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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 60, pp. 1--10.\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-2007/60\hfil Transmission problem for waves]
{Transmission problem for waves with frictional damping}

\author[ W. D. Bastos, C. A. Raposo\hfil EJDE-2007/60\hfilneg]
{Waldemar D. Bastos, Carlos A. Raposo}  % in alphabetical order

\address{Waldemar D. Bastos \newline
Universidade Estadual Paulista - UNESP, Departamento de
Matem\'atica, 15054-000 - S\~ao Jos\'e do Rio Preto - SP, Brazil}
\email{waldemar@ibilce.unesp.br}

\address{Carlos A. Raposo \newline
Universidade Federal de S\~ao Jo\~ao del-Rei - UFSJ, Departamento
de Matem\'atica, 36300-000 - S\~ao Jo\~ao del-Rei, MG, Brazil}
\email{raposo@ufsj.edu.br}

\thanks{Submitted June 23, 2006. Published April 22, 2007.}
\subjclass[2000]{35B35, 35B40, 35L05, 35L20}
\keywords{Exponential stability; dissipative system; transmission problem}

\begin{abstract}
 In this paper we consider the transmission problem, in one space
 dimension, for linear dissipative waves with frictional damping.
 We study the wave propagation in a medium with a component with
 attrition and another simply elastic. We show that for this type
 of material, the dissipation produced by the frictional part is
 strong enough to produce exponential decay of the solution,
 no matter how small is its size.
\end{abstract}

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

\section{Introduction}

A number of authors have studied the wave equation with dissipation.
We mention for example, the work of Zuazua \cite{z1} where it was
obtained the uniform rate of decay of the solution for a large class
of nonlinear wave equation with frictional damping acting in the
whole domain. In this direction, the natural question that arises is
about the rate of decay when the dissipation is effective only in a
part of the domain. It is the purpose of this investigation, at
least in part, to answer this question. We consider the wave
propagation over a body consisting of two different type of
materials. This is a transmission (or diffraction) problem. It
happens frequently in applications where the domain is occupied by
several materials, whose elastic properties are different, joined
together over the whole of a surface. From the mathematical point of
view a transmission problem for wave propagation consists on a
hyperbolic equation for which the corresponding elliptic operator
has discontinuous coefficients. Even though we consider a case of
space dimension one and linear equations with constant coefficients,
the problem studied here is interesting by its own.

Existence, regularity,  as well as the exact
controllability for the transmission problem for the pure wave
equation was studied in \cite{l1}. The transmission problem for
viscoelastic waves was studied by Rivera and Oquendo \cite{r2} who
proved the exponential decay of solution using regularity results of
the Volterra's integral equations and regularizing properties of the
viscosity. The asymptotic behavior for a coupled system of equations
of waves was studied by Raposo \cite{r1} by the same method used in
this paper.

 Let $k_{1}, k_{2}$ and $\alpha$ be positive real numbers and
$0<L_{0}< L$. The system
considered here is
\begin{gather}
 u_{tt}-k_{1}u_{xx}+\alpha u_{t}
=0,\quad x\in (0,L_{0}),\; t>0,  \label{1.1} \\
v_{tt}-k_{2}v_{xx} = 0,\quad x\in (L_{0},L),\; t>0,
 \label{1.2}
\end{gather}
 satisfying the boundary conditions
\begin{equation} \label{1.3}
 u(0,t)=v(L,t)=0,\quad t>0,
\end{equation}
 the transmission conditions
\begin{equation} \label{1.4}
 u(L_{0},t)=v(L_{0},t),\quad
k_{1}u_{x}(L_{0},t)=k_{2}v_{x}(L_{0},t),\quad t>0, \end{equation}
and initial conditions
\begin{equation} \label{1.5}
\begin{gathered}
 u(x,0) = u^{0}(x),\quad u_{t}(x,0)=u^{1}(x),\quad x\in (0,L_{0}),   \\
 v(x,0) = v^{0}(x),\quad v_{t}(x,0)=v^{1}(x),\quad x\in (L_{0},L).
\end{gathered}
\end{equation}
We are  concerned with the asymptotic properties of the
 system above. The main result of this paper is Theorem \ref{thm3.1}
which shows that the
solution of the transmission problem  \eqref{1.1}--\eqref{1.5}
decays exponentially to zero as time goes to infinity, no matter how
large is the difference $L-L_{0}$. The approach we use consists of
choosing appropriate multipliers to build a functional of Lyapunov
for the system.

 The notation used throughout this work is the standard one.
For instance $H^{m}$, $L^{2}=H^{0}$,
$W^{m,\,p}$  and  $W^{m,\,\infty }$ denote the usual Sobolev Spaces
(see Adams \cite{a1}). By $\mathcal{V}$  we denote the space
$$
 \mathcal{V}:=\{(u,v)\in H^{1}(0,L_{0})\times
H^{1}(L_{0},L):u(0)=v(L)=0,\;u(L_{0})=v(L_{0})\}
$$
which together with the inner product
$$
\langle (u^{1},v^{1})\,,\,(u^{2},v^{2})\rangle
:=\int_{0}^{L_{0}}u_{x}^{1}u_{x}^{2}\,dx
+ \int_{L_{0}}^{L}v_{x}^{1}v_{x}^{2}\,dx
$$
is a Hilbert space. The energies associated to the equations
(\ref{1.1}) and (\ref{1.2}) are:
\begin{gather*}
E_{1}(t) = \frac{1}{2}\int_{0}^{L_{0}}[|u_{t}|^{2}+k_{1}|u_{x}|^{2}]\,dx, \\
E_{2}(t) = \frac{1}{2}\int_{L_{0}}^{L}[|v_{t}|^{2}+k_{2}|v_{x}|^{2}]\,dx
\end{gather*}
respectively. We denote $E(t)= E_{1}(t) + E_{2}(t)$ the total energy
associated to the system (\ref{1.1})--(\ref{1.5}).

The remainder of this paper is organized as follows. In Section 2 we
show the existence of weak and strong solutions for the system
(\ref{1.1})--(\ref{1.5}), and in Section 3 we show the exponential
decay of such solutions.

 \section{Existence of solutions}

We begin this section defining what is meant by weak solution to our
transmission problem.
\begin{definition} \label{def2.1} \rm
 The couple $ ( u(x,t),v(x,t))$ is a weak solution of the system
(\ref{1.1})--(\ref{1.5})  when
 $$
(u,v) \in \, L^{\infty}(0,T;{\mathcal{V}}) \cap
W^{1,\infty}(0,T;L^{2}(0,L_{0}) \times L^{2}(L_{0},L)), $$
 and satisfies
\begin{gather*}
-\int_{0}^{L_{0}}u^{1}\phi
(0)\,dx-\int_{L_{0}}^{L}v^{1}\psi
(0)\,dx-\int_{0}^{T}\int_{0}^{L_{0}}u_{t}\phi
_{t}\,dx\,dt-\int_{0}^{T}\int_{L_{0}}^{L}v_{t}\psi _{t}\,dx\,dt \\
+ \,\,k_{1}\int_{0}^{T}\int_{0}^{L_{0}}u_{x}\phi _{x}\,dx\,dt+
k_{2}\int_{0}^{T}\int_{L_{0}}^{L}v_{x}\psi _{x}\,dx\,dt+\alpha
\int_{0}^{T}\int_{0}^{L_{0}}u_{t}\phi \,dx\,dt =0
\end{gather*}
for any
\[
(\phi ,\psi )\in \,L^{\infty }(0,T;{})\cap
W^{1,\infty }(0,T;L^{2}(0,L_{0})\times L^{2}(L_{0},L)),
\]
such that
\[
(\phi (T),\psi (T))=(0,0).
\]
\end{definition}

\begin{theorem} \label{thm2.1}
Let us take $(u^{0},v^{0}) \in \, (H^{2}(0,L_{0}) \times
H^{2}(L_{0},L)) \cap \mathcal{V}$ and $(u^{1},v^{1}) \in
\,\mathcal{V} $ verifying the transmission conditions.  Under this
conditions the solution $(u,v)$ of (\ref{1.1})--(\ref{1.5})
satisfies
$$
(u,v) \in  \bigcap_{k=0}^{2} W^{k,\infty}(0,T;H^{2-k}(0,L_{0}))
\times H^{2-k}(L_{0},L).
$$
\end{theorem}
\begin{proof}
The existence is proved using Galerkin method. In
order to do so we take a basis
$\{(\phi^{0},\psi^{0}),\,(\phi^{1},\psi^{1}),\,(\phi^{2},\psi^{2}) ,
\cdot \cdot \cdot \}$ of $\mathcal{V}$
 and let
 $$
(u^{0}_{m},v^{0}_{m}),(u^{1}_{m},v^{1}_{m}) \in \, span\{
(\phi^{0},\psi^{0}),(\phi^{1},\psi^{1})\cdot \cdot \cdot
\,(\phi^{m},\psi^{m})\}
 $$
 be a  projection of the initial state on a finite dimensional subspace
of $\mathcal{V} $. Standard results on
 ordinary differential equations guarantee that there exists one and
 only one solution
 $$
(u^{m}(t),v^{m}(t)) := \sum_{j=1}^{m}\,
h_{j,m}(t)(\phi^{j},\psi^{j})
 $$
of the approximated system,
 \begin{equation} \label{2.1}
\int_{0}^{L_{0}}    u_{tt}\phi^{i}\,dx +
\int_{L_{0}}^{L}    v_{tt}\psi^{i}\,dx +
k_{1}\int_{0}^{L_{0}}    u_{x}\phi_{x}^{i}\,dx +
k_{2}\int_{L_{0}}^{L}    v_{x}\psi_{x}^{i}\,dx + \alpha
\int_{0}^{L_{0}}    u_{t}\phi^{i}\,dx = 0\,\,
 \end{equation}
  $ i =  0, 1, 2, \dots,m$,
with initial data
 $$
(u^{m}(0),v^{m}(0))=(u^{0}_{m},v^{0}_{m}), \quad
(u_{t}^{m}(0),v_{t}^{m}(0))=(u^{1}_{m},v^{1}_{m}).
 $$
 We show next that the above solution remain bounded for any
$m \in \mathbf{N}$. In order to do
 so, we first multiply equation (\ref{2.1}) by $h'_{j,m}(t)$ and then
sum up in $i$,
 to obtain
 $$
\frac{d}{dt}E^{m}(t) = -\alpha \int_{0}^{L_{0}}|u^{m}_{t}|^{2}\,dx.
 $$
 Integrating the identity above from $0$ to $t$, we get
 $$
E^{m}(t) \leq E^{m}(0)
 $$
 showing that the first order energy $ E^{m}(t)$ is uniformly bounded
for $m \in \mathbf{N}$.

 Now we denote the second order energy by
$$
{\mathcal{E}}^{m}(t)= \frac{1}{2}\int_{0}^{L_{0}}[ |u_{tt}^{m}|^{2}
+
 k_{1}|u_{xt}^{m}|^{2} ]\,dx  + \frac{1}{2}\int_{L_{0}}^{L}[ |v_{tt}^{m}|^{2} +
 k_{2}|v_{xt}^{m}|^{2} ]\,dx.
$$
Differentiating equation (\ref{2.1}) with respect to $t$, we get
 \begin{equation} \label{2.2}
\begin{aligned}
&\int_{0}^{L_{0}} u_{ttt}\phi^{i}\,dx
+ \int_{L_{0}}^{L} v_{ttt}\psi^{i}\,dx
+ k_{1}  \int_{0}^{L_{0}} u_{xt}\phi_{x}^{i}\,dx \\
&+ k_{2}  \int_{L_{0}}^{L}    v_{xt}\psi_{x}^{i}\,dx + \alpha
\int_{0}^{L_{0}}    u_{tt}\phi^{i}\,dx = 0.
\end{aligned}
 \end{equation}
Multiplying equation (\ref{2.2}) by $h''_{j,m}(t)$ and summing up in
$i$, we obtain
$$
\frac{d}{dt}{\mathcal{E}}^{m}(t) = -\alpha
\int_{0}^{L_{0}}|u^{m}_{tt}|^{2}\,dx
 $$
 which integrated from $0$ to $t$ furnishes
 $$
{\mathcal{E}}^{m}(t) \leq {\mathcal{E}}^{m}(0).
 $$
 The next step is to estimate the second order energy. Letting $t \rightarrow 0^{+}$ in equation (\ref{2.1}),
 multiplying the limit result by $h''_{j,m}(t)$ we get
\begin{align*}
&\int_{0}^{L_{0}} |u_{tt}^{m}(0)|^{2}\,dx +
\int_{L_{0}}^{L}    |v_{tt}^{m}(0)|^{2}\,dx\\
&= -k_{1}\int_{0}^{L_{0}}    u_{x}^{m}(0)u_{xtt}^{m}(0)\,dx
- k_{2}\int_{L_{0}}^{L}    v_{x}^{m}(0)v_{xtt}^{m}(0)\,dx
 - \alpha \int_{0}^{L_{0}}u_{t}^{m}(0)u_{tt}^{m}(0)\,dx .
 \end{align*}
 Integrating by parts the equation above, we get
\begin{equation} \label{Y}
\begin{aligned}
&\int_{0}^{L_{0}}    |u_{tt}^{m}(0)|^{2}\,dx
+ \int_{L_{0}}^{L}    |v_{tt}^{m}(0)|^{2}\,dx \\
&= k_{1}\int_{0}^{L_{0}}    u_{xx}^{m}(0)u_{tt}^{m}(0)\,dx
+ k_{2}\int_{L_{0}}^{L}    v_{xx}^{m}(0)v_{tt}^{m}(0)\,dx
- \alpha \int_{0}^{L_{0}}    u_{t}^{m}(0)u_{tt}^{m}(0)\,dx .
\end{aligned}
 \end{equation}
 After application of Young's inequality in equation (\ref{Y})
 we find
\begin{align*}
&\int_{0}^{L_{0}}|u_{tt}^{m}(0)|^{2}\,dx +
\int_{L_{0}}^{L}|v_{tt}^{m}(0)|^{2}\,dx \\
&\leq c\Big\{\int_{0}^{L_{0}}|u_{xx}^{m}(0)|^{2}\,dx
+ \int_{L_{0}}^{L}|v_{xx}^{m}(0)|^{2}\,dx\Big\}
+ c\int_{0}^{L_{0}}|u_{t}^{m}(0)|^{2}\,dx .
 \end{align*}
 which implies that the initial data
$$
(u_{tt}^{m}(0),v_{tt}^{m}(0)) \quad\mbox{is bounded in }
L^{2}(0,L_{0}) \times L^{2}(L_{0},L)),
$$
and so is ${\mathcal{E}}^{m}(0)$. Whence we have
\[ %\label{2.5}
{\mathcal{E}}^{m}(t) \quad\mbox{is bounded for every } m \in
\mathbf{N}.
\]
The first and second order energy  boundedness implies that there
exists a subsequence of $(u^{m},v^{m})$, which we still denote in
the same way, such that
\begin{gather*}
(u^{m},v^{m}) \stackrel{*}{\rightharpoonup} (u,v) \quad \mbox{in }
  L^{\infty}(0.T;\mathcal{V}),\\
(u^{m}_{t},v^{m}_{t}) \stackrel{*}{\rightharpoonup} (u_{t},v_{t}) \quad
  \mbox{in }  L^{\infty}(0.T;\mathcal{V}),\\
(u^{m}_{tt},v^{m}_{tt}) \stackrel{*}{\rightharpoonup}
(u_{tt},v_{tt})
 \quad \mbox{in } L^{\infty}(0.T;L^{2}(0,L_{0}) \times L^{2}(L_{0},L))).
\end{gather*}
Therefore the couple $(u,v)$ satisfies
\begin{gather*}
u_{tt} - k_{1}u_{xx} + \alpha u_{t} = 0 \\
v_{tt} - k_{2}v_{xx} = 0.
\end{gather*}
The rest of the proof is a matter of routine.
\end{proof}

 \section{Exponential stability}

With a view toward proving the main result of this paper we
formulate and prove a series of five lemmas. They will provide some
technical inequalities which play fundamental role in the proof of
Theorem \ref{thm3.1}.

\begin{lemma} \label{lem3.1}
The total energy $E(t)$ satisfies
\[
 \frac{d}{dt}E(t) = - \alpha \int_{0}^{L_{0}}|u_{t}|^{2} \, dx.
\]
 \end{lemma}

\begin{proof}
 Multiplying equation (\ref{1.1}) by $u_{t}$ and
 integrating in $(0,L_{0})$ we have
 \[
 \int_{0}^{L_{0}} u_{t}u_{tt}\,dx - k_{1}\int_{0}^{L_{0}}
 u_{t}u_{xx}\,dx = - \alpha \int_{0}^{L_{0}}| u_{t}|^{2}\,dx
 \]
 which integrated by parts leads to
\begin{equation} \label{3.1}
 \frac{d}{dt}\frac{1}{2}\int_{0}^{L_{0}}[ |u_{t}|^{2} +
 k_{1}|u_{x}|^{2} ]\,dx  = - \alpha \int_{0}^{L_{0}}| u_{t}|^{2}\,dx
 + k_{1}u_{x}(L_{0})u_{t}(L_{0}).
 \end{equation}
 Multiplying equation (\ref{1.2}) by $v_{t}$ and performing an
 integration in $(L_{0},L)$ we get
 \[
 \int_{L_{0}}^{L} v_{t}v_{tt}\,dx - k_{2}\int_{L_{0}}^{L}
 v_{t}v_{xx}\,dx = 0.
 \]
 After integrating by parts we arrive at
\begin{equation} \label{3.2}
 \frac{d}{dt}\frac{1}{2}\int_{L_{0}}^{L}[ |v_{t}|^{2} +
 k_{2}|v_{x}|^{2} ]\,dx  =
 - k_{2}v_{x}(L_{0})v_{t}(L_{0}).
 \end{equation}
  Adding (\ref{3.1}) with (\ref{3.2}) and using the transmission
  conditions (\ref{1.4}) we conclude
  \begin{equation} \label{3.3}
 \frac{d}{dt}E(t)
= - \alpha \int_{0}^{L_{0}}|u_{t}|^{2} \, dx. \end{equation}
 \end{proof}

\begin{lemma}  \label{lem3.2}
There exist positive constants $C_{0}$ and $C_{1}$,
independent of initial data, such that the functional defined by
$$
J_{1}(t) = \int_{0}^{L_{0}}( x - L_{0} ) u_{t}u_{x} \, dx
$$
satisfies
\[
 \frac{d}{dt}J_{1}(t) \leq - C_{1}
E_{1}(t)  + C_{0}\int_{0}^{L_{O}}|u_{t}|^{2} \, dx  +
 \frac{k_{1}L_{0}}{2}|u_{x}(0)|^{2}.
\]
 \end{lemma}

\begin{proof}
 Multiplying equation (\ref{1.1}) by $(x - L_{0})u_{x}$ and performing an
 integration in $(0,L_{0})$ we get
 \begin{equation} \label{3.4}
 \int_{0}^{L_{0}} (x - L_{0})u_{x}u_{tt}\,dx - k_{1}\int_{0}^{L_{0}}
 (x - L_{0})u_{x}u_{xx}\,dx = - \alpha \int_{0}^{L_{0}}(x - L_{0})u_{x} u_{t}\,dx.
 \end{equation}
 Note that
  \begin{equation} \label{3.5}
\frac{d}{dt}(x - L_{0})u_{x}u_{t} = (x - L_{0})u_{x}u_{tt} + (x -
L_{0})u_{xt}u_{t}.
 \end{equation}
 Now using (\ref{3.5}) in (\ref{3.4}) we get
\begin{align*}
 \frac{d}{dt}\int_{0}^{L_{0}} (x - L_{0})u_{x}u_{t}\,dx
&= \int_{0}^{L_{0}} (x - L_{0})\frac{1}{2}[\frac{d}{dx}|u_{t}|^{2}]\,dx\\
&\; + k_{1}\int_{0}^{L_{0}}
 (x - L_{0})\frac{1}{2}[\frac{d}{dx}|u_{x}|^{2}]\,dx
 - \alpha \int_{0}^{L_{0}}(x - L_{0})u_{x} u_{t}\,dx
\end{align*}
and performing integration by parts we get
 \begin{align*}
 \frac{d}{dt}\int_{0}^{L_{0}} (x - L_{0})u_{x}u_{t}\,dx.
&= -  \frac{1}{2}\int_{0}^{L_{0}}|u_{t}|^{2}\,dx
  - \frac{k_{1}}{2}\int_{0}^{L_{0}} |u_{x}|^{2}\,dx \\
&\quad -  \alpha \int_{0}^{L_{0}}(x - L_{0})u_{x} u_{t}\,dx
 + \frac{k_{1}L_{0}}{2}|u_{x}(0)|^{2}
\end{align*}
  from which it follows that
\[
 \frac{d}{dt}J_{1}(t) \leq - C_{1}
E_{1}(t)
 + C_{0}\int_{0}^{L_{0}}|u_{t}|^{2} \, dx  +
 \frac{k_{1}L_{0}}{2}|u_{x}(0)|^{2}.
\]
 \end{proof}

\begin{lemma} \label{lem3.3}
There exists a positive constant $C_{2}$, independent of initial
data, such that the functional defined by
$$
J_{2}(t) = \int_{L_{0}}^{L}( x - L_{0} ) v_{t}v_{x} \, dx
$$
satisfies
\[ %\label{J_{2}}
\frac{d}{dt}J_{2}(t) \leq - C_{2} E_{2}(t)
 +  \frac{k_{2}(L -L_{0})}{2}|v_{x}(L)|^{2}.
\]
\end{lemma}

\begin{proof}
 Multiplying equation (\ref{1.2}) by $(x - L_{0})v_{x}$ and performing an
 integration in $(L_{0},L)$ we get
 \begin{equation} \label{3.6}
 \int_{L_{0}}^{L} (x - L_{0})v_{x}v_{tt}\,dx - k_{2}\int_{L_{0}}^{L}
 (x - L_{0})v_{x}v_{xx}\,dx =0.
 \end{equation}
 Notice that
  \begin{equation} \label{3.7}
\frac{d}{dt}(x - L_{0})v_{x}v_{t} = (x - L_{0})v_{x}v_{tt} + (x -
L_{0})v_{xt}v_{t}.
 \end{equation}
 Now using (\ref{3.7}) in (\ref{3.6}) we get
\[
 \frac{d}{dt}\int_{L_{0}}^{L} (x - L_{0})v_{x}v_{t}\,dx
= \int_{L_{0}}^{L}
 (x - L_{0})\frac{1}{2}[\frac{d}{dx}|v_{t}|^{2}]\,dx
 + k_{2}\int_{L_{0}}^{L}
 (x - L_{0})\frac{1}{2}[\frac{d}{dx}|v_{x}|^{2}]\,dx
\]
 and performing integration by parts we get
 \[
 \frac{d}{dt}\int_{L_{0}}^{L} (x - L_{0})v_{x}v_{t}\,dx.
= -  \frac{1}{2}\int_{L_{0}}^{L}
|v_{t}|^{2}\,dx - \frac{k_{2}}{2}\int_{L_{0}}^{L}
 |v_{x}|^{2}\,dx +  \frac{k_{2}(L -L_{0})}{2}|v_{x}(L)|^{2}
\]
  from which it follows that
  \[
 \frac{d}{dt}J_{2}(t) \leq - C_{2} E_{2}(t)
 +  \frac{k_{2}(L -L_{0})}{2}|v_{x}(L)|^{2}.
 \]
 \end{proof}

Now we must control the punctual terms $|u_{x}(0)|^{2}$ and
$|v_{x}(L)|^{2}$ present in the inequalities given by the lemmas
\ref{lem3.2} and \ref{lem3.3} respectively. In order to do so we introduce the two
following lemmas.

\begin{lemma} \label{lem3.4}
Let us take $ p \in \, C^{1}(0,L_{0})$ with $p(0) >
0$ and $p(L_{0}) = 0$. Then, there exist positive constants
$C_{0},C_{4},N_{0} $  independent of initial data, such that the
functional defined by
$$
J_{3}(t) = N_{0}J_{1}(t) + \int_{0}^{L_{0}}p u_{t}u_{x} \, dx
 $$
 satisfies
\[
\frac{d}{dt}J_{3}(t) \leq - C_{4} E_{1}(t)
 + N_{0}C_{0}\int_{0}^{L_{0}}| u_{t}|^{2} \, dx.
\]
\end{lemma}

\begin{proof}
 Multiplying equation (\ref{1.1}) by $p\,u_{x}$ and performing an
 integration in $(0,L_{0})$ we get
 \begin{equation} \label{3.8}
 \int_{0}^{L_{0}} p\,u_{x}u_{tt}\,dx - k_{1}\int_{0}^{L_{0}}
 p\,u_{x}u_{xx}\,dx = - \alpha \int_{0}^{L_{0}}p\,u_{x} u_{t}\,dx.
 \end{equation}
 Notice that
  \begin{equation} \label{3.9}
\frac{d}{dt}p\,u_{x}u_{t} = p\,u_{x}u_{tt} + p\,u_{xt}u_{t}.
 \end{equation}
 Now using (\ref{3.9}) in (\ref{3.8}) we get
\begin{align*}
&\frac{d}{dt}\int_{0}^{L_{0}} p\,u_{x}u_{t}\,dx \\
&= \int_{0}^{L_{0}} p\,\frac{1}{2}[\frac{d}{dx}|u_{t}|^{2}]\,dx
 + k_{1}\int_{0}^{L_{0}}
 p\,\frac{1}{2}[\frac{d}{dx}|u_{x}|^{2}]\,dx
 - \alpha \int_{0}^{L_{0}}p\,u_{x} u_{t}\,dx
 \end{align*}
and performing integration by parts we get
 \begin{align*}
&\frac{d}{dt}\int_{0}^{L_{0}} p\,u_{x}u_{t}\,dx \\
&= -  \frac{1}{2}\int_{0}^{L_{0}} p'\,|u_{t}|^{2}\,dx
 - \frac{k_{1}}{2}p(0)\,|u_{x}(0)|^{2} -
\frac{k_{1}}{2}\int_{0}^{L_{0}}  p'\,|u_{x}|^{2}\,dx
 -  \alpha \int_{0}^{L_{0}}p\,u_{x} u_{t}\,dx,
\end{align*}
from which it follows that
\[
 \frac{d}{dt}\int_{0}^{L_{0}} p\,u_{x}u_{t}\,dx \leq  -\frac{k_{1}}{2}p(0)\,|u_{x}(0)|^{2} + C_{3}
E_{1}(t).
\]
  Denoting
$$
J_{3}(t) = N_{0}J_{1}(t) + \int_{0}^{L_{0}}p u_{t}u_{x} \, dx,
$$
  we have
\begin{align*}
\frac{d}{dt}J_{3}(t)
&\leq -N_{0}C_{1}\,E_{1}(t) + C_{3}\,E_{1}(t)
 + \frac{N_{0}k_{1}L_{0}}{2}\,|u_{x}(0)|^{2} -\frac{k_{1}}{2}p(0)\,|u_{x}(0)|^{2}
  \\
&\quad + N_{0}\,C_{0}\int_{0}^{L_{0}}| u_{t}|^{2} \, dx.
 \end{align*}
Now taking $N_{0} $ such that $N_{0}\,C_{1} > C_{3}$
 and choosing $ p(0) = N_{0}L_{0}$ we conclude that
\[
 \frac{d}{dt}J_{3}(t) \leq - C_{4} E_{1}(t)
 + N_{0}C_{0}\int_{0}^{L_{0}}| u_{t}|^{2} \, dx.
\]
 \end{proof}

\begin{lemma} \label{lem3.5}
Let us take $ q \in  C^{1}(L_{0},L)$ with $q(L_{0})= 0$ and
$q(L) < 0$. Then, there exist positive constants $C_{5}$ and $N_{1} $
 independent of initial data such that the functional defined
by
$$
J_{4}(t) = N_{1}J_{2}(t) + \int_{L_{0}}^{L}q v_{t}v_{x} \, dx
$$
satisfies
$\frac{d}{dt}J_{4}(t) \leq - C_{5} E_{2}(t)$.
\end{lemma}

\begin{proof}
 Multiplying equation (\ref{1.2}) by $q\,v_{x}$ and performing an
 integration in $(L_{0},L)$ we get
 \begin{equation} \label{3.10}
 \int_{L_{0}}^{L} q\,v_{x}v_{tt}\,dx - k_{2}\int_{L_{0}}^{L}
 q\,v_{x}v_{xx}\,dx = 0.
 \end{equation}
 Notice that
  \begin{equation} \label{3.11}
\frac{d}{dt}q\,v_{x}v_{t} = q\,v_{x}v_{tt} + q\,v_{xt}v_{t}.
 \end{equation}
 Now using (\ref{3.11}) in (\ref{3.10}) we get
\[
 \frac{d}{dt}\int_{L_{0}}^{L} q\,v_{x}v_{t}\,dx
= \int_{L_{0}}^{L}  q\,\frac{1}{2}[\frac{d}{dx}|v_{t}|^{2}]\,dx
 + k_{2}\int_{L_{0}}^{L}
 q\,\frac{1}{2}[\frac{d}{dx}|v_{x}|^{2}]\,dx
\]
 and performing integration by parts we arrive at
 \[
 \frac{d}{dt}\int_{L_{0}}^{L} q\,v_{x}v_{t}\,dx
= -  \frac{1}{2}\int_{L_{0}}^{L} q'\,|v_{t}|^{2}\,dx
+ \frac{k_{2}}{2}q(L)\,|v_{x}(L)|^{2} -
\frac{k_{2}}{2}\int_{L_{0}}^{L}
 q'\,|v_{x}|^{2}\,dx,
\]
  from which it follows that
\[
 \frac{d}{dt}\int_{L_{0}}^{L} q\,v_{x}v_{t}\,dx \leq  \frac{k_{2}}{2}q(L)\,|v_{x}(L)|^{2} + C_{4}
E_{2}(t).
\]
  Denoting
  $$
J_{4}(t) = N_{1}J_{2}(t) + \int_{L_{0}}^{L}q v_{t}v_{x} \, dx,
$$
  we have
\[
  \frac{d}{dt}J_{4}(t)
\leq -N_{1}C_{2}\,E_{2}(t) + C_{4}\,E_{2}(t)
  + \frac{N_{1}k_{2}(L -L_{0})}{2}\,|v_{x}(L)|^{2}
  + \frac{k_{2}}{2}q(L)\,|v_{x}(L)|^{2}.
\]
Now  taking $N_{1} $ such that $N_{1}\,C_{2} > C_{4}$ and choosing
 $ q(L) = -N_{1}(L - L_{0})$ we conclude that
\[ %\label{J_{3}}
\frac{d}{dt}J_{4}(t) \leq - C_{5} E_{2}(t).
\]
 \end{proof}

Now we are in position to show the main result of this paper.

\begin{theorem} \label{thm3.1}
Let us denote by $(u,v)$ a strong solution of system
\eqref{1.1}--\eqref{1.5}, as in Theorem \ref{thm2.1}. Then there exist
positive constants $C$ and $\omega$, such that
\[
E(t) \leq C \, E(0) e^{- \omega \, t}.
\]
\end{theorem}

\begin{proof} Let us define
\[
 {\mathcal{L}}(t)= N_{2}\,E(t) + J_{3}(t) + J_{4}(t).
\]
 From Lemma \ref{lem3.1} we have
\[
 \frac{d}{dt}E(t) = - \alpha \int_{0}^{L_{0}}|u_{t}|^{2} \, dx.
\]
 From Lemma \ref{lem3.4} we have
 \[
\frac{d}{dt}J_{3}(t) \leq - C_{4} E_{1}(t)
 + N_{0}C_{0}\int_{0}^{L_{0}}| u_{t}|^{2} \, dx.
\]
 From Lemma \ref{lem3.5} we have
 \[
\frac{d}{dt}J_{4}(t) \leq - C_{5} E_{2}(t).
\]
 In fact we have
$$
\frac{d}{dt}{\mathcal{L}}(t) \leq - C_{4} E_{1}(t) -
C_{5} E_{2}(t)  + (N_{0}C_{0} - N_{2}\alpha)\int_{0}^{L_{0}}| u_{t}|^{2} \, dx.
$$
Taking $N_{2}$ large enough it follows
$$
\frac{d}{dt}{\mathcal{L}}(t) \leq - C_{6}\,E(t)
$$
Since $\mathcal{L}(t)$ is equivalent to $E(t)$, we conclude that
there exist positive constants $C$ and $\omega$, such that
\[
E(t) \leq C \, E(0) e^{- \omega \, t}.
\]
\end{proof}


Theorem \ref{thm3.1} can be extended easily  to weak solutions by using
density arguments and the lower semicontinuity of the energy
functional $E(t)$. This is the content of the following corollary
whose proof is omitted.

\begin{corollary} \label{coro3.1}
 Under the same hypotheses of Theorem \ref{thm3.1},  there exists
positive constants $\bar{C}$ and $\bar{\omega}$, such that
\[
E(t) \leq \bar{C} \, E(0) e^{- \bar{\omega}  t}.
\]
\end{corollary}

 \subsection*{Acknowledgements}
The authors are grateful to the anonymous referee for giving us valuable
suggestions that improved our paper.


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