\documentclass[twoside]{article}
\usepackage{amssymb}
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\markboth{\hfil On a Mixed Problem \hfil EJDE--1998/04}%
{EJDE--1998/04\hfil H.R. Clark, L.P.S.G. Jutuca, \& M.M. Miranda \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Eletronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~04, pp. 1--20. \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 a mixed problem for a linear coupled system with variable coefficients
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35F15, 35N10, 35B40.
\hfil\break\indent
{\em Key words and phrases:} Mixed problem, Boundary damping, Exponential stability.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of North Texas.
\hfil\break\indent
Submitted August 24, 1997. Published February 13, 1998.}}
\date{}
\author{H. R. Clark, L. P. San Gil Jutuca, \& M. Milla Miranda}
\maketitle

\begin{abstract}
We prove existence, uniqueness and exponential decay of solutions to the mixed 
problem
\begin{eqnarray*}
&u''(x,t)-\mu(t)\Delta u(x,t)+\sum_{i=1}^n{\frac{\partial \theta}{\partial x_i}(x,t)}=0\,,&\\
&\theta'(x,t)-\Delta \theta(x,t) +\sum_{i=1}^n{\frac{\partial u'}{\partial x_i}(x,t)}=0\,,&
\end{eqnarray*} 
with a suitable boundary damping, and a positive real-valued function $\mu$.
\end{abstract}  

\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\newtheorem{myth}{Theorem}[section]
\newtheorem{mylem}{Lemma}[section]
\newtheorem{mypro}{Proposition}[section]
\newtheorem{mydef}{Definition}[section]
\newtheorem{myrem}{Remark}[section]
\newtheorem{mycor}{Corollary}[section]
\newcommand{\cqd}{\hfill\fbox{\thinspace}\medskip}

\section{Introduction}

Let $\Omega$ be a bounded and open set in ${\mathbb R}^n$ $(n\geq 1)$ with boundary
 $\Gamma$ of class $C^2$. 
Assumed that there exists a partition $\{\Gamma_0,\Gamma_1\}$ of $\Gamma$ such 
that $\Gamma_0$ and $\Gamma_1$ each has positive induced Lebesgue measure, and 
that $\overline\Gamma_0\cap \overline\Gamma_1$ is empty. We consider the 
linear system
\begin{eqnarray}
&u''(x,t)-\mu(t)\Delta u(x,t)+\sum_{i=1}^n{\frac{\partial \theta}{\partial x_i}
(x,t)}=0\hspace{7pt}\mbox{in}\hspace{7pt}\Omega\times]0,\infty[
\label{eq:1.1}&\\
&\theta'(x,t)-\Delta \theta(x,t) +\sum_{i=1}^n{\frac{\partial u'}{\partial x_i}
(x,t)}=0\hspace{7pt}\mbox{in}\hspace{7pt}\Omega \times ]0,\infty[
\label{eq:1.2}&\\
&u(x,t)=0,\;\;\theta(x,t)=0\hspace{7pt}\mbox{on}\hspace{7pt}\Gamma_0\times]0,
\infty[
\label{eq:1.3}&\\
&\frac{\partial u}{\partial \nu}(x,t)+\alpha(x)u'(x,t)=0\hspace{7pt}\mbox{on}
\hspace{7pt}\Gamma_1\times ]0,\infty[
\label{eq:1.4}&\\
&\frac{\partial \theta}{\partial \nu}(x,t)+\beta \theta(x,t)=0\hspace{7pt}
\mbox{on}\hspace{7pt}\Gamma_1\times ]0,\infty[
\label{eq:1.5}&\\
&u(x,0)=u^0(x),\hspace{4pt}u'(x,0)=u^1(x),\hspace{4pt}\theta(x,0)=\theta^0(x)
\hspace{7pt}\mbox{on}\hspace{7pt}\Omega,
\label{eq:1.6}
\end{eqnarray}
where $\mu$ is a function of $W^{1,\infty}_{\rm loc}(0,\infty)$, such that 
$\mu(t)\geq \mu_0>0.$ By $\alpha $ we represent a function of 
$W^{1,\infty}(\Gamma_1)$ such that $\alpha (x)\geq \alpha_0>0$, and by $\beta $ 
a positive real number. The prime notation denotes time derivative, 
and $\frac{\partial}{\partial \nu}$ denotes derivative in the direction of the 
exterior normal to $\Gamma$.

The above system is physically meaningful only in one dimension. For which 
there exists an extensive literature on existence, uniqueness and stability 
when $\mu \equiv 1$. See the recent papers of Mu\~nhoz Rivera \cite{rivera}, 
Henry, Lopes, Perisinotto \cite{henry}, and Scott Hansen \cite{scott}.    

The paper of Milla Miranda and L. A. Medeiros \cite{mi-me} on wave equations 
with variable coefficients has a particular relevance to this work. In that 
paper, due to the boundary condition of feedback type, the authors introduced 
a special basis necessary to apply the Galerkin method. This is the natural 
method solving problems with variable coefficients.

In this article, we show the existence of a strong global solution of 
(\ref{eq:1.1})--(\ref{eq:1.6}), when $u^0,\;u^1$ and $\theta^0$ satisfy 
additional regularity hypotheses. Then this result is used for finding a weak 
global solution to (\ref{eq:1.1})--(\ref{eq:1.6}) in the general case. 
By the use of a method proposed in \cite{kom-zua}, we study the asymptotic 
behavior of an energy determined by solutions. 

The paper is organized as follows:
In \S2 notation and basic results, in \S3 strong solutions,
in \S4 weak solutions, and in \S5 asymptotic behavior.      

\section{Notation and Basic Results} \setcounter{equation}{0}

Let the Hilbert space
$$V=\{v\in H^1(\Omega);v=0\quad\mbox{on }\Gamma_0\}$$
be equipped with the inner product and norm given by
$$((u,v))={\sum_{i=1}^n\int_{\Omega}}\frac{\partial u}{\partial x_i}(x)
\frac{\partial v}{\partial x_i}(x)\,dx\,, \quad \|v\|=\left({\sum_{i=1}^n}
{\int_{\Omega}}\left(\frac{\partial u}{\partial x_i}(x)\right)^2\,dx\right)^{1/2}\,.$$
While in $L^2(\Omega)$, $(.,.)$ and $|.|$ represent the inner product 
and norm, respectively.

\begin{myrem} Milla Miranda and Medeiros \cite{mi-me} showed that in 
$V\cap H^2(\Omega)$ the norm $ {\left(|\Delta u|^2+\left\|
\frac{\partial u}{\partial \nu}\right\|^2_{H^{1/2}(\Gamma_1)}\right)}^{1/2}$ 
is equivalent to the norm $\|.\|_{H^2(\Omega)}$.
\end{myrem}

We assume that
\begin{equation}
\beta\geq \frac{n}{2\alpha_0\mu_0}\,.
\label{eq:2.1}
\end{equation}
To obtain the strong solution and consequently weak solution for system
 (1.1)--(1.6), we need the following results.

\begin{mypro} Let $u_1\in V\cap H^2(\Omega),\;u_2\in V$ and $\theta\in 
V\cap H^2(\Omega)$ satisfy 
\begin{equation}
\frac{\partial u_1}{\partial \nu}+\alpha (x)u_2=0\quad \mbox{on }
\Gamma_1\quad\mbox{and}\quad\frac{\partial \theta}{\partial \nu}+\beta \theta=0
\quad\mbox{on }\Gamma_1\,.
\label{eq:2.2}
\end{equation}
Then, for each $\varepsilon >0$, there exist w, y and z in $V\cap H^2(\Omega)$, 
such that
$$\|w-u_1\|_{V\cap H^2(\Omega)}<\varepsilon\,,\;\|z-u_2\|<\varepsilon\,,\;
\|y-\theta\|_{V\cap H^2(\Omega)}<\varepsilon\,,$$
with
$$\frac{\partial w}{\partial \nu}+\alpha (x)z=0\quad\mbox{on }\Gamma_1\quad
\mbox{and}\quad\frac{\partial y}{\partial \nu}+\beta y=0\quad\mbox{on }\Gamma_1\,.$$
\end{mypro}

\paragraph{Proof.} We assume the conclusion of Proposition 3 in \cite{mi-me}. 
So, it suffices to prove the existence of $y$. 

By the hypothesis $\Delta \theta \in L^2(\Omega)$, for each $\varepsilon >0$ 
there exists $y\in {\cal D} (\Omega)$ such that $|y-\Delta \theta|<\varepsilon$. 
Let q be solution of the elliptic problem
\begin{eqnarray*}
&-\Delta q=-y\quad\mbox{in }\Omega &\\
&q=0\quad\mbox{on }\Gamma_0 &\\
&\frac{\partial q}{\partial \nu}+\beta q=0\quad\mbox{on }\Gamma_1\,.&
\end{eqnarray*}  

On the other hand, we observe that $\theta $ is the solution of the above problem 
with $y=\Delta \theta$. Using results of elliptic regularity, cf. H. Brezis \cite{bre},
 we conclude that $q-\theta \in V\cap H^2(\Omega)$ and that there exists a positive 
constant C such that
$$\|q-\theta\|_{V\cap H^2(\Omega)}\leq C\,|y-\Delta \theta|\,.$$

\begin{mypro} If $\theta \in V$, then for each $\varepsilon >0$ there exists 
$q\in V\cap H^2(\Omega)$ satisfying $ \frac{\partial q}{\partial \nu}+\beta q=0$ on 
$\Gamma_1$ such that $\|\theta -q\|<\varepsilon $.
\end{mypro}
\paragraph{Proof.} Observe that the set
$$W=\left\{q\in V\cap H^2(\Omega); \frac{\partial q}{\partial \nu}+\beta q=0
\quad\mbox{on }\Gamma_1\right\}$$
is dense in $V$. This is so because $W$ is the domain of the operator 
$A=-\Delta $ determined by the triplet $\left\{V,L^2(\Omega),a(u,v)\right\}$, where
$$a(u,v)=((u,v))+(\beta u,v)_{L^2(\Gamma_1)}\,.$$
See for example J. L. Lions \cite{lions}. Hence, the result follows.

\section{Strong Solutions} \setcounter{equation}{0} 

In this section, we prove existence and uniqueness of a solution to (1.1)--(1.6) 
when $u^0, u^1$ and $\theta^0$ are smooth. First, we have the following result.

\begin{myth} Suppose that $u^0 \in V\cap H^2(\Omega)$, $u^1 \in V$, and $\theta^{0} 
\in V\cap H^2(\Omega)$ satisfy 
$$\frac{\partial u^0}{\partial \nu}+\alpha (x)u^1=0\quad\mbox{on }\Gamma_1\quad
\mbox{and}\quad\frac{\partial \theta^0}{\partial \nu}+\beta \theta^0=0\quad\mbox{on }\Gamma_1\,.$$
Then there exists a unique pair of real functions $\{u,\theta\}$ such that
\begin{eqnarray}
&u\in L^{\infty}_{\rm loc}(0,\infty;V\cap H^2(\Omega)),\;u'\in L^{\infty}_{\rm loc}
(0,\infty;V), \label{eq:3.1}&\\
&u''\in L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))
\label{eq:3.2}&\\
&\theta\in L^{\infty}_{\rm loc}(0,\infty;V\cap H^2(\Omega)),\;\;\theta'\in 
L^{\infty}_{\rm loc}(0,\infty;V)
\label{eq:3.3}&\\
&u''-\mu\Delta u+\displaystyle\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}=0
\hspace*{7pt}\mbox{in}\hspace*{7pt}L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))
\label{eq:3.4}&\\
&\frac{\partial u}{\partial \nu}+\alpha u'=0\hspace*{7pt}\mbox{in}\hspace*{7pt}
L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:3.5}&\\
&\theta'-\Delta \theta+\displaystyle\sum_{i=1}^n\displaystyle\frac{\partial u'}
{\partial x_i}=0\hspace*{7pt}\mbox{in}\hspace*{7pt}L^{\infty}_{\rm loc}(0,\infty;
L^2(\Omega))
\label{eq:3.6}&\\
&\frac{\partial \theta}{\partial \nu}+\beta \theta=0\hspace*{7pt}\mbox{in}
\hspace*{7pt}L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:3.7}&\\
&u(0)=u^0,\;\; u'(0)=u^1,\;\;\theta(0)=\theta^0\,.&
\label{eq:3.8}
\end{eqnarray}
\end{myth}

\paragraph{Proof.} We use the Galerkin method with a special basis in $V\cap 
H^2(\Omega)$. Recall that from Proposition 2.1 there exist sequences 
$(u_{\ell}^0)_{\ell\in {\mathbb N}}$, $(u_{\ell}^1)_{\ell\in {\mathbb N}}$ and 
$(\theta_{\ell}^0)_{\ell\in {\mathbb N}}$ of vectors in $V\cap H^2(\Omega)$ such that:
\begin{eqnarray}
&u_{\ell}^0\longrightarrow u^0\hspace*{7pt}\mbox{strongly}\hspace*{7pt}\mbox{in}
\hspace*{7pt}V\cap H^2(\Omega)
&\label{eq:3.9}\\
&u_{\ell}^1\longrightarrow u^1\hspace*{7pt}\mbox{strongly}\hspace*{7pt}\mbox{in}
\hspace*{7pt}V 
&\label{eq:3.10}\\
&\theta_{\ell}^0\longrightarrow \theta_0\hspace*{7pt}\mbox{strongly}\hspace*{7pt}
\mbox{in}\hspace*{7pt}V\cap H^2(\Omega)
&\label{eq:3.11}\\
&\frac{\partial u_{\ell}^0}{\partial \nu}+\alpha u_{\ell}^1=0\hspace*{7pt}\mbox{on}
\hspace*{7pt}\Gamma_1
&\label {eq:3.12}\\
&\frac{\partial \theta_{\ell}^0}{\partial \nu}+\beta \theta_{\ell}^0=0\hspace*{7pt}
\mbox{on}\hspace*{7pt}\Gamma_1\,.
&\label{eq:3.13}
\end{eqnarray}

For each $\ell \in{\mathbb N}$ pick $u_{\ell}^0$, $u_{\ell}^1$ and $\theta_{\ell}^0$ 
linearly independent, then define the vectors $w_1^{\ell}=u_{\ell}^0,\;w_2^{\ell}=
u^1_{\ell}$ and $w_3^{\ell}=\theta_{\ell}^0$, and then construct an orthonormal basis 
in $V\cap H^2(\Omega)$, 
$$\{w_1^{\ell},w_2^{\ell},...,w_j^{\ell},...\}\hspace{7pt}\mbox{for each}\hspace{4pt}
\ell\in{\mathbb N}\,.$$
For $\ell$ fixed and each $m \in{\mathbb N}$, we consider the subspace $W^{\ell}_{m}=
[w^{\ell}_1,w^{\ell}_2,...,w^{\ell}_ m]$ generated by the m-first vectors of the basis.
 Thus for $u_{\ell m}(t),\;\theta_{\ell m}(t)\in W^{\ell}_{m}$ we have
$$u_{\ell m}(t)=\sum_{j=1}^{m}g_{\ell jm}(t)w^{\ell }_j(x)\quad\mbox{and}\quad
\theta_{\ell m}(t)=\sum_{j=1}^{m}h_{\ell jm}(t)w^{\ell}_j(x)\,.$$

For each $m \in{\mathbb N}$, we find pair of functions $\{u_{\ell m}(t),
\theta_{\ell m}(t)\}$ in $W^{\ell}_{m}\times W^{\ell}_{m}$, such that for all 
$v\in W^{\ell}_{m}$ and all $w\in W^{\ell}_{m}$, 
\begin{eqnarray}
&(u''_{\ell m}(t),v)+\mu(t)((u_{\ell m}(t),v))+\mu(t)\int_{\Gamma_1}{\alpha (x)
u'_{\ell m}(t)vd\Gamma}& \nonumber \\
&+\sum_{i=1}^{n}\left(\frac{\partial \theta_{\ell m}}{\partial x_{i}}(t),v\right)=0\,,& 
\label{eq:3.14} \\
&(\theta'_{\ell m}(t),w)+((\theta_{\ell m}(t),w)) +\beta\int_{\Gamma_1}{\theta_{\ell m}
(t)w\,d\Gamma}
+\sum^{n}_{i=1}\left(\frac{\partial u'_{\ell m}}{\partial x_{i}}(t),w\right)=0\,,
\nonumber \\
&u_{\ell m}(0)=u_{\ell}^0,\;\;\;u'_{\ell m}(0)=u_{\ell}^1\quad \mbox{and}\quad 
\theta_{\ell m}(0)=\theta^0\,. &\nonumber
\end{eqnarray}

The solution $\left\{u_{\ell m}(t),\theta_{\ell m}(t)\right\}$ is defined on a certain 
interval $[0,t_m[$. This interval will be extended to any interval $[0,T]$, with $T>0$,
 by the use of the following a priori estimate.

\paragraph{Estimate I.} In (\ref{eq:3.14}) we replace  $v$ by $u_{\ell m}'(t)$ and 
$w$ by $\theta_{\ell m}(t)$. 
Thus
\begin{eqnarray*}
&\frac{1}{2}\frac{d}{dt}|u'_{\ell m}(t)|^2+\frac{1}{2}\frac{d}{dt}\left\{\mu(t)
\|u_{\ell m}(t)\|^2\right\}+\mu(t)\displaystyle\int_{\Gamma_1}{\alpha (x)(u'_{\ell m}
(t))^2d\Gamma}&\\
&+\sum_{i=1}^n{\left(\frac{\partial \theta_{\ell m}}{\partial x_i}(t),
u'_{\ell m}(t)\right)\;\leq \;|\mu'(t)|\|u_{\ell m}(t)\|^2}\,,&\\
&\frac{1}{2}\frac{d}{dt}|\theta_{\ell m}(t)|^2+\|\theta_{\ell m}(t)\|^2+
\beta\int_{\Gamma_1}{(\theta_{\ell m}(t))^2d\Gamma}+
\sum_{i=1}^{n}{\left(\frac{\partial {u'_{\ell m}}}{\partial x_i}(t),\theta_{\ell m}(t)
\right)}=0.&
\end{eqnarray*}
Define
$$E_1(t)=\frac{1}{2}\left\{|u'_{\ell m}(t)|^2+\mu(t)\|u_{\ell m}(t)\|^2+
|\theta_{\ell m}(t)|^2\right\}.$$
and we make use of the Gauss identity
$$\sum_{i=1}^n\left(\frac{\partial u_{\ell m}'}{\partial x_i}(t),\theta_{\ell m}(t)
\right)=-\sum_{i=1}^n\left(u_{\ell m}'(t),\frac{\partial \theta_{\ell m}}
{\partial x_i}(t)\right)+\sum_{i=1}^n\int_{\Gamma_1}{u'_{\ell m}(t)
\theta_{\ell m}(t)\nu_{i}d\Gamma}$$
to obtain
\begin{eqnarray*}
\lefteqn{ \frac{d}{dt}E_1(t)+\|\theta_{\ell m}(t)\|^2+\mu(t)\int_{\Gamma_1}{\alpha 
(x)(u'_{\ell m}(t))^2d\Gamma} }&&\\
\lefteqn{ +\sum_{i=1}^{n}\left(\frac{\partial \theta_{\ell m}}{\partial x_i}(t),
u'_{\ell m}(t)\right)
+\beta\int_{\Gamma_1}{(\theta_{\ell m}(t))^2d\Gamma} }&&\\
&\leq& \sum_{i=1}^{n}
{\int_{\Gamma_1}{u'_{\ell m}(t)\theta_{\ell m}(t)\nu_id\Gamma}+
\frac{|\mu'(t)|}{\mu(t)}E_1(t).}
\end{eqnarray*}
%
By the Cauchy-Schwarz inequality it follows that 
$${\sum_{i=1}^{n}\int_{\Gamma_1}{u'_{\ell m}(t)\theta_{\ell m}(t)\nu_id\Gamma}\leq 
\frac{n}{2\alpha_0\mu_0}\int_{\Gamma_1}{(\theta_{\ell m}(t))^2d\Gamma}+\frac{\alpha_0
\mu(t)}{2}\int_{\Gamma_1}{(u'_{\ell m}(t))^2d\Gamma}},$$ 
and this yields
\begin{eqnarray}
& {\frac{d}{dt}E_1(t)+\|\theta_{\ell m}(t)\|^2+\mu(t)\frac{\alpha_0}{2}\int_{\Gamma_1}
{(u'_{\ell m}(t))^2d\Gamma}}
+\left(\beta-\frac{n}{2\alpha_0\mu_0}\right)\int_{\Gamma_1}{(\theta_{\ell m}(t))^2d
\Gamma} & \nonumber \\ \label{eq:3.15}
&\leq\quad \frac{|\mu'(t)|}{\mu(t)}E_1(t)\,.&
\end{eqnarray}  
Integrating (\ref{eq:3.15}) over [0,t[, $0\leq t\leq t_m$, using (\ref{eq:2.1}) and 
applying Gronwall inequality, we conclude that there is a positive constant $C>0$, 
independent of $\ell$ and $m$, such that
\begin{equation}
E_1(t)+\int_0^t{\|\theta_{\ell m}(s)\|^2ds}\leq C.
\label{eq:3.16}
\end{equation}
Then there exists a subsequence  still denoted by $(u_{\ell m})_{m\in{\mathbb N}}$ and 
a subsequence still denoted by $(\theta_{\ell m})_{m\in{\mathbb N}}$, such that
\begin{eqnarray}
& (u_{\ell m})_{m\in{\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^{\infty}_{loc}(0,\infty;V)
&\label{eq:3.17}\\
& (u'_{\ell m})_{m\in{\mathbb N}} \;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^{\infty}_{loc}(0,\infty;L^2(\Omega))
&\label{eq:3.18}\\
& (\theta_{\ell m})_{m\in{\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^2_{loc}(0,\infty;V).
\label{eq:3.19}
&\end{eqnarray}

\paragraph{Estimate II.} Differentiating in (\ref{eq:3.14}) with respect to t, taking 
$v=u''_{\ell m}(t)$ and $w=\theta'_{\ell m}(t)$, we obtain 
\begin{eqnarray}
\lefteqn{\frac{d}{dt}E_2(t)+\mu(t)\int_{\Gamma_1}{\alpha (x)(u''_{\ell m}(t))^2d\Gamma}
+\mu'(t)\int_{\Gamma_1}{\alpha(x) u'_{\ell m}(t)u''_{\ell m}(t)\,d\Gamma} }&&
\nonumber \\
\lefteqn{+\|\theta'_{\ell m}(t)\|^2
+ \beta\int_{\Gamma_1}{ (\theta'_{\ell m}(t))^2d\Gamma} } &&\label{eq:3.20}\\
&=&\frac{1}{2}\mu'(t)\|u'_{\ell m}(t)\|^2
-\mu'(t)((u_{\ell m}(t),u''_{\ell m}(t)))
+\sum_{i=1}^n\int_{\Gamma_1}{\theta'_{\ell m}(t)u''_{\ell m}(t)\nu_i \,d\Gamma},
\nonumber 
\end{eqnarray}
where
$$E_2(t)=\frac{1}{2}\left\{|u''_{\ell m}(t)|^2+\mu(t)\|u'_{\ell m}(t)\|^2+|
\theta'_{\ell m}(t)|^2\right\}.$$

Put $v=\frac{\mu'(t)}{\mu(t)}u''_{\ell m}(t)$ in $(\ref{eq:3.14})_1$, to obtain
\begin{eqnarray*}
\mu'(t)((u_{\ell m}(t),u''_{\ell m}(t)))&=&-\frac{\mu'(t)}{\mu(t)}|u''_{\ell m}(t)|^2+
\mu'(t)\int_{\Gamma_1}{\alpha (x)u'_{\ell m}(t)u''_{\ell m}(t)d\Gamma}\\
&&-\frac{\mu'(t)}{\mu(t)}\sum_{i=1}^{n}\left(\frac{\partial \theta_{\ell m}}
{\partial x_i}(t),u''_{\ell m}(t)\right).
\end{eqnarray*}
Replacing this last expression in (\ref{eq:3.20}) we obtain
\begin{eqnarray}
\lefteqn{ \frac{d}{dt}E_2(t)+\mu(t)\int_{\Gamma_1}{\alpha (x) (u''_{lm}(t))^2d\Gamma}+
\|\theta'_{\ell m}(t)\|^2
+ \beta \int_{\Gamma_1}{(\theta'_{\ell m}(t))^2d\Gamma} }&&\nonumber\\
&=&\frac{1}{2}\mu'(t)\|u'_{\ell m}(t)\|^2+\frac{\mu'(t)}{\mu(t)}|u''_{\ell m}(t)|^2+
\frac{\mu'(t)}{\mu(t)}\sum_{i=1}^{n}\left(\frac{\partial \theta_{\ell m}}{\partial x_i}(t),u''_{\ell m}(t)\right) \nonumber \\
&&+\sum_{i=1}^n\int_{\Gamma_1}{\theta'_{\ell m}(t)u''_{\ell m}(t)\nu_i d\Gamma}\,.
\label{eq:3.21}
\end{eqnarray}
Making use of the Cauchy-Schwarz inequality in the last two terms of the 
right-hand-side of (\ref{eq:3.21}), we obtain 
\begin{equation}
\frac{\mu'(t)}{\mu(t)}\sum_{i=1}^n\left|\left(\frac{\partial \theta_{\ell m}}
{\partial x_i}(t),u''_{\ell m}(t)\right)\right|\leq \frac{1}{2}\frac{|\mu'(t)|}
{\mu(t)}|u''_{\ell m}(t)|^2+\frac{n}{2}\frac{|\mu'(t)|}{\mu(t)}\|\theta_{\ell m}(t)\|^2
\label{eq:3.22}
\end{equation}
and
\begin{equation}
\sum_{i=1}^{n}\int_{\Gamma_1}{\theta'_{\ell m}(t)u''_{\ell m}(t)\nu_i\,d\Gamma}\leq 
\frac{\mu_0\alpha_0}{2}\int_{\Gamma_1}{(u''_{\ell m}(t))^2d\Gamma}
+\frac{n}{2\mu_0\alpha_0}\int_{\Gamma_1}(\theta'_{\ell m}(t))^2d\,\Gamma.
\label{eq:3.23}
\end{equation}
Combining (\ref{eq:3.21}), (\ref{eq:3.22}) and (\ref{eq:3.23}) we obtain
\begin{eqnarray}
& \frac{d}{dt}E_2(t)+\mu(t)\frac{\alpha_0}{2}\int_{\Gamma_1}{(u''_{\ell m}(t))^2d\Gamma}
+\|\theta'_{\ell m}(t)\|^2
+\left(\beta-\frac{n}{2\mu_0\alpha_0}\right)\int_{\Gamma_1}{(\theta'_{\ell m}(t))^2d
\Gamma} & \nonumber \\ 
&\leq\quad \frac{1}{2}\frac{|\mu'(t)|}{\mu_0}\mu(t)\|u'(t)\|^2
+\frac{3}{2}\frac{|\mu'(t)|}{\mu_0}|u''_{\ell m}(t)|^2 
+\frac{n|\mu'(t)|}{2\mu_0}\|\theta_{\ell m}(t)\|^2\,.&
\end{eqnarray}
From (\ref{eq:2.1}) it follows that
$$
\frac{d}{dt}E_2(t)+\|\theta'_{\ell m}(t)\|^2 
+\leq 4
\frac{|\mu'(t)|}{\mu_0}E_2(t)+\frac{n|\mu'(t)|}{2\mu_0}\|\theta_{\ell m}(t)\|^2.$$

To complete this estimate, we integrate the above inequality over [0,t], $t\leq T$. 
Now we show that $u''_{\ell m}(0)$ and $\theta'_{\ell m}(0)$ are bounded in 
$L^2(\Omega)$. For this end put $v=u''_{\ell m}(t)$, $w=\theta'_{\ell m}(t)$, and 
$t=0$. Because of the choice of basis we have
\begin{eqnarray*}
\lefteqn{|u''_{\ell m}(0)|^2} &&\\ 
&\leq&\left(\mu(0)|\Delta u^0_{\ell}|+\sum_{i=1}^n{\left|
\frac{\partial \theta_{\ell}^0}{\partial x_i}\right|}\right)|u''_{\ell m}(0)|+\mu(0)
\int_{\Gamma_1}{\left(\frac{\partial u^0_{\ell}}{\partial \nu}+\alpha(x)
u^1_{\ell}\right)u''_{\ell m}(0)d\Gamma}
\end{eqnarray*}
and
$$|\theta'_{\ell m}(0)|^2\leq \left(|\Delta \theta^0_{\ell }|+\sum_{i=1}^{n}\left|
\frac{\partial u^1_{\ell}}{\partial x_i}\right|\right)|\theta'_{\ell m}(0)|+
\int_{\Gamma_1}{\left(\frac{\partial \theta^0_{\ell}}{\partial \nu}+\beta 
\theta^0_{\ell}\right)\theta'_{\ell m}(0)d\Gamma }\,.$$

Since by hypothesis $\frac{\partial u^0_{\ell}}{\partial \nu}+\alpha(x) u^1_{\ell}=0$ 
and $\frac{\partial \theta^0_{\ell}}{\partial \nu}+\beta \theta^0_{\ell}=0\;\mbox{in}\;
\Gamma_1$, it follows that $(u''_{\ell m}(0))_{m\in{\mathbb N}}\;\mbox{and}\;
(\theta'_{\ell m}(0))_{m\in{\mathbb N}}$ are bounded in $L^2(\Omega)$.
Consequently for a fixed $\ell$,
\begin{eqnarray}
&(u'_{\ell m})_{m\in{\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^{\infty}_{\rm loc}(0,\infty;V),
&\label{eq:3.24}\\
&(u''_{\ell m})_{m\in{\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega)),
&\label{eq:3.25}\\
&(\theta'_{\ell m})_{m\in{\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))
&\label{eq:3.26}\\
&(\theta'_{\ell m})_{m\in {\mathbb N}}\;\;\mbox{is}\;\;\mbox{bounded}\;\;\mbox{in}\;\;
L^2_{\rm loc}(0,\infty;V)
&\label{eq:3.27}
\end{eqnarray}

From (\ref{eq:3.17})--(\ref{eq:3.19}) and (\ref{eq:3.24})--(\ref{eq:3.27}), by 
induction and the diagonal process, we obtain subsequences, denoted with the same 
symbol as the original sequences, $(u_{\ell m_{n}})_{n\in{\mathbb N}}$ and 
$(\theta_{\ell m_{n}})_{n\in{\mathbb N}}$; and functions $u_{\ell}:\Omega \times ]0,
\infty[\longrightarrow {\mathbb R}$ and $\theta_{\ell}:\Omega \times]0,\infty[
\longrightarrow {\mathbb R}$ such that:
\begin{eqnarray}
&&u_{\ell m}\longrightarrow u_{\ell}\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}
(0,\infty;V)
\label{eq:3.28}\\
&&u'_{\ell m}\longrightarrow u'_{\ell}\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}
(0,\infty;V)
\label{eq:3.29}\\
&&u''_{\ell m}\longrightarrow u''_{\ell}\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}
(0,\infty;L^2(\Omega))
\label{eq:3.30}\\
&&u'_{\ell m}\longrightarrow u'_{\ell}\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}
(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:3.31}\\
&&\theta_{\ell m}\longrightarrow \theta_{\ell}\;\;\mbox{weakly  in}\;\;L^2_{\rm loc}
(0,\infty;V)
\label{eq:3.32}\\
&&\theta'_{\ell m}\longrightarrow \theta'_{\ell}\;\;\mbox{weak star in}\;\;
L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))
\label{eq:3.33}\\
&&\theta_{\ell m}\longrightarrow \theta_{\ell}\;\;\mbox{weak star in}\;\;
L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1)).
\label{eq:3.34}
\end{eqnarray}

Next, we multiply both sides of (\ref{eq:3.14}) by $\psi \in {\cal D}(0,\infty)$ and 
integrate with respect to $t$.
From (\ref{eq:3.28})--(\ref{eq:3.34}), for all $v,w\in V^{\ell}_m$ we obtain
\begin{eqnarray}
&&\int_0^\infty{(u''_{\ell}(t),v)\psi(t) \,dt}+\int_0^\infty{\mu(t)((u_{\ell}(t),v))
\psi(t) \,dt}\label{eq:3.35}\\
&&+ \int_0^\infty\int_{\Gamma_1}{\alpha (x) u'_{\ell}(t)\,v\,\psi(t)\,d\Gamma}\,dt+
\sum_{i=1}^n\int_0^\infty\left(\frac{\partial \theta_{\ell}}{\partial x_i}(t),v\right)
\psi(t)\,dt=0,\nonumber \\
&&\int_0^\infty{(\theta'_{\ell},w)\psi(t)\,dt}+\int_0^\infty{((\theta_{\ell}(t),w))
\psi(t)\,dt}\label{eq:3.36}\\
&&+\beta\int_0^\infty{\int_{\Gamma_1} \theta_{\ell}(t)w\psi(t) d\Gamma}
+\sum_{i=1}^n\int_0^{\infty}{\left(\frac{\partial u'_{\ell}}{\partial x_i}(t),w\right)
\psi(t) \,dt}=0.\nonumber
\end{eqnarray}

Since $\{w_{1}^{\ell},w_{2}^{\ell},...\}$ is a basis of $V\cap H^2(\Omega)$,
then by denseness it follows that the last two equalities are true for all $v$ and $w$
 in $V\cap H^2(\Omega)$. Also notice that (\ref{eq:3.17})--(\ref{eq:3.19}) and 
(\ref{eq:3.24})--(\ref{eq:3.27}) hold for all $\ell \in{\mathbb N}$. Then by the same 
process used in obtaining of (\ref{eq:3.28})--(\ref{eq:3.34}), we find diagonal 
subsequences denoted as the original sequences, $(u_{\ell_{\ell}})_{\ell 
\in{\mathbb N}}$ and $\theta_{\ell_{\ell}})_{\ell \in{\mathbb N}}$, and functions 
$u:\Omega \times ]0,\infty[\longrightarrow {\mathbb R},\theta:\Omega \times ]0,
\infty[\longrightarrow {\mathbb R}$ such that:
\begin{eqnarray}
&&u_{\ell}\longrightarrow u\;\;\mbox{weak star in}\;\;L^\infty_{\rm loc}(0,\infty;V)
\label{eq:3.37}\\
&& u'_{\ell}\longrightarrow u'\;\;\mbox{weak star in }\;\;L^\infty_{\rm loc}(0,\infty;V)
\label{eq:3.38}\\
&& u''_{\ell}\longrightarrow u''\;\;\mbox{weak star in}\;\;L^\infty_{\rm loc}(0,\infty;
L^2(\Omega))
\label{eq:3.39}\\
&& u'_{\ell}\longrightarrow u'\;\;\mbox{weak star in}\;\;L^\infty_{\rm loc}(0,\infty;
H^{1/2}(\Gamma_1))
\label{eq:3.40}\\
&& \theta_{\ell}\longrightarrow \theta\;\;\mbox{weakly in}\;\;L^2_{\rm loc}(0,\infty;V)
\label{eq:3.41}\\
&&\theta'_{\ell}\longrightarrow \theta'\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}
(0,\infty;L^2(\Omega))
\label{eq:3.42}\\
&&\theta_{\ell}\longrightarrow \theta\;\;\mbox{weak star in}\;\;L^2_{\rm loc}(0,\infty
;H^{1/2}(\Gamma_1))
\label{eq:3.43}
\end{eqnarray}

Taking limits in (\ref{eq:3.35}) and in (\ref{eq:3.36}), using the convergences showed 
in (\ref{eq:3.37})--(\ref{eq:3.43}), and using the fact that $V\cap H^2(\Omega)$ is 
dense in V, we obtain that for all $\psi$ in ${\cal D}(0,\infty)$ and $v,w\in V$,
\begin{eqnarray}
&&\int_0^{\infty}{(u''(t),v)\psi(t)\, dt}+\int_0^{\infty}{\mu(t)((u(t),v))\psi(t)\, dt}
\label{eq:3.44}\\         && +\int_0^{\infty}{\int_{\Gamma_1}{\alpha (x)u'(t)v\psi(t)\,
 d\Gamma} dt}+\sum_{i=1}^n\int_0^\infty\left(\frac{\partial \theta}{\partial x_i}(t),
v\right)\psi(t)\,dt=0,\nonumber \\
&&\int_0^{\infty}{(\theta'(t),w)\psi(t)\,dt}+\int_0^\infty{((\theta(t),w))\psi(t)\,dt}
\label{eq:3.45}\\
&&+\beta\int_0^{\infty}{\int_{\Gamma_1}{\theta(t)w\psi(t)d\Gamma}dt}+\sum_{i=1}^n
\int_0^\infty\left(\frac{\partial u'}{\partial x_i}(t),w\right)\psi(t)\,dt=0\,. 
\nonumber
\end{eqnarray}

Since ${\cal D}(\Omega)\subset V$, by (\ref{eq:3.44}) and (\ref{eq:3.45}) it follows
 that
\begin{eqnarray}
&u''-\mu\,\Delta u+\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}=0\;\;\mbox{in}\;\;
L^{2}_{\rm loc}(0,\infty;L^2(\Omega))\,, &\label{eq:3.46}\\
&\theta'-\Delta \theta+\sum_{i=1}^n\frac{\partial u'}{\partial x_i}=0\;\;\mbox{in}\;\;
L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))\,.
\label{eq:3.47}
\end{eqnarray}
Since $u\in L^{\infty}_{\rm loc}(0,\infty;V)$ and $\theta \in L^{2}_{\rm loc}(0,\infty;
V)$, we take into account (\ref{eq:3.46}) and (\ref{eq:3.47}) to deduce that $\Delta u,
\;\Delta \theta\in L^{2}_{\rm loc}(0,\infty;L^2(\Omega))$. Therefore
\begin{equation} 
\frac{\partial u}{\partial \nu},\frac{\partial \theta}{\partial \nu}\in 
L^{2}_{\rm loc}
(0,\infty;H^{-{1/2}}(\Gamma_1))
\label{eq:3.48}
\end{equation}

Multiply (\ref{eq:3.46}) by $v\psi$ and (\ref{eq:3.47}) by $w\psi$ with $v,\,w\in V$ 
and $\psi \in {\cal D}(0,\infty)$. By integration and use of the Green's formula, we 
obtain
\begin{eqnarray}
&&\int_0^{\infty}{(u''(t),v)\psi(t)\, dt}+\int_0^{\infty}{\mu(t)((u(t),v))\psi(t)\, dt}
\label{eq:3.49}\\
&&-\int_0^{\infty}{\langle\mu(t)\frac{\partial u}{\partial \nu}(t),v\rangle\psi(t)\, dt}
+\sum_{i=1}^n\int_0^{\infty}{\left(\frac{\partial \theta}{\partial x_i}(t),v\right)
\psi(t)\,dt}=0\,, \nonumber\\
&&\int_0^{\infty}{(\theta'(t),w)\psi(t)\,dt}+\int_0^{\infty}{((\theta(t),w))\psi(t)\,dt}
\label{eq:3.50}\\
&&-\int_0^{\infty}{\langle\frac{\partial \theta}{\partial \nu}(t),w\rangle\psi(t)\,dt}+
\sum_{i=1}^n\int_0^{\infty}{\left(\frac{\partial u'}{\partial x_i}(t),w\right)\psi(t)\,
dt=0},\nonumber
\end{eqnarray}
where $\langle.,.\rangle$ denotes the duality pairing of $H^{-{1/2}}(\Gamma_1)\times 
H^{1/2}(\Gamma_1)$.

Comparing (\ref{eq:3.44}) with (\ref{eq:3.49}) and (\ref{eq:3.45}) with 
(\ref{eq:3.50}), we obtain that for all $\psi$ in ${\cal D}(0,\infty)$ and for all 
$v,w\in V$,
$$\int _0^{\infty}{\langle\frac{\partial u}{\partial \nu}(t)+\alpha (x) u'(t),
v\rangle\psi(t)\, dt} =0,\qquad \int_0^{\infty}{\langle\frac{\partial \theta}{\partial
 \nu}(t)+\beta \theta(t),w\rangle\psi(t)\,dt}=0.$$
From (\ref{eq:3.38}), (\ref{eq:3.43}) and (\ref{eq:3.48}) it follows that
\begin{eqnarray*}
&\frac{\partial u}{\partial \nu}+\alpha u'=0\;\;\mbox{in}\;\;L^{\infty}_{\rm loc}
(0,\infty;H^{-{1/2}}(\Gamma_1)),&\\
&\frac{\partial \theta}{\partial \nu}+\beta\theta=0\;\;\mbox{in}\;\;L^2_{\rm loc}
(0,\infty;H^{-{1/2}}(\Gamma_1)).&
\end{eqnarray*}
Since $\alpha u'\in L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))$ and $\beta 
\theta\in L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))$, it follows that
\begin{eqnarray}
&&\frac{\partial u}{\partial \nu}+\alpha u'=0\;\;\;\mbox{in}\;\;\;L^2_{\rm loc}
(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:3.51}\\
&& \frac{\partial \theta}{\partial \nu}+\beta\,\theta=0\;\;\mbox{in}\;\;
L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1)).
\label{eq:3.52}
\end{eqnarray}

To complete the proof of the Theorem 3.1, we shall show that $u$ and $\theta$ are in 
$L^{\infty}_{\rm loc}(0,\infty;H^2(\Omega))$. In fact, for all $T>0$ the pair 
$\{u,\theta\}$ is the solution to
\begin{eqnarray}
&-\Delta u=-\frac{1}{\mu}\left(u''+\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}
\right)\;\;\;\mbox{in}\;\;\;\Omega \times ]0,T[&\nonumber\\
&-\Delta \theta=-\theta'-\displaystyle \frac{\partial u'}{\partial x_i}\quad\mbox{in}
\quad\Omega \times ]0,T[\nonumber\\
&u=\theta=0\;\quad\mbox{on}\;\quad\Gamma_0\times ]0,T[\nonumber\\
&\frac{\partial u}{\partial \nu}=-\alpha u'\;\;\;\mbox{on}\;\;\;\Gamma_1\times]0,T[&\\
&\frac{\partial \theta}{\partial \nu}=-\beta \theta\;\quad\mbox{on}\;\quad\Gamma_1
\times]0,T[.&\nonumber
\label{eq:3.53}
\end{eqnarray}
In view of (\ref{eq:3.39}), (\ref{eq:3.41}) and (\ref{eq:3.38}) we have $u''$ and 
$\frac{\partial \theta}{\partial x_i}$ are in $L^{\infty}_{\rm loc}(0,\infty;
L^2(\Omega))$ and $\alpha u'$ is in $L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))$.
 Thus by results on elliptic regularity, it follows that $u\in L^{\infty}_{\rm loc}
(0,\infty;V\cap H^2(\Omega))$. In the same manner it follows that $\theta \in 
L^{\infty}_{\rm loc}(0,\infty;H^2(\Omega))$.

Uniqueness of the solution $\{u,\theta\}$ is showed by the standard energy method. 
The verification of the initial conditions is done through the convergences in 
(\ref{eq:3.37})--(\ref{eq:3.43}). \cqd

Next, we establish a result on existence and uniqueness of global solutions.

\begin{mycor} Under the supplementary hypothesis $\mu'\in L^1(0,\infty)$, the pair of 
functions $\{u,\theta\}$ obtained by Theorem 3.1 satisfies   
\begin{eqnarray*}
& u\in L^{\infty}(0,\infty;V\cap H^2(\Omega)),\;\;u'\in L^{\infty}(0,\infty;V),\;\;
\theta\in L^{\infty}(0,\infty;V\cap H^2(\Omega))&\\
&\frac{\partial u}{\partial \nu}+\alpha u'=0\hspace{4pt}\mbox{and}\hspace{4pt} 
\frac{\partial \theta}{\partial \nu}+\beta \theta=0\hspace{4pt}\mbox{in}\hspace{4pt}
L^2(0,\infty;L^2(\Gamma_1))& \\
&u(0)=u^0,\;\;u'(0)=u^1\;\;\mbox{and}\;\;\theta(0)=\theta^0\,.& 
\end{eqnarray*}
\end{mycor}

\section{Weak Solutions} \setcounter{equation}{0}

In this section, we find a solution for the system (\ref{eq:1.1})--(\ref{eq:1.6}) 
with initial data $u^0\in V$, $u^1\in L^2(\Omega)$ and $\theta^0\in V$. To reach this
 goal we approximate $u^0$, $u^1$ and $\theta^0$ by sequences of vectors in 
$V\cap H^2(\Omega)$, and we use the Theorem~3.1. 
\begin{myth}
If $\{u^0,u^1,\theta^0\}\in V\times L^2(\Omega)\times V$, then for each real number $
T>0$ there exists a unique pair of real functions $\{u,\theta\}$ such that:
\begin{eqnarray}
&u\in C([0,T];V)\cap C^1([0,T];L^2(\Omega)),\;\;\;\theta\in C([0,T];L^2(\Omega))
&\label{eq:4.1}\\
&u''-\mu\,\Delta u+\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}=0\hspace*{7pt}
\mbox{in}\hspace*{7pt}L^2(0,T;V')
&\label{eq:4.2}\\
&\theta'-\Delta \theta+\sum_{i=1}^n\frac{\partial u'}{\partial x_i}=0\hspace*{7pt}
\mbox{in}\hspace*{7pt}L^2(0,T;V')
&\label{eq:4.3}\\
&\frac{\partial u}{\partial \nu}+\alpha u'=0\hspace*{7pt}\mbox{in}
\hspace*{7pt}L^2(0,T;L^2(\Gamma_1))
&\label{eq:4.4}\\
&\frac{\partial \theta}{\partial \nu}+\beta \theta=0\hspace*{7pt}
\mbox{in}\hspace*{7pt}L^2(0,T;L^2(\Gamma_1))
&\label{eq:4.5}\\
&u(0)=u^0,\;\;u'(0)=u^1,\;\;\mbox{and}\;\;\theta(0)=\theta^0.
&\label{eq:4.6}
\end{eqnarray}
\end{myth}

\paragraph{Proof.} Let $(u^0_p)_{p\in{\mathbb N}},\;(u^1_p)_{p\in {\mathbb N}},\;
(\theta^0_p)_{p\in {\mathbb N}}$ be sequences in $V\cap H^2(\Omega)$ such that
$$u^0_p\longrightarrow u^0\;\;\mbox{in}\;\;V,\;\;
u^1_p\longrightarrow u^1\;\;\mbox{in}\;\;L^2(\Omega)\;\mbox{and}\;
\theta^0_p\longrightarrow \theta^0\;\;\mbox{in}\;\;V$$
with  $$\frac{\partial u^0_p}{\partial \nu}+\alpha (x) u^1_p=0\;\;\mbox{on}\;\;
\Gamma_1\;\;\;\mbox{and}\;\;\; \frac{\partial \theta^0_p}{\partial \nu}+\beta 
\theta^0_p=0\;\;\mbox{on}\;\;\Gamma_1.$$
Let $\{u_p,\theta_p\}_{p\in {\mathbb N}}$ be a sequence of strong solutions to 
(\ref{eq:1.1})--(\ref{eq:1.6}) with initial data 
$\{u^0_p,u^1_p,\theta^0_p\}_{p\in{\mathbb N}}$. 
Using the same arguments as in the preceding section, we obtain the following estimates
\begin{eqnarray}
&(u_p)_{p\in{\mathbb N}}\;\;\mbox{is bounded in}\;\;L^{\infty}_{\rm loc}(0,\infty;V)
&\label{eq:4.7}\\
&(u'_p)_{p\in{\mathbb N}}\;\;\mbox{is bounded in}\;\;L^{\infty}_{\rm loc}(0,\infty;V)
&\label{eq:4.8}\\
&(u'_p)_{p\in{\mathbb N}}\;\;\mbox{is bounded in}\;\;L^{\infty}_{\rm loc}(0,
\infty;H^{1/2}(\Gamma_1))
&\label{eq:4.9}\\
&\left(\frac{\partial u_p}{\partial \nu}\right)_{p\in{\mathbb N}}\;\;\mbox{is bounded 
in}\;\;L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
&\label{eq:4.10}\\
&(\theta_p)_{p\in{\mathbb N}}\;\;\mbox{is bounded in}\;\;L^2_{\rm loc}(0,\infty;V)
&\label{eq:4.11}\\
&(\theta_p)_{p\in{\mathbb N}}\;\;\mbox{is bounded in}\;\;
L^{\infty}_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
&\label{eq:4.12}\\
& \left(\frac{\partial \theta_p}{\partial \nu}\right)_{p\in{\mathbb N}}\;\;
\mbox{is bounded in }\;\;L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1)).
&\label{eq:4.13}
\end{eqnarray}
Note that (\ref{eq:4.10}) and (\ref{eq:4.13}) follow as a consequence of 
\begin{equation}
\begin{array}{l}
\frac{\partial u_p}{\partial \nu}+\alpha u_p'=0\;\;\mbox{in}\;\;
L^{\infty}(0,\infty;H^{1/2}(\Gamma_1))\\[8pt]
\frac{\partial \theta_p}{\partial \nu}+\beta\theta_p=0\;\;\mbox{in}\;\;L^{\infty}
(0,\infty;H^{1/2}(\Gamma_1)).
\end{array}
\label{eq:4.14}
\end{equation}

From (\ref{eq:4.7})--(\ref{eq:4.13}) there exist subsequences of $(u_p)_{p
\in{\mathbb N}}$ and $(\theta_p)_{p\in{\mathbb N}}$, still denoted as the original 
sequences, and functions $u:\Omega\times ]0,\infty[\to {\mathbb R}$, 
$\theta:\Omega\times ]0,\infty[\to {\mathbb R}$, 
$\varphi_1:\Gamma_1\times ]0,\infty[\to {\mathbb R}$, $\varphi_2:\Gamma_1\times 
]0,\infty[\to {\mathbb R}$, 
$\chi_1:\Gamma_1\times ]0,\infty[\to {\mathbb R}$, and 
$\chi_2:\Gamma_1\times ]0,\infty[\to {\mathbb R}$, such that
\begin{eqnarray}
&&u_p\to u\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}(0,\infty;V)
\label{eq:4.15}\\
&&u'_p\to u'\;\;\mbox{weak star in}\;\;L^{\infty}_{\rm loc}(0,\infty;L^2(\Omega))
\label{eq:4.16}\\
&&u'_p\to \varphi_1\;\;\mbox{weakly in}\;\;L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:4.17}\\
&&\frac{\partial u_p}{\partial \nu}\to \varphi_2\;\;\mbox{weakly in}\;\;
L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:4.18}\\
&&\theta_p\to \theta\;\;\mbox{weakly in}\;\;L^2_{\rm loc}(0,\infty;V)
\label{eq:4.19}\\
&&\theta_p\to \chi_1\;\;\mbox{weakly in}\;\;L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1))
\label{eq:4.20}\\
&&\frac{\partial \theta_p}{\partial \nu}\to \chi_2\;\;\mbox{weakly in}\;\;
L^2_{\rm loc}(0,\infty;H^{1/2}(\Gamma_1)).
\label{eq:4.21}
\end{eqnarray}
Moreover, from Theorem 3.1,
\begin{eqnarray}
&u''_p-\mu\Delta u_p+\sum _{i=1}^n\frac{\partial \theta_p}{\partial x_i}=0\;\;
\mbox{in}\;\;L^{\infty}_{loc}(0,\infty;L^2(\Omega)),&
\label{eq:4.22}\\
&\theta_p'-\Delta \theta_p +\sum_{i=1}^n\frac{\partial u_p'}{\partial x_i}=0\;\;
\mbox{in}\;\;L^{\infty}_{loc}(0,\infty;L^2(\Omega))\,.&
\label{eq:4.23}
\end{eqnarray}
Multiplying (\ref{eq:4.22}) and (\ref{eq:4.23}) by $v\psi$ and $w\phi$ respectively, 
with $v$ and $w$  in V and $\phi\;\mbox{in}\;{\cal D}(0,\infty)$ , we deduce the 
equalities
\begin{eqnarray*}
&&-\int_0^{\infty}{(u'_p(t),v)\phi'(t)}dt+\int_0^{\infty}{\mu(t)((u_p(t),v))\phi(t)}dt\\
&&+\int_0^{\infty}{\int_{\Gamma_1}{\alpha(x) u'_p(t)v\phi(t) d\,\Gamma}dt}+\sum_{i=1}^n\int_0^{\infty}\left(\frac{\partial \theta_p}{\partial x_i}(t),v\right)\phi dt=0\\
&&-\int_0^{\infty}(\theta_p(t),w)\phi'(t)dt+\int_0^{\infty}((\theta_p(t),w))dt\\
&&+\beta\int_0^{\infty}\int_{\Gamma_1}\theta_p(t)w\phi(t)d\,\Gamma dt+\sum_{i=1}^n
\int_0^{\infty}\left(\frac{\partial u'_p}{\partial x_i}(t),w\right)\phi(t)dt=0.
\end{eqnarray*}
Taking the limit, as $p\longrightarrow \infty$, from (\ref{eq:4.15})--(\ref{eq:4.21}) 
we conclude that
\begin{eqnarray}
&&-\int_0^{\infty}(u'(t),v)\phi'(t)dt+\int_0^{\infty}\mu(t)((u(t),v))\phi(t)
\label{eq:4.24}\\
&&+\int_0^{\infty}\int_{\Gamma_1}\alpha(x) u'(t)v\phi(t)d\,\Gamma dt
+\sum_{i=1}^n\int_0^{\infty}\left(\frac{\partial \theta}{\partial \nu}(t),v\right)
\phi(t)dt=0\nonumber \\
&&-\int_0^{\infty}(\theta(t),w)\phi'(t)dt+\int_0^{\infty}((\theta(t),w))\phi(t)dt 
\label{eq:4.25} \\
&&+\beta\int_0^{\infty}\int_{\Gamma_1}\theta(t)w\phi(t)d\,\Gamma dt +\sum_{i=1}^n
\int_0^{\infty}\left(\frac{\partial u'}{\partial x_i},w \right) \phi(t)dt=0.\nonumber
\end{eqnarray}
In view of (\ref{eq:4.24}) and (\ref{eq:4.25}), for $v$ and $w\in {\cal D}(\Omega),$ 
we obtain
\begin{equation}
\begin{array}{c}
u''-\mu\Delta u+\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}=0\;\;\mbox{in}\;\;
H^{-1}_{loc}(0,\infty;L^2(\Omega))\\[8pt]
\displaystyle \theta'-\Delta \theta+\sum_{i=1}^n\frac{\partial u'}{\partial x_i}=0\;\;
\mbox{in}\;\;H^{-1}_{loc}(0,\infty;L^2(\Omega)).
\end{array}
\label{eq:4.26}
\end{equation}
As shown in M. Milla Miranda \cite{milla}, from (\ref{eq:4.8}) follows that for 
$T>0$
\begin{equation}
u_p''\longrightarrow u''\;\;\mbox{weakly in}\;\;H^{-1}(0,T;L^2(\Omega)).
\label{eq:4.27}
\end{equation}
Thus, from (\ref{eq:4.19}), (\ref{eq:4.22}) and (\ref{eq:4.27}) we conclude that
\begin{equation}
\Delta u_p\longrightarrow \Delta u\;\;\mbox{weakly in}\;\;H^{-1}(0,T;L^2(\Omega)).
\label{eq:4.28}
\end{equation}
Furthermore, from (\ref{eq:4.15}) and (\ref{eq:4.28}) we obtain
$\frac{\partial u}{\partial \nu}\;\mbox{in}\; H^{-1}(0,T;H^{-{1/2}}(\Gamma_1))$
 and 
\begin{equation}
\frac{\partial u_p}{\partial \nu}\to \frac{\partial u}{\partial \nu}\;\;
\mbox{weakly in}\;\;H^{-1}(0,T;H^{-{1/2}}(\Gamma_1)).
\label{eq:4.29}
\end{equation}
To prove that $\varphi_1=u'$ and $\varphi_2=\frac{\partial u}{\partial \nu}$, we use 
(\ref{eq:4.18}) and the fact that 
\begin{equation}
\frac{\partial u_p}{\partial \nu}\to \varphi_2\;\;\mbox{weakly in}\;\;H^{-1}
(0,T;H^{1/2}(\Gamma_1)).
\label{eq:4.30}
\end{equation}
Whence we conclude that $\varphi_2=\frac{\partial u}{\partial \nu}$ is in 
$L^2(0,T;L^2(\Gamma_1)),$ for all $T>0$. Also from (\ref{eq:4.15}), cf. M. 
Milla Miranda \cite{milla}, we get 
\begin{equation}
u_p'\longrightarrow u'\;\;\mbox{weakly in}\;\;H^{-1}(0,T;H^{1/2}(\Gamma_1));
\label{eq:4.31}
\end{equation}
and from (\ref{eq:4.17}) and (\ref{eq:4.31}) we have $u'=\varphi_1$ in 
$L^{\infty}(0,T;H^{1/2}(\Gamma_1))$.

Next, we shall prove that $\chi_1=\theta$ and $\chi_2=\frac{\partial \theta}
{\partial \nu}$. In fact, from 
\begin{equation}
\begin{array}{l}
\displaystyle \frac{\partial u_p'}{\partial x_i}\to \frac{\partial u}{\partial x_i}\;\;
\mbox{weakly in}\;\;H^{-1}(0,T;L^2(\Omega))\\[5pt]
\displaystyle \theta_p'\to \theta'\;\;\mbox{weakly in}\;\;H^{-1}(0,T;V)
\end{array}
\label{eq:4.32}
\end{equation}
and (\ref{eq:4.30}) it follows that
\begin{equation}
\Delta \theta_p\longrightarrow \Delta \theta\;\;\mbox{weakly in}\;\;H^{-1}(0,T;
L^2(\Omega)).
\label{eq:4.33}
\end{equation}
%
From (\ref{eq:4.19}) and (\ref{eq:4.33}) it results that 
$$\frac{\partial \theta_p}{\partial \nu}\longrightarrow \frac{\partial \theta}
{\partial \nu}\;\;\mbox{weakly in}\;\;H^{-1}(0,T;H^{-{1/2}}(\Gamma_1)).$$
On the other hand, by (\ref{eq:4.21})
$$\frac{\partial \theta_p}{\partial \nu}\longrightarrow \chi_2\;\;\mbox{weakly in}
\;\;H^{-1}(0,T;H^{-{1/2}}(\Gamma_1)),$$
whence we conclude that $\frac{\partial \theta}{\partial \nu}=\chi_2$. We deduce that 
$\displaystyle \chi_1=\theta$ in $L^2(0,T;H^{1/2}(\Gamma_1))$ through of the 
convergences showed in (\ref{eq:4.19}) and (\ref{eq:4.20}). Therefore we obtain
\begin{equation}
\begin{array}{l}
\displaystyle \frac{\partial u}{\partial \nu}+\alpha u'=0\;\;\mbox{in}\;\;
L^2(0,T;L^2(\Gamma_1))\\[5pt]
\displaystyle \frac{\partial \theta}{\partial \nu}+\beta \theta=0\;\;\mbox{in}\;\;
L^2(0,T;L^2(\Gamma_1)).
\end{array}
\label{eq:4.34}
\end{equation}
%
To prove (\ref{eq:4.2}) and (\ref{eq:4.3}) we remark that for all $v,\;w\in V$, 
\begin{eqnarray*}
&|\langle-\Delta u,v\rangle|\leq \|u\|.\|v\|+{\left\|\frac{\partial u}{\partial \nu}
\right\|}_{H^{-{1/2}}(\Gamma_1)}.\|v\|_{H^{1/2}(\Gamma_1)},&\\
&|\langle-\Delta \theta,v\rangle|\leq \|\theta\|.\|w\|+{\left\|\frac{\partial \theta}
{\partial \nu}\right\|}_{H^{-{1/2}}(\Gamma_1)}.\|w\|_{H^{1/2}(\Gamma_1)}&
\end{eqnarray*}
and by continuity of the trace operator we deduce to inequalities:
$$|\langle-\Delta u,v\rangle|\leq C(u)\|v\|\;\mbox{and}\; |\langle-\Delta \theta ,
w\rangle|\leq C(\theta )\|w\|,$$
whence for all $T>0$ we obtain that
\begin{equation}
-\Delta u\in L^2(0,T;V')\;\;\;\mbox{and}\;\;-\Delta \theta \in L^2(0,T;V').
\label{eq:4.35}
\end{equation}
So, by (\ref{eq:4.24}), (\ref{eq:4.25}), (\ref{eq:4.35}) and Green's formula, for all 
$\psi$ in ${\cal D}(0,T)$, for all $v\;\mbox{and}\; w\;\mbox{in}\; V$ we get
\begin{eqnarray*}
&&-\int_0^T{(u'(t),v)\psi'(t)dt}+\int_0^T{\mu(t)\langle-\Delta u(t),v\rangle\psi(t)dt}\\
&&+\sum_{i=1}^n\int_0^T{\left(\frac{\partial \theta}{\partial x_i}(t),v\right)\psi(t)dt}
=0\\
&&-\int_0^T{(\theta(t),w)\phi'(t)dt}+\int_0^T\langle-\Delta \theta (t),w\rangle\psi(t)
dt\\
&&+\sum_{i=1}^n\int_0^T{\left(\frac{\partial u'}{\partial x_i}(t),w\right)}\psi(t)dt=0
\,.\\
\end{eqnarray*}
From these two inequalities and (\ref{eq:4.35}), we obtain that for each $T>0$  
\begin{eqnarray*}
&u''-\mu \Delta u+\sum_{i=1}^n \frac{\partial \theta}{\partial x_i}=0\;\;\mbox{in}\;\;
L^2(0,T;V')&\\
&\theta'-\Delta \theta +\sum_{i=1}^n \frac{\partial u'}{\partial  x_i}=0\;\;\mbox{in}
\;\;L^2(0,T;V')&
\end{eqnarray*}

The regularity in (\ref{eq:4.1}) follows from $\{u_p,\theta_p\}$ being a Cauchy 
sequence.
The initial data considerations follow from the analysis of the Galerkin approximation.
The uniqueness of the weak solution is proved by the method of Lions Magenes 
\cite{lima}, see also Visik-Ladyzhenskaya \cite{vi}.
\cqd

Now, we give a result which assures the existence and uniqueness of a weak global 
solution for (\ref{eq:1.1})--(\ref{eq:1.6}).
\vspace*{10pt}
\begin{mycor} Under the supplementary hypothesis $\mu'\in L^1(0,\infty)$, the pair of 
functions $\{u,\theta\}$ obtained by Theorem 4.1 satisfies the following properties:  
\begin{eqnarray*}
&u\in L^{\infty}(0,\infty;V),\;\;\;\theta\in L^{\infty}(0,\infty;L^2(\Omega))&\\
&\frac{\partial u}{\partial \nu}+\alpha u'=0\hspace*{7pt}and\hspace*{7pt} 
\frac{\partial \theta}{\partial \nu}+\beta \theta=0\hspace*{7pt}\mbox{in}\hspace*{7pt}
L^2(0,\infty;L^2(\Gamma_1))&\\
&u(0)=u^0,\;\;u'(0)=u^1\quad\mbox{and}\quad\theta(0)=\theta^0\,.&
\end{eqnarray*}
\end{mycor}

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

This section concerns the behavior of the solutions obtained in the preceding sections,
 as $t\to +\infty$.
First note that for strong solutions and weak solutions to 
(\ref{eq:1.1})--(\ref{eq:1.6}), the energy 
\begin{equation}
E(t)=\frac{1}{2}\left\{\mu(t)\|u(t)\|^2+|u'(t)|^2+|\theta(t)|^2\right\}.
\label{eq:5.1}
\end{equation}
does not increase. In fact, we can easily see that
\begin{eqnarray*}
E'(t)&=&\frac{\mu'(t)}{2}\|u(t)\|^2-\mu(t)\int_{\Gamma_1}\alpha(x)(u'(t))^2\,d
\Gamma-\|\theta(t)\|^2\\
&&-\beta \int_{\Gamma_1}(\theta(t))^2\,d\Gamma-\sum_{i=1}^n\int_{\Gamma_1}u'(t)
\theta(t)\nu_{i}\,d\Gamma \,.
\end{eqnarray*}
Also observe that
$$-\sum_{i=1}^n\int_{\Gamma_1}u'(t)\theta(t)\nu_i\,d\Gamma\leq\frac{\mu(t)}{2}
\int_{\Gamma_1}\alpha(x)(u'(t))^2\,d\Gamma+\frac{n}{2\mu(t)}\int_{\Gamma_1}\frac{1}
{\alpha(x)}(\theta(t))^2\,d\Gamma \,.$$
Because $\mu'(t)\leq 0$ and the hypothesis (2.1), we can conclude that
\begin{equation}
E'(t)\leq -\frac{\mu(t)}{2}\int_{\Gamma_1}\alpha(x) (u'(t))^2\,d\Gamma-\|\theta(t)\|^2. 
\label{eq:5.2}
\end{equation}



To estimate $E(t)$ we put $\alpha(x)=m(x).\nu(x)$ and use the representation 
$$\Gamma_0=\{x\in \Gamma;\;m(x).\nu(x)\leq 0\},\quad\Gamma_1=\{x\in \Gamma;\;\;
m(x).\nu(x)>0\}\,,$$
where $m(x)$ is the vectorial function $x-x^0$, for $x\in{\mathbb R}^n$ and \lq\lq.'' 
denotes scalar product in ${\mathbb R}^n$. We also use 
\begin{equation}
R(x^0)=\|m\|_{L^{\infty}(\Omega)}\,,
\label{eq:5.3}
\end{equation}
and positive constants $\delta_0$, $\delta_1$, $k$ such that
\begin{eqnarray}
&|v|^2\leq \delta_0\|v\|^2,\;\;\;\mbox{for all}\;\; v\in V
&\label{eq:5.4}\\
&\|v\|^2\leq \delta_1\|v\|^2_{V\cap H^2(\Omega)},\;\;\;\mbox{for all}\;\; v\in V
\cap H^2(\Omega)
&\label{eq:5.5}\\
&\int_{\Gamma_1}(m.\nu)v^2\,d\Gamma \leq k \|v\|^2,\;\;\;\mbox{for all}\;\; v\in V.
&\label{eq:5.6}
\end{eqnarray}

\begin{myth}
If $\{u^0,u^1,\theta^0\}\in V\times L^2(\Omega)\times V$, 
$\mu \in W^{1,\infty}(0,\infty)$ with $\mu'(t)\leq 0$ on $]0,\infty[$, then 
there exists a positive constant $\omega$ such that
\begin{equation}
E(t)\leq 3E(0)e^{-\omega t},\;\;\;\;\mbox{for all}\;\; t\geq 0.
\label{eq:5.7}
\end{equation}
\end{myth}
\paragraph{Proof.} As a first step, we consider the strong solution. Let
\begin{equation}
\rho(t)=2(u'(t),m.\nabla u(t))+(n-1)(u'(t),u(t)).
\label{eq:5.8}
\end{equation}
Then 
\begin{equation}
|\rho(t)|\leq (n-1)|u(t)|^2+n|u'(t)|^2+R^2(x^0)\|u(t)\|^2.
\label{eq:5.9}
\end{equation}
Let $\varepsilon_1$, $\varepsilon_2$, $\varepsilon$ be positive real numbers such that
\begin{eqnarray}
&\varepsilon_1\leq \min\left\{\displaystyle \frac{1}{4n},\frac{\mu_0}{12nR^2(x^0)+
12n^3\delta_0}\right\}
&\label{eq:5.10}\\
&\varepsilon_2\leq \min\left\{\frac{1}{2\left(R^2(x^0)+\displaystyle \frac{1}{\mu_0}+
6kn^2\right)},\displaystyle\frac{2}{\delta_0}\right\}
&\label{eq:5.11}\\
&\varepsilon \leq \min\left\{\varepsilon_1,\varepsilon_2\right\}.
&\label{eq:5.12}
\end{eqnarray}
%
Also let the perturbed energy given by
\begin{equation}
E_{\varepsilon}(t)=E(t)+\varepsilon \rho(t).
\label{eq:5.13}
\end{equation}
Then from (\ref{eq:5.13}), (\ref{eq:5.4}), and (\ref{eq:5.9}) we get
$$E_{\varepsilon}(t)\leq E(t)+\left(\varepsilon n\delta_0+\varepsilon R^2(x^0)\right)
\|u(t)\|^2+\varepsilon n|u'(t)|^2,$$
whence by (\ref{eq:5.12}) it follows that
$$E_{\varepsilon}(t)\leq E(t)+\varepsilon_1\left( n\delta_0+R^2(x^0)\right)\|u(t)\|^2+
\varepsilon_1 n|u'(t)|^2.$$
By (\ref{eq:5.1}) and (\ref{eq:5.10}) we obtain $E_{\varepsilon}\leq \frac{3}{2}E(t).$
 On the other hand, using similar arguments, from (\ref{eq:5.9}) and (\ref{eq:5.13}) 
we deduce that
$\frac{1}{2}E(t)\leq E_{\varepsilon}$. In summary,
\begin{equation}
\frac{1}{2}E(t)\leq E_{\varepsilon}\leq \frac{3}{2}E(t),\quad\mbox{for all } t\geq 0.
\label{eq:5.14}
\end{equation}
To estimate $E'_{\varepsilon}(t)$ we differentiate $\rho(t)$,
\begin{eqnarray}
\rho'(t)&=&2(u''(t),m.\nabla(t))+2(u'(t),m.\nabla u'(t))\\
&&+(n-1)(u''(t),u(t))+(n-1)|u'(t)|^2\,.\nonumber
\label{eq:5.15}
\end{eqnarray}
Since $u''=\mu \Delta u-\sum_{i=1}^n\frac{\partial \theta}{\partial x_i}(t)$ we have 
\begin{eqnarray}
\rho'(t)&=&2\mu(t)(\Delta u(t),m.\nabla u(t)){-2\sum_{i=1}^n}\left(\frac{\partial 
\theta}{\partial x_i}(t),m.\nabla u(t)\right)\nonumber\\
&&+2\left(u'(t),m.\nabla u'(t)\right)+(n-1)\mu(t)(\Delta u(t),u(t))\\
&&-(n-1)\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),u(t)\right)+(n-1)|
u'(t)|^2.\nonumber
\label{eq:5.16}
\end{eqnarray}
our next objective is to find bounds for the right-hand-side terms of the equation 
above.

\begin{myrem}
For all $v\in V\cap H^2(\Omega)$,
\begin{equation}
2\,(\Delta v,m.\nabla v)\leq (n-2)\|v\|^2
+R^2(x^0){\displaystyle\int_{\Gamma_1}}\frac{1}{m.\nu}{\left|\frac{\partial v}
{\partial \nu}\right|}^2d\,\Gamma\,.
\label{eq:5.17}
\end{equation}
\end{myrem}
In fact, the Rellich's identity, see V. Komornik and E. Zuazua \cite{kom-zua}, gives
\begin{equation}
2\,(\Delta v,m.\nabla v)=(n-2)\|v\|^2-\int_{\Gamma}(m.\nu)|\nabla v|^2d\,\Gamma+2\,
\int_{\Gamma}\frac{\partial v}{\partial \nu}m.\nabla v\,d\,\Gamma.
\label{eq:5.18}
\end{equation}
Note that
\begin{eqnarray}
-\int_{\Gamma}(m.\nu)|\nabla v|^2\,d\,\Gamma & =&-\int_{\Gamma_0}(m.\nu)\left(
\frac{\partial v}{\partial \nu}\right)^2d\,\Gamma-\int_{\Gamma_1}(m.\nu)|\nabla v|^2d\,
\Gamma \nonumber\\
&\leq&-\int_{\Gamma_0}(m.\nu){\left (\frac{\partial v}{\partial \nu}\right)}^2d\,
\Gamma \,,
\label{eq:5.19}
\end{eqnarray}
because $\;\frac{\partial v}{\partial x_i}=\nu_i\frac{\partial v}{\partial \nu}$ on 
$\Gamma_0$ and $m.\nu >0$ on  $\Gamma_1$. Also note that
\begin{equation}
2\int_{\Gamma}\frac{\partial v}{\partial \nu}m.\nabla v\,d\,\Gamma=2\int_{\Gamma_0}
(m.\nu)\left(\frac{\partial v}{\partial \nu}\right)^2d\,\Gamma+2\int_{\Gamma_1}
\frac{\partial v}{\partial \nu}m.\nabla v\,d\,\Gamma,
\label{eq:5.20}
\end{equation}
and by (5.3)
\begin{eqnarray*}
2\int_{\Gamma_1}\frac{\partial v}{\partial \nu}m.\nabla v\,d\,\Gamma 
&\leq &\displaystyle 2\int_{\Gamma_1}\left|\frac{\partial v}{\partial \nu}
\right|R(x^0)|\nabla v|\,d\,\Gamma \\
&\leq &{R^2(x^0)\int_{\Gamma_1}}\frac{1}{m.\nu}\left(\frac{\partial v}{\partial \nu}
\right)^2d\,\Gamma 
+\int_{\Gamma_1}(m.\nu)|\nabla v|^2d\,\Gamma.
\end{eqnarray*}
This inequality with (\ref{eq:5.20}) yields
\begin{eqnarray}\label{eq:5.21}
\lefteqn{2\int_{\Gamma}\frac{\partial v}{\partial \nu}m.\nabla v\,d\,\Gamma }\\
&\leq &
2\int_{\Gamma_0}(m.\nu){\left(\frac{\partial v}{\partial \nu}\right)}^2\,d\,\Gamma 
+R^2(x^0)\int_{\Gamma_1}\frac{1}{m.\nu}\left(\frac{\partial v}{\partial\nu}\right)^2d\,
\Gamma+\int_{\Gamma_1}(m.\nu)|\nabla v|^2d\,\Gamma.
\nonumber
\end{eqnarray}
Combining (\ref{eq:5.18}), (\ref{eq:5.19}), and (\ref{eq:5.21}), we come to the 
inequality 
\begin{eqnarray*}
2(\Delta v,m.\nabla v)&\leq &(n-2)\|v\|^2+\int_{\Gamma_0}(m.\nu){\left(
\frac{\partial v}{\partial \nu}\right)}^2\,d\,\Gamma \\[10pt] 
&&+R^2(x^0)\int_{\Gamma_1}\frac{1}{m.\nu}\left(\frac{\partial v}{\partial \nu}\right)^2
d\,\Gamma .
\end{eqnarray*}
Recall that $ m.\nu \leq 0$ on $\Gamma_0$; therefore, (\ref{eq:5.17}) follows.
Now, we shall analyze each term in (\ref{eq:5.16}).

\paragraph{Analysis of $2\mu(t)(\Delta u(t),m.\nabla u(t))$:} Thanks to Remark 5.1 and
 (\ref{eq:3.5}) we have
\begin{equation}
2\mu(t)(\Delta u(t),m.\nabla u(t))\leq \mu(t)(n-2)\|u(t)\|^2
+\mu(t)R^2(x^0)\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma.
\label{eq:5.22}
\end{equation}
\paragraph{Analysis of $-2\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),
m.\nabla u(t)\right)$:}
\begin{eqnarray*}
-2\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),m.\nabla u(t)\right)&
\leq &2\sum_{i=1}^n\left|\frac{\partial \theta}{\partial x_i}(t)\right|R(x^0)\|u(t)\|\\
&\leq &\sum_{i=1}^n \frac{6nR^2(x^0)}{\mu_0}{\left|\frac{\partial \theta}{\partial x_i}
(t)\right|}^2+\sum_{i=1}^n \frac{1}{6n}\mu_0\|u(t)\|^2.
\end{eqnarray*}
Thus
\begin{equation}
-2\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),m.\nabla u(t)\right)\leq 
\frac{6nR^2(x^0)}{\mu_0}\|\theta(t)\|^2+\frac{\mu(t)}{6}\|u(t)\|^2.
\label{eq:5.23}
\end{equation}
%
\paragraph{Analysis of $2(u'(t),m.\nabla u'(t))$:}
\begin{eqnarray}
2(u'(t),m.\nabla u'(t))&=& 2\int_{\Omega}u'(t)m_j\frac{\partial u'}{\partial x_j}(t)
\,dx \nonumber \\
&=& \int_{\Omega}m_j\frac{\partial (u')^2}{\partial x_j}(t)\,dx \nonumber \\
&=& -\int_{\Omega}\frac{\partial m_j}{\partial x_j}(u'(t))^2dx+\int_{\Gamma_1}
(m_j{\nu}_j)(u'(t))^2d\Gamma \label{eq:5.24}
\\
&=&- n|u'(t)|^2+\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma\,.\nonumber
\end{eqnarray}
%
\paragraph{Analysis of $\mu(t)(n-1)(\Delta u(t),u(t))$:} Applying Green's theorem and
 (\ref{eq:3.5}), we get
$$\mu(t)(n-1)(\Delta u(t),u(t))
=-\mu(t)(n-1)\left[\|u(t)\|^2+\int_{\Gamma_1}(m.\nu)u'(t)u(t)\,d\Gamma\right].$$
By the Cauchy-Schwarz inequality 
\begin{eqnarray*}
\mu(t)(n-1)(\Delta u(t),u(t))
&\leq& -\mu(t)(n-1)\|u(t)\|^2\\
&&+6k\mu(t)(n-1)^2\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma \\
&&+\frac{\mu(t)}{6k}\int_{\Gamma_1}(m.\nu)(u(t))^2d\Gamma, 
\end{eqnarray*}
and by (\ref{eq:5.6})
\begin{eqnarray*}
\mu(t)(n-1)(\Delta u(t),u(t))
&\leq& -\mu(t)(n-1)\|u(t)\|^2\\
&&+6k\mu(t)(n-1)^2\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma
+\frac{\mu(t)}{6}\|u(t)\|^2\,.
\end{eqnarray*}
%
Hence 
\begin{eqnarray}
\mu(t)(n-1)(\Delta u(t),u(t))&\leq & 
-\mu(t)(n-\frac{7}{6})\|u(t)\|^2
\label{eq:5.25}\\
&&+6k\mu(t)(n-1)^2\int_{\Gamma_1}(m.\nu)(u'(t)^2d\Gamma\,.\nonumber
\end{eqnarray}
%
\paragraph{Analysis of $ -(n-1)\left(\sum_{i=1}^n\frac{\partial \theta}{\partial x_i},
u(t)\right)$:}
\begin{eqnarray*}
-(n-1)\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),u(t)\right)&\leq 
&(n-1)\sum_{i=1}^n \left|\frac{\partial \theta}{\partial x_i}(t)\right||u(t)|
\\
&\leq &\frac{6n\delta_0(n-1)^2}{\mu_0}\|\theta(t)\|^2+\sum_{i=1}^n
\frac{\mu_0}{6n\delta_0}|u(t)|^2,
\end{eqnarray*}
whence by (\ref{eq:5.4})
\begin{equation}
-(n-1)\sum_{i=1}^n\left(\frac{\partial \theta}{\partial x_i}(t),u(t)\right)\leq 
\frac{6n\delta_0(n-1)^2}{\mu_0}\|\theta(t)\|^2+\frac{\mu(t)}{6}\|u(t)\|^2.
\label{eq:5.26}
\end{equation}
%
Using (\ref{eq:5.22})--(\ref{eq:5.6}) in (\ref{eq:5.16}) we conclude that
\begin{eqnarray}
\rho'(t)&\leq& -\frac{\mu(t)}{2}\|u(t)\|^2+\left[\frac{6nR^2(x^0)+6n^3\delta_0}
{\mu_0}\right]\|\theta(t)\|^2-|u'(t)|^2 \nonumber \\
&&+\mu(t)\left[R^2(x^0)+\frac{1}{\mu_0}+6kn^2\right]\int_{\Gamma_1}(m.\nu)(u'(t))^2
d\Gamma\,.
\label{eq:5.27}
\end{eqnarray}
Combining (\ref{eq:5.2}), (\ref{eq:5.13}) and (\ref{eq:5.27}), we get
\begin{eqnarray*}
E'_{\varepsilon}(t)&\leq& -\|\theta(t)\|^2
-\frac{\mu(t)}{2}\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma\\
&&-\frac{\varepsilon}{2}\mu(t)\|u(t)\|^2
+\varepsilon\left[\frac{6nR^2(x^0)+6n^3\delta_0}{\mu_0}\right]\|\theta(t)\|^2-
\varepsilon|u'(t)|^2\\
&&+\varepsilon \mu(t)\left[R^2(x^0)+\frac{1}{\mu_0}+6kn^2\right]\int_{\Gamma_1}
(m.\nu)(u'(t))^2d\Gamma\,.
\end{eqnarray*}
Then, by (\ref{eq:5.4}) and (\ref{eq:5.12}), it results that 
\begin{eqnarray*}
E'_{\varepsilon}(t)&\leq &-\|\theta(t)\|^2
-\frac{\mu(t)}{2}\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma\\
&&-\frac{\varepsilon}{2}\mu(t)\|u(t)\|^2
+\varepsilon_1\left[\frac{6nR^2(x^0)+6n^3\delta_0}{\mu_0}\right]\|\theta(t)\|^2-
\varepsilon|u'(t)|^2\\ &&+\varepsilon_2\mu(t)\left[R^2(x^0)+\frac{1}{\mu_0}+
6kn^2\right]\int_{\Gamma_1}(m.\nu)(u'(t))^2d\Gamma.
\end{eqnarray*}
Using (\ref{eq:5.10}) and (\ref{eq:5.11}) we obtain
$$E'_{\varepsilon}(t)\leq -\frac{1}{2}\|\theta(t)\|^2-\frac{\varepsilon}{2}\mu(t)
\|u(t)\|^2-\frac{\varepsilon}{2}|u'(t)|^2.$$
Also, from (\ref{eq:5.4}), (\ref{eq:5.11}) and (\ref{eq:5.12}) we obtained 
$$E'_{\varepsilon}(t)\leq -\frac{1}{\delta_0}|\theta(t)|^2-\frac{\varepsilon}{2}
\mu(t)\|u(t)\|^2-\frac{\varepsilon}{2}|u'(t)|^2.$$
By (\ref{eq:5.11}) and (\ref{eq:5.12}) we have $-\frac{\varepsilon}{2}\geq -
\frac{1}{\delta_0}$, then
\begin{eqnarray}
E'_{\varepsilon}(t)&\leq & -\frac{\varepsilon}{2}|\theta(t)|^2-\frac{\varepsilon}{2}
\mu(t)\|u(t)\|^2-\frac{\varepsilon}{2}|u'(t)|^2\nonumber \\
&=&-\frac{\varepsilon }{2}E(t)\,.
\label{eq:5.28}
\end{eqnarray}
From (\ref{eq:5.14}), we obtain 
$E'_{\varepsilon}(t)\leq -\frac{2\varepsilon}{3}E_{\varepsilon}(t)$. In turn this 
inequality implies
$E_{\varepsilon}(t)\leq E_{\varepsilon}(0)e^{-\frac{2}{3}\varepsilon t}$. From (5.14), 
we obtain exponential decay for strong solutions
$$E(t)\leq 3E(0)e^{-\frac{2}{3}\varepsilon t},\quad \mbox{for all } t\geq 0.$$


\paragraph{Remark} Using a denseness argument, we prove the same behavior for weak 
solutions.


\paragraph{Acknowledgments.} The authors would like to thank Professor Luiz Adauto 
Medeiros for his suggestions and comments.

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\end{thebibliography}
\medskip


{\sc H. R. Clark}\\
Universidade Federal Fluminense, RJ, Brazil\\
E-mail address: ganhrc@vm.uff.br\medskip

{\sc L. P. San Gil Jutuca}\\
Universidade do Rio de Janeiro, RJ, Brazil\\
E-mail address: rsangil@iq.ufrj.br\medskip

{\sc M. Milla Miranda}\\
Universidade Federal do Rio de Janeiro, RJ, Brazil\\
Instituto de Matem\'atica
CP 68530  -  CEP 21949-900 

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