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\markboth{\hfil On reaction-diffusion systems \hfil EJDE--1998/24}%
{EJDE--1998/24\hfil Luiz Augusto F. de Oliveira \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~24, pp. 1--10. \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} \\
On reaction-diffusion systems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K57, 35B35, 35B65.
\hfil\break\indent
{\em Key words and phrases:} reaction-diffusion systems, analytic semigroup,
\hfil\break\indent exponential decay, global attractor.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted May 18, 1998. Published October 6, 1998.
\hfil\break\indent
Partially supported by CNPq, Proc. 300385/95-1 - Brazil
} }
\date{}
\author{Luiz Augusto F. de Oliveira}
\maketitle
\begin{abstract}
We consider reaction-diffusion systems which are strongly coupled.
we prove that they generate analytic semigroups, find a characterization for
the spectrum of the generator, and present some examples.
\end{abstract}
\newtheorem{lem}{Lemma}
\newtheorem{thm}[lem]{Theorem}
\newtheorem{cor}[lem]{Corollary}
\section{Introduction}
The investigation of qualitative properties of solutions to systems of partial
differential equations is a fascinating and challenging subject.
Even though a systematic study of this subject is yet in its infancy,
some results have been appeared in the literature since the early 1950's
with classical and pioneer work of Fichera \cite{Fichera1, Fichera2}
on elliptic systems. More recently, a great contribution to the study of
quasilinear parabolic systems was given by Amann
\cite{Amann1, Amann2, Amann3, Amann4} and references therein.
In this note we consider systems of reaction-diffusion equations of the form
\begin{equation}
u_t = D\Delta u +f(u),
\label{originalsystem}
\end{equation}
where $D$ is an $N\times N$ real matrix and $f:{\mathbb R}^N\to {\mathbb R}^N$
is a $C^2$ function.
Except for some publications on the subject, such as the searching for
traveling waves solutions and some problems in ecology and epidemic theory,
most of the authors assume that the diffusion matrix $D$ is {\it diagonal}
with positive entries, so that the coupling between the equations in
(\ref{originalsystem}) is present only on the nonlinearity of the reaction
term $f$. However, cross-diffusion phenomena are not uncommon
(see, e.g. \cite{Capasso} and references therein) and even certain mathematical
models of vibrations of plates (see the examples in Section 4) can be treated
as equations like (\ref{originalsystem}) in which $D$ is not even
diagonalizable. It is the main subject of this note to consider the case in
which the matrix $D$ is not necessarily diagonal but has eigenvalues in the
half plane $\{z\in {\bf C}:\mbox{Re}\,z >0\}$. We prove in this case that the
semigroup generated by the linear part of (\ref{originalsystem}) is an
{\it analytic} semigroup. We shall consider only the case of Dirichlet
boundary conditions, but the method should extend to some other cases without
major modifications. For the case of the entire space, see \cite{Luiz2}.
\section{The linear semigroup}
\setcounter{equation}{0}
Let $\Omega$ be a bounded region in ${\mathbb R}^n$ with smooth boundary and
let $D$ be an $N\times N$ real matrix. In this section we are concerned with
the system
\begin{equation}
u_t=D\Delta u
\label{linear}
\end{equation}
subjected to Dirichlet boundary condition $u=0$ on $\partial \Omega$.
Our main objective in this section is to give a condition on $D$ in order
that (\ref{linear}) generate an analytic semigroup in a Hilbert space, and
derive some of its properties. To put this problem into the framework of
\cite{Henry}, we need some notation.
Let $X=L^2(\Omega)^N$ be the Hilbert space of square integrable functions
$u:\Omega\to {\mathbb R}^N$ with the inner product
$$\langle u,v\rangle = \int_\Omega (u_1(x)\bar v_1(x) +...+
u_N(x)\bar v_N(x))\,dx\,.$$
Let $A:D(A)\subset X\to X$ be the linear operator given by
$D(A)=(H^2(\Omega)\cap H_0^1(\Omega))^N$ and $Au = -D\Delta u$. Our main
result is the following
\begin{thm}
Assume that all eigenvalues of $D$ have positive real part. Then $A$ is
sectorial and therefore, $-A$ is the generator of an analytic semigroup
$\{e^{-At}:t\geq 0\}$ in $X$.
\end{thm}
\noindent {\bf Proof}. Let $\theta \in (0,{\pi \over 2})$ such that
$| \arg \lambda | < \theta$ for any eigenvalue $\lambda$ of $D$. We prove that
the sector
$$S = \{z\in {\bf C}:\theta\leq |\arg z|\leq \pi, z\neq 0\}$$
is in the resolvent set of $A$ and there exists a constant $C$ such that
for any $z\in S$,
$$\| (z-A)^{-1}\| \leq {C \over {|z|}}\,.$$
Let $\{\lambda_j\}_{j=1}^\infty$ and $\{\phi_j\}_{j=1}^\infty$ be the
eigenvalues and eigenfunctions of $-\Delta$ in $\Omega$:
\[
\left\{ \begin{array}{l}
\Delta \phi_j + \lambda_j \phi_j = 0 \mbox{ in } \Omega \\
\phi_j = 0 \mbox{ on } \partial \Omega.
\end{array}
\right.
\]
We may assume that $\{\phi_j\}$ is an orthonormal basis of $L^2(\Omega)$.
For $z\in S$ and $f\in X$, let $u$ be given by
$$u = \sum_{j=1}^\infty (z-\lambda_j D)^{-1} f_j \phi_j,$$
where $f_j:=\int_\Omega f(x)\phi_j(x)\,dx$. Since $z\in S$ implies
$z \over \lambda_j$ is not an eigenvalue of $D$, the matrix $z-\lambda_j D$
is invertible and there exists a constant $C>0$ such that
$\|(z-\lambda_j D)^{-1}\| \leq {C\over {|z|}}$, for all $j\geq 1$.
It follows that the above series is convergent in $X$, so $u$ is well defined
and $\|u\| \leq {C\over {|z|}}\|f\|$.
Also,
\begin{eqnarray*}
z u+D\Delta u & = & \sum_{j=1}^\infty \left[ z(z-\lambda_j D)^{-1} f_j - \lambda_j Df_j\right]\phi_j \\
&=& \sum_{j=1}^\infty (z-\lambda_j D)^{-1} (z-\lambda_j D)f_j \phi_j \\
&=& \sum_{j=1}^\infty f_j \phi_j = f\,,
\end{eqnarray*}
so, $u=(z-A)^{-1}f$. Therefore, $z$ is in the resolvent set of $A$,
$$ \| (z-A)^{-1} \| \leq {C \over {|z|}},$$
and the proof is complete. \hfill$\diamondsuit$ \medskip
For $z=0$ the equation $-Au=f$ is equivalent to the system
\[
\left\{ \begin{array}{l}
\Delta u = D^{-1}f \mbox{ in } \Omega \\
u = 0 \mbox{ on } \partial \Omega
\end{array}
\right.
\]
Since the solution of this latter is a compact map of $f$, it follows that
$A^{-1}$ is a compact operator. Therefore, the spectrum of $A$ consists only
of eigenvalues of finite multiplicity and $\mu$ is in the spectrum of $A$ if
and only if the equation
\[
\left\{ \begin{array}{l}
D\Delta u + \mu u = 0 \mbox{ in } \Omega \\
u = 0 \mbox{ on } \partial \Omega
\end{array}
\right.
\]
has nontrivial solution. Writing $u=\sum_{j=1}^\infty u_j \phi_j$, this is
equivalent to the requirement that the equation
$$\sum_{j=1}^\infty (\mu - \lambda_j D)u_j \phi_j = 0$$
has nontrivial solution which in turn is equivalent to
$\det (\mu I - \lambda_j D)=0$ for some $j\geq 1$. Therefore $\mu$ is in the
spectrum of $A$ if and only if there exist $j\geq 1$ and $1\leq k \leq N$ such
that $\mu = \lambda_j d_k$, where $d_1, ...,d_N$ are the eigenvalues of $D$.
In short:
$\sigma(A) = \cup_{j=1}^\infty \lambda_j \sigma(D)$, so the spectrum of $A$
is a countable set located on the rays drawn from the origin to the set of the
eigenvalues of $D$.
\begin{cor}
$\{e^{-At}:t\geq 0\}$ is a compact semigroup in ${\cal L}(X)$ and there exist
constants $C\geq 1$ and $\alpha>0$ such that
$$\|e^{-At}\|_{{\cal L}(X)} \leq Ce^{-\alpha t}\,,$$
for all $t\geq 0$.
\end{cor}
Using Fourier series and the orthonormal basis $\{\phi_j\}_{j=1}^\infty$ of
$L^2(\Omega)$, it is easy to compute the semigroup generated by $-A$:
$$e^{-At}f = \sum_{j=1}^\infty e^{-\lambda_jDt}f_j \phi_j\,,$$
for all $f\in X$, where $f_j:=\int_\Omega f(x)\phi_j(x)\,dx$.
\medskip
Next we study the spaces of fractional powers $X^\alpha=D(A^\alpha)$ of $A$.
As expected, we prove that each $X^\alpha$ is a product of similar spaces.
\begin{lem}
Let $\alpha\geq 0$. Then $X^\alpha = D((-\Delta)^\alpha)^N$. In particular,
$X^{1/2} = H_0^1(\Omega)^N$.
\end{lem}
\noindent {\bf Proof}. Let $\alpha>0$ and $f\in X$. Then
\begin{eqnarray*}
A^{-\alpha} f & = & {1\over {\Gamma(\alpha)}}\int_0^\infty t^{\alpha-1} e^{-At}f\,dt \\
& = & {1\over {\Gamma(\alpha)}} \int_0^\infty t^{\alpha-1} \sum_{j=1}^\infty e^{-\lambda_jDt} f_j \phi_j \,dt \\
& = & {1\over {\Gamma(\alpha)}} \sum_{j=1}^\infty \left( \int_0^\infty t^{\alpha-1}e^{-\lambda_jDt}\,dt \right) f_j \phi_j \\
& = & {1\over {\Gamma(\alpha)}} \sum_{j=1}^\infty \lambda_j^{-\alpha}\left( \int_0^\infty s^{\alpha-1}e^{-Ds}\,ds \right) f_j \phi_j \\
& = & \sum_{j=1}^\infty \lambda_j^{-\alpha} D^{-\alpha} f_j\phi_j\,.
\end{eqnarray*}
Therefore, $g\in R(A^{-\alpha})=D(A^\alpha)$ if and only if $\displaystyle{\sum_{j=1}^\infty \lambda_j^\alpha \|D^\alpha g_j\|^2}$ is convergent, so
\begin{eqnarray*}
D(A^\alpha) & = & \{ g\in L^2(\Omega)^N: \sum_{j=1}^\infty
\lambda_j^{2\alpha}\|g_j\|^2 < \infty \} \\
& = & D((-\Delta)^\alpha)\times D((-\Delta)^\alpha)\times ... \
times D((-A)^\alpha)\,,
\end{eqnarray*}
and $A^\alpha g = \sum_{j=1}^\infty \lambda_j^\alpha D^\alpha g_j \phi_j$.
We have $X^{1/2}= H_0^1(\Omega)^N$ since $D((-\Delta)^{1/2}=H_0^1(\Omega)$.
The proof is complete.
\section{Nonlinear problems}
Given a vector field $f:{\mathbb R}^N\to {\mathbb R}^N$ satisfying certain
growth and regularity assumptions, we can define a map $f^e:X^\alpha \to X$
for some $\alpha>0$ in such a way that the resulting map $f^e$ is locally
Lipschitz continuous. In these cases, the study of well-posedness of the
initial value problem for (\ref{originalsystem}) can be put into the
framework of abstract evolution equations. We first write (\ref{originalsystem}) as the evolution equation
\begin{equation}
\dot u + Au = f^e(u)
\label{eveq}
\end{equation}
and then we look for mild solutions of (\ref{eveq}), defined as continuous solutions of the integral equation
$$u(t)=e^{-At}u(0)+\int_0^t e^{-A(t-s)}f^e(u(s))\,ds\,.$$
We refer the reader to Henry's book \cite{Henry} for general results on
existence, uniqueness and continuation.
Instead of giving general conditions on $f$ such that (\ref{originalsystem})
defines a local dynamical system on $X^\alpha$, we consider in the next
section some examples where this property can be easily verified.
\section{Examples}
\noindent {\bf Example 1.} Let $D$ and $f$ satisfy the hypotheses stated in
the introduction, and assume also that $\mbox{Re }\sigma(D)>0$. If $n$
(the dimension of the space variable) is 1, 2 or 3, then, by Theorem 1.6.1 in
\cite{Henry}, we have $X^\alpha \subset C^\nu(\Omega)$ for
${3\over 4}<\alpha<1$ and therefore the map $f^e:X^\alpha \to X$ defined by
$f^e(u)(x)=f(u(x))$, $x\in \Omega$ is well defined and it is $C^1$ with $f$
and $f'$ bounded on bounded sets. It follows from the previous results that
the system
\begin{equation}
\left\{ \begin{array}{l}
u_t = D\Delta u + f(u), x\in \Omega, t>0\\
u = 0 \mbox{ on } \partial \Omega
\end{array}
\right.
\label{diff}
\end{equation}
defines a local dynamical system on $X^\alpha$. Stability of periodic solutions
of system (\ref{diff}) was considered by Leiva \cite{Leiva} for a diagonal
matrix $D$ and Neumann boundary conditions.
\bigskip
\noindent {\bf Example 2.} Let $\beta>0$ and consider the equation
\begin{equation}
u_{tt}-2\beta\Delta u_t -f\left(\int_\Omega | \nabla u|^2\,dx \right) \Delta
u + \Delta^2 u=0, x\in \Omega, t>0
\label{Sev}
\end{equation}
with boundary conditions
$$u = \Delta u = 0 \mbox{ for } x\in \partial\Omega.$$
Equation (\ref{Sev}) arise in the mathematical study of structural damped
nonlinear vibrations of a string or a beam and was considered in
\cite{Sevicovic} and references therein.
As it is well known (see, e.g. \cite{Sevicovic}), the linear part of
(\ref{Sev}) generates an exponentially stable analytic semigroup in the space
$Y=H^2(\Omega)\cap H_0^1(\Omega)\times L^2(\Omega)$. It is our purpose here to
obtain the same result as a consequence of Theorem 2.1. To this task, we first
change variables $w=\Delta u$, $v=u_t$. In this new variables, the linear part
of equation (\ref{Sev}) becomes
\begin{equation}
\left\{ \begin{array}{l}
w_t = \Delta v \\
v_t = - \Delta w + 2\beta\Delta v,
\end{array}
\right.
\end{equation}
or, $z_t=D\Delta z$, with Dirichlet boundary conditions, where
$z= \left( \begin{array}{c} w \\ v \end{array} \right)$ and
$D= \left( \begin{array}{cc} 0 & 1 \\ -1 & 2\beta \end{array} \right) $.
Since the eigenvalues of $D$ are $d_{1,2}=\beta \pm \sqrt{\beta^2-1}$, the
spectral condition on $D$ in Theorem 2.1 is satisfied.
Since $(u,v)\mapsto (w,v):H^2(\Omega)\cap H_0^1(\Omega)\times L^2(\Omega)
\to L^2(\Omega)\times L^2(\Omega)$ is an isomorphism, it follows that the
operator $L(u,v)=(v,-\Delta^2 u+2\beta \Delta v)$,
$D(L)=\{u\in H^4(\Omega):u=\Delta u = 0 \mbox{ on }\partial \Omega\}
\times H^2(\Omega)\cap H_0^1(\Omega)$ is also the generator of an
exponentially stable analytic semigroup on
$H^2(\Omega)\cap H_0^1(\Omega)\times L^2(\Omega)$.
Now we consider the initial value problem for (\ref{Sev}). As usual, we
introduce the variable $u_t=v$ and write (\ref{Sev}) as the following system
\begin{equation}
\left\{ \begin{array}{l}
u_t = v \\
v_t = - \Delta^2 u + 2\beta\Delta v + f\left(\int_\Omega | \nabla u |^2\, dx
\right)\Delta u
\end{array}
\right.
\label{Sevsys}
\end{equation}
in the space $Y$. Letting $\hat f :H^2(\Omega)\cap H_0^1(\Omega) \to
L^2(\Omega)$ be defined by
$$\hat f (u) = f\left(\int_\Omega | \nabla u |^2\,dx \right)\Delta u$$
and $f^e(u,v)=(0,\hat f (u))$, system (\ref{Sevsys}) is in the
form (\ref{eveq}), where $A=-L$.
\begin{lem}
Assume that $f:[0,\infty)\to {\mathbb R}$ is $C^1$. Then $f^e$ is locally
Lipschitz continuous on $Y$.
\label{Lip}
\end{lem}
The proof is straightforward. If we assume that $f$ also satisfies some kind
of dissipation condition, then it follows
from arguments contained, for example, in Hale \cite{Hale} and Henry
\cite{Henry}) that (\ref{Sevsys}) defines a dynamical system in $Y$ which has
a global attractor (see \cite{Sevicovic} for more details). \medskip
\noindent {\bf Example 3.} Let $M:{\mathbb R}\to {\mathbb R}$ be a $C^1$
function, $\alpha>0$, $m\neq 0$ be real constants and consider the system
\begin{equation}
\left\{ \begin{array}{l}
u_{tt}+\alpha \Delta^2 u - m\Delta\theta=M\left(\int_\Omega |\nabla u|^2\,dx\right)\Delta u, x\in \Omega, t>0 \\
\theta_t - \Delta\theta + m\Delta u_t=0 \,,
\end{array}
\right.
\label{thermo}
\end{equation}
defined on a bounded region $\Omega\subset {\mathbb R}^n$ with boundary
conditions
$$u=\Delta u = \theta = 0, x\in \partial \Omega\,.$$
System (\ref{thermo}) comes from the thermoelasticity theory and is a model of
deflection $u$ of a plate submitted to local variation of its temperature
$\theta$. The linear version of (\ref{thermo}) was considered in
\cite{Kim, Luiz1, LiuRen}.
To put (\ref{thermo}) as an evolution equation, we first write (\ref{thermo})
as the first order system
\begin{equation}
\left\{ \begin{array}{l}
u_t = v \\
v_t = -\alpha \Delta^2 u + m\Delta\theta + M\left(\int_\Omega |\nabla u|^2\,dx\right)\Delta u, x\in \Omega, t>0 \\
\theta_t = \Delta\theta - m\Delta v .
\end{array}
\right.
\label{thermosys}
\end{equation}
Let $Y=H^2(\Omega)\cap H_0^1(\Omega) \times L^2(\Omega)\times L^2(\Omega)$ and
define $L:D(L)\subset Y \to Y$ and $f:Y\to Y$ setting
$$D(L)=\{u\in H^4(\Omega):u=\Delta u = 0 \mbox{ on }\partial \Omega \}
\times H^2(\Omega)\cap H_0^1(\Omega) \times H^2(\Omega)\cap H_0^1(\Omega)\,,$$
$$L(u,v,\theta)=(-v,\alpha\Delta^2 u - m\Delta\theta, -\Delta\theta +
m\Delta v)\,,$$
and
$$f(u,v,\theta)= (0, M\left( \int_\Omega |\nabla u|^2\,dx \right)\Delta u,0)\,.$$
Letting $z=(u,v,\theta)$, we rewrite (\ref{thermosys}) as the system
$$\dot z + L z = f(z).$$
The proof of the following lemma was pointed out in \cite{LiuRen} and can be
found in \cite{Luiz1}. Here we give another proof which is an application of
Theorem~1.
\begin{lem}
$-L$ is the generator of an analytic semigroup in $Y$ with compact resolvent
and there exist constants $C\geq 1$ and $\sigma>0$ such that
$$\| e^{-Lt} \|_{{\cal L}(Y)} \leq Ce^{-\sigma t},$$
for all $t\geq 0$.
\end{lem}
\noindent{\bf Proof}. With the change of variable $w=\Delta u$, the equation
$\dot z + Lz=0$ becomes
\begin{equation}
\left\{ \begin{array}{l}
w_t = \Delta v \\
v_t = -\alpha \Delta w + m\Delta\theta, x\in \Omega, t>0 \\
\theta_t = \Delta\theta - m\Delta v ,
\end{array}
\right.
\label{thermodif}
\end{equation}
or, equivalently, $z_t=D\Delta z$ with Dirichlet boundary conditions.
Here, $z=(w,v,\theta)$ and $D$ is the matrix
$$D=\left( \begin{array}{ccc} 0 & 1 & 0 \\ -\alpha & 0 & m \\ 0 & -m & 1
\end{array} \right) .$$
Since the eigenvalues of $D$ are the roots of the characteristic equation
$$z^3-z^2+(\alpha+m^2)z-\alpha = 0\,,$$
a simple application of the Routh-Hurwitz criterion shows that
$\mbox{Re }\lambda >0$ for any eigenvalue $\lambda$ of $D$. Therefore
Theorem~1 applies and we have the result. \hfill$\diamondsuit$
\medskip
If $M$ satisfy the same assumptions as $f$ in the previous example, then the
initial value problem for (\ref{thermosys}) is well posed and, once again
using results in \cite{Hale} and \cite{Henry}, we can prove
that (\ref{thermosys}) has a global attractor in $Y$.
\bigskip
\noindent {\bf Example 4.} The following example was considered by M. Alves
in \cite{Alves} where she proves the existence of a global attractor of the
dynamical system generated by the system
\begin{equation}
\left\{ \begin{array}{l}
u_{tt}-cv_{tt}+\alpha \Delta^2 u - \Delta u_t - M(\int_\Omega
| \nabla u|^2\,dx) \Delta u=0 \\
-c u_{tt}+\gamma v_{tt} + \delta \Delta^2 v - \beta_0\Delta v - \Delta v_t =0
\end{array}
\right.
\label{Alves}
\end{equation}
defined on a bounded smooth region $\Omega\subset {\mathbb R}^n$ together
with the boundary conditions
$$u=\Delta u = v = \Delta v = 0\,.$$
Here, $M$ satisfies the same hypotheses as in Example~3
and $\alpha$, $\delta$, $\gamma$, $c$ and $\beta_0$ are positive constants
such that $\gamma>c^2$.
In order to put this system as an evolution equation, we proceed in the usual
way of writing it as a first order system. Let $u_t=w$ and $v_t=z$, so that
(\ref{Alves}) becomes
\begin{equation}
\left\{ \begin{array}{l}
u_t=w \\
w_t = {1\over d} \left[-\alpha \Delta^2 u - \delta c \Delta^2 v +
\gamma\Delta w + c\Delta z + c\beta_0\Delta v + \gamma M\left(\int_\Omega
|\nabla u|^2\,dx\right)\Delta u \right] \\
v_t=z \\
z_t = {1\over d} \left[ -\alpha c\Delta^2 u - \delta \Delta^2 v + c \Delta w
+ \Delta z + \beta_0\Delta v + c M\left(\int_\Omega |\nabla u|^2\,dx\right)
\Delta u \right], \\
\end{array}
\right.
\label{Alvessys}
\end{equation}
where $d=\gamma - c^2$.
Let $Y=H^2(\Omega)\cap H_0^1(\Omega) \times L^2(\Omega) \times H^2(\Omega)
\cap H_0^1(\Omega) \times L^2(\Omega)$ and let $p=(u,w,v,z)$.
Then (\ref{Alvessys}) can be written as the evolution equation
$$\dot p = Lp+f(p)\,,$$
where
\begin{eqnarray*}
\lefteqn{ L(u,w,v,z)} \\
&=& (w,{1\over d}(-\alpha \gamma\Delta^2u - \delta c \Delta^2 v +
\gamma\Delta w + c\Delta z), z,
{1\over d}(-\alpha c\Delta^2u - \delta \Delta^2 v + c\Delta w + \Delta z )
\end{eqnarray*}
with domain
$D(L)=\{(u,w,v,z): u,v\in H^4(\Omega), w,z\in H^2(\Omega)\cap H_0^1(\Omega)
\mbox{ and } u=v=\Delta u=\Delta v = 0 \mbox{ on } \partial\Omega \}$
and $f:Y\to Y$ is given by
\begin{eqnarray*}
\lefteqn{f(u,w,v,z)} \\
&=&{1\over d}\left(0, c\beta_0\Delta v +
\gamma M(\int_\Omega | \nabla u|^2\,dx)\Delta u, 0, \beta_0 \Delta v +
c M(\int_\Omega | \nabla u|^2\,dx)\Delta u \right)\,.
\end{eqnarray*}
Now we consider the change of variables
$(u_1,u_2,v_1,v_2) = (\Delta u,w,\Delta v,z)$, which transform the equation
$\dot p = Lp$ into the equation
$$q_t = D\Delta q$$
with Dirichlet boundary conditions, where $q=(u_1,u_2,v_1,v_2)$ and $D$ is the
matrix
$$D={1\over d}\left( \begin{array}{cccc}0 & d & 0 & 0 \\
-\alpha\gamma & \gamma & -\delta c & c \\
0 & 0 & 0 & d \\
-\alpha c & c & -\delta & 1
\end{array} \right)$$
The characteristic equation of $D$ is
$$(\gamma - c^2)z^4 - (\gamma+1)z^3 + (\alpha\gamma+\delta+1)z^2 -
(\alpha + \delta)z + \alpha\delta = 0\,.$$
Now, Routh-Hurwitz criterion can be applied to show that $\mbox{Re }z>0$ for
any eigenvalue $z$ of $D$ and therefore,
Theorem~1 can be applied to get the following result
\begin{thm}
Let $Y=\left( H^2(\Omega)\cap H_0^1(\Omega) \times L^2(\Omega) \right)^2$
and $L$ as above. Then, $-L$ is the generator of an analytic semigroup
$\{e^{-Lt}:t\geq 0\}$ in $Y$ and there exist constants $C\geq 1$ and
$\sigma>0$ such that $\|e^{-Lt}\|_{{\cal L}(Y)} \leq Ce^{-\sigma t}$
for all $t\geq 0$. Moreover, for each $t>0$, $e^{-Lt}$ is a compact operator.
\end{thm}
It is now a simple matter to verify that $f$ satisfies the sufficient
hypotheses for (\ref{Alvessys}) to be a well posed problem in $Y$ and we
quote the reference \cite{Alves} for the proof that (\ref{Alvessys}) has a
global attractor.
\paragraph{Remark}
All the previous results obtained in this note remain true if we replace
$-\Delta$ in (\ref{originalsystem}) by a positive selfadjoint linear operator
with compact resolvent. This consideration should be useful to study abstract
versions of the above examples.
\paragraph{Acknowledgment}
\noindent The author would like to thank the anonymous referee for his
suggestions.
\begin{thebibliography}{99}
\bibitem{Alves} M.S. Alves, Atrator global para o sistema de Timoshenko,
atas do 44o. Semin\'ario Brasileiro de An\'alise, 597-602, 1996.
\bibitem{Amann1} H. Amann, Global existence for semilinear parabolic systems,
{\it J. Reine Angew. Math.}, 360, 47-83, 1985.
\bibitem{Amann2} H. Amann, Dynamic theory of quasilinear parabolic systems. III.
Global existence, {\it Math. Z.} 202, 2, 219-250, 1989 and {\it Math. Z.}
205, 2, 231, 1990.
\bibitem{Amann3} H. Amann, Highly degenerate quasilinear parabolic systems,
{\it Ann. Scuola Sup. Pisa, Cl. Sci.} (4) 18, 135-166, 1991.
\bibitem{Amann4} H. Amann, Hopf bifurcation in quasilinear reaction-diffusion
systems, {\it Delay differential equations and dynamical systems
(Claremont, CA, 1990)}, Lecture Notes in Math. 1475, Springer-Verlag, 1991.
\bibitem{Capasso} V. Capasso and A. Di Liddo, Global attractivity for
reaction-diffusion systems. The case of nondiagonal matrices, {\it J. Math.
Anal. Appl.} 177, 510-529, 1993.
\bibitem{Fichera1} G. Fichera, Analisi esistenziali per le soluzioni dei
problemi al contorno misti, relativi all'equazioni e ai sistemi di equazioni
del secondo ordine di tipo ellittico, autoaggiunti, {\it Ann. Scuola Norm.
Sup. Pisa} (3) 1 (1947) 75-100, 1949.
\bibitem{Fichera2} G. Fichera, Linear elliptic differential systems and
eigenvalue problems, Lecture Notes in Math. 8, Springer-Verlag, 1965.
\bibitem{Hale} J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS
Mathematical Surveys and Monographs 25, Providence, RI, 1988.
\bibitem{Henry} D.B. Henry, {\it Geometric Theory of Semilinear Parabolic
Equations}, Lecture Notes in Mathematics 840, Springer-Verlag, 1981.
\bibitem{Kim} J.U. Kim, On the energy decay of a linear thermoelastic bar and
plate, {\it SIAM J. Math. Anal.}, vol. 23, no. 4, 889-899, 1992.
\bibitem{Leiva} H. Leiva, Stability of a periodic solution for a system of
parabolic equations, {\it Applicable Analysis}, vol. 60, 277-300, 1996.
\bibitem{LiuRen} Z.-Y. Liu and M. Renardy, A note on the equations of a
thermoelastic plate, {\it Appl. Math. Lett}, Vol. 8, 1-6, 1995.
\bibitem{Luiz1} L.A.F. de Oliveira, Exponential decay in thermoelasticity,
{\it Comm. in Appl. Analysis}, vol.1, 113-118, 1997.
\bibitem{Luiz2} L.A.F. de Oliveira, Instability of homogeneous periodic
solutions of parabolic-delay equations, {\it J. Diff. Eq.}, Vol. 110, 42-76,
1994.
\bibitem{Sevicovic} D. Sevicovic, Existence and limiting behaviour for
damped nonlinear evolution equations with nonlocal terms,
{\it Comment. Math. Univ. Carolinae} 31, 2, 283-293, 1990.
\end{thebibliography}
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{\sc Luiz Augusto F. de Oliveira}\\
Instituto de Matem\'atica e Estat\'\i stica \\
Universidade de S\~ao Paulo - S\~ao Paulo, SP 05508-900- Brazil.\\
E-mail address: luizaug@ime.usp.br
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