\documentclass[twoside]{article}
\usepackage{amssymb} % font used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil On a nonlinear coupled system with internal damping
\hfil EJDE--2000/64}
{EJDE--2000/64\hfil R. F. Apolaya,  H. R. Clark, \&  A. J. Feitosa \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~64, pp.~1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  On a nonlinear coupled system \\ with internal damping
% 
\thanks{ {\em Mathematics Subject Classifications:} 35L70, 35B40.
\hfil\break\indent
{\em Key words:} Nonlinear system, internal damping, asymptotic behavior.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted July 24, 2000. Published October 18, 2000.} }
\date{}
%
\author{R. F. Apolaya, H. R. Clark, \&  A. J. Feitosa }
\maketitle

\begin{abstract}
 The purpose of this paper is to establish existence, uniqueness, and 
 asymptotic behavior of solutions of a non-linear coupled system with variable 
 coefficients, with coupling of a  non-linear wave equation,  and a linear 
 heat equation in a smooth-bounded-open domain of ${\mathbb R}^n$. 
\end{abstract} 

%
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

The nonlinear wave equation with thermo-elastic coupling is given by the 
system of equations
\begin{eqnarray} 
& u''(x,t)-\mu (t)\Delta u(x,t)+
\displaystyle \sum_{i=1}^n{\frac{\partial \theta}{\partial x_i}(x,t)}
+\gamma |u(x,t)|^{\rho}u(x,t)=0 &\label{eq:1.1}\\
&\theta'(x,t)-\Delta \theta(x,t) +
\displaystyle \sum_{i=1}^n{\frac{\partial u'}{\partial x_i}(x,t)}=0\quad \mbox{in}\quad Q,
\label{eq:1.2}
\end{eqnarray} 
with initial and boundary conditions
\begin{eqnarray} 
&u(x,0)=u^0(x),\quad u'(x,0)=u^1(x),\quad \theta(x,0)=\theta^0(x)\quad \mbox{in}\quad \Omega,
\label{eq:1.3}&\\[10pt]
&u(x,t)=0,\quad \theta(x,t)=0\quad \mbox{on}\quad \Gamma\times]0,\infty[,
\label{eq:1.4}
\end{eqnarray} 
where $u$ is  displacement, $\theta$ is absolute temperature, $\Delta$ 
denotes the Laplace operator, $\mu$ is a positive real function of $t$, 
$\gamma$ and  $\rho$ are positive real numbers, the temporal partial 
derivative is represented by $'$, $\Omega$ is a smooth-bounded-open set in  
${\mathbb R}^n$ with $C^2$ boundary $\Gamma$, and $Q=\Omega \times ]0,\infty[$.


The non-linearity $|v|^\rho v$ usually appears in relativistic quantum 
mechanic 
(see Segal \cite{Segal} or  Schiff \cite{Schiff}), and has been considered by 
various authors for hyperbolic, parabolic and elliptic equations.  
Lions \cite{Lions} studied the wave equation with the same non-linearity, 
\,i.e.,\, $|v|^\rho v$, in a smooth-bounded-open domain $\Omega$ of 
${\mathbb R}^n$  with $n \in {\mathbb N}$, and proved existence and uniqueness 
of solution using both Faedo-Galerkin's  and Compactness' methods. 

Clark at al \cite{Clark} investigated the system (\ref{eq:1.1})-(\ref{eq:1.2}) 
with $\gamma$ equal to zero and feedback-homogeneous conditions over a part of 
the boundary $\Gamma$. They established global existence of strong and weak 
solutions by Faedo-Galerkin's method using a particular basis of the space 
$H^1_0(\Omega)\cap H^2(\Omega)$ intruduced by  Medeiros \& Milla Miranda 
\cite{LAM-MMM-1} and  the exponential stability of total energy associated 
to the weak solution using Komornik-Zuazua's method \cite{kom-zua}. 

Based in the theory developed in the papers of Clark at al \cite{Clark} and 
Lions \cite{Lions} (Theorems 1.1, 1.2 and 1.3),  we will prove that the system 
(\ref{eq:1.1})-(\ref{eq:1.4}) has a unique global strong solution, a unique 
global weak solution, and the total energy associated to these solutions is 
asymptotically stable.
 
The outline of this article is as follows. In Section 2, the basic notations 
are laid out and global existence of strong and weak solutions are issued, 
whilst exponential decay is aired in Section 3. 

\section{Notation and strong-weak solutions}
\setcounter{equation}{0}

In the sequel $L^p(\Omega)$, $1 \leq p < \infty $, will denote the collection 
of L-functions which are $p$th-integrable over $\Omega$. For 
$m \in {\mathbb N}$, the space $H^m(\Omega)$ is the Sobolev class of the 
functions of the spatial variable $x$ which along with their first $m$ 
derivatives belong to $L^2(\Omega)$ (see, for exemple,  
Sobolev \cite{Sobolev} or Medeiros \& Milla Miranda  \cite{LAM-MMM})  and the 
closure in $H^m(\Omega)$ of the space ${\cal D}(\Omega)$ of test functions on 
$\Omega$ is denoted by $H^m_0(\Omega)$. The inner product and norm of 
$H^m_0(\Omega)$ and $L^2(\Omega)$ are represented by $((\cdot,\cdot))$, 
$\|\cdot\|$ and $(\cdot,\cdot)$, $|\cdot|$, respectively. 
By $W^{m,p}(0,\infty)$ we denote the Sobolev space of the functions of the 
temporal variable $t$.

Let $X$ be a Banach space, $T$ a positive real number or 
$T=+\infty$ and 
\linebreak
$1\leq p \leq \infty$, denotes by $L^p(0,T; X)$ 
the Banach space of all measurable functions $u:]0,T[ 
\longrightarrow X$, such that $t \mapsto \|u(t)\|_X$ is in 
$L^p(0,T)$, with norm 
\[
\|v\|_p=\Big(\int_0^T\|v(t)\|^p_Xdt\Big)^{1/p},\quad 
\mbox{if}\quad 1\leq p < \infty, 
\]
and if $p=\infty,$ then
\[
\|v\|_{\infty}= \mathop{\rm ess}\sup_{\hspace*{-10pt} t \in [0,T]} 
\|v(t)\|_X. 
\]

To obtain the existence and uniqueness of global solution of the 
mixed problem (\ref{eq:1.1})-(\ref{eq:1.4}) we suppose the additional 
hypotheses about $\mu$ and $\rho$:
\begin{eqnarray}
&\mu \in W^{1,1}(0,\infty),\quad \mu(t) \geq \mu_0>0\quad \mbox{ and }\quad  
\mu'(t)\leq 0.& \label{eq:2.1} \\
&\rho \mbox{ is any finite-positive real number if }\; n=1, 2 \mbox{ and } &
\nonumber \\
&\rho \leq \frac{2}{n-2} 
\quad \mbox{ if }\quad  n \geq 3 . &
\label{eq:2.2}
\end{eqnarray}
Note, for later use, that if $\rho$ satisfies (\ref{eq:2.2}) then  
$H^1_0(\Omega)$ has continuous injection into $L^{\rho+2}(\Omega)$.

The constant $\gamma$ is considered only to get the asymptotic behavior of the total energy
of solutions. Thus, it imposes the hypothesis (\ref{eq:3.1}) fixed in the section 3.     
 
Now we can present the existence results of the initial- and boundary-value problem (\ref{eq:1.1})-(\ref{eq:1.4}). Thus, the global strong solution is guaranteed by

%Teorema 2.1

\begin{theorem}{\bf (Strong solution)}
Let $\mu$ be the fuction defined by (\ref{eq:2.1}). Let $\rho$ be the given real-positive constant satisfying (\ref{eq:2.2}) and suppose also $\gamma$ a real-positive number. If  
\[
u_0, \;\theta_0\; \in\; H^1_0(\Omega) \cap H^2(\Omega)
\quad \mbox{and}\quad 
u_1 \;\in\; H^1_0(\Omega),
\]
then the system (\ref{eq:1.1})-(\ref{eq:1.4}) has a unique solution $\{u,\theta\}$ such that
\begin{equation}
\begin{array}{c}
u \;\mbox{ and }\; \theta \in L^{\infty}\left(0,\infty; H^1_0(\Omega)
\cap H^2(\Omega)\right), 
\end{array}
\label{eq:2.3}
\end{equation}
\begin{equation}
\begin{array}{c}
u' \in L^{\infty}\left(0,\infty; H^1_0(\Omega)\right) \;\mbox{ and }\;\; \theta' \in L^2\left(0,\infty; H^1_0(\Omega)\right),
\end{array}
\label{eq:2.4}
\end{equation}
\begin{equation}
\begin{array}{c}
u''\in L^{\infty}\left(0,\infty;L^2(\Omega)\right),
\end{array}
\label{eq:2.5}
\end{equation} 
and  the equations  (\ref{eq:1.1}) and (\ref{eq:1.2})  are given in the sense of 
$L^{\infty}\left(0,\infty; L^2(\Omega)\right)$. 
\end{theorem}
\paragraph{Proof.}{\it Existence.}  To show the global existence of 
solution we will use both the Faedo-Galerkin's and Compactness' methods. 
We consider $(w_j)_{j\in {\mathbb N}}$ an orthonormal basis  of 
$ H^1_0 (\Omega)\cap H^2 (\Omega)$,  and denote by 
$V_m=[w_1,w_2,w_3, \cdots ,w_m]$ the subspace of 
$H^1_0 (\Omega)\cap H^2(\Omega)$ spanned by the $m$ first vectors of 
$(w_j)_{j\in {\mathbb N}}$. In these conditions, the approximated system 
associated to system (\ref{eq:1.1}) (\ref{eq:1.2}) is given by
\begin{eqnarray}
&
\hspace{-30pt}
\displaystyle 
(u''_m(t), v) + \mu(t)((u_m(t),v))+\sum_{i=1}^n\left(\frac{\partial \theta_m}{\partial x_i}(t),v\right)+\gamma \left(|u_m(t)|^{\rho}u_m(t), v\right)= 0, 
\label{eq:2.6}&\\
&
\displaystyle 
(\theta'_m(t),w) + ((\theta_m(t),w))+\sum_{i=1}^n\left(\frac{\partial u'_m}{\partial x_i}(t),w\right) = 0, 
\label{eq:2.7}
\end{eqnarray}
where $v $ and  $w$ belong to $V_m$.

Let $u_m(0)=u_{0m}$, $u'_m(0)=u_{1m}$, and  $\theta_m (0)=\theta_{0m}$ be. 
Hence,  $u_{0m}$, $u_{1m}$, and  $\theta_{0m}$ belong to $V_m$, and satisfy
\begin{eqnarray}
&u_{0m}\longrightarrow u_0
\quad \mbox{strongly}\quad \mbox{in}\quad  
H^1_0(\Omega)\cap H^2(\Omega),\hspace{2pt}
&\label{eq:2.8}\\
&\theta_{0m} \longrightarrow \theta_0 
\quad \mbox{strongly}\quad \mbox{in}\quad 
H^1_0(\Omega)\cap H^2(\Omega),
&\label{eq:2.9}\\
&\ u_{1m}\longrightarrow u_1 
\quad \mbox{strongly}\quad \mbox{in}\quad 
H^1_0(\Omega).\hspace{44pt}
&\label{eq:2.10}
\end{eqnarray}
Under these conditions, the system (\ref{eq:2.6}) (\ref{eq:2.7}) 
has a local solution $\left\{u_{ m}(t),\theta_{ m}(t)\right\}$ 
over the interval $[0,t_m[$. This interval will be extended to any 
interval $[0, \infty[$ thanks to the first estimate  below. 

\paragraph {Estimate I.}  Substituting $v$ by $2u_{m}'(t)$ and $w$ by $2\theta_{m}(t)$ in (\ref{eq:2.6})  and  (\ref{eq:2.7}) respectively, using Green's formula in the term 
$\sum_{i=1}^n\left(\frac{\partial u'_m}{\partial x_i}(t),\theta_{m}(t)\right) $
 and integrating over [0,t[, $0\leq t\leq t_m$, we get 
\begin{eqnarray*}
E_1(t)+2\int_0^t\|\theta_{ m}(s)\|^2ds
&=&\int_0^t\mu ' (s)\|u_{ m}(s)\|^2ds +|u_{1m}|^2+|\theta_{0m}|^2 \\
&&+\mu(0)\|u_{0m}\|^2+
\frac{2\gamma}{\rho+2}\int_{\Omega}|u_{0m}|^{\rho+2}dx,
\end{eqnarray*}
where
\[
E_1(t)=|u'_{m}(t)|^2+|\theta_{m}(t)|^2+\mu(t)\|u_{m}(t)\|^2+
\frac{2\gamma}{\rho+2}\int_{\Omega}|u_{m}(t)|^{\rho+2}dx.
\]

From (\ref{eq:2.8})-(\ref{eq:2.10}) and (\ref{eq:2.1})  there is a positive 
constant $k$, independent of $m$ such that
\begin{equation}
E_1(t)+2\int_0^t\|\theta_{ m}(s)\|^2ds \leq k\quad \mbox{for all}\quad t \geq 0.
\label{eq:2.11}
\end{equation}
Hence, we can extent the approximate solutions 
$\left\{u_m(t),\, \theta_{ m}(t)\right\}$  on the whole interval 
$[0, \infty[$ independent of $m$  and  $t$. 

\paragraph{Estimate II.} 
Differentiating the equations (\ref{eq:2.6}) and (\ref{eq:2.7}) with respect 
to $t$, replacing   $v$ by $2u_{m}''(t)$ and $w$ by $2\theta'_{m}(t)$, we have 
according to (\ref{eq:2.1}) that
\begin{eqnarray}
\lefteqn{ \frac{d}{dt}\left\{|u''_{m}(t)|^2+\mu(t)\|u'_{m}(t)\|^2+
|\theta^{'}_{m}(t)|^2\right \}+2\|\theta^{'}_{ m}(t)\|^2 }\label{eq:2.12}\\
&\leq&
- 2\gamma(\rho+1)\int_{\Omega}|u_{m}(t)|^{\rho}u'_m(t)u''_m(t)dx
-2\mu'(t)((u_{m}(t), u''_m(t))). \nonumber
\end{eqnarray}
Now, let us make an analysis about the term $-2\mu'(t)((u_{m}(t), u''_m(t)))$.  In the equation (\ref{eq:2.6}) replacing $v$ by 
$2u_m''(t)$ we can write
\begin{eqnarray*}
-2\mu(t)((u_{m}(t), u''_m(t))) &=&2|u''_m(t)|^2
+2\sum_{i=1}^n\int_{\Omega}\frac{\partial \theta _m}{\partial x_i}(t)u''_m(t)dx
\\
&&+2\gamma \int_{\Omega}|u_{m}(t)|^\rho u_{m}(t) u''_m(t)dx\,.
\end{eqnarray*}
Multiplying both sides by $\frac{\mu'(t)}{\mu(t)}$ and using (\ref{eq:2.1}) we obtain
\begin{eqnarray}
-2\mu'(t)((u_{m}(t), u''_m(t))) &\leq&
2\frac{\mu'(t)}{\mu(t)}\sum_{i=1}^n\int_{\Omega}\frac{\partial \theta _m}{\partial x_i}(t)u''_m(t)dx
\label{eq:2.13}\\
&&+2\gamma \frac{\mu'(t)}{\mu(t)}\int_{\Omega}|u_{m}(t)|^\rho u_{m}(t)u''_m(t)dx.
\nonumber\end{eqnarray}
Substituting (\ref{eq:2.13})  in (\ref{eq:2.12}) it follows 
\begin{eqnarray*}
 \frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2 &\leq&
- 2\gamma(\rho+1)\int_{\Omega}|u_{m}(t)|^\rho u'_m(t)u''_m(t)dx\\
&&+\frac{2 \mu'(t)}{\mu(t)}\sum_{i=1}^n\int_{\Omega}
\frac{\partial \theta _m}{\partial x_i}(t)u''_m(t)dx  \\
&&+ \frac{2\gamma \mu'(t)}{\mu(t)}\int_{\Omega}|u_{m}(t)|^\rho u_{m}(t)u''_m(t)dx,
\end{eqnarray*}
where
\[
E_2(t)=|u''_{m}(t)|^2+\mu(t)\|u'_{m}(t)\|^2+
|\theta^{'}_{m}(t)|^2.
\]
Hence, we can write 
\begin{eqnarray}
 \frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2 &\leq&
2\gamma(\rho+1)\int_{\Omega}|u_{m}(t)|^\rho |u'_m(t)||u''_m(t)|dx \nonumber\\
&&+\frac{2|\mu'(t)|}{\mu(t)}\sum_{i=1}^n\int_{\Omega}\left|\frac{\partial 
\theta_m}{\partial x_i}(t)\right||u''_m(t)|dx \label{eq:2.14}\\
&&+\frac{2\gamma |\mu'(t)|}{\mu(t)}
\int_{\Omega}|u_{m}(t)|^{\rho+1}|u''_m(t)|dx. \nonumber
\end{eqnarray}
Now, we are going to make an analysis on the second member of (\ref{eq:2.14}) 
in order to apply the Gronwall's inequality.  By using the H\"{o}lder's 
generalized inequality for the case
\(\frac{1}{q}+\frac{1}{n}+\frac{1}{2}= 1\) 
we get  from  hypothesis (\ref{eq:2.2}) the following continuous injections: 
\begin{itemize}
\item
If $n=1, 2$ then $H^1_0(\Omega)\hookrightarrow L^q(\Omega)$ for all  $q >0$.

\item
If $n \geq 3$ then  $\rho \leq 
\frac{2}{n-2}.$ Thus,  
$H^1_0(\Omega)\hookrightarrow L^q(\Omega)$ where $q=\frac{2n}{n-2}$.
Consequently, as $q\geq \rho+2$ then 
$H^1_0(\Omega)\hookrightarrow L^{\rho+2}(\Omega)$,
and since $n\rho \leq q$ 
then it also follows that $H^1_0(\Omega)\hookrightarrow L^{n\rho}(\Omega)$.  
\end{itemize}
\noindent
Using the two items above and the estimate I in the first term of right-hand side of (\ref{eq:2.14}) yields
\begin{eqnarray}
\lefteqn{ \int_{\Omega}|u_{m}(t)|^{\rho}|u'_m(t)||u''_m(t)|dx }\nonumber\\
&\leq& 
\left[\int_{\Omega}|u_{m}(t)|^{\rho n}dx\right]^
{1/n}\left[\int_{\Omega}|u'_{m}(t)|^qdx\right]^{1/q}
\left[\int_{\Omega}|u''_m(t)|^2dx\right]^{1/2} \nonumber \\
&=&
\|u_{m}(t)\|^{\rho}_{L^{\rho n}(\Omega)}\|u_{m}'(t)\|_{L^{q}(\Omega)}
|u_{m}''(t)|_{L^{2}(\Omega)} \label{eq:2.15}\\
&\leq&
c_0\|u_{m}(t)\|^{\rho}_{H^{1}_{0}(\Omega)}\|u_{m}'(t)\|_{H^{1}_{0}(\Omega)}
|u_{m}''(t)| \nonumber \\
&\leq& 
c_1\|u_{m}'(t)\|\hspace{.5pt}|u_{m}''(t)|. \nonumber
\end{eqnarray}
Substituting (\ref{eq:2.15}) and applying Cauchy-Schwarz's inequality in the 
last two integrals in the right-hand side of (\ref{eq:2.14}) we have
\begin{eqnarray*}
\frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2 
&\leq& 2c_1\gamma(\rho+1)\|u_{m}'(t)\||u_{m}''(t)| \\
&&+\frac{2\sqrt n}{\mu(t)}|\mu'(t)|\|\theta_m(t)\||u_{m}''(t)| \\
&&+\frac{2\gamma}{\mu(t)}|\mu'(t)|\|u_{m}(t)\|^{\rho+1}|u_{m}''(t)|\,.
\end{eqnarray*}
Hence, (\ref{eq:2.1}), and usual inequalities yields
\begin{eqnarray*}
\lefteqn{ \frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2 }\\
&\leq& \frac{c_1\gamma (\rho+1)}{\mu_0}\mu(t)\|u_{m}'(t)\|^2
+\frac{c_1\gamma (\rho+1)}{\mu_0}\mu(t)|u_{m}''(t)|^2
+\frac{\sqrt n}{\mu_0}|\mu'(t)|\|\theta_m(t)\|^2 \\
&&+\frac{\sqrt n}{\mu_0}|\mu'(t)||u_{m}''(t)|^2
+\frac{\gamma}{\mu_0}|\mu'(t)|\|u_{m}(t)\|^{2(\rho+1)}
+\frac{\gamma}{\mu_0}|\mu'(t)||u''_{m}(t)|^2.
\end{eqnarray*}
Let $\nu(t)$ be the function given by
\[
\nu(t)=\frac{1}{\mu_0}\left\{c_1\gamma(\rho+1)\mu(t)+(\sqrt{n}+\gamma)|\mu'(t)|\right\},
\]
then it implies
\begin{eqnarray*}
\frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2
&\leq&\nu(t)\left\{\|u_{m}'(t)\|^2+|u_{m}''(t)|^2\right\}\\
&&+\frac{\sqrt n}{\mu_0}|\mu'(t)|\|\theta _m(t)\|^2+
\frac{\gamma}{\mu_0}|\mu'(t)|\|u_{m}(t)\|^{2(\rho+1)}.
\end{eqnarray*}
 From (\ref{eq:2.11}) there exists a constant $c_2$ independent of $m$ and 
 $t$ such that
\begin{eqnarray*}
\frac{d}{dt}E_2(t)+2\|\theta^{'}_{ m}(t)\|^2
&\leq& \nu(t)\left\{\|u_{m}'(t)\|^2+|u_{m}''(t)|^2\right\} \\
&&+\frac{\sqrt n}{\mu_0}|\mu'(t)|\|\theta _m(t)\|^2+\frac{c_2\gamma}
{\mu_0}|\mu'(t)|.
\end{eqnarray*}
Now, integrating over the interval $[0,t[$, using again (\ref{eq:2.1}), (\ref{eq:2.11}), and  
observing that $u''_{m}(0)$ and $\theta'_{m}(0)$ are bounded in $L^2(\Omega)$ we conclude that there exists a constant $c$ independent of $m$ and $t$ such that
$$
E_2(t)+2\int_0^t\|\theta^{'}_{ m}(s)\|^2ds \leq
c+ \int _0^t\nu(s)\left\{\|u_{m}'(s)\|^2+|u_{m}''(s)|^2\right\}ds.
$$ 
From (\ref{eq:2.1}) it follows that $\nu$ belongs to $L^1(0, \infty)$. Thus, 
 applying the Gronwall's inequality  we obtain
\begin{equation}
E_2(t)+2\int_0^t\|\theta^{'}_{ m}(s)\|^2ds \leq k\quad \mbox{for all}
\quad t\geq 0.
\label{eq:2.16}
\end{equation}

\paragraph{Limit of the approximated solutions.}
From (\ref{eq:2.11}) and (\ref{eq:2.16}) it is  possible to take the limit in the nonlinear system (\ref{eq:2.6}) (\ref{eq:2.7}).  In fact, from (\ref{eq:2.11}) we obtain
\begin{eqnarray*}
\begin{array}{cc}
(u_m)\quad \mbox{is bounded in}\quad L^{\infty}\left(0,\infty; H^1_0(\Omega)\right),
 \\[4pt]
(u'_m)\quad \mbox{is bounded in}\quad L^{\infty}\left(0,\infty; L^2(\Omega)\right).
\end{array}
\end{eqnarray*}  
Hence, we have in particular, that the sequences $\left(u_m\right)_{m \in {\mathbb N}}$ and 
$\left(u'_m\right)_{m \in {\mathbb N}}$ are bounded in  
$L^2\left(0, T; H^1_0(\Omega)\right)$ and $L^2\left(0, T; L^2(\Omega)\right)$ 
respectively. Thus, by compact injection  of $H^1(\Omega \times ]0,T[)$ into 
$L^2(\Omega \times ]0,T[)$ it follows by Lions-Aubin's 
theorem that there exists a subsequence of $(u_m)$, which we denote as the 
original sequence such that
\begin{eqnarray*}
\begin{array}{cc}
u_m \longrightarrow u\quad \mbox{strong in}\quad L^{2}\left(0, T; L^2(\Omega)\right),
\end{array}
\end{eqnarray*}  
whence,
$u_m \longrightarrow u$ in $\Omega \times ]0,T[$ almost everywhere.
Hence, 
\begin{eqnarray}
\begin{array}{c}
|u_m|^{\rho}u_m \longrightarrow |u|^{\rho}u 
\quad \mbox{in}\quad \Omega \times ]0,T[\quad \mbox{almost everywhere}. 
\end{array}
\label{eq:2.17}
\end{eqnarray}
From (\ref{eq:2.11}) we also have that
\begin{eqnarray}
\begin{array}{c}
(u_m)\quad \mbox{is bounded in}\quad L^{\infty}\left(0, T; L^{\rho+2}(\Omega)\right). 
\end{array}
\label{eq:2.18}
\end{eqnarray} 
From (\ref{eq:2.17}), (\ref{eq:2.18}) and Lions' lemma (cf. Lions \cite{Lions},  lemma 1.3)  we obtain
\begin{equation}
|u_m|^{\rho}u_m \longrightarrow |u|^{\rho}u\quad \mbox{weak star in }\quad 
L^{\infty}(0, T;L^{p^{\prime}}(\Omega)),
\label{eq:2.19}
\end{equation}
where  $p=\rho+2$. Since $T$ any positive real number then the convergence (\ref{eq:2.19}) is also held for all $t \in [0, \infty[$.  

We still can obtain from (\ref{eq:2.11}) and (\ref{eq:2.16}) the following convergences
\begin{equation}
\begin{array}{l}
u_m \longrightarrow u\quad\hspace{5pt}
\mbox{weak star in }\quad 
L^{\infty}(0,\infty;H^1_0(\Omega)),
\\[4pt]
u'_m \longrightarrow u'\quad\hspace{2pt}
\mbox{weak star in }\quad 
L^{\infty}(0,\infty;H^1_0(\Omega)),
\\[4pt]
u''_m \longrightarrow u''\quad
\mbox{weak star in }\quad 
L^{\infty}(0,\infty;L^2(\Omega)),
\\[4pt]
\theta_m \longrightarrow \theta\quad\hspace{7.2pt}
\mbox{weak in }\quad\hspace{19pt} 
L^{2}(0,\infty;H^1_0(\Omega)),
\\[4pt]
\theta'_m \longrightarrow \theta'\quad\hspace{4.5pt}
\mbox{weak in }\hspace{30pt}
L^2(0,\infty;H^1_0(\Omega)).
\end{array}
\label{eq:2.20}
\end{equation}

By using (\ref{eq:2.19}) and (\ref{eq:2.20}), we can take the limit in 
the system (\ref{eq:2.6}) (\ref{eq:2.7}), and  the 
statement (\ref{eq:2.4}) and (\ref{eq:2.5}) are given by these
convergences.

The regularity (\ref{eq:2.3}) is guaranteed by results on elliptic 
regularity (see, for instance, Medeiros \& Milla Miranda \cite{LAM-MMM} 
or Nirenberg \cite{Nirenberg}) in view of 
\[
\Delta u= \frac{1}{\mu}\left\{u''+ 
\sum_{i=1}^n{\frac{\partial \theta}{\partial x_i}}+\gamma |u|^{\rho}u\right\}
\quad \mbox{belongs to}\quad L^2(0,\infty; L^2(\Omega))
\]  
and 
\[
\Delta \theta=\theta +\sum_{i=1}^n{\frac{\partial u'}{\partial x_i}}
\quad \mbox{belongs to}\quad L^2(0,\infty; L^2(\Omega)).
\]

From the two previous results we can conclude that equations (\ref{eq:1.1})
and (\ref{eq:1.2}) are given in the sense of $L^{\infty}(0,\infty; L^2(\Omega))$
As a consequence of (\ref{eq:2.3})-(\ref{eq:2.5}),  the functions 
$u$, $u'$ and $\theta$ are continuous. 
(see Lions \& Magenes \cite{Lions-Magenes} Vol. I, Cap. 1). Therefore,
the initial conditions (\ref{eq:1.3}) are well defined. 
\hfill $\diamondsuit$\medskip

\paragraph{Uniqueness.} 
If  $\{u,\theta\}$  and  $\{\hat u, \hat {\theta}\}$ are solutions of 
(\ref{eq:1.1})-(\ref{eq:1.4}), then for all $\varphi$ and $\xi$ belong to
$L^2\left(0, \infty; L^2(\Omega)\right)$ the functions $v=u-\hat{u}$ and $w=\theta-\hat{\theta}$ satisfy
$$\displaylines{
(v''(t),\varphi )+\mu(t)((v(t),\varphi ))+
\sum_{i=1}^n\left(\frac{\partial w}{\partial x_i}(t),\varphi \right)\cr
+\gamma \left(|u(t)|^\rho u(t)-|\hat{u}(t)|^\rho \hat{u}(t),\varphi 
\right)=0, \cr
(w'(t),\xi )+((w(t),\xi )) + 
\sum_{i=1}^n \left(\frac{\partial v'}{\partial x_i}(t),\xi \right) = 0,
 \cr
v(0)=0, \quad v'(0)=0 \quad \mbox{and} \quad w(0)=0\,. 
}$$ 
When we replace $\varphi$ for $2v_t$ and $\xi$ for $2w$ we obtain 

\begin{eqnarray*}
\frac{d}{dt}\left\{|v'(t)|^2+\mu(t)\|v(t)\|^2+|w(t)|^2\right\}+2\|w(t)\|^2
&& \\
+2\gamma \int _{\Omega}\left(|u(t)|^\rho u(t)-|\hat{u}(t)|^\rho \hat{u}(t)
\right)v'(t)dx&=&2\mu'(t)\|v(t)\|^2.
\end{eqnarray*}
Using the hypothesis (\ref{eq:2.1}) we can write
\begin{eqnarray}
\lefteqn{\frac{d}{dt}\left\{|v'(t)|^2+\mu(t)\|v(t)\|^2+|w(t)|^2\right\}
+2\|w(t)\|^2 } \label{eq:2.21}\\
&\leq& 2\gamma \int _{\Omega}\left||u(t)|^\rho u(t)-|\hat{u}(t)|^\rho 
\hat{u}(t)\right||v'(t)|dx\,. \nonumber
\end{eqnarray}
Applying the mean value theorem we get
\begin{eqnarray*}
\lefteqn{ \int _{\Omega}\left||u(t)|^\rho u(t)-|\hat{u}(t)|^\rho 
\hat{u}(t)\right||v'(t)|dx }\\
&\leq&  (\rho+1)\int _{\Omega}\;\,\sup_{0\leq t \leq T}
\left\{|u(t)|^{\rho},|\hat{u}(t)|^{\rho}\right\}|v(t)||v'(t)|dx.
\end{eqnarray*}
By using H\"{o}lder's generalized inequality for the case 
$\frac{1}{q}+\frac{1}{n}+\frac{1}{2}=1,$ there exists a constant $c>0$
 such that 
\begin{eqnarray*}
\lefteqn{ \int _{\Omega}\left||u(t)|^\rho u(t)-|\hat{u}(t)|^\rho
 \hat{u}(t)\right||v'(t)|dx}\\
&\leq &
c\left\{\||u|^{\rho}\|_{L^{n}(\Omega)}+\||\hat {u}|^{\rho}\|_{L^{n}(\Omega)}\right\}\|v(t)\|_
{L^{q}(\Omega)}\|v'(t)\|_{L^{2}(\Omega)}.
\end{eqnarray*}
As $\|\,|z|^{\rho}\,\|_{L^{n}(\Omega)}=\|z\|^{\rho}_{L^{n\rho}(\Omega)}$ and 
$n\rho \leq q$, then using the same argument of the estimate II  we obtain
$\|\,|z|^{\rho}\,\|_{L^{n}(\Omega)}\leq c \|z\|^{\rho}_{H^{1}_{0}(\Omega)}$ for all 
$z \in  H^{1}_{0}(\Omega)$. 

On the other hand, the estimate (\ref{eq:2.11}) implies that  $\|z(t)\|^{\rho}_{H^{1}_{0}(\Omega)}$ is bounded for all $t\geq 0$. Therefore, a positive real constant $c$ exists such that
\begin{equation}
\int _{\Omega}\left||u(t)|^\rho u(t)-|\hat{u}(t)|^\rho \hat{u}(t)\right||v'(t)|dx
\leq c\|v\|_{H^{1}_{0}(\Omega)}|v'|_{L^{2}(\Omega)}.
\label{eq:2.22}
\end{equation}
Substituting (\ref{eq:2.22}) in (\ref{eq:2.21}) and integrating from $0$ to $t$ we get
\begin{eqnarray*}
\lefteqn{ 
|v'(t)|^2+\mu(t)\|v(t)\|^2+|w(t)|^2+2\int_0^t\|w(s)\|^2ds }\\
&\leq& 
\gamma \int_0^t\left\{|v'(s) |^2 +c^2\|v(s)\|^2 \right\}ds.
\end{eqnarray*}
As $\frac{\mu(t)}{\mu_o}\geq 1$ then choosing 
$k=\max\left\{\gamma,\;\;\frac{c^2}{\mu_0} \right\}$ yields
\begin{eqnarray*}
\lefteqn{|v'(t)|^2+\mu(t)\|v(t)\|^2+|w(t)|^2+2\int_0^t\|w(s)\|^2ds}\\
&\leq&
k\int_0^t\left\{|v'(s) |^2 +\mu(s)\|v(s)\|^2 \right\}ds.
\end{eqnarray*}
Finally, by Gronwall's inequality it follows that 
$v(t)=w(t)=0$ on $[0,\infty [$. 
Hence, Theorem 2.1 is established\hfill$\diamondsuit$\medskip

Now, we shall present a result of the global existence and uniqueness of 
weak solution for the system (\ref{eq:1.1})-(\ref{eq:1.4}).  

%Teorema 2.2

\begin{theorem}{\bf (Weak solution)}
Let the assumptions of Theorem 2.1 about $\mu$, $\rho$, and $\gamma$ be 
satisfied.  If $p=\rho+2$ and 
\[
u_0 \in H^1_0(\Omega)\cap L^p(\Omega),\quad  u_1 \in L^2(\Omega) 
\quad \mbox{and}\quad 
\theta_0 \;\in\; H^1_0(\Omega),
\]
then the system (\ref{eq:1.1})-(\ref{eq:1.4}) has a unique solution 
$\{u,\theta\}$ such that
\begin{equation}
u \in L^{\infty}\left(0,\infty; H^1_0(\Omega) \cap L^p(\Omega) )\right),
\quad  u' \in L^{\infty}\left(0,\infty; L^2(\Omega)\right),
\label{eq:2.23}
\end{equation}
\vspace{-15pt}
\begin{equation}
u''\in L^2\left(0,\infty;H^{-1}(\Omega)+L^{p \prime}(\Omega)\right),
\label{eq:2.24}
\end{equation}
\begin{equation}
\theta \in L^{\infty}\left(0,\infty;L^2(\Omega)\right)  
\cap L^2\left(0,\infty;H^1_0(\Omega)\right),   
\label{eq:2.25}
\end{equation} 
\begin{equation}
\theta'\in L^2\left(0,\infty;H^{-1}(\Omega)\right),  
\label{eq:2.26}
\end{equation} 
and the equation (\ref{eq:1.1}) is given in the sense of 
$L^2\left(0,\infty; H^{-1}(\Omega)+L^{p \prime}(\Omega)\right)$
and (\ref{eq:1.2}) is given in the sense of
$L^2\left(0,\infty; H^{-1}(\Omega)\right)$. 
\end{theorem}
\paragraph{Proof.} {\it Existence.} The demonstration of existence of a weak solution is made by the same method utilized to get the strong solution. However, we now consider the basis $(w_j)_{j \in {\mathbb N}}$ in the space $H^1_0(\Omega) \cap L^p(\Omega)$, and the initial conditions sastisfying  
\begin{eqnarray}
&u_{0m}\longrightarrow u_0 
\quad \mbox{strongly}\quad \mbox{in}\quad 
H^1_0(\Omega)\cap L^p(\Omega),
&\label{eq:2.27}\\
&\theta_{0m} \longrightarrow \theta_0\quad \mbox{strongly}\quad \mbox{in}\quad 
H^1_0(\Omega),\hspace*{36pt}
&\label{eq:2.28}\\
&u_{1m}\longrightarrow u_1\quad \mbox{strongly}\quad \mbox{in}\quad 
L^2(\Omega).\hspace*{38pt}
&\label{eq:2.29}
\end{eqnarray}
Under these conditions, we have that the system (\ref{eq:2.6}) (\ref{eq:2.7}) has a local solution $\left\{u_{ m}(t),\theta_{ m}(t)\right\}$ on the interval $[0,t_m[$, and this solution will be extended over the whole interval $[0, \infty[$ independent of $m$ and $t$ thanks to estimate 
(\ref{eq:2.30}) above.

Using the same procedure used to get estimate I of Theorem 2.1,  we get 
\begin{eqnarray}
|u'_{m}(t)|^2+|\theta_{m}(t)|^2+\mu(t)\|u_{m}(t)\|^2 && \label{eq:2.30}\\
+\frac{2\gamma}{p}\int_{\Omega}|u_{m}(t)|^pdx+2\int_0^t\|\theta_{ m}(s)\|^2
ds &\leq& k\quad \mbox{for all}\quad t \geq 0.\nonumber
\end{eqnarray}
Hence, there exist subsequences of $(u_m)_{m \in {\mathbb N}}$ and 
$(\theta _m)_{m \in {\mathbb N}}$, which we denote as the original 
sequences, and functions $u:\Omega\times ]0,\infty[\to {\mathbb R}$ and
$\theta:\Omega\times ]0,\infty[\to {\mathbb R}$ such that
\begin{eqnarray}
&&u_m \longrightarrow u\quad\; \mbox{weak star in}\quad 
L^{\infty}(0,\infty;H^1_0(\Omega)\cap L^p(\Omega)),
\label{eq:2.31}\\
&&u'_m \longrightarrow u'\quad \mbox{weak star in}\quad 
L^{\infty}(0,\infty;L^2(\Omega)),
\label{eq:2.32}\\
&&\theta_m \longrightarrow \theta \quad\;\, \mbox{weakly in}\quad\quad\,  
L^2(0,\infty;H^1_0(\Omega)).
\label{eq:2.33}
\end{eqnarray}
From (\ref{eq:2.31})-(\ref{eq:2.33}) we can take the limit in (\ref{eq:2.6}) (\ref{eq:2.7}) yielding
\begin{eqnarray}
-\int_0^{\infty}(u'(t),v)\phi'(t)dt+\int_0^{\infty}\mu(t)((u(t),v))
\phi(t)&& \label{eq:2.34}\\
+\sum_{i=1}^n\int_0^{\infty}\left(\frac{\partial \theta}{\partial x_i}(t)
,v\right)\phi(t)dt+
\gamma\int_0^{\infty}\left(|u(t)|^\rho 
u(t),v\right)\phi(t)dt&=&0,\nonumber
\end{eqnarray}
\begin{eqnarray}
-\int_0^{\infty}(\theta(t),w)\phi'(t)dt+\int_0^{\infty}((\theta(t),w))
\phi(t)dt && \label{eq:2.35}\\
+\sum_{i=1}^n\int_0^{\infty}\left(\frac{\partial u'}{\partial x_i}(t),
w \right) \phi(t)dt&=&0, \nonumber
\end{eqnarray}
where $v$ and $w$  belong to $H^1_0(\Omega)$ and $\phi$ belongs 
to ${\cal D}(0, \infty)$.

Hence, following the same procedure of Clark at al \cite{Clark} 
(Theorem 4.1) we can conclude
\begin{eqnarray}
&
\displaystyle  
u''-\mu\Delta u+\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}
+\gamma|u|^\rho u=0
\quad \mbox{in}\quad L^2\left(0,\infty;H^{-1}(\Omega)+ L^{p'}(\Omega)
\right),&\nonumber \\
&
\displaystyle 
\theta'-\Delta \theta+\sum_{i=1}^n\frac{\partial u'}{\partial x_i}=0
\quad \mbox{in}\quad L^2\left(0,\infty;H^{-1}(\Omega)\right).&
\label{eq:2.36}
\end{eqnarray}
The statements (\ref{eq:2.23})-(\ref{eq:2.26}) are consequences of 
(\ref{eq:2.36}) and of convergences (\ref{eq:2.31})-(\ref{eq:2.33}).

The functions $u$, $u'$ and $\theta$ are continuous, thanks to 
(\ref{eq:2.23})-(\ref{eq:2.26}). Thus, the initial conditions 
(\ref{eq:1.3}) are verified.

\paragraph{\it Uniqueness.} From (\ref{eq:2.23})$_2$ and (\ref{eq:2.24}) 
the duality 
$\left <u'', u'\right>$
does not make sense. Thus, it is necessary to utilize the regularization's
 method of Ladyzhenskaya-Visik \cite{L-Visik}, see also Lions
 \cite{Lions} (Theorem 1.2, pp. 14), and we proceed such as in the 
demonstration of uniqueness of Theorem 2.1 
Therefore, Theorem 2.2 has been proven\hfill$\diamondsuit$\medskip 

\section{Asymptotic behavior}
\setcounter{equation}{0} 

The aim of this section is to prove that the total energy associated to 
solutions of  (\ref{eq:1.1})-(\ref{eq:1.4}) has exponential decay when 
the time $t$ goes to $+\infty$.
To reach our goal we will utilize the method of Komornik-Zuazua 
\cite{kom-zua}.

The total energy of the system (\ref{eq:1.1})-(\ref{eq:1.4}) is given by
\[
E(t)=|u'(t)|^2+\mu(t)\|u(t)\|^2+|\theta (t)|^2+\frac{2\gamma}{\rho+2} \int_{\Omega}|u(t)|^{\rho+2}dx.
\] 
To obtain the asymptotic behavior of the energy $E(t)$ it is necessary to 
consider the following hypothesis about the constant $\gamma$,
\begin{equation}
\gamma \leq \frac{\mu_0^{\rho+1}}{2ck^\rho}\,,
\label{eq:3.1}
\end{equation}
where $\mu_0$ and $k$ are fixed in (\ref{eq:2.1}) and (\ref{eq:2.11}) respectively, and $c$ is the constant of continuous injection of $H^1_0(\Omega)$ into $L^{\rho+2}(\Omega)$.
From estimate I and hypothesis (\ref{eq:2.1}) it is easy to see that the energy $E(t)$ is not increasing in view of
\begin{eqnarray}
\frac{d}{dt}E(t)\leq-\|\theta(t)\|^2\quad \mbox{for all}\quad t\geq 0.
\label{eq:3.2}
\end{eqnarray}
The asymptotic behavior of the energy of our system is given by
%
%Teorema 3.1
%
\begin{theorem}If the constant $\gamma$ satisfies the hypothesis (\ref{eq:3.1}) then the energy 
$E(t)$ associated to weak solution of the system (\ref{eq:1.1})-(\ref{eq:1.4}), guaranteed by 
Theorem 2.2, satisfies
\begin{equation}
E(t)\leq E(0)\exp(-\omega t)\quad \mbox{for all}\quad t\geq 0,
\label{eq:3.3}
\end{equation}
where $\omega$ is a real-positive number defined in (\ref{eq:3.23}).
\end{theorem}

\paragraph{Proof.}  First we get (\ref{eq:3.3}) for the energy $E(t)$ 
given by the strong solutions of (\ref{eq:1.1})-(\ref{eq:1.4}) 
 guaranteed by  Theorem 2.1. Thus, the stability result for the energy
 associated to the weak solution is established without difficulty by 
density arguments.
Let us consider the auxiliary real function
\begin{equation}
\psi(t)=2\left(u'(t),\left(x\cdot\nabla\right) u(t)\right)+(n-1)\left(u'(t),u(t)\right)
\quad \mbox{for all}\quad t\geq 0,
\label{eq:3.4}
\end{equation}
where $x \in {\mathbb R}^n$ is nonzero, $``\cdot"$ denotes the usual scalar product in 
${\mathbb R}^n$, and $n \in {\mathbb N}$.  

Differentiating $\psi(t)$ with respect to $t$ we obtain
\begin{eqnarray*}
\psi'(t)&=&2\left(u''(t),\left(x\cdot\nabla\right)u(t)\right)+
2\left(u'(t),\left(x\cdot\nabla\right) u'(t)\right)\\
&&+(n-1)\left(u''(t),u(t)\right)+(n-1)\left|u'(t)\right|^2.
\end{eqnarray*}
Replacing  $u''(t)$  by  
$\mu(t)\Delta u(t)-\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}(t)-
 \gamma|u(t)|^{\rho}u(t)$ in the identity above it implies
\begin{eqnarray}
 \psi'(t)&=&2\mu(t)\int_{\Omega}\Delta u(t)\left(x\cdot\nabla\right)u(t) dx-2\sum_{i=1}^n\int_{\Omega}\frac{\partial \theta}
{\partial x_i}(t)\left(x\cdot\nabla\right) u(t)dx \nonumber \\
&&-
2\gamma \int_{\Omega} |u(t)|^{\rho}u(t)\left(x\cdot\nabla\right) u(t)dx+
2\int_{\Omega} u'(t)\left(x\cdot\nabla\right) u'(t)dx \nonumber\\
&&+(n-1)|u'(t)|^2+
(n-1)\mu(t)\int_{\Omega}\Delta u(t)u(t)dx \label{eq:3.5}\\
&&- (n-1)\sum_{i=1}^n\int_{\Omega} \frac{\partial \theta}{\partial x_i}(t)
u(t)dx- \gamma (n-1)\int_{\Omega} |u(t)|^{\rho+2}dx. \nonumber
\end{eqnarray}

Our next objective is to limit each term of the right-hand side of (\ref{eq:3.5}).
In the following steps we will use the Green's formula and the boundary conditions (\ref{eq:1.4}) several times.

{\bf Step 1:} First term of (\ref{eq:3.5}). Since 
$u(t) \in H^1_0(\Omega) \cap H^2(\Omega)$, the Green's formula we have
\begin{eqnarray}
2\int_{\Omega}\Delta u \left(x\cdot\nabla\right)u dx
&=& 2\sum_{i,j=1}^n \int_{\Omega}\frac{\partial }{\partial x_i}\left(\frac{\partial u }
{\partial x_i}\right)x_j\frac{\partial u}{\partial x_j}dx\nonumber\\
&=& -2\sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}
{\partial x_i}\frac{\partial  }{\partial x_i}\left(x_j\frac{\partial u}
{\partial x_j}\right)dx \label{eq:3.6}\\
&=& -2\sum_{i=1}^n \int_{\Omega}\left(\frac{\partial u}{\partial x_i}
\right)^2dx-2\sum_{i,j=1}^n  \int_{\Omega}
\frac{\partial u}{\partial x_i}x_j
\frac{\partial^2 u}{\partial x_i \partial x_j}dx.\nonumber
\end{eqnarray}
However,
\begin{eqnarray*}
\lefteqn{ -2 \sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}
{\partial x_i}x_j\frac{\partial^2 u}{\partial x_i \partial x_j}dx}\\
&=& 2\sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}{\partial x_i}
\frac{\partial }{\partial x_j}\left(x_j\frac{\partial u}{\partial x_i}
\right)dx\\
&=& 2\sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}{\partial x_i}
\frac{\partial x_j}{\partial x_j}\frac{\partial u}{\partial x_i}dx
+2\sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}
{\partial x_i}x_j\frac{\partial^2 u}{\partial x_j \partial x_i}dx.
\end{eqnarray*}
Thus,
\begin{eqnarray}
-2 \sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}
{\partial x_i}x_j\frac{\partial^2 u}{\partial x_i \partial x_j}dx
&=& \sum_{i,j=1}^n \int_{\Omega}\frac{\partial u}
{\partial x_i}
\frac{\partial x_j}{\partial x_j}\frac{\partial u}{\partial x_i}dx
\label{eq:3.7}\\
&=& n\sum_{i=1}^n \int_{\Omega}\left(\frac{\partial u}
{\partial x_i}\right)^2dx.\nonumber
\end{eqnarray} 
Substituting (\ref{eq:3.7}) in (\ref{eq:3.6}) yields
\begin{equation}
 2\mu(t)\int_{\Omega}\Delta u(t)\left(x\cdot\nabla\right)u(t)dx
=-(2-n)\mu(t)\|u(t)\|^2.
\label{eq:3.8}
\end{equation}

{\bf Step 2} Second term of (\ref{eq:3.5}).
\begin{eqnarray}\label{eq:3.9}
-2\sum_{i=1}^n\int_{\Omega}\frac{\partial \theta}
{\partial x_i}(t)\left(x\cdot\nabla\right)u(t)dx
&\leq& \frac{nr^2_{\Omega}}{\mu_o}\|\theta(t)\|^2+
\sum_{i=1}^n\frac{\mu_o}{4n}\|u(t)\|^2\\
&=&\frac{nr^2_{\Omega}}{\mu_o}\|\theta(t)\|^2+
\frac{\mu(t)}{4}\|u(t)\|^2\,, \nonumber
\end{eqnarray}
where  $r_{\Omega}=\|x\|_{L^{\infty}(\Omega)}.$

{\bf Step 3} Third and eighth terms of (\ref{eq:3.5}). 
\begin{eqnarray*} 
\lefteqn{-2\gamma\int_{\Omega} |u(t)|^{\rho}u(t)\left(x\cdot\nabla\right)
u(t)dx } \\
&=& -2\gamma\sum_{i=1}^n \int_{\Omega}|u(t)|^{\rho}u(t)x_i
\frac{\partial u}{\partial x_i}(t)dx \\
&=& 
2\gamma\sum_{i=1}^n \int_{\Omega}\frac{\partial }{\partial x_i}
\left(|u(t)|^{\rho}u(t)x_i\right) u(t)dx \\
&=& 2\gamma\rho \sum_{i=1}^n \int_{\Omega}|u(t)|^{\rho+1}x_i
\frac{\partial }{\partial x_i}|u(t)|dx\\
&&+ 
2\gamma\sum_{i=1}^n \int_{\Omega}|u(t)|^{\rho}u(t)x_i
\frac{\partial u}{\partial x_i}(t)dx+
2\gamma\sum_{i=1}^n \int_{\Omega}|u(t)|^{\rho+2}dx.
\end{eqnarray*}
Observe that 
\begin{eqnarray*}
2\gamma\rho \sum_{i=1}^n \int_{\Omega}|u(t)|^{\rho+1}x_i
\frac{\partial }{\partial x_i}|u(t)|dx&=&
\frac{2\gamma\rho}{\rho+2}\sum_{i=1}^n \int_{\Omega}\frac{\partial }
{\partial x_i}\left(|u(t)|^{\rho+2}\right)x_i\,dx \\
&=& -\frac{2n\gamma\rho}{\rho+2}\int_{\Omega}|u(t)|^{\rho+2}dx\,.
\end{eqnarray*}
Thus,
\[
-2\gamma\int_{\Omega} |u(t)|^{\rho}u(t)\left(x\cdot\nabla\right)u(t)dx =
-\frac{n\gamma\rho}{\rho+2}\int_{\Omega} |u(t)|^{\rho+2}dx+n\gamma
\int_{\Omega} |u(t)|^{\rho+2}dx.
\]
Hence, adding the third and eighth terms of (\ref{eq:3.5}) yields
\begin{eqnarray} \label{I}
\lefteqn{-2\gamma\int_{\Omega} |u(t)|^{\rho}u(t)\left(x\cdot\nabla\right)
u(t)dx-\gamma(n-1)\int_{\Omega}|u(t)|^{\rho+2}dx} \\
& =&
-\frac{n\gamma\rho}{\rho+2}\int_{\Omega} |u(t)|^{\rho+2}dx+
\gamma\int_{\Omega} |u(t)|^{\rho+2}dx.\nonumber\hspace{2cm}
\end{eqnarray}
As $H^1_0(\Omega)$ is continuously embedded into $L^{\rho+2}(\Omega)$ then there exists a real positive constant $c$ such that
\[
\gamma\int_{\Omega}|u(t)|^{\rho+2}dx
\leq \gamma c\|u(t)\|^{\rho+2}=\gamma c\|u(t)\|^{\rho}\|u(t)\|^2.
\]
From (\ref{eq:2.11}) we have
$\|u(t)\|^{ \rho} \leq \frac{k^\rho}{\mu_0^\rho}$ for all $\,t \geq 0$. 
Therefore,
\[
\gamma\int_{\Omega}|u(t)|^{\rho+2}dx
\leq \frac{\gamma c k^\rho}{\mu_0^\rho}\|u(t)\|^2.
\]
As $\frac{\mu (t)}{\mu_o}\geq 1$ then
\[
\gamma\int_{\Omega}|u(t)|^{\rho+2}dx
\leq \frac{\gamma c k^\rho}{\mu_0^{\rho+1}}\mu(t)\|u(t)\|^2.
\]
Substituting this expression in (\ref{I}), we obtain 
\begin{eqnarray}\label{eq:3.10}
\lefteqn{ -2\gamma\int_{\Omega} |u(t)|^{\rho}u(t)\left(x\cdot
\nabla\right)u(t)dx-\gamma(n-1)\int_{\Omega}|u(t)|^{\rho+2}dx }\\
&\leq&
-\frac{n\gamma\rho}{\rho+2}\int_{\Omega}|u(t)|^{\rho+2}dx+
\frac{\gamma c k^\rho}{\mu_0^{\rho+1}}\mu(t)\|u(t)\|^2. \hspace{2cm}\nonumber
\end{eqnarray}

{\bf Step 4:} Fourth term of (\ref{eq:3.5}).
\[
2\int_{\Omega} u'(t)\left(x\cdot\nabla\right)u'(t)dx =\sum_{i=1}^n\int_{\Omega}x_i\frac{\partial }{\partial x_i}[u'(t)]^2dx=
-\sum_{i=1}^n\int_{\Omega}[u'(t)]^2dx.
\]
Hence,
$$2\int_{\Omega} u'(t)\left(x\cdot\nabla\right)u'(t)dx =-n|u'(t)|^2\,.
$$

{\bf Step 5:} Sixth term of (\ref{eq:3.5}).
\begin{equation}
(n-1)\mu(t)\int_{\Omega} \Delta u(t)u(t)dx=(1-n)\mu(t)\|u(t)\|^2.
\label{eq:3.12}
\end{equation} 

{\bf Step 6:} Seventh term of (\ref{eq:3.5}).
\begin{eqnarray}
-(n-1)\sum_{i=1}^n\int_{\Omega}
 \frac{\partial \theta}{\partial x_i}(t)u(t)dx 
&\leq& \frac{\lambda_1n(n-1)^2}{\mu_o}\|\theta(t)\|^2+
\sum_{i=1}^n\frac{\mu_o}{4n\lambda_1}|u(t)|^2 \nonumber \\
&\leq& \frac{\lambda_1n(n-1)^2}{\mu_o}\|\theta(t)\|^2+
\frac{\mu(t)}{4}\|u(t)\|^2, \label{eq:3.13}
\end{eqnarray}
where $\lambda_1$ is defined by $|v|^2\leq \lambda_1\|v\|^2$ for all 
$v \in H_0^1(\Omega)$.
Substituting (\ref{eq:3.8})-(\ref{eq:3.13}) in (\ref{eq:3.5}) yields
\begin{eqnarray} \label{eq:3.14}
\psi'(t)&\leq& -|u'(t)|^2-
\left(\frac{1}{2}-\frac{\gamma ck^\rho}{\mu_0^{\rho+1}}\right)\mu(t)\|u(t)\|^2
\\
&&+\left(\frac{nR^2_{\Omega}+\lambda_1n(n-1)^2}{\mu_o}\right)\|\theta(t)\|^2
-\frac{n\gamma \rho}{\rho+2} 
\int_{\Omega}|u(t)|^{\rho+2}dx.\nonumber
\end{eqnarray}
The constant $\frac{1}{2}-\frac{\gamma ck^\rho}{\mu_0^{\rho+1}}$ is positive thanks to the hypothesis (\ref{eq:3.1}). Now, multiplying (\ref{eq:3.14}) for a suitable $\epsilon$ and taking into account that
$-\|\theta\|^2\leq-\frac{1}{\lambda_1}|\theta|^2,$  for all $\theta \in H^1_0(\Omega)$ then we have from (\ref{eq:3.2}) and (\ref{eq:3.14}) that   
\begin{eqnarray} 
E'(t)+\epsilon \psi'(t)&\leq&  -\epsilon|u'(t)|^2-
\epsilon\left( \frac{1}{2}- \frac{\gamma ck^\rho}{\mu_0^{\rho+1}}\right)
\mu(t)\|u(t)\|^2 \nonumber\\
&&-\frac{1}{\lambda_1}\left[1-\epsilon \left(\frac{nr^2_{\Omega}+
\lambda_1n(n-1)^2}{\mu_o}\right)\right]|\theta(t)|^2  \label{eq:3.15}\\
&&-\epsilon\frac{n\gamma \rho}{\rho+2} 
\int_{\Omega}|u(t)|^{\rho+2}dx.\nonumber
\end{eqnarray}
Let $\epsilon$ and $\tau$ be real positive numbers such that
\begin{equation}
\epsilon =\min\left\{\frac{\mu_0}{nr^2_{\Omega}+
\lambda_1n(n-1)^2},\quad\frac{1}{\tau_0}\right\},
\label{eq:3.16}  
\end{equation}
\begin{equation}
\tau=\min\left\{\frac{\mu_0^{\rho+1}-2\gamma ck^\rho}{2\mu_0^{\rho+1}},\;
\frac{\mu_0-\epsilon \left[nr^2_{\Omega}+\lambda_1n(n-1)^2\right]}
{\mu_o\lambda_1\epsilon},\;\frac{n\rho}{2} \right\}, 
\label{eq:3.17}  
\end{equation}
where $\tau_0$ is defined in (\ref{eq:3.19}).
Taking into account (\ref{eq:3.16}) and (\ref{eq:3.17}) in (\ref{eq:3.15}) we get 
\begin{equation}
E'_{\epsilon}(t) \leq -\epsilon \tau E(t)
\quad \mbox{for all}\quad t\geq 0,
\label{eq:3.18}
\end{equation}
where  $E_{\epsilon}(t)=E(t)+\epsilon \psi(t)$.

On the other hand,
\begin{eqnarray*}
|\epsilon \psi (t)| &\leq& \epsilon \left\{\frac{1}{\mu_o} |u'(t)|^2+r^2_{\Omega}\mu_o\|u(t)\|^2+\frac{n-1}{2\mu_o}|u'(t)|^2+ 
\frac{(n-1)\mu_o}{2}|u(t)|^2\right\}
\\
&\leq& \epsilon \left\{ \frac{n+1}{2\mu_o}|u'(t)|^2+\left[\frac{2r^2_{\Omega}+
\lambda_1 (n-1)}{2}\right]\mu(t)\|u(t)\|^2\right\}.
\end{eqnarray*}
Choosing 
\begin{equation}
\tau _0 = \max \left\{\frac{2\mu_o}{n+1}\,,\quad \frac{2}{2r^2_{\Omega}+\lambda_1(n-1)}\right\},
\label{eq:3.19}
\end{equation}
it follows
\begin{equation}
|\epsilon \psi (t)| \leq \epsilon \tau _0E(t)\quad \mbox{for all}\quad t \geq 0.
\label{eq:3.20}
\end{equation}
Note that
\begin{equation}
|E_{\epsilon}(t)-E(t)| \leq \epsilon |\psi(t)|\quad \mbox{for all}\quad t \geq 0.
\label{eq:3.21}
\end{equation}
 From (\ref{eq:3.20}) and (\ref{eq:3.21})  we have
\begin{eqnarray}
(1-\epsilon \tau _0)E(t) \leq E_{\epsilon}(t) \leq (1+\epsilon \tau _0)E(t)
\quad \mbox{for all}\quad t \geq 0.
\label{eq:3.22}
\end{eqnarray}
 From (\ref{eq:3.18}) and (\ref{eq:3.22}) we obtain
\begin{equation} \label{eq:3.23}
E'(t) \leq -\omega E(t)\quad\mbox{or all } t \geq 0\,,
\end{equation}
where $\omega=\epsilon \tau /(1-\epsilon \tau_0)$.
From (\ref{eq:3.16}) it follows that $1-\epsilon \tau_0$ is a real positive number. Thus,
there exists a real positive number $\omega$ 
satisfying (\ref{eq:3.3}). Therefore, the proof of Theorem 3.1 
is complete \hfill$\diamondsuit$

\paragraph{Acknowledgement}  We want to thank the anonymous referee of 
this journal and Professor Julio G. Dix for pointing out some corrections
on the first version of this paper.  

\begin{thebibliography}{00}

\bibitem{Clark} Clark, H. R., San Gil Jutuca,  L. P. \& Milla Miranda, M., 
{\it On a mixed problem for a linear coupled system with variable coefficients}, 
Eletronic Journal of Differential Equations, Vol. {\bf 1}(1998), No.~04, pp. 1-20. 

\bibitem{kom-zua} Komornik, V., Zuazua, E.,  
{\it A direct method for boundary stabilization of the wave equation}, 
J. Math. Pure et Appl., {\bf 69}(1990), pp. 33-54.

\bibitem{L-Visik} Ladyzhenskaia,  O.  A., Visik,  M. I.,  
{\it Boundary value problems for partial differential equations and certain classes of operator equations}, 
A.M.S. Translations Series 2 10, (1958), pp. 223-281.

\bibitem{Lions}  Lions,  J. L., 
{\it Quelques m\'ethodes de r\'esolution des probl\`emes aux limites non lin\'eares.}, 
Dunod-Gauthier Villars, Paris, First edition, 1969.

\bibitem{Lions-Magenes}  Lions,  J. L.,  Magenes, E., 
{\it Non-homogeneous  boundary value problems and applications.}, 
Vol. I, Springer-Verlag, New York, 1972.

\bibitem{LAM-MMM-1} Medeiros, L. A., Milla Miranda, M., 
{\it On a boundary value problem for wave equations: Existence, uniqueness-asymptotic behavior}, 
Revista de Matem¤ticas Aplicadas, Univerdidade de Chile, {\bf 17}(1996), 47-73.

\bibitem{LAM-MMM}  Medeiros, L. A., Milla Miranda, M., 
{\it Espa\c{c}o de Sobolev - Inicia\c{c}\~ao aos problemas el\'{i}pticos n\~ao homog\^eneos.}, Instituto de Matem\'atica - UFRJ, Rio de Janeiro, First edition, 1977.

\bibitem{Nirenberg}  Nirenberg, L., 
{\it Remarks on strongly elliptic partial differential equations.}, 
Comm. Pure Applied Math., {\bf 8}(1977), pp. 648-674.

\bibitem{Schiff} Schiff, L. I., 
{\it Non linear meson theory of nuclear forces.}, 
I. Physic. Rev., {\bf 84}(1951), pp. 1-9.

\bibitem{Segal} Segal, I. E., 
{\it The global Cauchy problem for a relativistic scalar field with power interaction.}, 
Bull. Soc. Math., France, {\bf 91}(1963), pp. 129-135.

\bibitem{Sobolev} Sobolev, S. I., 
{\it Applications de analyse functionnelle aux \'equations de la physique math\'ematique.}, L\'eningrad, (1950).

\end{thebibliography} \smallskip
 
\noindent{\sc R. F. Apolaya }\\
Universidade Federal Fluminense, RJ, Brasil. \\
e-mail: ganrefa@vm.uff.br  \smallskip

\noindent{\sc H. R. Clark }\\
Universidade Federal Fluminense, RJ, Brasil. \\
e-mail: ganhrc@vm.uff.br \smallskip

\noindent{\sc A. J. Feitosa }\\
Universidade Federal da Para\'{\i}ba, PB, Brasil.\\
e-mail: joaquim@mat.ufpb.br
\end{document}