for the thermoelastic system
has been proved in the one-dimensional case, and only for special domains 
in higher dimensions. 

\section{Derivation of the model}

Consider the planar motion of a beam that occupies, in the
reference position, the region
\[
U=\{(x,y,z);\quad 0\leq x\leq L,\;-1\leq y\leq 1, \;\frac{-h}{2}\leq z\leq 
\frac{h}{2}\}\,. 
\]
In this setting, $L$ is the length of the beam, and the segment \{$0\leq x\leq L,\,\,y=z=0\}$ is called the 
medium line of the beam.

The fact that the beam is stretchable implies the existence of nonlinear 
terms in the equations describing the motion.
In addition to the mechanical load, we assume that the body is subjected to
an unknown heat distribution, $\tau $, that
vanishes at the boundary of the beam.

Let the displacement be denoted by $(u,w)=((u_1,u_2),w)$, and the domain 
by 
\[
 \Omega =\{(x,y,0),\,0<x<L,\,-1<y<1\}\,. 
\]
It is known \cite{La-Le,La-Li} that, up to a normalization of both
the physical constants and $h$, the mechanical energy of the system is
given by
\begin{eqnarray*}
K(t)&=&\frac{1}{2}\{\int_\Omega \left| \partial _{t}u\right|
^2\,dxdy+\int_\Omega \left| \partial _{t}w\right| ^2\,dx\,dy+\gamma
^2\int_\Omega \left| \partial _{t}\nabla w\right| ^2\,dx\,dy\\
&&+(C(\varepsilon (u(t)+f(\nabla w(t)),\varepsilon (u(t)+f(\nabla w(t)))_0\\
&&+\int_\Omega \left| \Delta w\right| ^2\,dx-\int_\Omega \alpha (%
\widetilde{\theta }\diver u+\theta \Delta w)\,dx\,dy \,\}\,,
\end{eqnarray*}
where $C(\varepsilon (u(t)+f(\nabla w(t))$ is the
strain tensor in the plane $(x,y)$, 
$\varepsilon $ is the tensor of deformations, $\alpha $ a positive
constant, and 
$\widetilde{\theta }$ and $\theta $ are
thermal strain resultants with
$$
f(\nabla w)=\frac{1}{2}\nabla w\otimes \nabla w\,, \qquad 
\widetilde{\theta }=\frac{1}{h}\int\!\!\int_{-\frac{h}{2}}^{\frac{h}{2}}\tau 
\,dz \,,\qquad 
\theta =\frac{12}{h^{3}}\int\!\!\int_{-\frac{h}{2}}^{\frac{h}{2}}z\tau\, dz\,.
$$

We also assume that the motion occurs in the $xz$-plane, in which case
the energy becomes 
\begin{eqnarray*}
K(t)&=&\int_0^{L}\left( \left| \partial _{t}u_1(t)\right| ^2+\left|
\partial _{t}w(t)\right| ^2+\gamma ^2\left| \partial _{t}\partial
_{x}w(t)\right| ^2\right) dx \\ 
&&+\int_0^{L}[\left| \partial _{xx}w\right| ^2+| \partial_xu_1
+\frac{1}{2}(\partial_{x}w)^2| ^2]\,dx 
-\int_0^{L}\alpha (\varphi \partial_xu_1+\psi \partial _{xx}w)\,dx\,,
\end{eqnarray*}
where 
\[
\varphi =\int_{-1}^1\widetilde{\theta }\,dy\,,\qquad 
\psi =\int_{-1}^1\theta \,dy\,. 
\]
Finally, we suppose that on the boundary, the displacement is only
horizontal, which implies 
\[
w(x,.)=\partial_xw(x,.)=0\,,\quad\mbox{for }x=0,\ x=L \,.
\]
Then the dynamical variation $\delta$ satisfies  
\[
\delta K=0\,, 
\]
and we deduce the following
equations (where $u_1$ is denoted by $u$).
\[
\left\{ 
\begin{array}{l}
\partial _{tt}u-\partial_x(\partial_xu+\frac{1}{2}(\partial
_{x}w)^2)=\partial_x\varphi ,\,\,\,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial _{tt}(I-\gamma ^2\partial _{xx})w+\partial _{xxxx}w\\
-\partial_{x}[(\partial_xu+\frac{1}{2}(\partial_xw)^2)\partial_xw]=-\alpha
\partial _{xx}\psi ,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial_xu(0,t)=\partial_xu(L,t)=w(0,t)=w(L,t)=\partial
_{x}w(0,t)=\alpha \partial_xw(L,t)=0\,.
\end{array}
\right. 
\]
The two heat equations have the following form. (see \cite{La-Li}) 
\[
\left\{ 
\begin{array}{l}
\partial _{t}\varphi -\partial _{xx}\varphi =\alpha \partial_x\partial
_{t}u,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial _{t}\psi -\partial _{xx}\psi =\alpha \partial _{xx}\partial
_{t}w,\,\,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt]
\varphi (0,t)=\varphi (L,t)=\psi (0,t)=\psi (L,t)=0,\,\,\,t\in {\Bbb R}^{+}
\end{array}\right. 
\]
So that for $\alpha =1$, we obtain
\begin{equation}
\left\{ 
\begin{array}{l}
\partial _{tt}u-\partial_x(\partial_xu+\frac{1}{2}(\partial
_{x}w)^2)=\partial_x\varphi ,\,\,\,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial _{tt}(I-\gamma ^2\partial _{xx})w+\partial _{xxxx}w 
\\[5pt]
-\partial_x[(\partial_xu+\frac{1}{2}(\partial_xw)^2)\partial
_{x}w]=-\partial _{xx}\psi ,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial _{t}\varphi -\partial _{xx}\varphi =\partial_x\partial
_{t}u,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial _{t}\psi -\partial _{xx}\psi =\partial _{xx}\partial
_{t}w,\,\,\,\,(x,t)\in ]0,L[\times {\Bbb R}^{+} 
\\[5pt] 
\partial_xu(0,t)=\partial_xu(L,t)=0 
\\[5pt] 
w(0,t)=w(L,t)=\partial_xw(0,t)=\partial_xw(L,t)=0,\,\,t\in {\Bbb R}^{+} 
\\[5pt]
\varphi (0,t)=\varphi (L,t)=\psi (0,t)=\psi (L,t)=0,\,\,\,t\in {\Bbb R}^{+}
\\[5pt] 
u(x,0)=u_0(x),\,\,\partial _{t}u(x,0)=u_1(x),x\in ]0,L[ 
\\[5pt] 
w(x,0)=w(x),\,\partial _{t}w(x,0)=w_1(x),\,\ x\in ]0,L[  
\\[5pt]
\varphi (x,0)=\varphi (x),\,\,\psi (x,0)=\psi _0(x),\,\,\,\ x\ \in ]0,L[
\end{array}
\right.  \label{eq1}
\end{equation}

\section{Existence and uniqueness of a solution}

Existence follows from the argument in the paper by
J. Lagnese and G. Leugering \cite{La-Le}. Nevertheless, we provide all the
details for the coupled equation.
Let $\Omega =(0,L)$, and rewrite the system above in the form 
\begin{eqnarray}  
&CY^{\prime }=AY+F(Y)& \label{eq2}\\ 
&Y(0)=Y_0\,,&\nonumber
\end{eqnarray}
where 
$$
 Y=\left[  \begin{array}{c}
u \\ 
v \\ 
\varphi \\ 
w \\ 
z \\ 
\psi \end{array}\right]
\,,\quad 
C=\left( 
\begin{array}{cccccc}
I & 0 & 0 & 0 & 0 & 0 \\ 
0 & I & 0 & 0 & 0 & 0 \\ 
0 & 0 & I & 0 & 0 & 0 \\ 
0 & 0 & 0 & I & 0 & 0 \\ 
0 & 0 & 0 & 0 & (I-\gamma ^2\partial _{xx}) & 0 \\ 
0 & 0 & 0 & 0 & 0 & I
\end{array}\right)\,,$$
$$
A=\left( 
\begin{array}{cccccc}
0 & I & 0 & 0 & 0 & 0 \\ 
\partial _{xx} & 0 & \partial_x & 0 & 0 & 0 \\ 
0 & \partial_x & \partial _{xx} & 0 & 0 & 0 \\ 
0 & 0 & 0 & 0 & I & 0 \\ 
0 & 0 & 0 & -\partial _{xxxx} & 0 & -\partial _{xx} \\ 
0 & 0 & 0 & 0 & \partial _{xx} & \partial _{xx}
\end{array}\right)\,, $$
and
$$F(Y)=\left( 
\begin{array}{c}
0 \\ 
\frac{1}{2}\partial_x((\partial_xw)^2) \\ 
0 \\ 
0 \\ 
\partial_x[\ (\partial_xu+\frac{1}{2}\ (\partial_xw)^2)\partial
_{x}w] \\ 
0
\end{array}\right)\,. 
$$

Let the energy space be  
\[
H=\widetilde{H}^1(\Omega )\times L^2(\Omega )\times L^2(\Omega
)\times H_0^2(\Omega )\times H_0^1(\Omega )\times L^2(\Omega )\,, 
\]
where $\widetilde{H}^1(\Omega )$ is the Sobolev space $H^1(\Omega )$%
\thinspace with null average, 
\[
\widetilde{H}^1(\Omega )=\{u\in H^1(\Omega );\,\int_{\Omega
}u(x)\,dx=0\}\,. 
\]
Let $|.|$ denote the norm in $L^2(\Omega )$, and $\|.\|$ denote the
 norm in $H$,
\[
\| Y\| ^2=\left| \nabla u\right| ^2+\left|
v\right| ^2+\left| \varphi \right| ^2+\left| \Delta w\right| ^2+\left|
L_\gamma ^{1/2}z\right| ^2+\left| \psi \right| ^2 \,.
\]
with $L_\gamma =(I-\gamma ^2\Delta )$ and $\Delta $ being the
 Dirichlet-Laplace operator.

\begin{theorem}
For all $Y_0\in H$ there exists a unique weak solution $Y$ of (\ref{eq2}) 
such that 
\[
Y\in C({\Bbb R}^{+},H)\,. 
\]
\end{theorem}

We prove this theorem as follows: First it is shown that the linear part
defines a semigroup of solutions, and the nonlinear part is Lipschitz.
From these two facts, we conclude the existence of a local solution.
The proof is then completed by establishing estimates, on the local solution,
that avoid blowup in finite time; hence, ensuring global existence.

\begin{lemma}
$C^{-1}A$ is a generator of a semigroup of contractions in $H$.
\end{lemma}

\noindent{\bf Proof.} \thinspace One has
\begin{eqnarray*}
D(C^{-1}A)&=&\big\{Y\in H\,:\, \Delta u\in L^2(\Omega ),\, 
v\in \widetilde{H}^1(\Omega),\,\varphi \in H^2(\Omega )\cap H_0^1(\Omega ), \\ 
&&\quad w\in H^4(\Omega )\cap H_0^2(\Omega )\,\,z\in H_0^1(\Omega ),\,\psi \in
H^2(\Omega )\cap H_0^1(\Omega )\big\}\,.
\end{eqnarray*}
The operator $C^{-1}A$ is a generator of a semigroup of contractions, because
it is the diagonal matrix of two operators that are generators of semigroups of 
contractions. Those two operators are: the thermoelasticity 
\[
A_1=\left( 
\begin{array}{ccc}
0 & I & 0 \\ 
\partial _{xx} & 0 & \partial_x \\ 
0 & \partial_x & \partial _{xx}
\end{array}
\right)\,, 
\]
and the thermoplates
\[
A_2=\left( 
\begin{array}{ccc}
0 & I & 0 \\ 
-L_\gamma ^{-1}\partial _{xxxx} & 0 & -L_\gamma ^{-1}\partial _{xx} \\ 
0 & \partial _{xx} & \partial _{xx}
\end{array}
\right)\,. 
\]

\subsection*{Existence and uniqueness of a local solution}

The nonlinear part $C^{-1}F$ of (\ref{eq2}) can be considered as a perturbation of the
operator $C^{-1}A$. So, to prove local existence and uniqueness of a
solution, we have to verify that\thinspace \thinspace \thinspace $%
C^{-1}F\ $is locally Lipschitz continuous in $H$. 
(See Theorem 4.3.4. p. 57 in \cite{Ca-Ha})

\begin{lemma}
The function $F$ is locally Lipschitz continuous in $H$
\end{lemma}

\noindent {\bf Proof.} \thinspace For $Y\in H$, define the energy  
\[
E(Y)=\frac{1}{2}\left\{ \left| \partial_xu+\frac{1}{2}(\partial
_{x}w)^2\right| ^2+\left| v\right| ^2+\left| \varphi \right|
^2+\left| \partial _{xx}w\right| ^2+\left| L_\gamma ^{1/2} z\right|^2
+\left| \psi \right| ^2\right\}\,.
\]
For $Y_1,Y_2\in B(0,R)$, one has
\begin{eqnarray} 
\lefteqn{ \| F(Y_1)-F(Y_2)\| } &&\nonumber \\
 &=&\frac{1}{2}\left|  \partial
_{x}((\partial_xw_1)^2)- \partial_x((\partial
_{x}w_2)^2)\right|  \label{lip1} \\ 
&&+\left| L_\gamma ^{-1/2}\left( \partial_x[\ (\partial_xu_1+%
\frac{1}{2}\ (\partial_xw_1)^2)\partial_xw_1]-\partial_x[\
(\partial_xu_2+\frac{1}{2}\ (\partial_xw_2)^2)\partial
_{x}w_2]\right) \right|\,. \nonumber
\end{eqnarray}
The first term on the right is estimated as follows: 
\begin{eqnarray*}
\lefteqn{ \left| \partial_x((\partial_xw_1)^2)-\ \partial_x((\partial
_{x}w_2)^2)\right| } &&\\
&=&\left| \partial_x\ \left( (\partial
_{x}w_1-\partial_xw_2)(\partial_xw_1+\partial_xw_2)\right)
\right| \\  
&\leq& \left| \partial _{xx}(w_1-w_2)\right| \left( \| \partial
_{x}w_1\| _{L^\infty (\Omega )}+\| \partial
_{x}w_2\| _{L^\infty (\Omega )}\right) \\
&&+ \| \partial_xw_1-\partial_xw_2\|
_{L^\infty (\Omega )}\left( \left| \partial _{xx}w_1\right| +\left|
\partial _{xx}w_2\right| \right)\,.
\end{eqnarray*}
As the space has dimension one, we have the embedding 
\[
H^1(\Omega )\subset L^\infty (\Omega )\,. 
\]
Therefore, there exist positive constants denoted by $C$ such that
\begin{eqnarray*}
&\left( \| \partial_xw_1\| _{L^\infty (\Omega)}
+\| \partial_xw_2\| _{L^\infty (\Omega)}\right) \leq 
C\left( \| Y_1\| +\| Y_2\| \right)\,, &
\\
&\| \partial_xw_1-\partial_xw_2\ \|_{L^\infty (\Omega )}\leq C\| Y_1-Y_2\|\,.& 
\end{eqnarray*}
Since $\left( \| Y_1\| +\| Y_2\| \right) \leq 2R $, 
the first term on the right-hand side of (\ref{lip1}) is bounded by
\[
K(R)\| Y_1-Y_2\|\,.
\]

Let's estimate the second term in the right-hand side of (\ref{lip1}). 
\begin{eqnarray*}
\lefteqn{ \left| L_\gamma ^{-1/2}\left( \partial_x[\ (\partial_xu_1+
\frac{1}{2} (\partial_xw_1)^2)\partial_xw_1]-
\partial_x[ (\partial_xu_2+\frac{1}{2} (\partial_xw_2)^2)
\partial_xw_2]\right) \right| }&& \\
&\leq & \left| (\partial_xu_1+\frac{1}{2} (\partial_xw_1)^2)-(\partial
_{x}u_2+\frac{1}{2} (\partial_xw_2)^2)\right| \|
\partial_xw_1\| _{L^\infty (\Omega )} \\
&& +\left| (\partial_xu_2+\frac{1}{2} (\partial_xw_2)^2)\right|
\| \partial_xw_1-\partial_xw_2\|_{L^\infty (\Omega )}\,.
\end{eqnarray*}
So that
\[
\begin{array}{c}
\left| L_\gamma ^{-1/2}\left( \partial_x[(\partial_xu_1+
\frac{1}{2}\ (\partial_xw_1)^2)\partial_xw_1]-\partial_x[
(\partial_xu_2+\frac{1}{2}\ (\partial_xw_2)^2)\partial_{x}w_2]\right) \right| 
\\[5pt]
\leq  E(Y_1-Y_2)\left( \| \partial_xw_1\|
_{L^\infty (\Omega )}+\left| (\partial_xu_2+\frac{1}{2}\ (\partial
_{x}w_2)^2)\right| \right)\,,
\end{array}
\]
where once again we have used the embedding of $H^1(\Omega )$ into 
$L^\infty (\Omega )$. 

Furthermore, we have 
\begin{eqnarray} 
\lefteqn{\| Y\| ^2} &&\nonumber\\
&=&\left| \partial_xu+\frac{1}{2}
(\partial_xw)^2-\frac{1}{2}(\partial_xw)^2\right| ^2+\left|
v\right| ^2+\left| \varphi \right| ^2+\left| \partial _{xx}w\right|
^2+\left| L_\gamma ^{1/2}z\right| ^2+\left| \psi \right| ^2
\label{eq3}  \\  
&\leq& 2\left| \partial_xu+\frac{1}{2}(\partial_xw)^2\right|
^2+\left| v\right| ^2+\left| \varphi \right| ^2+\left| \partial
_{xx}w\right| ^2+\left| L_\gamma ^{1/2}z\right| ^2+\left| \psi
\right| ^2+2\left| \frac{1}{2}(\partial_xw)^2\right| ^2 
\nonumber \\ 
&\leq& 2\sqrt{E(Y(t))}\,E(Y(t))\,,\nonumber 
\end{eqnarray}
and 
\begin{equation}\label{eq4}
E(Y(t))\leq 2\| Y(t)\| ^2\| Y(t)\|\,.
\end{equation}
Since for all $Y\in B(0,R)$, there exist constants $C_1(R),C_2(R)$ such that 
\[
C_1\| Y\| ^2\leq E(Y)\leq C_2\| Y\| ^2 \,.
\]
From the previous estimates, we deduce 
\[
\| F(Y_1)-F(Y_2)\| \leq C_3\| Y_1-Y_2\|\,, 
\]
where $C_3$ is a constant depending on $R$. This proves the
existence of a local solution to (\ref{eq2}).

\subsection*{Existence of a global solution}

Existence of a global solution follows from the decay of the energy $E(Y)$. 
First, we notice that for initial data in the domain of $C^{-1}A,$ the local 
solution of (\ref{eq2}) remains in the same domain. To see this,
we have to verify only that
\[
C^{-1}F(D(C^{-1}A)\cap B(O,R))\subset D(C^{-1}A)\,, 
\]
which is obtained from calculations similar to the ones above, and by
 the embedding of $H^1(\Omega )$ into $L^\infty (\Omega )$.

For $Y_0\in D(C^{-1}A)$, the corresponding solution of (\ref{eq2}) satisfies 
\begin{equation} \label{eq5}
\frac{d}{dt}E(Y)=-\left| \partial_x\varphi \right| ^2-\left| \partial
_{x}\psi \right| ^2\,.
\end{equation}
So  that
\[
E(Y(t))\leq E(Y(0)) \,,
\]
and using  (\ref{eq4}) and (\ref{eq5}), one gets 
\[
\| Y(t)\| ^2\leq 2E(Y(0))^{3/2}\,. 
\]
Which proves boundedness of $Y$ in the $H$-norm, and therefore, 
global existence is proven. (see Theorem 4.3.4 page 57 in \cite{Ca-Ha})

\section{Exponential decay}

\begin{theorem}
For all $R>0$ and all $Y_0\in B(0,R)$ there exist positive constants
$M(R)$ and $\omega (R)$ such that solutions to (\ref{eq2}) satisfy
\[
E(Y(t))\leq M(R)e^{-\omega (R)\,t}E(Y_0) \,.
\]
\end{theorem}

\noindent{\bf Proof.} \thinspace
Our argument is based on the choice of a suitable Lyapunov function, 
\begin{eqnarray*}
\sigma _{\varepsilon }(t)&=&E(Y(t))+\varepsilon \left( \int_\Omega \psi
(-\partial _{xx})^{-1}L_{\gamma \,}zdx+\frac{1}{2}\left( \int_{\Omega
}v\,u\,dx+\frac{1}{2}\int_\Omega L_\gamma z\,w\,dx\right)\right) \\ 
 &&-\varepsilon \ \left( \alpha \int_\Omega L_\gamma ^{\ }z\,\
(h(x)\partial_xw)\,dx-\ \frac{1}{2}\int_\Omega \varphi q\,dx\right)\,,
\end{eqnarray*}
where 
\[
(-\partial _{xx})^{-1}:L^2(\Omega )\rightarrow H^2(\Omega )
\cap H_0^1(\Omega )\,, \quad h(x)=\frac{2}{L}x-1\,, 
\quad  q(x)=\int_0^x v(y,t)\,dy\,,
\]
and $\varepsilon$ and $\alpha $ are positive constants which will be
chosen later.

This Lyapunov function consists of two parts: One concerns the
thermoelastic equations and the other the thermoplates. For the
thermoelasticity, J.S. Gibson, G.Rosen and Tao \cite{G.R.T} have constructed
the same multiplier, but it does not work for the thermoplates equations. For
this system, we use the multiplier introduced by F.Ammar Khodja and
A.Benabdallah \cite{A-B} and prove that it works for the nonlinear term.

Our purpose is to show that 
\[
\frac{d}{dt}\sigma _{\varepsilon }(t)\leq -c\sigma _{\varepsilon
}(t)\,,\quad c>0 \,,
\]
from which we will deduce that
\begin{equation} \label{eq6}
\sigma _{\varepsilon }(t)\leq \sigma _{\varepsilon }(0)e^{-ct}\,.
\end{equation}
Then, noticing that there exist two positive constants $a_1,a_2$
such that 
\[
a_1E(Y(t))\leq \sigma _{\varepsilon }(t)\leq a_2E(Y(t)) 
\]
we conclude the theorem.
Inequality (\ref{eq6}) is obtained in the following 5 steps.


\paragraph{1.) Estimate for $\frac{d}{dt}\int_\Omega \psi (-\partial
_{xx})^{-1}L_\gamma z\,dx$:}
\[
\frac{d}{dt}\int_\Omega \psi (-\partial _{xx})^{-1}
L_{\gamma}zdx=\int_\Omega \psi _{t}(-\partial _{xx})^{-1}
L_{\gamma}zdx+\int_\Omega \psi (-\partial_{xx})^{-1}L_{\gamma}z_{t\,}dx\,. 
\]
But 
\[
\psi _{t}=\partial _{xx}\psi -\partial _{xx}z \,.
\]
So
\begin{eqnarray*}
\int_\Omega \psi _{t}(-\partial _{xx})^{-1}L_{\gamma \,}zdx
&=&\int_\Omega \psi L_\gamma z\,dx-\int_\Omega zL_\gamma z\,dx \\
&\leq &-| L_\gamma ^{1/2}z| ^2+| L_\gamma ^{1/2}\psi | | L_\gamma ^{1/2}z| \\
&\leq &-(1-\delta _1)| L_\gamma ^{1/2}z| ^2+\frac{1}{%
4\delta _1}| L_\gamma ^{1/2}\psi | ^2 \\
&\leq &-(1-\delta _1)| L_\gamma ^{1/2}z| ^2+\frac{c_1}{%
\delta _1}| \partial_x\psi | ^2 \,,
\end{eqnarray*}
where $\delta _1$ is an arbitrary positive constant which will be chosen
later.

On the other hand 
\[
\int_\Omega \psi (-\partial _{xx})^{-1}L_\gamma z_{t}\,dx=\int_{\Omega
}(-\partial _{xx})^{-1}\psi \,\,L_\gamma z_{t}\,dx\,, 
\]
but 
\begin{equation} \label{eq7}
L_\gamma z_{t}=-\partial _{xxxx}w+\partial_x[(\partial_xu+\frac{1}{2}%
(\partial_xw)^2)\partial_xw]-\partial _{xx}\psi
\end{equation}
and 
\begin{eqnarray*}
-\int_\Omega (-\partial _{xx})^{-1}\psi \,\,\partial _{xxxx}wdx
&=&\int_\Omega \psi \,\partial _{xx}w\,dx-\partial_x(-\partial
_{xx})^{-1}\psi (L)(\partial _{xx}w)(L) \\
&&+\partial_x(-\partial _{xx})^{-1}\psi (0)(\partial _{xx}w)(0)\,.
\end{eqnarray*}
So
\begin{eqnarray*}
-\int_\Omega (-\partial _{xx})^{-1}\psi \,\,\partial _{xxxx}w\,dx 
&\leq &| \psi | | \partial _{xx}w| +| \partial_x(-\partial
_{xx})^{-1}\psi (L)| | \partial _{xx}w(L)| \\
&& +| \partial_{x}(-\partial _{xx})^{-1}\psi (0)|\, | \partial _{xx}w(0)|\,.
\end{eqnarray*}
But  
\begin{eqnarray*}
\lefteqn{ \left| \int_\Omega \partial_x(-\partial _{xx})^{-1}\psi [(\partial
_{x}u+\frac{1}{2}(\partial_xw)^2)\partial_xw]\,dx\right| }   \\
&\leq &\| \partial_xw\| _{L^\infty (\Omega )}|
\partial_xu+\frac{1}{2}(\partial_xw)^2| | \partial
_{x}(-\partial _{xx})^{-1}\psi |   \\
&\leq &\delta _2R^2| \partial_xu+\frac{1}{2}(\partial
_{x}w)^2| ^2+\frac{1}{4\delta _2}| \partial_x(-\partial
_{xx})^{-1}\psi | ^2\,.
\end{eqnarray*}
So, it follows 
\begin{eqnarray*}
\frac{d}{dt}\int_\Omega \psi (-\partial _{xx})^{-1}L_\gamma z\,dx 
&\leq&-(1-\delta _1)| L_\gamma ^{1/2}z| ^2+(\frac{c_1}{%
\delta _1}+\frac{c_2}{\delta _2})| \partial_x\psi | ^2 \\
&&+\delta _2R^2 | \partial_xu+\frac{1}{2}(\partial_xw)^2|
^2\ +| \psi | | \partial _{xx}w| \\
&&+ | \psi | | \partial _{xx}w| +| \partial_x(-\partial
_{xx})^{-1}\psi (L)| | \partial _{xx}w(L)| \\
&& + |\partial_x(-\partial _{xx})^{-1}\psi (0)| | \partial_{xx}w(0)| +
| \psi | ^2\,.
\end{eqnarray*}


\paragraph{2.) Estimate for $\frac{d}{dt}\int_\Omega vu\,dx$:}
\[
\frac{d}{dt}\int_\Omega v\,u\,dx\ =\left| v\right| ^2+\int_{\Omega
}v_{t}u\,dx\ 
\]
and 
\begin{eqnarray*}
\int_\Omega v_{t}u\,dx &=&\int_\Omega \partial_x(\partial_xu+\frac{%
1}{2}(\partial_xw)^2)u\,dx-\int_\Omega \partial_x\varphi
\,u\,,dx \\
&\leq &-\int_\Omega \ (\partial_xu+\frac{1}{2}(\partial
_{x}w)^2)\partial_xu\,dx+\left| \partial_x\varphi \right| \left|
u\right|\,.
\end{eqnarray*}
Here we have used the boundary condition on $u$, 
$\partial_xu(L)=\partial_{x}u(0)=0$.
So 
\[
\frac{d}{dt}\int_\Omega vu\,dx\leq \left| v\right| ^2-\int_\Omega \
(\partial_xu+\frac{1}{2}(\partial_xw)^2)\partial_xu\,dx+\left|
\partial_x\varphi \right| \left| u\right|\,. 
\]

\paragraph{3.) Estimate for $\frac{d}{dt}\int_\Omega L_\gamma w\,dx $:}
One has 
\[
\frac{d}{dt}\int_\Omega L_\gamma z\,w\,dx=\int_\Omega L_{\gamma
}z\,z\,dx+\int_\Omega L_\gamma z_{t}w=\left| L_\gamma ^{1/2}z\right|^2
+\int_\Omega L_\gamma z_{t}w\,dx\,.
\]
Using (\ref{eq7}) we obtain 
$$
\int_\Omega L_\gamma z_{t}wdx =-\left| \partial _{xx}w\right|
^2-\int_\Omega (\partial_xu+\frac{1}{2}(\partial_xw)^2)(\partial
_{x}w)^2\,dx 
+\int_\Omega \partial_x\psi \partial_xw\,dx\,.
$$
So 
\begin{eqnarray*}
\frac{d}{dt}\int_\Omega L_\gamma z\,w\,dx- &=&-\left| \partial
_{xx}w\right| ^2-\int_\Omega (\partial_xu+\frac{1}{2}(\partial
_{x}w)^2)(\partial_xw)^2\,dx \\
&&+\int_\Omega \partial_x\psi \partial_xw\,dx+\left| 
L_\gamma ^{1/2}z\right| ^2\,.
\end{eqnarray*}

\paragraph{4.) Estimate for $\frac{d}{dt}\int_\Omega 
L_{\gamma}zh(x)\partial_xw\,dx$:}
\[
-\frac{d}{dt}\int_\Omega L_\gamma z\,(h(x)\partial
_{x}w)\,dx=-\int_\Omega L_\gamma z_{t}h(x)\partial
_{x}w\,dx-\int_\Omega L_\gamma zh(x)\partial_xz\,dx\,. 
\]
An integration by parts of the second term of the right member of the
previous equality gives 
\[
\ \int_\Omega L_\gamma z\,(h(x)\partial_xz)\,dx\leq c\left| L_{\gamma
}^{1/2}z\right| ^2\,.
\]
Furthermore, (\ref{eq7}) implies
\begin{eqnarray*}
\int_\Omega L_\gamma z_{t}h(x)\partial_xw\,dx &\leq &c\left|
(\partial_xu+\frac{1}{2}(\partial_xw)^2)\right| \left| \partial
_{xx}w\,\ \right| \| \partial_xw\,\| \ _{L^\infty (\Omega
)} \\
&&+(\delta _{3}-\frac{3}{L})\left| \partial _{xx}w\,\ \right| ^2+\
c(\delta _{3})\left| \partial_x\psi \right| \ ^2 \\
&&+\frac{1}{2}\left( \left| \partial _{xx}w\,\ (0)\right| ^2+\left|
\partial _{xx}w\,\ (L)\right| ^2\right)\,.
\end{eqnarray*}

\paragraph{5.) Estimate for $\frac{d}{dt}\int_\Omega \varphi\,q\,dx$:}
\begin{eqnarray*}
\frac{d}{dt}\int_\Omega \varphi q\,dx&=&\left| v\right| ^2-\int_{\Omega
}\partial_x\varphi v\,dx+\int_\Omega \varphi q_{t}\,dx \\  
&\leq& \left| v\right| ^2+\left| \partial_x\varphi \right| \left|
v\right| +\int_\Omega \varphi q_{t}\,dx\,.
\end{eqnarray*}
To simplify notation, let  
\[
k(x,t)=\int_0^x \varphi (y,t)\,dy\,. 
\]
So  that
\[
\int_\Omega \varphi q_{t}\,dx=-\int_\Omega k\partial
_{x}q_{t}\,dx=-\int_\Omega kv_{t}\,dx\,. 
\]
But 
\begin{eqnarray*}
\int_\Omega kv_{t}dx &=&\int_\Omega \partial_x(\partial_xu+\frac{1%
}{2}(\partial_xw)^2)k\,dx+\int_\Omega \partial_x\varphi k\,dx \\
&=&-\int_\Omega \ (\partial_xu+\frac{1}{2}(\partial_xw)^2)\partial
_{x}k\,dx-\int_\Omega \varphi \partial_xk\,dx\,.
\end{eqnarray*}
So 
\begin{eqnarray*}
-\frac{d}{dt}\int_\Omega \varphi q\,dx &\leq &-(1-2\delta _{4})\left|
v\right| ^2\ +\delta _{5}\left| (\partial_xu+\frac{1}{2}(\partial
_{x}w)^2)\right| ^2 \\
&&+(\frac{1}{4\delta _{4}}+\frac{1}{4\delta _{5}}+c_0)\left| \partial
_{x}\varphi \right| ^2 \,.
\end{eqnarray*}

\subsection*{Conclusion}

Gathering all the above calculations and using Cauchy-Schwarz inequality, 
we obtain
\begin{eqnarray*}
\frac{d}{dt}\sigma _{\varepsilon }(t) &\leq &-\ [1-\varepsilon (c(\delta
_1)+c(\delta _2)+c_1+c(\delta _{3}))]\left| \partial_x\psi \right|
^2 \\
&&-[1-\varepsilon (c(\ \delta _{4})+c(\delta _{5})+c_2]\left| \partial
_{x}\varphi \right| ^2 \\
&&-\varepsilon [(\frac{3}{4}-\delta _1)\left| L_\gamma ^{\frac{1}{2}%
}z\right| ^2+(\frac{1}{2}-\delta _{4})\left| v\right| ^2] \\
&&-\varepsilon [(1-\delta _2R^2-\frac{4}{L^2}\alpha )\left| \partial
_{x}u+\frac{1}{2}(\partial_xw)^2\right| ^2] \\
&&-\varepsilon [(\frac{1}{4}-((\frac{3}{L}+R^2)\alpha -\delta _{3})\left|
\partial _{xx}w\right| ^2] \\
&&+\varepsilon \frac{\alpha }{2}[\ (\left| \partial _{xx}w(0)\right|
^2+\left| \partial _{xx}w(L)\right| ^2)] \\
&&\frac{\varepsilon }{2\alpha }[\left| \partial_x(\partial
_{xx})^{-1}\psi (0)\right| ^2+\left| \partial_x(\partial
_{xx})^{-1}\psi (L)\right| ^2] \\
&&+\varepsilon \delta _{6}\left| u\right| ^2+\frac{\varepsilon }{4\delta
_{6}}\left| \partial_x\varphi \right| ^2 \\
&&-\frac{\varepsilon \alpha }{2\ }(\left| \partial _{xx}w(0)\right|
^2+\left| \partial _{xx}w(L)\right| ^2)\,.
\end{eqnarray*}
It remains  to choose, in the above steps, the constants $\delta _i$,
$\alpha$, $\varepsilon$ sufficiently small to make negative the constants 
before the energy. This is always possible, and then we obtain
\[
\frac{d}{dt}\sigma _{\varepsilon }(t)\leq -cE(Y(t)). 
\]
This gives (\ref{eq6}) and the theorem is proved. Notice that the previous
constant $c$ depends explicitly on $R$.

\paragraph{Acknowledgment} The authors are indebted to the referee for all 
the comments, remarks, and bibliographical suggestions.

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\smallskip

{\sc Assia Benabdallah}\\ 
Laboratoire de Calcul Scientifique, 
16 route de Gray, 25000, Besan\c{c}on, France.\\
e-mail address: assia@math.univ-fcomte.fr\smallskip

{\sc Djamel Teniou}\\
Institut de Math\'{e}matiques, 
Universit\'{e} Houari Boumediene, B.P.32.\\
El Alia 16111. Alg\'{e}rie

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