\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 85, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/85\hfil Stability for a coupled system of 
wave equations]
{Stability for a coupled system of wave equations of Kirchhoff
type with nonlocal boundary conditions}

\author[J. Ferreira, D. C. Pereira, \& M. L. Santos\hfil EJDE--2003/85\hfilneg]
{Jorge Ferreira, Ducival C. Pereira, \& Mauro L. Santos} % In alphabetical order

\address{Jorge Ferreira \newline
Departamento de Matem\'atica-DMA, Universidade Estadual de
Maring\'a-UEM \newline
Av. Colombo, 5790-Zona 7, CEP 87020-900, Maring\'a-Pr., Brazil}
\email{jferreira@bs2.com.br}

\address{Ducival C. Pereira \newline
Instituto de Estudos Superiores da Amaz\^onia (IESAM)\\
Av. Gov. Jos\'e Malcher 1148, CEP 66.055-260, Bel\'em-Pa.,
Brazil\\
Faculdade Ideal(FACI) \\
Rua dos Mundurucus, 1427, CEP 66025-660, Bel\'em-Pa., Brazil}
\email{ducival@aol.com}

\address{Mauro L. Santos \newline
Departamento de Matem\'atica, Universidade Federal do Par\'a\\
Campus Universitario do Guam\'a \\
Rua Augusto Corr\^ea 01, Cep 66075-110, Par\'a, Brazil}
\email{ls@ufpa.br}


\date{}
\thanks{Submitted April 2, 2003. Published August 14, 2003.}
\subjclass[2000]{34A34, 34M30, 35B05}
\keywords{Coupled system, wave equation, Galerkin method, \hfill\break\indent
asymptotic behavior, boundary value problem}


\begin{abstract}
 We consider a coupled system of two nonlinear wave equations of
 Kirchhoff type with nonlocal boundary condition and we study the
 asymptotic behavior of the corresponding solutions. We prove that
 the energy decay at the same rate of decay of the relaxation
 functions, that is, the energy decays exponentially when the
 relaxation functions decay exponentially and polynomially when the
 relaxation functions decay polynomially.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

The main purpose of this article is to study the existence of
global solutions and the asymptotic behavior of the energy related
to a coupled system of two nonlinear wave equations of Kirchhoff
type with nonlocal boundary condition. Consider the system of equations
 \begin{gather}
 \label{1eq1-1} u_{tt} - M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2}) \Delta u
-\Delta u_{t}+ f_{1}(u)= 0 \quad\mbox{in }{\Omega\times(0,\infty)},\\
\label{1eq1-2}
 v_{tt} - M(\|\nabla u\|^{2}_{2}+\|\nabla
 v\|^{2}_{2})\Delta v -\Delta v_{t}+ f_{2}(v) = 0
 \quad\mbox{in }{\Omega\times(0,\infty)},\\
\label{1eq1-3}
 u =v= 0\quad\mbox{on }\Gamma_{0}\times(0,\infty), \\
\label{1eq1-4} \begin{gathered}
u +\int^{t}_{0}g_{1}(t-s) ((M(\|\nabla
u(s)\|^{2}_{2}+\|\nabla v(s)\|^{2}_{2})\frac{\partial u}{\partial
\nu}(s)+\frac{\partial
u_{t}}{\partial \nu}(s))ds = 0\\
\quad\mbox{on } \Gamma_{1}\times(0,\infty),  \end{gathered} \\
\label{1eq1-5} \begin{gathered}
v +\int^{t}_{0}g_{2}(t-s) ((M(\|\nabla
u(s)\|^{2}_{2}+\|\nabla v(s)\|^{2}_{2})\frac{\partial v}{\partial
\nu}(s)+\frac{\partial v_{t}}{\partial \nu}(s))ds = 0\\
\quad\mbox{on }\Gamma_{1}\times(0,\infty), \end{gathered}\\
\label{1eq1-6}
(u(0,x),v(0,x)) =(u_{0}(x),v_{0}(x)),\quad
(u_{t}(0,x),v_{t}(0,x))=(u_{1}(x),v_{1}(x))
\quad\mbox{in }{\Omega},
\end{gather}
where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\geq 1$, with
smooth boundary $\Gamma=\Gamma_0 \cup \Gamma_1$. Here, $\Gamma_0$
and $\Gamma_1$ are closed, disjoint, $\Gamma_0\neq \emptyset$ and
$\nu$ is the unit normal vector pointing towards the exterior of
$\Omega$. The equations (\ref{1eq1-4})-(\ref{1eq1-5}) are nonlocal
boundary conditions responsible for the memory effect.
Concerning the history condition, we must add the condition
$$
u=v=0 \quad \mbox{on } \Gamma_0 \times ] -\infty, 0].
$$
We observe that, $u$ and $v$ represent transverse
displacements. The relaxation functions $g_i$ are positive and non
decreasing; while the functions $f_{i}\in C^{1}(\mathbb{R})$, $i=1,2$, satisfy
\[
f_{i}(s)s \ge 0 \quad \forall s \in \mathbb{R}
\]
Additionally, we suppose that $f_{i}$ is superlinear, that is
\[
 f_{i}(s)s \ge (2+\delta)F_{i}(s),\quad F_{i}(z) = \int^{z}_{0} f_{i}(s)ds
 \quad \forall s \in \mathbb{R},\quad i=1,2,
\]
for some $\delta >0$. Also the following growth conditions are satisifed:
\begin{eqnarray*}
|f_{i}(x)-f_{i}(y)| \leq c( 1 + |x|^{\rho -1} + |y|^{\rho -1} )|x-y|, \quad \forall x,y \in \mathbb{R}, \quad i=1,2,
\end{eqnarray*}
for some $c>0$ and $\rho\geq 1$ such that $(n-2)\rho \leq n$. We
shall assume that the function $M\in C^{1}([0,\infty[)$ satisfies
\begin{equation}\label{1eq1-7}
M(\lambda)\geq m_{0}>0, \quad
M(\lambda)\lambda\geq\widehat{M}(\lambda), \quad \forall
\lambda\geq 0,
\end{equation}
where $\widehat{M}(\lambda)=\int^{\lambda}_{0}M(s)ds$. Also, we
shall assume that there exists $x_{0}\in \mathbb{R}^{n}$ such that
\begin{gather*}
\Gamma_{0}=\{x \in \Gamma : \nu (x)\cdot(x-x_{0})\leq0\}, \\
\Gamma_{1}=\{x \in \Gamma : \nu (x)\cdot(x-x_{0})>0\}.
\end{gather*}
Let us denote by $m(x)=x-x_{0}$. Note that by the compactness of
$\Gamma_{1}$, there exist a small positive constant
$\delta_{0}$ such that
\begin{equation}\label{delta-0}
0 < \delta_{0} \leq m(x) \cdot \nu (x), \quad \forall x \in \Gamma_{1}.
\end{equation}
The existence of global solutions and exponential decay to the
problem (\ref{1eq1-1}), (\ref{1eq1-3}) with $\partial
\Omega=\Gamma_{0}$ and frictional dissipative damping has been
investigated by many authors (see, e.g. \cite{Biler, Brito,
Ikerata-1, Ikerata-2, Matos-Pereira, Matsuyama, Nishihara, Yamada}
). There exists a large body of literature regarding viscoelastic
problems with the memory term acting in the domain or in the
boundary. Among the numerous works in this direction, we can cite
Rivera \cite{Rivera} and M. L. Santos \cite{Santos-1, Santos-2}.
Park \& Bae \cite{Park} studied the existence and uniform decay of
strong solutions of the coupled wave equations
(\ref{1eq1-1})-(\ref{1eq1-2}) with nonlinear boundary damping and
memory source term and $M(s)=1+s$. In the present paper, we
obtained respectively besides the exponential decay and uniform
rate of polynomial decay. Moreover, the system
(\ref{1eq1-1})-(\ref{1eq1-6}) is more general than the system
considered in \cite{Park}, because they only consider the case in
that $M(s)=1+s$. As we have said before we study the asymptotic
behavior of the solutions of system (\ref{1eq1-1})-(\ref{1eq1-6}).
We show that the energy of the coupled system
(\ref{1eq1-1})-(\ref{1eq1-6}) decays uniformly in time with the
same rate of decay of the relaxation functions. More precisely,
denoting by $k_{1}$ and $k_{2}$ the resolvent kernels of
$-g_{1}'/g_{1}(0)$ and $-g_{2}'/g_{2}(0)$ respectively, we show
that the energy decays exponentially to zero provided $k_{1}$ and
$k_{2}$ decays exponentially to zero. When the resolvent kernels
$k_{1}$ and $k_{2}$ decays polynomially, we show that energy also
decays polynomially to zero. This means that the memory effect
produces strong dissipation capable of making a uniform rate of
decay for the energy. The method used is based on the construction
of a suitable Lyapunov functional $\mathcal{L}$ satisfying
\[
\frac{d}{dt}\mathcal{L}(t)\leq - c_{1}\mathcal{L}(t)+c_{2}e^{- \gamma t}
\quad \mbox {or} \quad
\frac{d}{dt}\mathcal{L}(t)\leq -c_{1}
\mathcal{L}(t)^{1+ \frac{1}{\alpha}}+ \frac{c_{2}}{(1+t)^{\alpha +1}}
\]
for some positive constants $c_{1},c_{2},\gamma$ and $\alpha$. Note that, because of condition (\ref{1eq1-3}) the solution of
the system (\ref{1eq1-1})-(\ref{1eq1-6}) must belong to the  space
\begin{eqnarray*}
V:=\{v \in H^{1}(\Omega): v = 0 \quad\mbox{on}\quad \Gamma_{0}\}.
\end{eqnarray*}
The notation used in this paper is standard and can be found
in Lion's book \cite{Lions}. In the sequel by $c$ (sometime
$c_{1}, c_{2},\dots$) we denote various positive constants
independent of $t$ and on the initial data. The organization of
this paper is as follows. In section 2 we establish the existence
and uniqueness of strong solutions for the
system (\ref{1eq1-1})-(\ref{1eq1-6}). In section 3 we prove the
uniform rate exponential decay. In section 4 we prove the
uniform rate of polynomial decay.

\section{Notation and Main Results}

In this section we shall study the existence and regularity of solutions
for the coupled system
(\ref{1eq1-1})-(\ref{1eq1-6}). First, we shall use equations
(\ref{1eq1-4})-(\ref{1eq1-5}) to estimate the terms
$M(\|\nabla u(t)\|^{2}_{2}+\|\nabla v(t)\|^{2}_{2})
\frac{\partial u}{\partial \nu}+\frac{\partial u_t}{\partial \nu}$ and
$M(\|\nabla u(t)\|^{2}_{2}+\|\nabla v(t)\|^{2}_{2})
\frac{\partial v}{\partial \nu}+\frac{\partial v_t}{\partial \nu}$.
Denoting by
\[
(g \ast \varphi)(t) = \int^{t}_{0}g(t-s)\varphi(s) ds,
\]
the convolution product operator and differentiating the equations
(\ref{1eq1-4}) and (\ref{1eq1-5}) we arrive at the
following Volterra equations:
\begin{align*}
& M(\|\nabla u(t)\|^{2}_{2}+\|\nabla v(t)\|^{2}_{2})\frac{\partial u}{\partial \nu}
 +\frac{\partial u_{t}}{\partial \nu} +
 \frac{1}{g_{1}(0)}g_{1}'\ast (M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
 \frac{\partial u}{\partial \nu}+\frac{\partial
 u_{t}}{\partial \nu})\\
&= -\frac{1}{g_{1}(0)}u_{t},\\
&M(\|\nabla u(t)\|^{2}_{2}+\|\nabla v(t)\|^{2}_{2})\frac{\partial v}{\partial \nu}
 +\frac{\partial v_{t}}{\partial \nu} +
 \frac{1}{g_{2}(0)}g_{2}'\ast (M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
 \frac{\partial v}{\partial \nu}+\frac{\partial
 v_{t}}{\partial \nu}) \\
&= -\frac{1}{g_{2}(0)}v_{t}.
\end{align*}
Applying the Volterra's inverse operator, we get
\begin{gather*}
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial u}{\partial \nu}
+\frac{\partial u_t}{\partial \nu}
= -\frac{1}{g_{1}(0)}\{u_{t} + k_{1} \ast u_{t}\}, \\
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial v}{\partial \nu}
+\frac{\partial v_t}{\partial \nu}
= -\frac{1}{g_{2}(0)}\{v_{t} + k_{2} \ast v_{t}\},
\end{gather*}
where the resolvent kernels satisfies
\[
k_{i}+ \frac{1}{g_{i}(0)}g_{i}' \ast k_{i}=-
\frac{1}{g_{i}(0)}g_{i}'\quad\mbox{for } i=1,2.
\]
Denoting  $\tau_{1}=\frac{1}{g_{1}(0)}$  and
$\tau_{2}=\frac{1}{g_{2}(0)}$, we obtain
\begin{gather}\label{2eq2-1}
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial u}{\partial \nu}
+\frac{\partial u_{t}}{\partial \nu}= -\tau_{1}
\{u_{t} + k_{1}(0)u - k_{1}(t)u_{0} + k_{1}' \ast u \}\\
\label{2eq2-2}
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial v}{\partial \nu}
+\frac{\partial v_t}{\partial \nu}= -\tau_{2}
\{v_{t} + k_{2}(0)v - k_{2}(t)v_{0} + k_{2}' \ast v \}.
\end{gather}
Reciprocally, taking initial data such that $u_{0}=v_{0}=0$ on $\Gamma_{1}$,
the identities (\ref{2eq2-1})-(\ref{2eq2-2})
imply (\ref{1eq1-4})-(\ref{1eq1-5}). Since we are interested in relaxation
functions of  exponential or polynomial type and
the identities (\ref{2eq2-1})-(\ref{2eq2-2}) involve the resolvent kernels $k_i$,
we want to know if $k_i$ has the same
properties.  The following Lemma answers this question. Let $h$ be a
relaxation function and $k$ its resolvent kernel, that is
\begin{equation}
k(t)-k*h(t)=h(t).
\label{eq-volt}
\end{equation}

\begin{lemma}\label{Lem2.1}
If $h$ is a positive continuous function, then $k$  is also a
positive continuous function. Moreover,
\begin{enumerate}
\item If there exist positive constants $c_0$ and $\gamma$ with $c_0<\gamma$
such that $h(t)\leq c_0 e^{-\gamma t}$,
then, the function $k$ satisfies
$$
k(t)\leq \frac{c_0(\gamma-\epsilon)}{\gamma-\epsilon-c_0} e^{-\epsilon t},
$$
for all $0<\epsilon<\gamma-c_0$.

\item Given $p>1$, let us denote by
$c_p:=\sup_{t\in\mathbb{R}^+} \int_0^t (1+t)^p (1+t-s)^{-p} (1+s)^{-p} \,ds$.
If there exists a
positive constant $c_0$ with $c_0c_p<1$ such that
$h(t)\leq c_0 (1+t)^{-p}$,
then, the function $k$ satisfies
$$
k(t)\leq \frac{c_0}{1-c_0c_p} (1+t)^{-p}.
$$
\end{enumerate}
\end{lemma}

\begin{proof}
Note that $k(0)=h(0)>0$. Now, we take $t_0=\inf \{ t\in\mathbb{R}^+: k(t)=0\}$, so
$k(t)>0$ for all $t\in [0,t_0[$. If $t_0\in\mathbb{R}^+$, from  equation (\ref{eq-volt})
we get that $-k*h(t_0)=h(t_0)$ but this is
contradictory. Therefore $k(t)>0$ for all $t\in\mathbb{R}^+_0$. Now, let us fix
 $\epsilon$, such that $0<\epsilon<\gamma-c_0$ and
denote by
$$
k_{\epsilon}(t):=e^{\epsilon t}k(t), \quad h_{\epsilon}(t):=e^{\epsilon t}h(t).
$$
Multiplying equation (\ref{eq-volt}) by $e^{\epsilon t}$ we get
$k_{\epsilon}(t)=h_{\epsilon}(t)+k_{\epsilon}*h_{\epsilon}(t)$, hence
\[
\sup_{s\in[0,t]} k_{\epsilon}(s)
\leq \sup_{s\in[0,t]} h_{\epsilon}(s)
+\Big(\int_0^{\infty}c_0 e^{(\epsilon-\gamma)s}\,ds\Big)
\sup_{s\in[0,t]}k_{\epsilon}(s)
 \leq  c_0 +\frac{c_0}{(\gamma-\epsilon)}\sup_{s\in[0,t]} k_{\epsilon}(s).
\]
Therefore,
$$
k_{\epsilon}(t) \leq \frac{c_0(\gamma-\epsilon)}{\gamma-\epsilon-c_0},
$$
which implies our first assertion. To show the second part we use
the notation
$$
k_p(t):=(1+t)^pk(t), \quad h_p(t):=(1+t)^ph(t).
$$
Multiplying equation (\ref{eq-volt}) by $(1+t)^p$ we get
\[
k_p(t)=h_p(t)+ \int_0^t k_p(t-s) (1+t-s)^{-p} (1+t)^p h(s) \,ds\,,
\]
hence
\[
\sup_{s\in[0,t]} k_p(s)
\leq \sup_{s\in[0,t]} h_p(s) + c_0c_p \sup_{s\in[0,t]} k_p(s) \ \leq\  c_0 +c_0c_p\sup_{s\in[0,t]} k_p(s).
\]
Therefore,
$$
k_p(t) \leq \frac{c_0}{1-c_0c_p},
$$
which proves our second assertion. \end{proof}

\noindent{\bf Remark: } The fact that the constant $c_p$ is finite
can be found in \cite[Lemma 7.4]{Racke}.

Due to this Lemma, in the remainder of this paper, we shall use
(\ref{2eq2-1})-(\ref{2eq2-2}) instead of
(\ref{1eq1-4})-(\ref{1eq1-5}). Let us denote
\begin{eqnarray*}
(g \Box \varphi)(t) := \int^t_0 g(t-s)|\varphi(t)-\varphi(s)|^2 ds.
\end{eqnarray*}
The following lemma states an important property of the convolution operator.

\begin{lemma}\label{Lem2.2}
For $g,\varphi \in C^{1}([0,\infty[:\mathbb{R})$ we have
\[
(g \ast \varphi)\varphi_{t}  =  -\frac{1}{2} g(t)| \varphi(t)
|^{2} + \frac{1}{2} g' \Box \varphi  - \frac{1}{2}
\frac{d}{dt}\Big[g \Box \varphi - (\int^{t}_{0}g(s) ds)
|\varphi|^{2}\Big].
\]
\end{lemma}

The proof of this lemma follows by differentiating the expression
$g \Box \varphi$.

 The first order energy of coupled system
(\ref{1eq1-1})-(\ref{1eq1-6}) is defined as
\begin{align*}
E(t)&:=E(t,u,v) \\
&=\frac{1}{2}\int_{\Omega}|u_{t}|^{2}dx+\frac{1}{2}\int_{\Omega}|v_{t}|^{2}dx
 +\frac{1}{2}\widehat{M}(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
 + \int_{\Omega}F_{1}(u) dx \\
&\quad + \int_{\Omega}F_{2}(v) dx
+\frac{\tau_{1}}{2}k_{1}(t) \int_{\Gamma_{1}}|u|^{2} d
 \Gamma_{1}- \frac{\tau_{1}}{2}\int_{\Gamma_{1}}k_{1}' \Box u d \Gamma_{1}\\
&\quad + \frac{\tau_{2}}{2}k_{2}(t) \int_{\Gamma_1}|v|^{2} d \Gamma_{1}
 - \frac{\tau_2}{2}\int_{\Gamma_1}k_{2}' \Box v d \Gamma_{1}.
\end{align*}
The main goal of this work is given by the following Theorem.

\begin{theorem}\label{teo2.1}
Let $k_{i}\in C^{2}(\mathbb{R}^{+})$ be such that
$k_{i}, -k_{i}', k_{i}'' \ge 0$ for $i=1,2$.
If $(u_{0},v_{0})\in (H^{2}(\Omega)\cap V)^{2}$ and
$(u_{1},v_{1})\in (H^{2}(\Omega)\cap V)^{2}$ satisfy the compatibility
conditions
\begin{gather}\label{2eq2-3}
M(\|\nabla u_0\|^{2}_{2}+\|\nabla v_0\|^{2}_{2})\frac{\partial
u_{0}}{\partial \nu}+\frac{\partial u_{1}}{\partial \nu} +
\tau_{1}u_{1} = 0 \quad {on } \Gamma_{1}, \\
\label{2eq2-4}
M(\|\nabla u_0\|^{2}_{2}+\|\nabla v_0\|^{2}_{2})
\frac{\partial v_{0}}{\partial \nu}+\frac{\partial
v_{1}}{\partial \nu} + \tau_{2} v_{1} = 0 \quad {on }\Gamma_{1}.
\end{gather}
Then there exists only one solution $(u,v)$ of the system
(\ref{1eq1-1})-(\ref{1eq1-6}) satisfying
\begin{gather*}
u,v \in L^{\infty}(0,T:V), \quad  u_t,v_t \in L^{\infty}(0,T:V)\,,\\
u_{tt},v_{tt}\in L^{\infty}(0,T:L^{2}(\Omega)), \quad  \Delta
u,\Delta v \in L^{\infty}(0,T:L^{2}(\Omega))\,,\\
\Delta u_{t}, \Delta v_{t} \in L^{2}(0,T:L^{2}(\Omega)).
\end{gather*}
In addition, considering (\ref{delta-0}) and assuming that there
exist positive constants $b_1$, $b_2$ such that
\begin{gather}\label{2eq2-5}
k_i(0)>0,\quad k_i'(t)\leq -b_1k_i(t), \quad k_i''(t)\geq-b_2k_i'(t),
\quad i=1,2, \quad \mbox{or} \\
 \label{2eq2-6}
k_i(0)>0, \quad k_i'(t)\leq-b_1k_i'(t)^{1+\frac{1}{p}}, \quad  k_i''(t)\geq
b_2[-k_i'(t)]^{1+\frac{1}{p+1}}, \quad p>1, \quad i=1,2
\end{gather}
then the energy $E(t)$ associated to problem
(\ref{1eq1-1})-(\ref{1eq1-6}) decays, respectively, a the
following rate
\begin{gather}\label{2eq2-7}
E(t)\leq \alpha_1 e^{-\alpha_2 t} E(0), \\
\label{2eq2-8}
E(t) \leq \frac{c}{(1+t)^{p+1}} E(0),
\end{gather}
where $\alpha_1$, $\alpha_2$ and $c$ are positive constants.
\end{theorem}

\subsection*{Proof of the existence of regular solutions}

The main idea is to use the Galerkin method. To do this let us take
a basis $\{w_{j}\}_{j\in \mathbb{N}}$ to
$V$ which is orthonormal in $L^{2}(\Omega)$ and we represent by
$V_{m}$ the subspace of $V$ generated by the first $m$ vectors.
Standard results on ordinary differential equations guarantee that
there exists only one local solution
$$
(u^{m}(t),v^{m}(t)):= \sum^{m}_{j=1}(g_{j,m}(t),h_{j,m}(t))w_{j},
$$
of the approximate systems
\begin{equation}\label{2eq2-10}
\begin{aligned}
&\int_{\Omega}u^{m}_{tt}wdx +M(\|\nabla
u^{m}(t)\|^{2}_{2}+\|\nabla v^{m}(t)\|^{2}_{2})\int_{\Omega}\nabla
u^{m} \cdot \nabla w\,dx\\
&+\int_{\Omega}\nabla u^{m}_{t} \cdot \nabla wdx +\int_{\Omega}f_{1}(u^{m}) w\,dx \\
& = - \tau_{1}\int_{\Gamma_{1}}\{u^{m}_{t} + k_{1}(0) u^{m} - k_{1}(t)
u^{m}(0) + k_{1}' \ast u^{m} \}w d
\Gamma_{1}
\end{aligned}
\end{equation}
and
\begin{equation} \label{2eq2-11}
\begin{aligned}
& \int_{\Omega}v^{m}_{tt}wdx + M(\|\nabla
u^{m}(t)\|^{2}_{2}+\|\nabla v^{m}(t)\|^{2}_{2})\int_{\Omega}\nabla
v^{m} \cdot\nabla w\,dx\\
&+\int_{\Omega}\nabla v^{m}_{t} \cdot \nabla w\,dx
+\int_{\Omega} f_{2}(v^{m})w\,dx \\
&= - \tau_{2}\int_{\Gamma_{1}}\{v^{m}_{t} + k_{2}(0) v^{m} - k_{2}(t)
v^{m}(0) + k_{2}' \ast v^{m} \}w d \Gamma_{1},
\end{aligned}
\end{equation}
for all $w\in V_{m}$ with the initial data
$$
(u^{m}(0),v^{m}(0))=(u_{0},v_{0}), \quad (u^{m}_{t}(0),v^{m}_{t}(0))=(u_{1},v_{1}).
$$
The extension of these solutions to the whole interval $[0,T]$,
$0<T<\infty$, is a consequence of the first estimate below.

\paragraph{A priori estimate I}
 Replacing $w$ by $u'_{m}(t)$ in (\ref{2eq2-10})
and $v'_{m}(t)$ in (\ref{2eq2-11}), respectively, and then adding
the results and using Lemma \ref{Lem2.2} we conclude that
\[
\frac{d}{dt}E(t,u^{m},v^{m}) \leq c E(0,u^{m},v^{m}).
\]
Integrating over $[0,t]$ and taking into account the definition
of the initial data of $(u^{m},v^{m})$ we conclude that
\begin{equation}\label{2eq2-12}
E(t,u^{m},v^{m}) \leq c, \quad \forall t \in[0,T], \quad  \forall m \in \mathbb{N}.
\end{equation}

\paragraph{A priori estimate II}
First, we estimate the initial data $u^{m}_{tt}(0)$ and
$v^{m}_{tt}(0)$ in the $L^{2}$-norm. Letting $t \to 0^{+}$ in the
equations (\ref{2eq2-10})-(\ref{2eq2-11}), replacing $w$ by
$u''_{m}(0)$ and $v''_{m}(0)$, respectively, and using the
compatibility conditions (\ref{2eq2-3})-(\ref{2eq2-4}) we get
\begin{align*}
&\|u^{m}_{tt}(0)\|^{2}_{2} \\
&=M(\|\nabla u_{0}\|^{2}_{2}+\|\nabla
v_{0}\|^{2}_{2})\int_{\Omega} \Delta u_{0}u^{m}_{tt}(0)
dx+\int_{\Omega} \Delta u_{1}u^{m}_{tt}(0) dx
- \int_{\Omega}f_{1}(u_{0})u^{m}_{tt}(0) dx,\\
&\|v^{m}_{tt}(0)\|^{2}_{2} \\
&= M(\|\nabla u_{0}\|^{2}_{2}+\|\nabla
v_{0}\|^{2}_{2}) \int_{\Omega} \Delta
v_{0}v^{m}_{tt}(0)dx+\int_{\Omega} \Delta v_{1}v^{m}_{tt}(0) dx -
\int_{\Omega}f_{2}(v_{0})v^{m}_{tt}(0) dx.
\end{align*}
Since $(u_{0},v_{0})\in [H^{2}(\Omega)]^{2}$, the growth
hypothesis for the functions $f_{1}$ and $f_{2}$ together with the
Sobolev's imbedding imply  $f_{1}(u_{0}),f_{2}(v_{0})\in
L^{2}(\Omega)$. Hence
\begin{equation}\label{2eq2-13}
\|u^{m}_{tt}(0)\|_{2} + \|v^{m}_{tt}(0)\|_{2} \leq c_{1}, \quad
\forall m \in \mathbb{N}.
\end{equation}
Differentiating the equations (\ref{2eq2-10})-(\ref{2eq2-11}) with
respect to the time, replacing $w$ by $u''_{m}(t)$ and
$v''_{m}(t)$, respectively, and summing the results we arrive at
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\Big \{\int_{\Omega}|u^{m}_{tt}|^{2}dx+
\int_{\Omega}|v^{m}_{tt}|^{2}dx \Big \} + \int_{\Omega}|\nabla
u^{m}_{tt}|^{2} dx + \int_{\Omega}|\nabla v^{m}_{tt}|^{2} dx \\
&=-M(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})\Big
\{\int_{\Omega}\nabla u^{m}_{t}\cdot \nabla
u^{m}_{tt}dx+\int_{\Omega}\nabla v^{m}_{t}\cdot
\nabla v^{m}_{tt}dx \Big \} \\
&\quad+2M'(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})\Big
\{\int_{\Omega}\nabla u^{m} \nabla u^{m}_{t}dx+\int_{\Omega}\nabla
v^{m} \nabla v^{m}_{t}dx\Big\}\\
&\quad\times \Big\{\int_{\Omega}\nabla u^{m}\cdot \nabla u^{m}_{tt}dx
+\int_{\Omega}\nabla v^{m}\cdot \nabla v^{m}_{tt}dx \Big\}
- \int_{\Omega} f_{1}'(u^{m})u^{m}_{t}u^{m}_{tt}dx\\
&\quad - \int_{\Omega} f_{2}'(v^{m})v^{m}_{t}v^{m}_{tt}dx
 - \tau_{1}\int_{\Gamma_{1}}|u^{m}_{tt}|^{2} dx -
\tau_{1}\int_{\Gamma_{1}}k_{1}(0)u^{m}_{t}u^{m}_{tt}d\Gamma_{1}\\
&\quad +\tau_{1}k_{1}'(t) \int_{\Gamma_1} u^{m}(0)u^{m}_{tt} d\Gamma_{1}
-\tau_{1}\int_{\Gamma_1}(k_{1}'\ast u^{m})_{t} u^{m}_{tt}d
\Gamma_{1} - \tau_{2}\int_{\Gamma_{1}}|v^{m}_{tt}|^{2} dx \\
&\quad -\tau_{2}\int_{\Gamma_{1}}k_{2}(0)v^{m}_{t}v^{m}_{tt}d\Gamma_{1}
+\tau_{2}k_{2}'(t) \int_{\Gamma_1} v^{m}(0)v^{m}_{tt} d
\Gamma_{1}-\tau_{2}\int_{\Gamma_1}(k_{2}'\ast v^{m})_{t}
v^{m}_{tt}d \Gamma_{1} .
\end{align*}
Noting that
\begin{gather*}
(k_{1}'\ast
u^{m})_{t}=k_{1}'(t)u^{m}_{0}+\int^{t}_{0}k_{1}'(t-s)u^{m}(\cdot,s)ds,\\
(k_{2}'\ast v^{m})_{t}=k_{2}'(t)v^{m}_{0}+\int^{t}_{0}k_{2}'(t-s)v^{m}(\cdot,s)ds
\end{gather*}
and using Lemma \ref{Lem2.2}, we obtain
\begin{equation}\label{2eq2-14}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\Big \{\int_{\Omega}|u^{m}_{tt}|^{2}dx+
 \int_{\Omega}|v^{m}_{tt}|^{2}dx
 +\tau_{1}k_{1}(t)\int_{\Gamma_{1}}|u^{m}_{t}|^{2}d
 \Gamma_{1}-\tau_{1}\int_{\Gamma_{1}}k_{1}'\Box u^{m}_{t}d \Gamma_{1}  \\
&-\tau_{2}k_{2}(t)\int_{\Gamma_{1}}|v^{m}_{t}|^{2}d
 \Gamma_{1}-\tau_{2}\int_{\Gamma_{1}}k_{2}'\Box v^{m}_{t}d
 \Gamma_{1}\Big\} + \int_{\Omega}|\nabla
 u^{m}_{tt}|^{2} dx + \int_{\Omega}|\nabla v^{m}_{tt}|^{2} dx\\
&=-M(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})\left
  \{\int_{\Omega}\nabla u^{m}_{t}\cdot \nabla
  u^{m}_{tt}dx+\int_{\Omega}\nabla v^{m}_{t}\cdot
  \nabla v^{m}_{tt}dx \right \} \nonumber \\
&\quad +2M'(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})\Big
 \{\int_{\Omega}u^{m}u^{m}_{t}dx+\int_{\Omega}v^{m}v^{m}_{t}dx\Big\}\\
&\quad\times\Big\{\int_{\Omega}\nabla u^{m}\cdot \nabla u^{m}_{tt}dx
 +\int_{\Omega}\nabla v^{m}\cdot \nabla v^{m}_{tt}dx \Big\}
 - \int_{\Omega} f_{1}'(u^{m})u^{m}_{t}u^{m}_{tt}dx\\
&\quad -\int_{\Omega} f_{2}'(v^{m})v^{m}_{t}v^{m}_{tt}dx
 - \tau_{1}\int_{\Gamma_{1}}|u^{m}_{tt}|^{2} dx +
 \tau_{1}k_{1}'(t) \int_{\Gamma_1} u^{m}(0)u^{m}_{tt} d \Gamma_{1}\\
&\quad +  \frac{\tau_{1}}{2}k_{1}'(t)\int_{\Gamma_{1}}|u^{m}_{t}|^{2}d \Gamma_{1}
 -\frac{\tau_{1}}{2}\int_{\Gamma_{1}}k_{1}''\Box u^{m}_{t}d\Gamma_{1}
 - \tau_{2}\int_{\Gamma_{1}}|v^{m}_{tt}|^{2} dx \\
&\quad + \tau_{1}k_{2}'(t) \int_{\Gamma_1} v^{m}(0)v^{m}_{tt} d \Gamma_{1}
 + \frac{\tau_{1}}{2}k_{2}'(t)\int_{\Gamma_{1}}|u^{m}_{t}|^{2}d
  \Gamma_{1}-\frac{\tau_{1}}{2}\int_{\Gamma_{1}}k_{1}''\Box
 u^{m}_{t} d\Gamma_{1}.
\end{aligned}
\end{equation}
Let us take $p_n=2n/(n-2)$. From the growth condition of the functions $f_i$
and from the Sobolev imbedding we have
\begin{align*}
&\int_{\Omega}f_{1}'(u^{m})u^{m}_{t}u^{m}_{tt}dx\\
& \leq  c \int_{\Omega}(1+2|u^{m}|^{\rho - 1})|u^{m}_{t}\|u^{m}_{tt}| dx \\
& \leq c \Big [ \int_{\Omega}(1+2 |u^{m}|^{\rho - 1})^{n}dx \Big ]^{1/n}
 \Big [ \int_{\Omega}|u^{m}_{t}|^{p_n}dx \Big ]^{1/p_n}
 \Big [ \int_{\Omega} |u^{m}_{tt}|^{2}dx \Big]^{1/2} \\
& \leq c \Big [\int_{\Omega}(1+|\nabla u^{m}|^{2})dx\Big ]^{(\rho - 1)/2}
 \Big [ \int_{\Omega} |\nabla u^{m}_{t}|^{2}dx \Big ]^{1/2}
 \Big [ \int_{\Omega} |u^{m}_{tt}|^{2}dx \Big ]^{1/2}.
\end{align*}
Taking into account the estimate (\ref{2eq2-12}) we conclude
that
\begin{equation}\label{2eq2-15}
\begin{aligned}
\int_{\Omega}f_{1}'(u^{m})u^{m}_{t}u^{m}_{tt} dx
& \leq  c \Big [ \int_{\Omega} |\nabla u^{m}_{t}|^{2}dx  \Big]^{1/2}
 \Big [ \int_{\Omega} |u^{m}_{tt}|^{2}dx  \Big ]^{1/2}  \\
& \leq c  \Big \{ \int_{\Omega} |\nabla u^{m}_{t}|^{2}dx
 +  \int_{\Omega} |u^{m}_{tt}|^{2}dx  \Big \}.
\end{aligned}
\end{equation}
Similarly  we get
\begin{equation}\label{2eq2-16}
\int_{\Omega}f_{2}'(v^{m})v^{m}_{t}v^{m}_{tt}dx
 \leq c  \Big \{ \int_{\Omega} |\nabla v^{m}_{t}|^{2}dx  +  \int_{\Omega}
|v^{m}_{tt}|^{2}dx  \Big \}.
\end{equation}
Note that Young's inequality, the first estimate and hypothesis on
$M$ give us
\begin{equation}\label{2eq2-17}
\begin{aligned}
&M(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})
\Big\{ \int_{\Omega}\nabla u^{m}_{t}\cdot \nabla
u^{m}_{tt}dx+\int_{\Omega}\nabla v^{m}_{t}\cdot \nabla
v^{m}_{tt}dx \Big\}  \\
&\leq c\Big\{\int_{\Omega}(|\nabla u^{m}_{t}|^{2}+|\nabla
v^{m}_{t}|^{2})dx + \frac{1}{4}\int_{\Omega}|\nabla
u^{m}_{tt}|^{2}dx + \frac{1}{4}\int_{\Omega}|\nabla
v^{m}_{tt}|^{2}dx \Big\}
\end{aligned}
\end{equation}
and similarly
\begin{equation}\label{2eq2-18}
\begin{aligned}
&2M'(\|\nabla u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})
\Big\{\int_{\Omega}u^{m}u^{m}_{t}dx+\int_{\Omega}v^{m}v^{m}_{t}dx\Big\}\\
&\times \Big\{\int_{\Omega}\nabla u^{m}\cdot \nabla u^{m}_{tt}dx
+\int_{\Omega}\nabla v^{m}\cdot \nabla v^{m}_{tt}dx\Big \} \\
&\leq c\Big\{\int_{\Omega}(|\nabla u^{m}_{t}|^{2}+|\nabla v^{m}_{t}|^{2})dx
+ \frac{1}{4}\int_{\Omega}|\nabla u^{m}_{tt}|^{2}dx +
\frac{1}{4}\int_{\Omega}|\nabla v^{m}_{tt}|^{2}dx\Big\}.
\end{aligned}\end{equation}
Substitution of inequalities (\ref{2eq2-15})-(\ref{2eq2-18})
into (\ref{2eq2-14}) yields
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\Big\{\int_{\Omega}|u^{m}_{tt}|^{2}dx+
\int_{\Omega}|v^{m}_{tt}|^{2}dx
+\tau_{1}k_{1}(t)\int_{\Gamma_{1}}|u^{m}_{t}|^{2}d
\Gamma_{1}-\tau_{1}\int_{\Gamma_{1}}k_{1}'\Box u^{m}_{t}d\Gamma_{1} \\
&+\tau_{2}k_{2}(t)\int_{\Gamma_{1}}|v^{m}_{t}|^{2}d
\Gamma_{1}-\tau_{2}\int_{\Gamma_{1}}k_{2}'\Box v^{m}_{t}d
\Gamma_{1}\Big\} + \frac{1}{2}\int_{\Omega}|\nabla
u^{m}_{tt}|^{2} dx + \frac{1}{2}\int_{\Omega}|\nabla v^{m}_{tt}|^{2} dx \\
&\leq
\frac{\tau_{1}c}{2}\int_{\Gamma_{1}}|u_{0}|^{2}d\Gamma_{1}+\frac{\tau_{2}c}{2}\int_{\Gamma_{1}}|v_{0}|^{2}d\Gamma_{1}+c\int_{\Omega}(|\nabla
u^{m}_{t}|^{2}+|\nabla v^{m}_{t}|^{2})dx \\
&+c\Big\{\int_{\Omega}|u^{m}_{tt}|^{2}dx+\int_{\Omega}|v^{m}_{tt}|^{2}dx
\Big\}.
\end{align*}
Integrating with respect to time and applying Gronwall's inequality we conclude
that for all $m \in \mathbb{N}$ and all $t\in [0,T]$,
\begin{equation}\label{2eq2-19}
\int_{\Omega}|u^{m}_{tt}|^{2}dx+ \int_{\Omega}|v^{m}_{tt}|^{2}dx
+\int^{t}_{0}\int_{\Omega}|\nabla u^{m}_{tt}|^{2} dx +
\int^{t}_{0}\int_{\Omega}|\nabla v^{m}_{tt}|^{2} dx \leq c\,.
\end{equation}

\paragraph{A priori estimate III}
Replacing $w$ by $-\Delta u^{m}_{t}$ in (\ref{2eq2-10}) and by $
-\Delta v^{m}_{t}$ in (\ref{2eq2-11}), respectively, and then
using the Green's formula and adding the results yields
\begin{align*}
&-\int_{\Omega}u^{m}_{tt}\Delta
u^{m}_{t}dx-\int_{\Omega}v^{m}_{tt}\Delta v^{m}_{t}dx+M(\|\nabla
u^{m}\|^2_{2}+\|\nabla v^{m}\|^2_{2})\{\int_{\Omega}\Delta
u^{m}\Delta u^{m}_{t}dx \\
&+\int_{\Omega}\Delta v^{m}\Delta v^{m}_{t}dx \}
+\int_{\Omega}|\Delta u^{m}_{t}|^{2}dx+\int_{\Omega}|\Delta
v^{m}_{t}|^{2}dx \\
&=\int_{\Omega}f_{1}(u^{m})\Delta
u^{m}_{t}dx+\int_{\Omega}f_{2}(v^{m})\Delta v^{m}_{t}dx.
\end{align*}
Using similar arguments as in (\ref{2eq2-19}) we conclude
that for all $m\in \mathbb{N}$ and all all $t\in [0,T]$,
\begin{equation}\label{2eq2-20}
\|\Delta u^{m}\|^{2}_{2}+\int^{T}_{0}\|\Delta
u^{m}_{t}(t)\|^{2}_{2}dt+\|\Delta
v^{m}\|^{2}_{2}+\int^{T}_{0}\|\Delta v^{m}_{t}(t)\|^{2}_{2}dt \leq
c\,.
\end{equation}
 Now, from  estimates (\ref{2eq2-12}), (\ref{2eq2-19}) and (\ref{2eq2-20}) and
of the Lions-Aubin's compactness Theorem we can pass to the limit
in (\ref{2eq2-10})-(\ref{2eq2-11}). The rest of the proof is a
matter of routine.

\section{Uniform Rate of Exponential Decay}

In this section we shall study the asymptotic behavior of the
solutions of system (\ref{1eq1-1})-(\ref{1eq1-6}) when the
resolvent kernels $k_{1}$ and $k_{2}$ satisfy (\ref{2eq2-5}). Our
point of departure will be to establish some inequalities for the
strong solution of coupled system (\ref{1eq1-1})- (\ref{1eq1-6}).
\begin{lemma}\label{Lem3.1}
Any strong solution $(u,v)$ of the system (\ref{1eq1-1})-(\ref{1eq1-6}) satisfy
\begin{align*}
\frac{d}{dt}E(t) & \leq
-\frac{\tau_{1}}{2}\int_{\Gamma_{1}}|u_{t}|^{2} d \Gamma_{1}
+\frac{\tau_{1}}{2}k^{2}_{1}(t)\int_{\Gamma_{1}}|u_{0}|^{2} d \Gamma_{1}
+ \frac{\tau_{1}}{2}k_{1}'(t)\int_{\Gamma_{1}}|u|^{2} d\Gamma_{1}\\
&\quad
-\frac{\tau_{1}}{2}\int_{\Gamma_{1}}k_{1}'' \Box u d \Gamma_{1}
- \frac{\tau_{2}}{2}\int_{\Gamma_{1}}|v_{t}|^{2} d \Gamma_{1}
+\frac{\tau_{2}}{2}k^{2}_{2}(t)\int_{\Gamma_{1}}|v_{0}|^{2} d \Gamma_{1} \\
&\quad
+ \frac{\tau_{2}}{2}k_{2}'(t)\int_{\Gamma_{1}}|v|^{2} d\Gamma_{1}
- \frac{\tau_{2}}{2}\int_{\Gamma_{1}}k_{2}'' \Box v d \Gamma_{1}
-\int_{\Omega}\{|\nabla u_{t}|^{2}+|\nabla v_{t}|^{2}\}dx.
\end{align*}
\end{lemma}

\begin{proof}
Multiplying the equation (\ref{1eq1-1}) by $u_{t}$ and integrating by parts over $\Omega$ we get
\begin{align*}
&\frac{1}{2}\frac{d}{dt} \int_{\Omega}|u_{t}|^{2}dx +
\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\frac{d}{dt}\int_{\Omega}|\nabla u|^{2}dx
+\int_{\Omega} F_{1}(u) dx +\int_{\Omega}|\nabla u_{t}|^{2} dx\\
&= \int_{\Gamma_{1}}\{(M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\frac{\partial u}{\partial \nu}+\frac{\partial
u_{t}}{\partial \nu}\}u_{t} d \Gamma_{1}.
\end{align*}
Similarly we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt} \int_{\Omega}|v_{t}|^{2} dx+
\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\frac{d}{dt}\int_{\Omega}|\nabla v|^{2}dx
+\int_{\Omega} F_{2}(v) dx +\int_{\Omega}|\nabla v_{t}|^{2} dx \\
&= \int_{\Gamma_{1}}\{(M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\frac{\partial v}{\partial \nu}+\frac{\partial
v_{t}}{\partial \nu}\}v_{t} d \Gamma_{1}.
\end{align*}
Summing the above identities, substituting the boundary terms by 
(\ref{2eq2-1})-(\ref{2eq2-2}) and using Lemma \ref{Lem2.1}
our conclusion follows. \end{proof}

Let us consider the  binary operator
$$
(k\diamond\varphi)(t):=\int^{t}_{0}k(t-s)(\varphi(t)-\varphi(s))ds.
$$
Then applying the H\"{o}lder's inequality for $0\leq\mu\leq1$ we
have
\begin{equation}\label{3eq3-2}
|(k\diamond\varphi)(t)|^{2}\leq
\Big[\int^{t}_{0}|k(s)|^{2(1-\mu)}ds\Big](|k|^{2\mu}\Box\varphi)(t).
\end{equation}
Let us introduce the  functionals
\begin{gather*}
\mathcal{N}(t):=\int_{\Omega}(|u_{t}|^{2}+|v_{t}|^{2}+F_{1}(u)+F_{2}(v))dx+\widehat{M}(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2}), \\
 \psi (t) = \int_{\Omega} \big\{m \cdot \nabla u + ( \frac{n}{2}- \theta)u\big\}u_{t} dx
 +\int_{\Omega} \big\{m \cdot \nabla v
+ \big( \frac{n}{2}- \theta)v\big\}v_{t} dx,
\end{gather*}
where $\theta$ is a small positive constant. The following Lemma
plays an important role for the construction of the Lyapunov
functional.

\begin{lemma}\label{Lem3.2}
For any strong solution of  the system (\ref{1eq1-1})-(\ref{1eq1-6}) we get
 \begin{align*}
\frac{d}{dt}\psi (t) & \leq  -\frac{\theta}{2}\mathcal{N}(t)+c
\int_{\Gamma_{1}}(|u_{t}|^{2}+|k_{1}(t)u|^{2}+|k_{1}'\diamond
u|^{2}+|k_{1}(t)u_{0}|^{2})d\Gamma_{1}\\
&\quad +c \int_{\Gamma_{1}}(|v_{t}|^{2}+|k_{2}(t)v|^{2}+|k_{2}'\diamond
v|^{2}+|k_{2}(t)v_{0}|^{2}) d \Gamma_{1} \\
&\quad +c_{\epsilon}\int_{\Omega}(|\nabla u_{t}|^{2}+|\nabla
v_{t}|^{2})dx ,
\end{align*}
for some positive constants $c$ and $\epsilon$.
\end{lemma}

\begin{proof}
From equation (\ref{1eq1-1}) we obtain
\begin{align*}
&\frac{d}{dt}\int_{\Omega}u_{t}\big\{m \cdot \nabla u
+ \big(\frac{n}{2} - \theta\big)u\big\} dx \\
&=\int_{\Omega}u_{t}m \cdot \nabla u_{t} dx
+ \big(\frac{n}{2}- \theta\big)\int_{\Omega}|u_{t}|^{2} dx \\
&\quad + \int_{\Omega}m \cdot \nabla u(M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\Delta u + \Delta u_{t}-f_{1}(u))dx \\
&\quad +(\frac{n}{2} - \theta)\int_{\Omega}u(M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\Delta u + \Delta
u_{t}-f_{1}(u))dx.
\end{align*}
Performing a integration by parts and using the Young's inequality,
we get
\begin{align*}
& \frac{d}{dt}\int_{\Omega}u_{t}\big\{m \cdot \nabla u
+ big(\frac{n}{2} - \theta\big)u\big\} dx \\
&\leq  \frac{1}{2} \int_{\Gamma_{1}}m \cdot \nu |u_{t}|^{2} d
\Gamma_{1} - \theta \int_{\Omega}|u_{t}|^{2} dx \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial u}{\partial \nu}+\frac{\partial u_{t}}{\partial
\nu})\left \{ m \cdot \nabla u +(\frac{n}{2}-\theta)u \right \} d \Gamma_{1} \\
&\quad -(1-\theta)M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla u|^{2}dx \\
&\quad +\epsilon c M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla
u|^{2}dx+c_{\epsilon}\int_{\Omega}|\nabla u_{t}|^{2}dx \\
&\quad -(\frac{n}{2}-\theta)\int_{\Omega}f_{1}(u)udx+n\int_{\Omega}F_{1}(u)dx\\
&\quad -\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu |\nabla u|^{2}d \Gamma_{1},
\end{align*}
where $\epsilon$ is a positive constant. Taking into account that
$f_i$ is superlinear we conclude that
\begin{align*}
& \frac{d}{dt}\int_{\Omega}u_{t}\big\{m \cdot \nabla u
+ \big(\frac{n}{2} - \theta\big)u\big\} dx \\
&\leq \frac{1}{2} \int_{\Gamma_{1}}m \cdot \nu |u_{t}|^{2} d
\Gamma_{1} - \theta \int_{\Omega}|u_{t}|^{2} dx \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial u}{\partial \nu}+\frac{\partial u_{t}}{\partial
\nu})\left \{ m \cdot \nabla u +(\frac{n}{2}-\theta)u \right \} d \Gamma_{1} \\
&\quad -(1-\theta)M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla u|^{2}dx \\
&\quad +\epsilon c M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla
u|^{2}dx+c_{\epsilon}\int_{\Omega}|\nabla u_{t}|^{2}dx \\
&\quad -(\frac{n}{2}-\theta)(2+\delta)\int_{\Omega}F_{1}(u)dx
-\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu |\nabla u|^{2}d \Gamma_{1}.
\end{align*}
Similarly, using equation (\ref{1eq1-2}) instead of (\ref{1eq1-1}) we get
\begin{align*}
& \frac{d}{dt}\int_{\Omega}v_{t}\big\{m \cdot \nabla v
+ \big(\frac{n}{2} - \theta\big) v\big\} dx \\
&\leq \frac{1}{2} \int_{\Gamma_{1}}m \cdot \nu |v_{t}|^{2} d
\Gamma_{1} - \theta \int_{\Omega}|v_{t}|^{2} dx \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial v}{\partial \nu}+\frac{\partial v_{t}}{\partial
\nu})\left \{ m \cdot \nabla v
+(\frac{n}{2}-\theta)v \right \} d \Gamma_{1} \\
&\quad -(1-\theta)M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla v|^{2}dx \\
&\quad +\epsilon c M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}|\nabla
v|^{2}dx+c_{\epsilon}\int_{\Omega}|\nabla v_{t}|^{2}dx \\
&-(\frac{n}{2}-\theta)(2+\delta)\int_{\Omega}F_{2}(v)dx
 -\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu |\nabla v|^{2}d \Gamma_{1}.
\end{align*}
Summing these two last inequalities we arrive at
 \begin{align*}
& \frac{d}{dt}\psi(t)\leq  \frac{1}{2}
\int_{\Gamma_{1}}m \cdot \nu (|u_{t}|^{2}+|v_{t}|^{2}) d
\Gamma_{1}- \theta \int_{\Omega}(|u_{t}|^{2}+|v_{t}|^{2}) dx \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial u}{\partial \nu}+\frac{\partial u_{t}}{\partial
\nu})\left \{ m \cdot \nabla u
+(\frac{n}{2}-\theta)u \right \} d \Gamma_{1} \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial v}{\partial \nu}+\frac{\partial v_{t}}{\partial
\nu})\left \{ m \cdot \nabla v
+(\frac{n}{2}-\theta)v \right \} d \Gamma_{1} \\
&\quad -(1-\theta)M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}(|\nabla u|^{2}+|\nabla v|^{2})dx \\
&\quad +\epsilon c M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Omega}(|\nabla
u|^{2}dx+|\nabla v|^{2})dx \\
&\quad +c_{\epsilon}\int_{\Omega}(|\nabla u_{t}|^{2}+|\nabla
v_{t}|^{2})dx
-(\frac{n}{2}-\theta)(2+\delta)\int_{\Omega}F_{1}(u)dx
\\
&\quad -(\frac{n}{2}-\theta)(2+\delta)\int_{\Omega}F_{2}(v)dx-\frac{1}{2}M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu
|\nabla u|^{2}d \Gamma_{1} \\
&\quad -\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu |\nabla v|^{2}d\Gamma_{1}.
\end{align*}
Using Poincar\'e's inequality and taking $\theta$ and $\epsilon$
small enough we obtain
\begin{equation}\label{3eq3-3}
\begin{aligned}
\frac{d}{dt}\psi(t)
&\leq -\theta\mathcal{N}(t)+c_{\epsilon}\int_{\Omega}(|\nabla u_{t}|^{2}+|\nabla
v_{t}|^{2})dx +\frac{1}{2} \int_{\Gamma_{1}}m \cdot \nu
(|u_{t}|^{2}+|v_{t}|^{2}) d \Gamma_{1}  \\
&\quad -\frac{1}{2}M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2})\int_{\Gamma_{1}}m \cdot \nu (|\nabla u|^{2}+|\nabla
v|^{2})d \Gamma_{1}  \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial u}{\partial \nu}+\frac{\partial u_{t}}{\partial
\nu})\left \{ m \cdot \nabla u
+(\frac{n}{2}-\theta)u \right \} d \Gamma_{1}  \\
&\quad + \int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})
\frac{\partial v}{\partial \nu}+\frac{\partial v_{t}}{\partial
\nu})\left \{ m \cdot \nabla v +(\frac{n}{2}-\theta)v \right \} d\Gamma_{1}.
\end{aligned}
\end{equation}
Now, we analyze some boundary term of the above inequality.
Applying Young and Poincar\'e's inequalities we have, for
$\epsilon_{1}>0$
\begin{align*}
&\int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2}) \frac{\partial u}{\partial \nu}+\frac{\partial
u_{t}}{\partial \nu})\big \{ m \cdot \nabla u
+(\frac{n}{2}-\theta)u \big \} d \Gamma_{1} \\
& \leq \epsilon_{1} \int_{\Gamma_{1}} \big\{|m \cdot \nabla u|^2
+\big(\frac{n}{2}-\theta\big)^2|u|^2\big\}  d \Gamma_{1} \\
&\quad + c_{\epsilon_{1}}\int_{\Gamma_{1}}|(M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial u}{\partial
\nu}+\frac{\partial u_{t}}{\partial \nu})|^{2}d \Gamma_{1} \\
&\leq \epsilon_{1} c \Big\{\int_{\Gamma_{1}} m\cdot\nu |\nabla
u|^2 d \Gamma_{1} + \mathcal{N}(t) \Big\} \\
&\quad + c_{\epsilon_{1}} \int_{\Gamma_{1}}|(M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial u}{\partial
\nu}+\frac{\partial u_{t}}{\partial \nu})|^{2}d \Gamma_{1}.
\end{align*}
Similarly, we obtain
\begin{align*}
&\int_{\Gamma_{1}}(M(\|\nabla u\|^{2}_{2}+\|\nabla
v\|^{2}_{2}) \frac{\partial v}{\partial \nu}+\frac{\partial
v_{t}}{\partial \nu})\left \{ m \cdot \nabla v
+(\frac{n}{2}-\theta)v \right \} d \Gamma_{1} \\
& \leq \epsilon_{1} \int_{\Gamma_{1}} \big\{|m \cdot \nabla v|^2
+\big(\frac{n}{2}-\theta\big)^2|v|^2\big\}  d \Gamma_{1} \\
&\quad + c_{\epsilon_{1}}\int_{\Gamma_{1}}|(M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial v}{\partial
\nu}+\frac{\partial v_{t}}{\partial \nu})|^{2}d \Gamma_{1} \\
& \leq \epsilon_{1} c \Big\{\int_{\Gamma_{1}} m\cdot\nu |\nabla
v|^2 d \Gamma_{1} + \mathcal{N}(t) \Big\} \\
&\quad + c_{\epsilon_{1}} \int_{\Gamma_{1}}|(M(\|\nabla
u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial v}{\partial
\nu}+\frac{\partial v_{t}}{\partial \nu})|^{2}d \Gamma_{1}.
\end{align*}
Substituting the two  inequalities above into (\ref{3eq3-3}),
choosing  $\epsilon_{1}$ small snough and taking into account that
the boundary conditions (\ref{2eq2-1})-(\ref{2eq2-2}) can be
written as
\begin{gather*}
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial
u}{\partial
\nu}+\frac{\partial u_{t}}{\partial \nu})= -\tau_{1} \{u_{t} + k_{1}(t)u -  k_{1}' \diamond u - k_{1}(t)u_{0} \},\\
M(\|\nabla u\|^{2}_{2}+\|\nabla v\|^{2}_{2})\frac{\partial
v}{\partial \nu}+\frac{\partial v_{t}}{\partial \nu})= -\tau_{2}
\{v_{t} + k_{2}(t)v - k_{2}' \diamond v - k_{2}(t)v_{0} \},
\end{gather*}
our conclusion follows. \end{proof}

To show that the energy decay exponentially we need of the
following Lemma whose proof can be found in \cite{Santos-2}.

\begin{lemma}\label{Lem3.3}
Let $f$ be a real positive function of class $C^1$. If there exists positive constants $\gamma_0, \gamma_1$ and $c_0$ such
that
$$
f'(t)\leq -\gamma_0f(t) +c_0e^{-\gamma_1 t},
$$
then there exist positive constants $\gamma$ and $c$ such that
$f(t)\leq (f(0) +c) e^{-\gamma t}$.
\end{lemma}

Next, we shall show inequality (\ref{2eq2-7}). We shall
prove this result for strong solutions, that is, for solutions
with initial data $(u_{0},v_{0})\in \left( H^{2}(\Omega) \cap V
\right)^{2}$ and $(u_{1},v_{1})\in \left( H^{2}(\Omega) \cap V
\right)^{2}$ satisfying the compatibility conditions
(\ref{2eq2-3})- (\ref{2eq2-4}). Our conclusion follow by standard
density arguments. Using hypothesis (\ref{2eq2-5}) in Lemma
\ref{Lem3.1} we get
\begin{align*}
\frac{d}{dt}E(t)
& \leq  - \frac{\tau_{1}}{2}\int_{\Gamma_{1}}
\big(|u_{t}|^{2} -b_2 k_{1}'
\Box u + b_1 k_1(t)|u|^2 - |k_1(t)u_0|^2 \big) d \Gamma \\
&\quad - \frac{\tau_{2}}{2}\int_{\Gamma_{1}}\big(|v_{t}|^{2} -b_2
k_2' \Box v +b_1 k_2(t)|v|^2 - |k_2(t)v_0|^2 \big) d \Gamma\\
&\quad -\int_{\Omega}(|\nabla u_{t}|^{2}+|\nabla v_{t}|^{2})dx .
\end{align*}
On the other hand applying inequality (\ref{3eq3-2}) with
$\mu=1/2$ in Lemma \ref{Lem3.2} we obtain
\begin{align*}
\frac{d}{dt}\psi (t)
& \leq -\frac{\theta}{2} \mathcal{N}(t)+ C\int_{\Gamma_{1}}
\big(|u_{t}|^{2} + k_1(t)|u|^2 -k_{1}' \Box u + |k_1(t)u_0|^2 \big) d \Gamma   \\
&\quad + C\int_{\Gamma_{1}}\big(|v_{t}|^{2} + k_2(t)|v|^2  -k_2'
 \Box v + |k_2(t)v_0|^2 \big) d \Gamma\\
&\quad +c_{\epsilon}\int_{\Omega}(|\nabla u_{t}|^{2}+|\nabla v_{t}|^{2})dx .
\end{align*}
Let us introduce the Lyapunov functional
\begin{equation}
\mathcal{L} (t) := N E(t) + \psi (t),\label{3eq3-4}
\end{equation}
with $N>0$. Taking $N$ large, the previous inequalities imply that
\[
\frac{d}{dt}\mathcal{L}(t)  \leq -\frac{\theta}{2} E(t)+ 2N R^2(t)E(0),
\]
where $R(t) = k_{1}(t) + k_{2}(t)$. Moreover, using Young's
inequality and taking $N$ large we find that
\begin{equation}\label{3eq3-5}
\frac{N}{2} E(t) \leq \mathcal{L} (t) \leq 2N E(t).
\end{equation}
 From this inequality we conclude that
\[
\frac{d}{dt}\mathcal{L}(t) \leq -\frac{\theta}{2} \mathcal{L}(t)+ 2N R^2(t)E(0),
\]
from where follows, in view of Lemma \ref{Lem3.3} and of the
exponential decay of $k_{1}$, $k_{2}$, that
\[
\mathcal{L} (t) \leq \{\mathcal{L}(0) + c\} e^{- {\gamma_{1}}t},
\]
for some positive constants $c, \gamma$. From the inequality
(\ref{3eq3-5}) our conclusion follows.

\section{Uniform Rate of Polynomial Decay}

Our attention will be focused on the uniform rate of decay
when the resolvent kernels $k_{1}$ and $k_{2}$ satisfy
(\ref{2eq2-6}). First of all we will prove the following three
lemmas that will be used in the sequel.

\begin{lemma}\label{Lem4.2}
Let $(u,v)$ be a solution of system (\ref{1eq1-1})-(\ref{1eq1-6})  and let us
denote by $(\phi_{1},\phi_{2})=(u,v)$. Then, for $p>1$, $0 < r < 1$ and
$t\ge0$, we have
\begin{align*}
&\Big(\int_{\Gamma_{1}}|k_{i}'| \Box \phi_{i} d \Gamma_{1}\Big)
^{\frac{1+(1-r)(p+1)}{(1-r)(p+1)}}\\
&\leq  2 \Big( \int^{t}_{0}|k_{i}'(s)|^{r}ds
\|\phi_{i}\|^{2}_{L^{\infty}(0,t;L^{2}(\Gamma_{1}))}\Big)^ {\frac{1}{(1-r)(p+1)}}
\int_{\Gamma_{1}}|k_{i}'|^{1+ \frac{1}{p+1}} \Box \phi_{i} d \Gamma_{1}
\end{align*}
while for $r=0$ we get
\begin{align*}
&\Big(\int_{\Gamma{1}}|k_{i}'| \Box \phi_{i} d \Gamma_{1}\Big)^{\frac{p+2}{p+1}}\\
&\leq 2 \Big(\int^{t}_{0}\|\phi_{i}(s,.)\|^{2}_{L^{2}(\Gamma_{1})} ds
+ t \|\phi_{i}(s,.)\|^{2}_{L^{2}(\Gamma_{1})}\Big)^{p+1}
\int_{\Gamma_{1}}|k_{i}'|^{1+ \frac{1}{p+1}}\Box \phi_{i} d \Gamma_{1},
\end{align*}
for $i=1,2$.
\end{lemma}
For the proof of this lemma see e. g. \cite{Santos-1}.

\begin{lemma}\label{Lem4.3}
Let $f \ge 0$ be a differentiable function satisfying
\[
f'(t) \leq -\frac{c_{1}}{f(0)^\frac{1}{\alpha}}f(t)^{1+\frac{1}{\alpha}} + \frac{c_{2}}{(1+t)^{\beta}}f(0) \quad \mbox{for}
\quad t \ge 0,
\]
for some positive constants $c_{1},c_{2}$, $\alpha$ and $\beta$ such that
$\beta \ge \alpha + 1$.
Then there exists a constant $c > 0$ such that
\[
f(t) \leq \frac {c}{(1+t)^{\alpha}}f(0) \quad \mbox{for} \quad t \ge 0.
\]
\end{lemma}
For the proof of this lemma see e. g. \cite{Santos-2}.

Next we show inequality (\ref{2eq2-8}). We shall
prove this result for strong solutions, that is, for solutions
with initial data $(u_{0},v_{0})\in \left( H^{2}(\Omega) \cap V
\right)^{2}$ and $(u_{1},v_{1})\in \left( H^{2}(\Omega) \cap V
\right)^{2}$ satisfying the compatibility conditions
(\ref{2eq2-3})- (\ref{2eq2-4}). Our conclusion will follow by
standard density arguments. We use some estimates of the previous
section which are independent of the behavior of the resolvent
kernels $k_1,\ k_2$. Using hypothesis (\ref{2eq2-6}) in
Lemma~\ref{Lem3.1} yields
\begin{align*}
\frac{d}{dt}E(t)
& \leq  - \frac{\tau_{1}}{2}\int_{\Gamma_{1}}\Big(|u_{t}|^{2} +b_2
[-k_{1}']^{1+\frac{1}{p+1}}
\Box u + b_1 k_1^{1+\frac{1}{p}}(t)|u|^2 - |k_1(t)u_0|^2 \Big) d \Gamma_{1}   \\
&\quad - \frac{\tau_{1}}{2}\int_{\Gamma_{1}}\big(|v_{t}|^{2} +b_2
[-k_2']^{1+\frac{1}{p+1}} \Box v +b_1 k_2^{1+\frac{1}{p}}(t)|v|^2
- |k_2(t)v_0|^2 \big) d \Gamma_{1}.
\end{align*}
Applying inequality (\ref{3eq3-2}) with $\mu=\frac{p+2}{2(p+1)}$
and using hypothesis (\ref{2eq2-6}) we obtain the estimates
\begin{eqnarray*}
|k_1'\diamond u|^2\leq c[-k_1']^{1+\frac{1}{p+1}}\Box u, \quad
|k_2'\diamond v|^2\leq c[-k_2']^{1+\frac{1}{p+1}}\Box v.
\end{eqnarray*}
The above inequalities in Lemma~\ref{Lem3.2} yields
\begin{align*}
\frac{d}{dt}\psi (t) & \leq -\frac{\theta}{2} \mathcal{N}(t)+
c\int_{\Gamma_{1}}\big(|u_{t}|^{2}+ k_1^{1+\frac{1}{p}}(t)|u|^2
+ [-k_{1}']^{1+\frac{1}{p+1}} \Box u + |k_1(t)u_0|^2 \big) d \Gamma_{1}   \\
&\quad + c\int_{\Gamma_{1}}\big(|v_{t}|^{2} +
k_2^{1+\frac{1}{p}}(t)|v|^2 + [-k_2']^{1+\frac{1}{p+1}} \Box v +
|k_2(t)v_0|^2 \big) d \Gamma_{1} .
\end{align*}
In this conditions, taking $N$ large the Lyapunov functional
defined in (\ref{3eq3-4}) satisfies
\begin{equation}
\begin{aligned}
\frac{d}{dt}\mathcal{L} (t)
&\leq  - \frac{\theta}{2}\mathcal{N}(t) +2N R^{2}(t)E(0) \nonumber \\
&\quad -\frac{Nc_2}{2}
\Big\{\int_{\Gamma_{1}}[-k_{1}']^{1+\frac{1}{p+1}}\Box u d
\Gamma_{1} +  \int_{\Gamma_{1}}[- k_{2}']^{1+\frac{1}{p+1}}\Box v
d \Gamma_{1} \Big\}.
\end{aligned}\label{lya-pol}
\end{equation}
Let us fix $0<r<1$ such that $ \frac{1}{p+1}<r<\frac{p}{p+1}$.
From (\ref{2eq2-6}) we have that
\[
\int^{\infty}_{0}|k_{i}'|^{r} \leq c
\int^{\infty}_{0}\frac{1}{(1+t)^{r(p+1)}} < \infty
\quad\mbox{for}\quad i=1,2.
\]
Using this estimate in Lemma \ref{Lem4.2} we get
\begin{gather}\label{4eq4-5}
 \int_{\Gamma_{1}}[-k_{1}']^{1+\frac{1}{p+1}}\Box u d \Gamma_{1}
\ge  cE(0)^{-\frac{1}{(1-r)(p+1)}}\Big(\int_{\Gamma_{1}}[-
k_{1}'] \Box u d \Gamma_{1} \Big)^{1+\frac{1}{(1-r)(p+1)}}, \\
\label{4eq4-6}
 \int_{\Gamma_{1}}[-k_{2}']^{1+\frac{1}{p+1}}\Box v\ d \Gamma_{1}
  \ge  cE(0)^{-\frac{1}{(1-r)(p+1)}}\Big(\int_{\Gamma_{1}}[-
k_{2}'] \Box v d \Gamma_{1} \Big)^{1+\frac{1}{(1-r)(p+1)}}.
\end{gather}
On the other hand, from the Trace theorem we have
\begin{equation}\label{4eq4-7}
E(t)^{1+ \frac{1}{(1-r)(p+1)}}
 \leq c E(0)^{\frac{1}{(1-r)(p+1)}}\mathcal{N}(t).
\end{equation}
Substitution of (\ref{4eq4-5})-(\ref{4eq4-7}) into (\ref{lya-pol})
we obtain
\begin{align*}
\frac{d}{dt}\mathcal{L} (t)
& \leq - cE(0)^{-\frac{1}{(1-r)(p+1)}} E(t)^{1+\frac{1}{(1-r)(p+1)}}
+ 2NR^{2}(t) E(0) \\
&\quad -cE(0)^{-\frac{1}{(1-r)(p+1)}} \Big\{ \Big(
\int_{\Gamma_{1}}[-k_{1}'] \Box u d \Gamma_{1}
\Big)^{1+\frac{1}{(1-r)(p+1)}}\\
&\quad + \Big( \int_{\Gamma_{1}}[-k_{2}']
\Box v d \Gamma_{1} \Big)^{1+\frac{1}{(1-r)(p+1)}} \Big\} .
 \end{align*}
Taking into account the inequality (\ref{3eq3-5}) we conclude that
\[
\frac{d}{dt}\mathcal{L} (t) \leq -\frac{c}{\mathcal{L}
(0)^{\frac{1}{(1-r)(p+1)}}} \mathcal{L} (t)^{1+ \frac{1}{(1-r)(p+1)}}
+ 2NR^{2}(t) E(0),
\]
for some $c>0$, from where follows, applying Lemma \ref{Lem4.3},
that
\begin{eqnarray*}
\mathcal{L} (t) \leq \frac{c}{(1+t)^{(1-r)(p+1)}}\mathcal{L}(0).
\end{eqnarray*}
Since $(1-r)(p+1)>1$ we get, for $t\geq 0$, the following bounds
\begin{gather*}
t\|u\|_{L^{2}(\Gamma_{1})}^2 + t\|v\|_{L^{2}(\Gamma_{1})}^2
\leq  t \mathcal{L}(t)  <  \infty, \\
\int^{t}_{0}\big(\|u\|_{L^{2}(\Gamma_{1})}^2 +
\|v\|_{L^{2}(\Gamma_{1})}^2\big)\,ds \leq c \int^{t}_{0} \mathcal{L}(t)\,ds
 <  \infty.
\end{gather*}
Using the above estimates in Lemma \ref{Lem4.2} with $r=0$ we get
\begin{gather*}
\int_{\Gamma_{1}}[-k_{1}']^{1+\frac{1}{p+1}}\Box u d \Gamma_{1}
\ge \frac{c}{E(0)^{\frac{1}{p+1}}}\Big(\int_{\Gamma_{1}}[-
k_{1}']\Box u d \Gamma  \Big)^{1+ \frac{1}{p+1}}, \\
\int_{\Gamma_{1}}[-k_{2}']^{1+\frac{1}{p+1}}\Box v d \Gamma_{1}
\ge \frac{c}{E(0)^{\frac{1}{p+1}}}\Big(\int_{\Gamma_{1}}[-
k_{2}']\Box v d \Gamma \Big)^{1+ \frac{1}{p+1}}.
\end{gather*}
Using these inequalities instead of (\ref{4eq4-5})-(\ref{4eq4-6})
and reasoning in the same way as above we conclude that
\[
\frac{d}{dt} \mathcal{L} (t) \leq -\frac{c}{\mathcal{L}
(0)^{\frac{1}{p+1}}}\mathcal{L} (t)^{1+ \frac{1}{p+1}} + 2NR^{2}(t)
E(0).
\]
Applying Lemma \ref{Lem4.3} again, we obtain
\[
\mathcal{L} (t) \leq \frac{c}{(1+t)^{p+1}} \mathcal{L}(0).
\]
Finally, from (\ref{3eq3-5}) we conclude
\[
E(t) \leq  \frac{c}{(1+t)^{p+1}}E(0),
\]
which completes the present proof.

\subsection*{Remark}We would like to mention that for
3 or more variables:u,v,..., the same procedure can be used to
obtain similar conclusions.


\subsection*{Acknowledgements} This research was written while the
first author was visiting, the Federal University of Par\'{a} -
UFPA-Brazil, during March 2003. He was partially supported by
CNPq-Bras\'{i}lia-Brazil under grant No. 300344/92-9. The authors
are thankful to the referee of this paper for valuable suggestions
which improved this paper. Also, the authors would like to thank
Professor Julio G. Dix for his valuable attention to our paper.


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\end{document}