\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2007(2007), No. 131, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
 http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/131\hfil A nonlinear transmission problem]
{A nonlinear transmission problem with time dependent coefficients}

\author[E. Cabanillas L., J. E. Mu\~noz R.\hfil EJDE-2007/131\hfilneg]
{Eugenio Cabanillas Lapa, Jaime E. Mu\~noz Rivera}  

\address{Eugenio Cabanillas Lapa \newline
Instituto de Investigaci\'on de Matem\'atica, 
Facultad de Ciencias Matem\'aticas, 
Universidad Nacional Mayor de San Marcos, Lima, Per\'u} 
\email{cleugenio@yahoo.com}

\address{Jaime E. Mu\~noz Rivera \newline
Laboratorio Nacional de Computa\c{c}ao Cientifica, 
Av. Get\'{u}lio Vargas, 333, 25651-070-Petropolis, Brazil}
\email{rivera@lncc.br}

\thanks{Submitted May 2, 2007. Published October 9, 2007.}
\subjclass[2000]{35B40, 35L70, 45K05} 
\keywords{Transmission problem; time dependent coefficients; stability}

\begin{abstract}
 In this article, we consider a nonlinear transmission problem
 for the wave equation with time dependent coefficients and linear
 internal damping. We prove the existence of a global solution and
 its exponential decay. The result is achieved by using the
 multiplier technique and suitable unique continuation
 theorem for the wave equation.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

In this work, we consider the transmission problem
\begin{gather}
\rho _{1}u_{tt}-bu_{xx}+f_{1}(u)=0\quad\text{in }
  ]0,L_{0}[\times \mathbb{R}^{+}, \label{e1.1} \\
\rho _{2}v_{tt}-(a(x,t)v_{x})_{x}+\alpha
v_{t}+f_{2}(v)=0\quad\text{in } ]L_{0},L[\times
\mathbb{R}^{+}, \label{e1.2} \\
u(0,t)=v(L,t), \quad  t>0 , \label{e1.3} \\
u(L_{0},t)=v(L_{0},t),\quad
  bu_{x}(L_{0},t)=a(L_{0},t)v_{x}(L_{0},t),\quad
t>0, \label{e1.4} \\
u(x,0)=u^{0}(x),\quad  u_{t}(x,0)=u^{1}(x),\quad  x\in ]0,L_{0}[,
 \label{e1.5} \\
v(x,0)=v^{0}(x),\quad  v_{t}(x,0)=v^{1}(x),\quad
 x\in ]L_{0},L[, \label{e1.6}
\end{gather}
where $\rho _{1}, \rho _{2}$ are constants; $\alpha ,b$ are positive
constants, $f, g$ are nonlinear functions and $a(x,t)$ is a positive
function. Controllability and Stability for transmission problem has
been studied by many authors (see for example Lions \cite{l5},
Lagnese \cite{l3},  Liu and  Williams \cite{l6},
  Mu\~{n}oz
Rivera and Portillo Oquendo \cite{m1},  Andrade,  Fatori and
 Mu\~{n}oz Rivera \cite{a1}).

The goal of this work is to study the existence and uniqueness of
 global
solutions of \eqref{e1.1}-\eqref{e1.6} and the asymptotic behavior
of the energy.

All the authors mentioned above established their results with
constant coefficients. To the best of our knowledge this is a first
publication
 on
transmission problem with time dependent coefficients and the
nonlinear terms. In general,the dependence on spatial and time
variables causes difficulties,semigroups arguments are not suitable
for finding
 solutions to
\eqref{e1.1}-\eqref{e1.6}; therefore,we make use of a Galerkin's
process. Note that the time-dependent coefficient also appear in the
second boundary condition, thus there are some technical
difficulties
 that
we need to overcome. To prove the exponential decay, the main
 difficulty is
that the dissipation only works in $[L_{0},L]$ and we need estimates
over the whole domain $[0,L]$; we overcome this problem introducing
suitable multiplicadors and a
 compactness/uniqueness
argument.

\section{Notation and statement of results}

We denote
\begin{equation*}
(w,z)=\int_{I}w(x)z(x)dx, \quad
 | z| ^{2}=\int_{I}| z(x)| ^{2}dx
\end{equation*}
where $I=]0,L_{0}[$ or $]L_{0},L[$ for $u$'s and $v$'s respectively.
Now, we state the general hypotheses.

\begin{itemize}
\item[(A1)] The functions $f_{i}\in C^{1}(\mathbb{R})$, $i=1,2$,
 satisfy
$f_{i}(s)s\geq 0$ for all $s\in \mathbb{R}$ and
\[
| f_{i}^{(j)}(s)| \leq c(1+| s| )^{\rho -j},\quad \forall  s\in
 \mathbb{R},
\; j=0,1
\]
for some $c>0$ and $\rho \geq 1$. We assume that $f_{1}(s)\geq
f_{2}(s)$ and set
\[
 F_{i}(s)=\int_{0}^{s}f_{i}(\xi )d\xi\,.
\]

\item[(A2)] We assume that the coefficient $a$ satisfies
\begin{gather*}
 a\in W^{1,\infty }(0,\infty ;C^{1}([L_{0},L]
))\cap W^{2,\infty }(0,\infty ;L^{\infty }(L_{0},L)) \\
 a_{t}\in L^{1}(0,\infty ;L^{\infty }(L_{0},L)) \\
 a(x,t)\geq a_{0}>0,\quad  \forall  (x,t)\in ]L_{0},L[\times ]0,\infty
 [\,.
\end{gather*}
\end{itemize}
We define the Hilbert space
\begin{equation*}
V=\{ (w,z)\in H^{1}(0,L_{0})\times H^{1}(L_{0},L):w(0)=z(L)=0;\ w(
L_{0})=z(L_{0})\}\,.
\end{equation*}
Also we define the first-order energy functionals associated to each
equation,
\begin{gather*}
E_{1}(t,u)=\frac{1}{2}\Big( \rho _{1}| u_{t}| ^{2}+b| u_{x}|
^{2}+2\int_{0}^{L_{0}}F_{1}(u)dx\Big)
\\
E_{2}(t,v)=\frac{1}{2}\Big( \rho _{2}| v_{t}| ^{2}+(a,v_{x}^{2})
+2\int_{L_{0}}^{L}F_{2}(v)dx\Big)
\\
E(t)=E_{1}(t,u,v)=E_{1}(t,u)+E_{2}(t,v).
\end{gather*}

We conclude this section with the following lemma which will play
 essential
role when establishing the asymptotic behavior of solutions.

\begin{lemma}[{\cite[Lemma 9.1]{k1}}] \label{lem2.1}
Let $E:\mathbb{R}_{0}^{+}\to \mathbb{R}_{0}^{+}$ be a non-increasing
function and assume that there exist two constants $p>0$ and $c>0$
such that
\begin{equation*}
\int_{s}^{+\infty }E^{(p+1)/2}(t)dt\leq cE(s),\quad 0\leq s<+\infty.
\end{equation*}
Then for all $t\geq 0$,
\[
E(t)\leq \begin{cases}
cE(0)(1+t)^{-2(p-1)} &\text{if } p>1, \\
cE(0)e^{1-wt} &\text{if } p=1,
\end{cases}
\]
where $c$ and $w$ are positive constants.
\end{lemma}

\section{Existence and uniqueness of solutions}

First, we define weak solutions of problem
\eqref{e1.1}-\eqref{e1.6}.

\begin{definition} \label{def3.1} \rm
We say that the pair $\{u,v\} $ is a weak solution of
\eqref{e1.1}-\eqref{e1.6} when
\begin{equation*}
\{ u,v\} \in L^{\infty }(0,T;V)\cap W^{1,\infty}(0,T;L^{2}(0,L_{0})
\times L^{2}(L_{0},L))
\end{equation*}
and satisfies
\begin{align*}
& -\rho _{1}\int_{0}^{L_{0}}u^{1}(x)\varphi ( x,0)dx+\rho
_{1}\int_{0}^{L_{0}}u^{0}(x)\varphi _{t}(x,0)dx-\rho
_{2}\int_{L_{0}}^{L}v^{1}(x)\psi
(x,0)dx \\
& +\rho _{2}\int_{L_{0}}^{L}v^{0}(x)\psi _{t}( x,0)dx+\rho
_{1}\int_{0}^{T}\int_{0}^{L_{0}}(u\varphi
_{tt}+bu_{x}\varphi _{x}+f_{1}(u)\varphi )\,dx\,dt \\
& +\rho _{2}\int_{0}^{T}\int_{L_{0}}^{L}(v\psi _{tt}+a(
x,t)v_{x}\psi _{x}+\alpha v_{t}\psi +f_{2}(v)\psi)\,dx\,dt=0
\end{align*}
for any $\{ \varphi ,\psi \} \in C^{2}(0,T;V)$ such that $\varphi
(T)=\varphi _{t}(T)=0=\psi (T)=\psi _{t}(T)$
\end{definition}

To show the existence of strong solutions we need a regularity
result for the elliptic system associated to the problem
\eqref{e1.1}--\eqref{e1.6} whose proof can be obtained, with little
modifications, in the book by Ladyzhenskaya and  Ural'tseva
\cite[theorem 16.2]{l1}.

\begin{lemma} \label{lem3.2}
For any given functions $F\in L^{2}(0,L_{0})$, $G\in
L^{2}(L_{0},L)$, there exists only one solution $\{ u,v\} $ to the
system
\begin{gather*}
-bu_{xx} =F \quad \text{in } ]0,L_{0}[,\\
-(a(x,t)v_{x})_{x} =G \quad\text{in }]L_{0},L[,\\
u(0)= v(L)=0, \\
u(L_{0}) = v(L_{0}),\quad  bu_{x}(L_{0})=a(L_{0},t)v_{x}(L_{0}),
\end{gather*}
with $t$ a fixed value in $[0,T]$, with $u$ in $H^{2}(0,L_{0})$ and
$v$ in $H^{2}(L_{0},L)$.
\end{lemma}

The existence result to the system \eqref{e1.1}--\eqref{e1.6} is
summarized in the following theorem.

\begin{theorem} \label{thm3.3}
Suppose that $\{u^{0},v^{0}\} \in V$, $\{ u^{1},v^{1}\} \in L^{2}(
0,L_{0})\times L^{2}(L_{0},L)$ and that assumptions (A1)--(A3) hold.
Then there exists a unique weak solution of
\eqref{e1.1}--\eqref{e1.6} satisfying
\begin{equation*}
\{ u,v\} \in C(0,T;V)\cap C^{1}(0,T;L^{2}(0,L_{0})\times
 L^{2}(L_{0},L)).
\end{equation*}
In addition, if $\{ u^{0},v^{0}\} \in H^{2}(0,L_{0})\times
 H^{2}(L_{0},L)$,
$\{u^{1},v^{1}\} \in V$, verifying the compatibility condition
\begin{equation}
bu_{x}^{0}(L_{0})=a(L_{0},0)\ v_{x}^{0}(L_{0})\,.  \label{eq3.1}
\end{equation}
Then
\begin{equation*}
\{ u,v\} \in \underset{k=0}{\overset{2}{\bigcap }}W^{k,\infty
}(0,T;H^{2-k}(0,L_{0})\times H^{2-k}(L_{0},L))
\end{equation*}
\end{theorem}

\begin{proof}
The main idea is to use the Galerkin Method. Let $\{ \varphi
^{i},\psi ^{i}\}$ , $i=1,2,\dots$ be a basis of $V$. Let us consider
the Galerkin approximation
\begin{equation*}
\{ u^{m}(t),v^{m}(t)\} =\sum_{i=1}^m h_{im}(t)\{ \varphi ^{i},\psi
^{i}\}
\end{equation*}
where $u^{m}$ and $v^{m}$ satisfy
\begin{equation}
\begin{aligned}
&\rho _{1}(u_{tt}^{m},\varphi ^{i})+b(u_{x}^{m},\varphi
_{x}^{i})+(f_{1}(u^{m}),\varphi ^{i})+\rho
_{2}(v_{tt}^{m},\psi ^{i})\\
&+(a(x,t) v_{x}^{m},\psi _{x}^{i}) +\alpha (v_{t}^{m},\psi
^{i})+(f_{2}(v^{m}),\psi ^{i})=0
\end{aligned}  \label{eq3.2}
\end{equation}
where $i=1,2,\dots$ With initial data
\begin{equation} \label{eq3.3}
\begin{gathered}
\{ u^{m}(0),v^{m}(0)\} \to \{ u^{0},v^{0}\} \quad\text{in }V , \\
\{ u_{t}^{m}(0),v_{t}^{m}(0)\} \to \{ u^{1},v^{1}\} \quad \text{in }
L^{2}(0,L)\times L^{2}(L_{0},L).
\end{gathered}
\end{equation}
Standard results about ordinary differential equations guarantee
that
 there
exists only one solution of this system on some interval
$[0,T_{m}[$. The priori estimate that follow imply that in fact
$T_{m}=+\infty$.
\end{proof}

subsection*{Existence of weak solutions}
 Multiplying (\ref{eq3.2}) by $h_{im}'(t)$ integrating by parts and
summing over $i$, we get
\begin{equation}
\frac{d}{dt}\ E(t,u^{m},v^{m})+\alpha | v_{t}^{m}| ^{2}\leq \frac{|
a_{t}(t) | _{L^{\infty }}}{a_{0}}\ E(t,u^{m},v^{m}). \label{eq3.4}
\end{equation}
From this inequality, the Gronwall's inequality and taking into
account
 the
definition of the initial data of $\{ u^{m},v^{m}\} $ we conclude
that
\begin{equation}
E(t,u^{m},v^{m})\leq C,\quad \forall  t\in [0,T], \; \forall  m\in
\mathbb{N}    \label{eq3.5}
\end{equation}
thus we deduce that
\begin{gather*}
\{ u^{m},v^{m}\} \text{  is bounded in  }L^{\infty }(
0,T;V)\\
\{ u_{t}^{m},v_{t}^{m}\} \text{  is bounded in  }L^{\infty
}(0,T;L^{2}(0,L_{0})\times L^{2}(L_{0},L))
\end{gather*}
which implies that
\begin{gather*}
\{ u^{m},v^{m}\} \to \{ u,v\} \text{ weak $\ast$  in  }
  L^{\infty }(0,T;V)\\
\{ u_{t}^{m},v_{t}^{m}\} \to \{u_{t},v_{t}\} \text{weak $\ast$  in
}
 L^{\infty }(0,T;L^{2}(0,L_{0})\times L^{2}(L_{0},L)).
\end{gather*}
In particular, by application of the Lions-Aubin's Lemma
\cite[Theorem 5.1]{l4}, we have $\{ u^{m},v^{m}\} \to \{ u,v\} $
strongly in $L^{2}(0,T;L^{2}(0,L_{0})\times L^{2}(L_{0},L))$ and
 consequently
\begin{gather*}
u^{m}\to u\text{ a.e in $]0,L_{0}[$  and  } f_{1}(u^{m})\to
f_{1}(u)\text{ a.e in $]0,L_{0}[$}
\\
v^{m}\to v\text{ a.e in $]L_{0},L[$  and  } f_{2}(v^{m})\to
f_{2}(v)\text{  a.e in $]L_{0},L[$.}
\end{gather*}
Also, from the growth condition in (A1) we have
\begin{gather*}
f_{1}(u^{m})\text{  is bounded in  }L^{\infty }( 0,T;L^{2}(0,L_{0}))
\\
f_{2}(v^{m})\text{  is bounded in  }L^{\infty }(
0,T;L^{2}(L_{0},L));
\end{gather*}
therefore,
\begin{equation*}
\{ f_{1}(u^{m}),f_{2}(v^{m})\} \rightharpoonup \{
f_{1}(u),f_{2}(v)\} \quad \text{in }L^{2}(0,T;L^{2}(0,L_{0})\times
L^{2}( L_{0},L)).
\end{equation*}
The rest of the proof of the existence of a weak solution is matter
of routine.

\subsection*{Regularity of solutions}
To get the regularity, we take a basis $B=\{\{ \varphi ^{i},\psi
^{i}\} ,i\in \mathbb{N}\} $ such that
\begin{equation*}
\{ u^{0},v^{0}\},\;\{ u^{1},v^{1}\}  \text{are in the span of } \{
\{ \varphi ^{0},\psi ^{0}\} ,\{ \varphi ^{1},\psi ^{1}\} \} .
\end{equation*}
Therefore, $\{ u^{m}(0),v^{m}(0)\} =\{ u^{0},v^{0}\} $ and $\{
u_{t}^{m}(0),v_{t}^{m}(0)\} =\{
 u^{1},v^{1}\} $.
Let us differentiate the approximate equation and multiply by
$h_{im}^{\prime \prime }(t)$. Using a similar argument as before we
obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt} E_{2}(t,u^{m},v^{m})+\alpha |v_{tt}^{m}| ^{2}&
=-(f_{1}'(u^{m})u_{t}^{m},u_{tt}^{m})-(f_{2}'(
v^{m})v_{t}^{m},v_{tt}^{m}) \\
&\quad -(a_{t}v_{x}^{m},v_{xtt}^{m})+\frac{1}{2}(
a_{t},(v_{xt}^{m})^{2})
\end{aligned}  \label{eq3.6}
\end{equation}
where
\begin{equation*}
E_{2}(t,u,v)=\frac{\rho _{1}}{2}| u_{tt}| ^{2}+\frac{b}{2}| u_{xt}|
^{2}+\frac{\rho _{2}}{2} | v_{tt}| ^{2}+\frac{1}{2}(a,v_{xt})^{2}.
\end{equation*}
Note that
\begin{equation}
-(a_{t}v_{x}^{m},v_{xtt}^{m}) =-(a_{t}v_{x}^{m},v_{xt}^{m})_{t}+(
a_{tt}v_{x}^{m},v_{xt}^{m})+\big(a_{t},(v_{xt}^{m})^{2}\big),
  \label{eq3.7}
\end{equation}
$E_{2}(0,u^{m},v^{m})$ is bounded, because of our choice of the
basis.

From the assumption (A1) and from the Sobolev imbedding we have
\begin{equation}
\int_{0}^{L_{0}}f_{1}'(u^{m})u_{t}^{m}u_{tt}^{m}dx\leq
C\Big[\int_{0}^{L_{0}}(1+| u_{x}^{m}| ) ^{2}dx\Big]^{(p-1)/2}|
u_{xt}^{m}| | u_{tt}^{m}|,   \label{eq3.8}
\end{equation}
and similarly
\begin{equation}
\int_{L_{0}}^{L}f_{2}'(v^{m})v_{t}^{m}v_{tt}^{m}dx\leq
C\Big[\int_{L_{0}}^{L}(1+| v_{x}^{m}| ) ^{2}dx\Big]^{(p-1)/2}|
v_{xt}^{m}| |v_{tt}^{m}|    \label{eq3.9}
\end{equation}
Substituting (\ref{eq3.7}), the inequalities
 (\ref{eq3.8})--(\ref{eq3.9}),
using the estimative (\ref{eq3.5}) in (\ref{eq3.6}) and applying
 Gronwall's
inequality we conclude that
\begin{equation}
E_{2}(t,u^{m},v^{m})\leq C   \label{eq3.10}
\end{equation}
which imply
\begin{gather*}
\{ u_{t}^{m},v_{t}^{m}\} \to \{ u_{t},v_{t}\} \quad \text{weak
$\ast$  in }L^{\infty }(
0,T;H^{1}(0,L_{0})\times H^{1}(L_{0},L))\\
\{ u_{tt}^{m},v_{tt}^{m}\} \to \{ u_{tt},v_{tt}\} \quad \text{weak
$\ast$ in  }L^{\infty }( 0,T;L^{2}(0,L_{0})\times L^{2}(L_{0},L)).
\end{gather*}
Therefore,  $\{ u,v\} $ satisfies \eqref{e1.1}--\eqref{e1.4} and we
 have
\begin{gather*}
-bu_{xx}=-\rho _{1}u_{tt}-f_{1}(u)\in L^{2}(0,L_{0}), \\
-(a(x,t)v_{x})_{x}=-\rho _{2}v_{tt}-f_{2}(
v)-\alpha v_{t}\in L^{2}(L_{0},L), \\
u(L_{0},t)=v(L_{0},t),\quad
bu_{x}(L_{0},t)=a(L_{0},t)v_{x}(L_{0},t).
 \\
u(0,t)=0=v(L,t)
\end{gather*}
Then using Lemma 3.2 we have the required regularity for $\{ u,v\}$.

\section{Exponential Decay}

In this section we prove that the solution of the system
\eqref{e1.1}--\eqref{e1.6} decay exponentially as time approaches
infinity. In the remainder of this
 paper we
denote by $c$ a positive constant which takes different values in
 different
places. We shall suppose that  $\rho _{1}\leq \rho _{2}$  and
\begin{gather*}
a(x,t)\leq b,\quad a_{t}(x,t)\leq 0,\quad \forall
(x,t)\in ]L_{0},L[\times ]0,\infty [\\
a_{x}(x,t)\leq 0\,.
\end{gather*}


\begin{theorem} \label{thm4.1}
Take $\{ u^{0},v^{0}\}$ in $V$ and $\{ u^{1},v^{1}\}$ in
$L^{2}(0,L_{0})\times L^{2}(L_{0},L)$ with
\begin{equation}
u_{x}^{0}(L_{0})=0.    \label{eq4.1}
\end{equation}
Then there exists positive constants $\gamma $ and $c$ such that
\begin{equation}
E(t)\leq cE(0)e^{-\gamma t},\quad  \forall t\geq 0.    \label{eq4.2}
\end{equation}
We shall prove this theorem for strong solutions; our conclusion
follow by standard density arguments.
\end{theorem}

The dissipative property of  \eqref{e1.1}--\eqref{e1.6} is given by
the following lemma.

\begin{lemma} \label{lem4.2}
The first-order energy satisfies
\begin{equation}
\frac{d}{dt}\ E_{1}(t,u,v)=-\alpha | v_{t}| ^{2}+(a_{t},v_{x}^{2}).
\label{eq4.3}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying equation \eqref{e1.1} by $u_{t}$, equation \eqref{e1.2}
by $v_{t}$ and performing an integration by parts we get the result.
\end{proof}

Let $\psi \in C_{0}^{\infty }(0,L)$ be such that $\psi =1$ in
$]L_{0}-\delta ,L_{0}+\delta [$ for some $\delta >0$, small
constant. Let us introduce the following functional
\begin{equation*}
I(t)=\int_{0}^{L_{0}}\rho _{1}u_{t}qu_{x}dx+\int_{L_{0}}^{L}\rho
_{2}v_{t}\psi qv_{x}dx
\end{equation*}
where $q(x)=x$.

\begin{lemma} \label{lem4.3}
There exists $c_{1}>0$ such that for all $\varepsilon >0$,
\begin{align*}
\frac{d}{dt} I(t) &\leq -\frac{L_{0}}{2}\{ (\rho _{2}-\rho
_{1})v_{t}^{2}(L_{0},t)+a(L_{0},t)
[1-\frac{a(L_{0},t)}{b}]v_{x}^{2}(L_{0},t)\}  \\
&\quad -L_{0}(F_{1}(u(L_{0},t))
-F_{2}(v(L_{0},t)))-\frac{1}{2}\int_{0}^{L_{0}}(
\rho _{1}u_{t}+bu_{x}^{2}+2F(u))dx \\
&\quad -\frac{1}{4}\int_{L_{0}}^{L_{0}+\delta }av_{x}^{2}dx
+c_{1}\Big(\int_{L_{0}+\delta }^{L_{0}}(v_{t}^{2}+av_{x}^{2})dx
+\int_{L_{0}}^{L}v_{t}^{2}dx+\int_{0}^{L_{0}}u^{2}dx \\
&\quad +\int_{L_{0}}^{L}v^{2}dx\Big)+\varepsilon E(t,u,v)\,.
\end{align*}
\end{lemma}

\begin{proof}
 Multiplying  \eqref{e1.1} by $qu_{x}$, equation
\eqref{e1.2} by $\psi qv_{x}$, integrating by parts and using the
 corresponding
boundary conditions we obtain
\begin{gather}
\begin{aligned}
\frac{d}{dt}(\rho _{1}u_{t},qu_{x}) & = \frac{L_{0}}{2} [\rho
_{1}u_{t}^{2}(L_{0},t)+bu_{x}^{2}(
L_{0},t)]-L_{0}F_{1}(u(L_{0},t)) \\
&\quad - \frac{1}{2}\int_{0}^{L_{0}}\rho
_{1}u_{t}^{2}+bu_{x}^{2}+2F_{1}(u)dx
\end{aligned} \label{eq4.4}
\\
\begin{aligned}
\frac{d}{dt}(\rho _{2}v_{t},\psi qv_{x}) &\leq -\frac{L_{0}}{2}
\big[\rho _{2}v_{t}^{2}(L_{0},t)+a(L_{0},t) v_{x}^{2}(L_{0},t)\big]\\
&\quad +L_{0}F_{2}(v(L_{0},t))
+\frac{1}{2}\int_{L_{0}}^{L_{0}+\delta }xa_{x}\psi v_{x}^{2}dx
-\frac{1}{4}\int_{L_{0}}^{L_{0}+\delta }av_{x}^{2}dx \\
&\quad +c_{1}[\int_{L_{0}+\delta }^{L}(v_{t}^{2}+av_{x}^{2})dx
+\int_{L_{0}}^{L}(v_{t}^{2}+F_{2}(v))dx]
\end{aligned}  \label{eq4.5}
\end{gather}
Summing up (\ref{eq4.4}) and (\ref{eq4.5}), and taking the
assumption
 on $a_{x}$
into account, we get
\begin{equation}
\begin{aligned}
\frac{d}{dt}I(t) &\leq -\frac{L_{0}}{2}[(\rho_{2}-\rho
 _{1})v_{t}^{2}(L_{0},t)+a(L_{0},t)
v_{x}^{2}(L_{0},t)-bu_{x}^{2}(L_{0},t)]
\\
&\quad  -L_{0}[F_{1}(u(L_{0},t)) -F_{2}(v(L_{0},t))]\\
&\quad -\frac{1}{2}\int_{0}^{L_{0}}(\rho
 _{1}u_{t}^{2}+bu_{x}^{2}+2F_{1}(u))dx
 -\frac{1}{4}\int_{L_{0}}^{L_{0}+\delta }av_{x}^{2}dx \\
&\quad +c_{1}\Big(\int_{L_{0}+\delta }^{L}(
v_{t}^{2}+av_{x}^{2})dx+\int_{L_{0}}^{L}(v_{t}^{2}+F_{2}(v))dx
+\int_{0}^{L_{0}}F(u)dx\Big)
\end{aligned} \label{eq4.6}
\end{equation}
According to (A1), we have $f_{i}(0)=0$ and
\begin{equation}
| f_{i}(s)| \leq c(|s| +| s| ^{\rho }) \label{eq4.7}
\end{equation}
this implies
\begin{equation}
| F_{i}(s)| \leq c(|s| ^{2}+| s| ^{\rho +1}) \leq c(| s| ^{2}+| s|
^{2\rho }).  \label{eq4.8}
\end{equation}
 From the interpolation inequality
\begin{equation*}
| y| _{p}\leq | y| _{2}^{\alpha}| y| _{q}^{1-\alpha },\quad
\frac{1}{p}=\frac{\alpha }{2}+\frac{1-\alpha }{q},\quad \alpha \in
[0,1]
\end{equation*}
and the immersion $H^{1}(\Omega )\hookrightarrow L^{2( 2p-1)}(\Omega
),\ \ \Omega =]0,L_{0}[,\ ]L_{0},L[,$ we obtain for all $t\geq 0$
\begin{equation*}
| u(t)| _{2\rho }^{2\rho }\leq c_{\varepsilon }[E(0)]^{2(\rho -1) }|
u(t)| _{2}^{2}+\frac{\varepsilon }{[ E(0)]^{2(\rho -1)}}| u_{x}( t)|
_{2}^{2(2\rho -1)},\text{ for all \ } \varepsilon >0.
\end{equation*}
Considering that
\begin{equation*}
| u_{x}(t)| _{2}^{2}\leq cE( 0,u,v)\equiv c_{1}E(0)
\end{equation*}
it follows that
\begin{equation}
| u(t)| _{2\rho }^{2\rho }\leq c_{\varepsilon }[E(0)]^{2(\rho -1) }|
u(t)| _{2}^{2}+\varepsilon E( t,u,v).    \label{eq4.9}
\end{equation}

Replacing the inequalities (\ref{eq4.7})--(\ref{eq4.9}) in (4.6) our
conclusion follows.
\end{proof}

Let $\varphi \in C^{\infty }(\mathbb{R})$ a nonnegative function
such that $\varphi =0$  in $I_{\delta /2}=]L_{0}-\frac{ \delta
}{2},\ L_{0}+\frac{\delta }{2}[$ and $\varphi =1$  in $
\mathbb{R}\backslash I_{\delta }$ and consider the functional
\begin{equation*}
J(t)=\int_{L_{0}}^{L}\rho _{2}v_{t}\varphi v\ dx.
\end{equation*}
We have the following lemma.

\begin{lemma} \label{lem4.4}
Given $\varepsilon >0$, there exists a positive constant
$c_{\varepsilon }$ such that
\begin{equation*}
\frac{d}{dt}\ J(t)\leq
-\frac{1}{2}\int_{L_{0}+\delta}^{L}av_{x}^{2}\,
 dx
+\varepsilon \int_{L_{0}}^{L_{0}+\delta }av_{x}^{2}\,dx
+c_{\varepsilon }\int_{L_{0}}^{L}(v^{2}+v_{t}^{2})dx
\end{equation*}
\end{lemma}

\begin{proof}
Multiplying equation \eqref{e1.2} by $\varphi v$ and integrating by
parts we get
\begin{equation*}
\frac{d}{dt} J(t)=-(av_{x},\varphi v_{x})-(av_{x},\varphi _{x}v)
-\alpha (v_{t},\varphi v)-(\varphi ,f_{2}(v)v)+(v_{t},\varphi
v_{t}).
\end{equation*}
Applying Young's Inequality and hypothesis (A1) we concludes our
assertion.
\end{proof}

Let us consider the  functional
\begin{equation*}
K(t)=I(t)+(2c_{1}+1)J(t)
\end{equation*}
and we take $\varepsilon =\varepsilon _{1}$ in lemma 4.4, where
$\varepsilon_{1}$ is the solution of the equation
\begin{equation*}
(2c_{1}+1)\varepsilon _{1}=\frac{1}{8}\,.
\end{equation*}
Taking in to consideration (A1) in lemma 4.3, we obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt}K(t) & \leq   -E_{1}(t,u)-
\frac{1}{8}\int_{L_{0}}^{L}(av_{x}^{2}+2F_{2}(v))
dx+\varepsilon E(t,u,v) \\
&\quad +c_{2}(\int_{L_{0}}^{L}(v_{t}^{2}+v^{2})
dx+\int_{0}^{L_{0}}u^{2}dx).
\end{aligned}  \label{eq4.10}
\end{equation}
Now in order to estimate the last two terms of (\ref{eq4.10}) we
need
 the
following result.

\begin{lemma} \label{lem4.5}
Let $\{ u,v\} $ be a solution in theorem 3.3. Then there exists
 $T_{0}>0$ such that if $T\geq T_{0}$ we have
\begin{equation}
\int_{S}^{T}(| v| ^{2}+|u| ^{2})ds \leq \varepsilon
\big[\int_{S}^{T}(b| u_{x}| ^{2}+|u_{t}| ^{2}) ds+\int_{S}^{T}|
a^{1/2}v_{x}| ^{2}ds\big] +c_{\varepsilon }\int_{S}^{t}| v_{t}|
^{2}ds
  \label{eq4.11}
\end{equation}
for any $\varepsilon >0$ and $c_{\varepsilon }$ is a constant
depending on $T$ and $\varepsilon $, by independent of
 $\{ u,v\} $, for any initial data $\{u^{0},v^{0}\} , \{ u^{1},v^{1}\}
 $
satisfying $E(0,u,v)\leq R$, where $R>0$ is fixed and $0<S<T<+\infty
$.
\end{lemma}

\begin{proof}
We use a contradiction argument. If (\ref{eq4.11}) were false, there
would exist a sequence of solutions $\{ u^{\nu },v^{\nu }\} $ such
that
\begin{equation*}
\int_{S}^{T}(| v^{\nu }| ^{2}+| u^{\nu }| ^{2})ds\geq \nu
\int_{S}^{t}| v_{t}^{\nu }| ^{2}ds+c_{0}\int_{S}^{T}(b| u_{x}^{\nu
}| ^{2}+| u_{t}| ^{2}+| a^{1/2}v_{x}| ^{2})ds
\end{equation*}
and  $E(0,u^{\nu },v^{\nu })\leq R$ for all $\nu$. Let
\begin{gather*}
 \lambda _{\nu }^{2}=\int_{S}^{T}(| v^{\nu }|^{2}+| u^{\nu }|
 ^{2})ds,\\
w^{\nu }(x,t)=\frac{u^{\nu }(x,t)}{\lambda _{\nu}} , \quad z^{\nu
}(x,t)=\frac{v^{\nu }(x,t)}{\lambda _{\nu }}, \quad 0\leq t\leq T.
\end{gather*}
Then we have
\begin{equation*}
\nu \int_{S}^{T}| z_{t}^{\nu }| ^{2}ds+c_{0}\int_{S}^{T}\Big(b|
w_{x}^{\nu }| ^{2}+| w_{t}^{\nu }| ^{2}+|a^{1/2}z_{x}^{\nu }|
^{2}\Big)ds\leq 1
\end{equation*}
and consequently
\begin{gather}
\int_{S}^{T}| z_{t}^{\nu }| ^{2}ds\to 0\quad \text{as }\nu \to
\infty,
\label{eq4.12} \\
\int_{S}^{T}(b| w_{x}^{\nu }|^{2}+| w_{t}^{\nu }| ^{2}
+|a^{1/2}z_{x}^{\nu }| ^{2})ds\leq c.  \label{eq4.13}
\end{gather}
Also we have
\begin{equation}
\int_{S}^{T}\big(| z^{\nu }| ^{2}+| w^{\nu }| ^{2}\big)ds=1\,.
\label{eq4.14}
\end{equation}
Since $S$ is chosen in the interval $[0,T[$, we obtain from
(\ref{eq4.12})--(\ref{eq4.13}) that, there exists a subsequence $\{
w^{\nu},z^{\nu }\} $ which we denote in the same way, such that
\begin{gather*}
w^{\nu } \to w\quad \text{in }L^{2}(0,T;H^{1}(0,L_{0})),\\
w_{t}^{\nu } \to w_{t}\quad \text{in  }L^{2}(0,T;L^{2}(0,L_{0})),\\
z^{\nu } \to z\quad \text{in }L^{2}(0,T;H^{1}(L_{0},L)),\\
z_{t}^{\nu } \to 0\quad \text{in  }L^{2}(0,T;L^{2}(L_{0},L)).
\end{gather*}
From which
\begin{gather*}
w^{\nu } \to w\quad \text{in }L^{2}(0,T;L^{2}(0,L_{0})),\\
z^{\nu } \to z\quad \text{in  }L^{2}(0,T;L^{2}(L_{0},L)).
\end{gather*}
This implies
\begin{equation}
\int_{0}^{T}\big(| z| ^{2}+| w|^{2}\big)ds=1\,.    \label{eq4.15}
\end{equation}
Besides, from the uniqueness of the limit we conclude that
$z_{t}(x,0)=0$ and therefore
\begin{equation}
z(x,t)=\varphi (x) \,. \label{eq4.16}
\end{equation}
\end{proof}

Note that $\{ w^{\nu },z^{\nu }\} $ satisfies
\begin{equation} \label{eq4.17}
\begin{gathered}
\rho _{1}w_{tt}^{\nu }-bw_{xx}^{\nu }+\frac{1}{\lambda _{\nu
}}f_{1}(
\lambda _{\nu }w^{\nu })=0\text{ \ in \ }]0,L_{0}[\times ]0,T[, \\
\rho _{2}z_{tt}^{\nu }-(a(x,t)z_{x}^{\nu })_{x}+ \frac{1}{\lambda
_{\nu }}f_{2}(\lambda _{\nu }z^{\nu })+\alpha z_{t}^{\nu }=0\text{ \
in \ }]L_{0},L[\times ]0,T[,
\\
w^{\nu }(0,t)=0=z^{\nu }(L,t), \\
w^{\nu }(L_{0},t)=z^{\nu }(L_{0},t), \\
bw_{x}^{\nu }(L_{0},t)=a(L_{0},t)z_{x}^{\nu}(L_{0},t), \\
w^{\nu }(x,0)=\frac{u^{\nu ,0}(x)}{\lambda _{\nu }},\quad
w_{t}^{\nu }(x,0)=\frac{1}{\lambda _{\nu }}u^{\nu ,1}(x), \\
z^{\nu }(x,0)=\frac{1}{\lambda _{\nu }}\ v^{\nu ,0}(x),\quad
z_{t}^{\nu }(x,0)=\frac{1}{\lambda_{\nu }}\ v^{\nu ,1}(x).
\end{gathered}
\end{equation}
Now, we observe that $\{ \lambda _{\nu }\} _{\nu \geq 1}$ is a
bounded sequence,
\begin{align*}
\lambda _{\nu } &=\Big[\int_{S}^{T}(| v^{\nu }|
^{2}+| u^{\nu }| ^{2})ds\Big]^{1/2}\\
&\leq c\Big[ \int_{S}^{T}(| v_{x}^{\nu }| ^{2}+|
u_{x}^{\nu }| ^{2})ds\Big]^{1/2} \\
&\leq cE(0,u,v)\leq cR,
\end{align*}
where $R$ is a fixed value, because the initial data are in the ball
$B(\theta ,R)$. Hence, there exists a subsequence of $\{ \lambda
_{\nu }\} _{\nu \geq 1}$ (still denoted by $(\lambda _{\nu
 })$
such that
\begin{equation*}
\lambda _{\nu }\to \lambda \in ]0,+\infty [.
\end{equation*}
In this case passing to limit in \eqref{eq4.17},
 when $\nu \to \infty $  for $\{ w,z\} $, we get
\begin{equation}
\begin{gathered}
\rho _{1}w_{tt}-bw_{xx}+\frac{1}{\lambda }\ f_{1}(\lambda w)=0
\text{ \ in \ }]0,L_{0}[\times ]0,T[, \\
(a(x,t)z_{x})_{x}+\frac{1}{\lambda }f_{2}(\lambda z)=0\quad
 \text{in  }]L_{0},L[\times ]0,T[, \\
w(0,t)=0=z(L,t) \\
w(L_{0},t)=z(L_{0},t) \\
bw_{x}(L_{0},t)=a(L_{0},t)z_{x}(L_{0},t), \\
z_{t}(x,0)=0 \quad  \text{in }]L_{0},L[\times ]0,T[,
\end{gathered}  \label{eq4.18}
\end{equation}
and for $y=w_{t}$,
\begin{equation}
\begin{gathered}
\rho _{1}y_{tt}-by_{xx}+f'(\lambda w)y=0\quad \text{in
}]0,L_{0}[\times
 ]0,T[,\\
y(0,t)=0=y(L_{0},t), \\
by_{x}(L_{0},t)=a_{t}(L_{0},t)z_{x}(L_{0},t).
\end{gathered}  \label{eq4.19}
\end{equation}
Here, we observe that
\begin{equation*}
\frac{w_{xt}(L_{0},t)}{w_{x}(L_{0},t)}=\frac{
a_{t}(L_{0},t)}{a(L_{0},t)}\,.
\end{equation*}
Then  after an integration, $w_{x}(L_{0},t)=k\ a(L_{0},t)$ whee $k$
is a constant. Using the hypotheses, we obtain
\begin{equation*}
0=\lim_{t\to 0^{+}} w_{x}(L_{0},t)=k a(L_{0},0).
\end{equation*}
Consequently  $k=0$  and $\ y_{x}(L_{0},t)=0$. Thus, the function
$y$ satisfies
\begin{gather*}
\rho _{1}y_{tt}-by_{xx}+f'(\lambda w)y=0 \quad \text{in
 }]0,L_{0}[\times ]0,T[,
 \\
y(0,t)=0=y(L_{0},t)\quad\text{on }]0,T[, \\
y_{x}(L_{0},t)=0 \quad \text{on }\ ]0,T[.
\end{gather*}
Then, using the results in \cite{l2} (based on Ruiz arguments
 \cite{r1})
adapted to our case we conclude that $y=0$, that is $w_{t}(x,t)=0$,
for $T$ suitable big.

Returning to (\ref{eq4.18}) we obtain the  elliptic system
\begin{gather*}
-bw_{xx}+\frac{1}{\lambda }\ f_{1}(\lambda w)=0, \\
(a(x,t)z_{x})_{x}+\frac{1}{\lambda }f_{2}(\lambda z)=0\,.
\end{gather*}
multiplying by $u$ and $v$ respectively and integrating, then
summing up we arrive at
\begin{equation*}
b\int_{0}^{L_{0}}w_{x}^{2}dx+\int_{L_{0}}^{L}a(x,t)z_{x}^{2}dx+
\frac{1}{\lambda }\int_{0}^{L_{0}}f_{1}(\lambda w)wdx+\frac{1}{
\lambda }\int_{L_{0}}^{L}f_{2}(\lambda z)zdx=0\,.
\end{equation*}
So we have $w=0$ and $z=0$, which contradicts (\ref{eq4.15}).

Suppose we are not in the above situation and there exists a
 subsequence
satisfying $\lambda _{\nu }\to 0$. Applying inequality
(\ref{eq4.10}) to the solutions $\{ u^{\nu },v^{\nu}\} $ we have
\begin{equation*}
\frac{d}{dt} K^{\nu }(t)\leq -\delta _{0} E(t,u^{\nu },v^{\nu
})+c_{3}\Big(\int_{L_{0}}^{L}((v_{t}^{\nu })^{2}+(v^{\nu
})^{2})dx+\int_{0}^{L_{0}}( u^{\nu })^{2}dx\Big),
\end{equation*}
integrating from $s$ to $T$, we obtain
\begin{equation*}
K^{\nu }(T)+\delta _{0}\int_{S}^{T}E(t,u^{\nu },v^{\nu })dt\leq
K(S)+c_{3}\Big(\int_{S}^{T}(| v_{t}^{\nu }| ^{2}+| v^{\nu }| ^{2}+|
u^{\nu }| ^{2})\Big)dt.
\end{equation*}
Since $K^{\nu }$ satisfies
\begin{equation*}
c_{0}E(t,u^{\nu },v^{\nu })\leq K^{\nu }(T)\leq
c_{1}E(t,u^{v},v^{\nu })
\end{equation*}
and $E$ is a decreasing function we have
\begin{align*}
&E(T,u^{v},v^{\nu })+\delta _{0}'\int_{S}^{T}E(t,u^{\nu },v^{\nu })dt\\
&\leq \frac{c_{1}'}{T}\int_{S}^{T}E(t,u^{\nu },v^{\nu })dt
 +c_{3}\int_{S}^{T}(| v_{t}^{\nu }|^{2}+| v^{\nu }| ^{2}+| u^{\nu
 }|^{2})dt;
\end{align*}
thus, we obtain
\begin{equation*}
E(T,u^{v},v^{\nu })+(\delta _{0}'-\frac{
c_{1}'}{T})\int_{S}^{T}E(t,u^{\nu },v^{\nu })dt \leq
c_{3}\int_{S}^{T}\big(| v_{t}^{\nu }| ^{2}+| v^{\nu }| ^{2}+| u^{\nu
}|^{2}\big)dt,
\end{equation*}
Dividing both sides of the above inequality by $\lambda _{\nu
}^{2}$, using (\ref{eq4.12}) and (\ref{eq4.14}), taking $T$ large
enough we
 conclude
that $(| w_{t}^{\nu }| ^{2}+| z_{t}^{\nu }| ^{2}+| w_{x}^{\nu }|
^{2}+|z_{x}^{\nu }| ^{2})(T)$ is bounded. Now, multiplying equations
\eqref{eq4.17}$_1$, \eqref{eq4.17}$_2$ by $w_{t}^{\nu }$ and
$z_{t}^{\nu }$ respectively, performing an
 integration by
parts we get
\begin{equation*}
E(t,w^{v},z^{\nu })\leq E(T,w^{v},z^{\nu })+\alpha \int_{S}^{T}|
z_{t}^{\nu }| ^{2}dt-\int_{S}^{T}( a_{t},(z_{x}^{\nu })^{2})dt.
\end{equation*}
 From (\ref{eq4.12}) , (\ref{eq4.13}) and Poincare Inequality we deduce
that $E(t,w^{v},z^{\nu })$ is bounded for all $t\in [S,T]$. Then in
particular, on a subsequence we obtain
\begin{gather*}
w^{\nu }\to w \quad\text{weak star in }L^{\infty }(
0,T;H^{1}(0,L_{0})), \\
w_{t}^{\nu }\to w_{t}\quad\text{weak star in }L^{\infty }(
0,T;L^{2}(0,L_{0})), \\
z^{\nu }\to z\quad\text{weak star in }L^{\infty }(
0,T;H^{1}(L_{0},L)), \\
z_{t}^{\nu }\to z_{t}\quad\text{weak star in } L^{\infty }(
0,T;L^{2}(L_{0},L)), \\
w^{\nu }\to w \quad\text{in } L^{2}(0,T;L^{2}(0,L_{0})), \\
z^{\nu }\to z \quad\text{in }L^{2}(0,T;L^{2}(L_{0},L)).
\end{gather*}
On the other hand,we note that
\begin{gather}
\frac{1}{\lambda _{\nu }}f_{1}(\lambda _{\nu }w^{\nu })
 \to f_{1}'(0)w \quad \text{in } L^{2}(0,T;L^{2}(0,L_{0})x]0,T[),
\label{eq*}\\
\frac{1}{\lambda _{\nu }}f_{2}(\lambda _{\nu }z^{\nu })
 \to f_{2}'(0)z \quad \text{in } L^{2}(0,T;L^{2}(L_{0},L)x]0,T[).
\label{eq**}
\end{gather}
Indeed
\begin{align*}
\Delta _{\nu } &=| f_{1}'(0)w^{\nu }-\frac{1}{\lambda _{\nu
}}f_{1}(\lambda _{\nu
 }w^{\nu })|
_{L^{2}((0,L_{0})x]0,T[)}^{2}
\\
&=\int_{| u^{\nu }| \leq  \epsilon }| f_{1}'(0)w^{\nu
}-\frac{1}{\lambda _{\nu }} f_{1}(u^{\nu })| ^{2}\,dx\,dt+\int_{|
u^{\nu }| >\epsilon }| f_{1}'(0)w^{\nu }- \frac{1}{\lambda _{\nu
}}f_{1}(u^{\nu })| ^{2}\,dx\,dt
\\
& \leq \int_{| u^{\nu }| \leq \epsilon}| w^{\nu }| ^{2}| f_{1}'(
0)-\frac{1}{\lambda _{\nu }w^{\nu }}f_{1}(u^{\nu }) |
^{2}\,dx\,dt+2| f_{1}'(0)| ^{2}\int_{| u^{\nu }| >\epsilon }| w^{\nu
}| ^{2}\,dx\,dt
\\
&\quad +2\int_{| u^{\nu }| >\epsilon }\frac{1}{\lambda _{\nu }^{2}}|
f_{1}(u^{\nu })| ^{2}\,dx\,dt
\\
&\leq M_{\varepsilon }^{2}| w^{\nu }| _{L^{2}((0,L_{0})x]0,T[)}^{2}
+C\int_{| u^{\nu}| >\epsilon }(\frac{1}{\lambda _{\nu }^{2}}| u^{\nu
}| ^{2}+\frac{1}{\lambda _{\nu }^{2}}| u^{\nu }| ^{2\rho })\,dx\,dt
\\
&\leq M_{\varepsilon }^{2}| w^{\nu }| _{L^{2}(
0,L_{0}x]0,T[)}^{2}+C\int_{| u^{\nu }| >\epsilon }\frac{1}{\lambda
_{\nu }^{2}}| u^{\nu }| ^{2\rho }(1+\frac{1}{\varepsilon ^{2\rho
-2}})\,dx\,dt
\\
&\leq M_{\varepsilon }^{2}| w^{\nu }| _{L^{2}((
0,L_{0})x]0,T[)}^{2}+C_{\varepsilon }\lambda _{\nu }^{2\rho -2}|
w^{\nu }| _{L^{2\rho }(( 0,L_{0})x]0,T[)}^{2\rho }
\end{align*}
where $M_{\varepsilon }=\sup_{| s| \leq \varepsilon }| f_{1}'(0)
-\frac{f_{1}(s)}{s}|$, $M_{\varepsilon }\to 0$  as $\varepsilon \to
0$.

 From \eqref{eq4.13}, $\{ w^{\nu }\}$  is bounded in
$L^{\infty }(0,T;H^{1}(0,L_{0}))\hookrightarrow L^{\infty
}(0,T;L^{2\rho }(0,L_{0}))$, and consequently
\[
\limsup_{\nu \to \infty }\Delta _{\nu }\leq \sup_{\nu }| w^{\nu }|
_{L^{2}((0,L_{0})x]0,T[ )}^{2}.M_{\varepsilon }^{2}
\]
Thus,taking $\varepsilon \to 0$  we obtain \eqref{eq*}. Applying a
similar method as that used for $\{ w^{\nu }\} $ we get
\eqref{eq**}.

Now, the limit function $\{ w,z\}$ satisfies
\begin{gather*}
\rho _{1}w_{tt}-bw_{xx}+f_{1}'(0)w=0 \quad\text{in }
]0,L_{0}[\times ]0,T[, \\
(a(x,t)z_{x})_{x}+f_{2}'(0)z=0 \quad\text{in }]L_{0},L[\times ]0,T[, \\
w(0,t)=0=z(L,t),   \\
w(L_{0},t)=z(L_{0},t),  \\
bw_{x}(L_{0},t)=a(L_{0},t)z_{x}(L_{0},t),  \\
z_{t}(x,t)=0 \quad\text{in } ]L_{0},L[\times ]0,T[
\end{gather*}
Repeating the above procedure we get $w=0$ and $z=0$ which is a
contradiction. The proof of lemma 4.5 is now complete.
%\end{proof}

\begin{proof}[Proof of theorem 4.1]
Let us introduce the functional
\begin{equation*}
L(t)=N\ E(t)+K(t)
\end{equation*}
with $N>0$. Using Young's Inequality and taking $N$ large enough we
 find that
\begin{equation}
\theta _{0}E(t)\leq L(t)\leq \theta _{1}E(t)  \label{eq4.20}
\end{equation}
for some positive constants $\theta _{0}$ and $\theta _{1}$.

Applying the inequalities (\ref{eq4.10}) and (\ref{eq4.20}), along
with
 the
ones in Lemma 4.5 and integrating from $S$ to $T$ where $0\leq S\leq
T<\infty $ we obtain
\begin{equation*}
\int_{S}^{T}E(t)dt\leq c\ E(S).
\end{equation*}
In this situation, lemma 2.1 implies
\begin{equation*}
E(t)\leq c\ E(0)e^{-rt}\ ,
\end{equation*}
this completes the proof.
\end{proof}

\noindent\textbf{Remark.} If in Equation \eqref{e1.2} we consider a
linear localized dissipation $\alpha =\alpha (x)$ in
$C^{2}(]L_{0},L[)$, with $\alpha (x)=1$  in $]L_{0},L_{0}+\delta [$
, $\alpha (x)=0$  in $]L_{0}+2\delta ,L[$, then  our situation is
very delicate and we need a new unique continuation theorem for the
wave equation with variable coefficients. This is a
 work
in preparation by the authors.

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\end{document}