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\markboth{\hfil Decay rates for a system of wave equations\hfil EJDE--2002/38}
{EJDE--2002/38\hfil Mauro de Lima Santos \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 38, 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)}
 \vspace{\bigskipamount} \\
 %
  Decay rates for solutions of a system of wave equations with memory
 %
\thanks{ {\em Mathematics Subject Classifications:} 35B40.
\hfil\break\indent
{\em Key words:} Asymptotic behavior, wave equation.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 18, 2002. Published May 6, 2002. \hfil\break\indent
Supported by CNPq/UFPa (Brazil)
} }
\date{}
%
\author{Mauro de Lima Santos}
\maketitle

\begin{abstract}
 The purpose of this article is to study the asymptotic 
 behavior of the solutions to a coupled system of wave 
 equations having integral convolutions as memory terms.
 We  prove that when the kernels of the convolutions 
 decay exponentially, the first and second order
 energy of the solutions  decay exponentially.
 Also we show that when the kernels decay polynomially, 
 these energies decay polynomially.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$
with smooth boundary $\Gamma$. In this domain, we consider the initial
boundary value problem
 \begin{gather}\label{1eq1-1}
u_{tt} - \Delta u + \int^{t}_{0}g_{1}(t-s) \Delta u(s) ds + \alpha (u-v)
= 0  \quad\mbox{in } {\Omega\times(0,\infty)},\\
\label{1eq1-2}
v_{tt} - \Delta v + \int^{t}_{0}g_{2}(t-s) \Delta v(s) ds - \alpha (u-v)
= 0  \quad\mbox{in }{\Omega\times(0,\infty)},\\
\label{1eq1-3}
u =v= 0  \quad\mbox{on }{\Gamma\times(0,\infty)},\\
\label{1eq1-4}
(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)) ,
\end{gather}
where $u$ and $v$ denote the transverse displacements of waves.
Here, $\alpha$ a non-negative constant and $g_i$ are
positive functions satisfy 
\begin{equation}\label{1eq1-5}
-c_{0}g_{i}(t)\leq g_{i}'(t)\leq -c_{1}g_{i}(t),
\quad 0\leq g_{i}''(t)\leq c_{2}g_{i}(t) \quad \mbox{ for } i=1,2,
\end{equation}
and for some positive constant $c_{j}$, $j=0,1,2$. We also assume that
\begin{equation}\label{1eq1-6}
\beta_{i}:= 1- \int^{\infty}_{0}g_{i}(s)ds>0, \quad \mbox{for } i=1,2.
\end{equation}
Dissipative coupled systems of the wave equations have been studied 
by several authors \cite{Aassila,Aassila-2,Alabau,Beyrath,Komornik-R}
whose results can be summarized as follows: 
 Komornik and Rao \cite{Komornik-R} studied a linear system of two
compactly coupled wave equations with boundary frictional damping in
both equations. They show the existence, regularity and stability of
the corresponding solutions. The stability results obtained in
\cite{Komornik-R} were extended by Aassila \cite{Aassila} for a coupled
system with weak frictional damping at the infinity.
In another work, Aassila \cite{Aassila-2} removes the dissipation of the
one equation and shows the strong asymptotic stability or the non uniform
stability for some particular cases depending on the coupling constant.
A similar coupled system with boundary frictional damping on only one of
the equations was studied by Alabau \cite{Alabau}. He shows the polynomial
decay of the corresponding strong solutions when the speed of wave propagation
of the both equations are the same. Some others coupled
systems with internal damping or with another coupling type can be found
in \cite{Alabau-C-K,Beyrath,Komornik-L,Soufyane}.

Our main result shows that the solution of system
(\ref{1eq1-1})-(\ref{1eq1-4}) decays uniformly in time,
with rates depending on the rate of decay of the kernel
of the convolutions. More precisely, the
solution decays exponentially to zero provided $g_{i}$
decays exponentially to zero. When $g_{i}$ decays polynomially,
we show that the corresponding solution also decays polynomially
to zero with the same rate of decay. The method used here is based on the
construction of suitable Lyapunov functionals, $\cal L$, satisfying
$$
\frac{d}{dt}{\cal L}(t)\leq - c_{1}{\cal L}(t)+c_{2}e^{- \gamma t} \quad
\mbox {or} \quad \frac{d}{dt}{\cal L}(t)\leq -c_{1}
{\cal L}(t)^{1+ \frac{1}{p}}+ \frac{c_{2}}{(1+t)^{p+1}}.
$$
for some positive constants $c_{1}$, $c_{2}$, $\gamma$ and $p>1$.
The notation we use in this paper is standard and can be
found in Lion's book \cite{Lions}. In the sequel, $c$
(some times $c_{1}, c_{2},\dots$) denote various positive constants
independent on $t$ and on the initial data. The organization of this paper
is as follows. In section 2 we establish a existence and regularity result.
In section 3 we prove the uniform rate of exponential decay. Finally in
section 4 we prove the uniform rate of polynomial decay.

\section{Existence and Regularity}

In this section we prove the existence and regularity of strong solutions of
the coupled system of wave equations with
memory. To simplify our analysis, we define the binary operator
$$
g \mathop{\Box} \nabla u(t)= \int^{t}_{0}g(t-s)\int_{\Omega}|\nabla u(t)
-\nabla u(s)|^{2}dx ds.
$$
With this notation we have the following statement.

\begin{lem}\label{Lem2.1} 
For $v \in C^{1}(0,T:H^{1}(\Omega))$,
\begin{eqnarray*}
\int_{\Omega}\int^{t}_{0}g(t-s)\nabla v ds \cdot \nabla v_{t} dx & = & -\frac{1}{2} g(t) \int_{\Omega}|\nabla v|^{2}dx +
\frac{1}{2}g' \mathop{\Box} \nabla v \\
&&- \frac{1}{2}\frac{d}{dt} \Big[ g \mathop{\Box} \nabla v 
- (\int^{t}_{0}g(s) ds)\int_{\Omega}|\nabla v|^{2}dx \Big].
\end{eqnarray*}
\end{lem}
The proof of this lemma follows by differentiating the term $g \mathop{\Box} \nabla v$.
The first order energy of system (\ref{1eq1-1})-(\ref{1eq1-4}) is 
\begin{eqnarray*}
E(t):=E(t;u,v)&=&\frac{1}{2}\int_{\Omega}|u_{t}|^{2}dx+\frac{1}{2}
\big(1-\int^{t}_{0}g_{1}(s)ds\big)\int_{\Omega}|\nabla
u|^{2}dx \\
&&+\frac{1}{2}\int_{\Omega}|v_{t}|^{2}dx+\frac{1}{2}
\big(1-\int^{t}_{0}g_{2}(s)ds\big)\int_{\Omega}|\nabla v|^{2}dx\\
&&+\frac{1}{2}g_{1}\mathop{\Box} \nabla u+\frac{1}{2}g_{2}\mathop{\Box}\nabla v
+ \frac{\alpha}{2}\int_{\Omega}|u-v|^{2}dx.
\end{eqnarray*}
The well-posedness of system (\ref{1eq1-1})-(\ref{1eq1-4}) is given by the
following theorem.

\begin{theorem}\label{teo2.1}
Assume $(u_{0},v_{0})\in (H^{2}(\Omega)\cap H^{1}_{0}(\Omega))^{2}$
 and $(u_{1},v_{1})\in (H^{1}_{0}(\Omega))^{2}$.
Then there exists only one strong solution $(u,v)$ to
(\ref{1eq1-1})-(\ref{1eq1-4}) satisfying
\begin{eqnarray*}
u,v \in L^{\infty}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap W^{1,\infty}(0,T;H^{1}_{0}(\Omega))\cap
W^{2,\infty}(0,T;L^{2}(\Omega)).
\end{eqnarray*}
\end{theorem}

\paragraph{Proof.}
Our starting point is to construct the Galerkin approximations $u^{m}$ and
$v^{m}$ of the solution. Let
$$
u^{m}(\cdot,t)=\sum^{m}_{j=1}h_{j,m}(t)w_{j}(\cdot), \quad v^{m}(\cdot,t)
=\sum^{m}_{j=1}f_{j,m}(t)w_{j}(\cdot)
$$
where the functions $f_{j,m}$ and $h_{j,m}$ are the solutions of the
approximated systems
\begin{eqnarray}\label{2eq2-1}
\int_{\Omega}u^{m}_{tt}w_{j}dx+\int_{\Omega}\nabla u^{m}\cdot \nabla w_{j}dx
&& \nonumber \\
-\int_{\Omega}\int^{t}_{0}g_{1}(t-s)\nabla u^{m}(s)ds \cdot \nabla w_{j}dx
+ \alpha \int_{\Omega}(u^{m}-v^{m})w_{j}dx&=&0 \\
\label{2eq2-2}
\int_{\Omega}v^{m}_{tt}w_{j}dx+\int_{\Omega}\nabla v^{m}\cdot \nabla w_{j}dx
&& \nonumber \\
-\int_{\Omega}\int^{t}_{0}g_{2}(t-s)\nabla
v^{m}(s)ds \cdot \nabla w_{j}dx - \alpha \int_{\Omega}(u^{m}-v^{m})w_{j}dx&=&0
\end{eqnarray}
with the initial conditions
$u^{m}(\cdot,0)=u_{0,m}$,  $u^{m}_{t}(\cdot,0)=u_{1,m}$,
$v^{m}(\cdot,0)=v_{0,m}$, and $v^{m}_{t}(\cdot,0)=v_{1,m}$,
where
$$\displaylines{
u_{0,m}=\sum^{m}_{j=1}\big\{\int_{\Omega}u_{0}w_{j}dx\big \}w_{j},\quad
u_{1,m}=\sum^{m}_{j=1}\big\{\int_{\Omega}u_{1}w_{j}dx\big\}w_{j}, \cr
v_{0,m}=\sum^{m}_{j=1}\big\{\int_{\Omega}v_{0}w_{j}dx\big\}w_{j},\quad
v_{1,m}=\sum^{m}_{j=1}\big\{\int_{\Omega}v_{1}w_{j}dx\big\}w_{j}.
}$$
The existence of the approximate solutions $u^{m}$ and $v^{m}$ are
guaranteed by standard results on ordinary differential
equations. Our next step is to show that the approximate solution
remains bounded for any $m>0$. To this end, let us
multiply equation (\ref{2eq2-1}) by $h_{j,m}'$ and (\ref{2eq2-2})
by $f_{j,m}'$. Summing up the product result in $j$ and
using Lemma \ref{Lem2.1} we arrive at
\begin{eqnarray*}
\frac{d}{dt}E(t;u^{m},v^{m})&=& - \frac{1}{2}g_{1}(t) \int_{\Omega}|\nabla u^{m}|^{2}dx -
\frac{1}{2}g_{2}(t)\int_{\Omega}|\nabla v^{m}|^{2}dx \\
&&+ \frac{1}{2}g_{1}'\mathop{\Box} \nabla u^{m}+\frac{1}{2}g_{2}'\mathop{\Box} \nabla v^{m}.
\end{eqnarray*}
Integrating from $0$ to $t$ the above relation follows that
$$
E(t;u^{m},v^{m}) \leq E(0;u^{m},v^{m}).
$$
>From our choice of $u_{0,m}$, $u_{1,m}$, $v_{0,m}$ and $v_{1,m}$ it follows that
\begin{eqnarray}\label{2eq2-3}
E(t;u^{m},v^{m})\leq c, \quad \forall t \in[0,T], \quad \forall m \in \mathbb{N}.
\end{eqnarray}
Next, we shall find an estimate for the second order energy. First, let us estimate the initial data $u^{m}_{tt}(0)$ and
$v^{m}_{tt}(0)$ in the $L^{2}$-norm. Letting $t\rightarrow0^{+}$ in the equations (\ref{2eq2-1}) and (\ref{2eq2-2}) and
multiplying the result by $h_{j,m}''(0)$ and $f_{j,m}''(0)$, respectively, we obtain
\begin{eqnarray}\label{2eq2-4}
\|u^{m}_{tt}(0)\|_{2}+\|v^{m}_{tt}(0)\|_{2}\leq c, \quad \forall m \in \mathbb{N}.
\end{eqnarray}
Differentiating the equations (\ref{2eq2-1}) and (\ref{2eq2-2}) with respect to
time, we obtain
\begin{eqnarray}\label{2eq2-5}
\int_{\Omega}u^{m}_{ttt}w_{j}dx+\int_{\Omega}\nabla u^{m}_{t} \cdot \nabla w_{j}dx + g_{1}(0)\int_{\Omega}\Delta
u^{m}_{0}w_{j}dx &&\nonumber \\
-\int_{\Omega}\int^{t}_{0}g_{1}'(t-s)\nabla u^{m}(s)ds \cdot \nabla w_{j}dx
+ \alpha \int_{\Omega}(u^{m}_{t}-
v^{m}_{t})w_{j}dx &=&0 \\
\label{2eq2-6}
\int_{\Omega}v^{m}_{ttt}w_{j}dx+\int_{\Omega}\nabla v^{m}_{t} \cdot \nabla w_{j}dx + g_{2}(0)\int_{\Omega}\Delta
v^{m}_{0}w_{j}dx &&\nonumber \\
-\int_{\Omega}\int^{t}_{0}g_{2}'(t-s)\nabla v^{m}(s)ds \cdot \nabla w_{j}dx - \alpha \int_{\Omega}(u^{m}_{t}-
v^{m}_{t})w_{j}dx &=&0.
\end{eqnarray}
Multiplying equation (\ref{2eq2-5}) by $h_{j,m}''$ and (\ref{2eq2-6}) 
by $f_{j,m}''$ and using similar arguments as above,
\begin{eqnarray*}
E(t;u^{m}_{t},v^{m}_{t}) \leq c, \quad \forall t \in [0,T], \quad \forall m \in \mathbb{N}.
\end{eqnarray*}
The rest of the proof is a matter of routine. \hfill $\diamondsuit$

\section{Exponential Decay}

In this section we study the asymptotic behavior of the solution of (\ref{1eq1-1})-(\ref{1eq1-4}). The point of departure of
this study is to establish the energy identities given in the next lemma.

\begin{lem}\label{Lem3.1}
Assume the initial data $\{(u_{0},v_{0}),(u_{1},v_{1})\}\in(H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))^{2}\times(H^{1}_{0}(\Omega))^{2}$. Then the solution of
(\ref{1eq1-1})-(\ref{1eq1-4}) satisfies
\begin{eqnarray*}
\frac{d}{dt}E(t;u,v)&=&-\frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx - \frac{1}{2}g_{2}(t)\int_{\Omega}|\nabla v|^{2}dx\\
&& + \frac{1}{2}g_{1}'\mathop{\Box}\nabla u + \frac{1}{2}g_{2}'\mathop{\Box} \nabla v, \\
\frac{d}{dt}E(t;u_{t},v_{t})&=&-\frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u_{t}|^{2}dx -
\frac{1}{2}g_{2}(t)\int_{\Omega}|\nabla v_{t}|^{2}dx + \frac{1}{2}g_{1}'\mathop{\Box} \nabla u_{t} \\
&& + \frac{1}{2}g_{2}'\mathop{\Box} \nabla v_{t} - g_{1}(t)\int_{\Omega}\Delta u_{0}u_{tt}dx - g_{2}(t)\int_{\Omega}\Delta
v_{0}v_{tt}dx.
\end{eqnarray*}
\end{lem}

\paragraph{Proof.}
Multiplying the equation (\ref{1eq1-1}) by $u_{t}$ and applying Green's formula, we get
\begin{eqnarray}\label{3eq3-1}
\frac{1}{2}\Big\{\int_{\Omega}|u_{t}|^{2}dx+\int_{\Omega}|\nabla u|^{2}dx\Big\}
&& \nonumber\\
-\int_{\Omega}\int^{t}_{0}g_{1}(t-s)\nabla u(s)ds \cdot \nabla u_{t}dx
+ \alpha \int_{\Omega}(u-v)u_{t}dx &=& 0.
\end{eqnarray}
Using Lemma \ref{Lem2.1} we obtain
\begin{align*}
&\int_{\Omega}\int^{t}_{0}g_{1}(t-s)\nabla u(s)ds \cdot \nabla u_{t}dx \\
&= - \frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx +
\frac{1}{2}g_{1}'\mathop{\Box} \nabla u
- \frac{1}{2}\frac{d}{dt}\Big[g_{1}\mathop{\Box} \nabla u
- \big(\int^{t}_{0}g_{1}(s)ds\big)\int_{\Omega}|\nabla u|^{2}dx \Big].
\end{align*}
Substituting the above identity into (\ref{3eq3-1}) we have
\begin{eqnarray}\label{3eq3-2}
\lefteqn{ \frac{1}{2}\frac{d}{dt}\Big\{\int_{\Omega}|u_{t}|^{2}dx
+\big(1-\int^{t}_{0}g_{1}(s)ds\big)\int_{\Omega}|\nabla
u|^{2}dx+g_{1}\mathop{\Box} \nabla u \Big\}
+\alpha\int_{\Omega}(u-v)u_{t}dx }\nonumber\\
&=&- \frac{1}{2}g_{1}(t) \int_{\Omega}|\nabla u|^{2}dx
+ \frac{1}{2}g_{1}'\mathop{\Box} \nabla u. \hspace{6cm}
\end{eqnarray}
Similarly we have
\begin{eqnarray}\label{3eq3-3}
\lefteqn{ \frac{1}{2}\frac{d}{dt}\Big\{\int_{\Omega}|v_{t}|^{2}dx
+\big(1-\int^{t}_{0}g_{2}(s)ds\big)\int_{\Omega}|\nabla
v|^{2}dx+g_{2}\mathop{\Box} \nabla v \Big\}
-\alpha\int_{\Omega}(u-v)v_{t}dx}\nonumber\\
&=&- \frac{1}{2}g_{2}(t) \int_{\Omega}|\nabla v|^{2}dx
+ \frac{1}{2}g_{2}'\mathop{\Box} \nabla v. \hspace{6cm}
\end{eqnarray}
Summing (\ref{3eq3-2}) and (\ref{3eq3-3}) it follows the first identity of Lemma. Differentiating the equation (\ref{1eq1-1})
with respect to the time we get
\begin{equation}\label{3eq3-4}
u_{ttt}-\Delta u_{t}+g_{1}(0)\Delta u + \int^{t}_{0}g_{1}'(t-s)\Delta u(s)ds
+ \alpha (u_{t}-v_{t}) = 0.
\end{equation}
Performing an integration by parts in the convolution term, we find that
$$
u_{ttt}-\Delta u_{t}+\int^{t}_{0}g_{1}(t-s)\Delta u_{t}(s)ds +\alpha(u_{t}-v_{t})= - g_{1}(t)\Delta u_{0}.
$$
Taking $\varphi=u_{t}$ we may use the same reasoning as above we have
\begin{eqnarray*}
\lefteqn{\frac{1}{2}\frac{d}{dt}\Big\{\int_{\Omega}|u_{tt}|^{2}dx
+\big(1-\int^{t}_{0}g_{1}(s)ds\big)\int_{\Omega}|\nabla
u_{t}|^{2}dx+g_{1}\mathop{\Box} \nabla u_{t} \Big\} }\\
\lefteqn{+\alpha\int_{\Omega}(u_{t}-v_{t})u_{tt}dx }\\
&=& - \frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u_{t}|^{2}dx
+ \frac{1}{2}g_{1}'\mathop{\Box}
\nabla u_{t}-g_{1}(t)\int_{\Omega}\Delta u_{0}u_{tt}dx.
\end{eqnarray*}
Similarly we obtain
\begin{eqnarray*}
\lefteqn{\frac{1}{2}\frac{d}{dt}\Big\{\int_{\Omega}|v_{tt}|^{2}dx
+\big(1-\int^{t}_{0}g_{2}(s)ds\big)\int_{\Omega}|\nabla
v_{t}|^{2}dx+g_{2}\mathop{\Box} \nabla v_{t} \Big\} }\\
\lefteqn{-\alpha\int_{\Omega}(u_{t}-v_{t})v_{tt}dx}\\
&=& - \frac{1}{2}g_{2}(t)\int_{\Omega}|\nabla v_{t}|^{2}dx
+ \frac{1}{2}g_{2}'\mathop{\Box} \nabla v_{t}-g_{2}(t)\int_{\Omega}\Delta v_{0}v_{tt}dx.
\end{eqnarray*}
Summing the two above identities it follows the conclusion of Lemma.
\hfill $\diamondsuit$ \smallskip

To prove the exponential decay of the solutions, we define the following
functionals:
\begin{eqnarray*}
K(t;u,v)&=&\frac{1}{2}\Big\{\int_{\Omega}|u_{tt}|^{2}dx
+\int_{\Omega}|v_{tt}|^{2}dx+\int_{\Omega}|\nabla u_{t}|^{2}dx\\
&&+ \int_{\Omega}|\nabla v_{t}|^{2}dx+2\int_{\Omega}\int^{t}_{0}f_{1}(t-s)
\nabla u(s)ds \cdot \nabla u_{t}dx \\
&&+2\int_{\Omega}\int^{t}_{0}f_{2}(t-s)\nabla v(s)ds \cdot \nabla v_{t}dx
\Big\},
\end{eqnarray*}
$$
I_{1}(t;u)=\int_{\Omega}u_{tt}u_{t}dx + \frac{1}{2}g_{1}'\mathop{\Box} \nabla u - g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx,
$$
$$
I_{2}(t;v)=\int_{\Omega}v_{tt}v_{t}dx + \frac{1}{2}g_{2}'\mathop{\Box} \nabla v - g_{2}(t)\int_{\Omega}|\nabla v|^{2}dx.
$$
where $f_{i}(t)=g_{i}(0)g_{i}(t)+g_{i}'(t)$, $i=1,2$.

\begin{lem}\label{Lem3.2}
Under the hypothesis of Lemma \ref{Lem3.1},
\begin{align*}
\frac{d}{dt}\big\{&K(t;u,v)+\frac{2}{3}g_{1}(0)I_{1}(t;u)
+\frac{2}{3}g_{2}(0)I_{2}(t;v)\big\}\\
\leq&-\frac{g_{1}(0)}{3}\big\{\int_{\Omega}|u_{tt}|^{2}dx
+\int_{\Omega}|\nabla u_{t}|^{2}dx\big\}
-\frac{g_{2}(0)}{3}\big\{\int_{\Omega}|v_{tt}|^{2}dx
+\int_{\Omega}|\nabla v_{t}|^{2}dx\big\}\\
&+\big\{\frac{3(c_{0}g_{1}(0)+c_{2})^{2}}{2}
+\frac{c_{0}g_{1}(0)}{3}\}g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx \\
&+\big\{\frac{3(c_{0}g_{1}(0)+c_{2})^{2}}{2}+\frac{c_{0}g_{2}(0)}{3}
\big\}g_{2}(t)\int_{\Omega}|\nabla v|^{2}dx\\
&+\big\{\frac{3(c_{0}g_{1}(0)+c_{2})^{2}}{2g_{1}(0)}
+\frac{c_{2}g_{1}(0)}{3}\big\}g_{1}\mathop{\Box}\nabla u \\
&+\big\{\frac{3(c_{0}g_{2}(0)+c_{2})^{2}}{2g_{2}(0)}
+\frac{c_{2}g_{2}(0)}{3}\big\}g_{2}\mathop{\Box}\nabla v
-\frac{2}{3}g_{1}(0)\alpha\int_{\Omega}(u_{t}-v_{t})u_{t}dx \\
&+\frac{2}{3}g_{2}(0)\alpha\int_{\Omega}(u_{t}-v_{t})v_{t}dx.
\end{align*}
\end{lem}

\paragraph{Proof.}
Substitution of the term $\Delta u$ given by equation (\ref{1eq1-1})
into (\ref{3eq3-4}) yields
\begin{eqnarray}\label{3eq3-5}
u_{ttt}-\Delta u_{t}+g_{1}(0)u_{tt}-g_{1}(0)\int^{t}_{0}g_{1}(t-s)\Delta u(s)ds
&&\nonumber \\
-g_{1}(0)\alpha(u-v)+\int^{t}_{0}g_{1}'(t-s)\Delta u(s)ds+\alpha(u_{t}-v_{t})
&=&0
\end{eqnarray}
Multiplying the above equation by $u_{tt}$ and integrating over $\Omega$, we get
\begin{eqnarray*}
\frac{1}{2}\frac{d}{dt}\big\{\int_{\Omega}|u_{tt}|^{2}dx+\frac{1}{2}
\int_{\Omega}|\nabla u_{t}|^{2}dx\big\}+g_{1}(0)\int_{\Omega}|u_{tt}|^{2}dx &&\\
+\int_{\Omega}\int^{t}_{0}f_{1}(t-s)\nabla u(s)ds\cdot \nabla u_{tt}dx &&\\
-g_{1}(0)\alpha \int_{\Omega}(u-v)u_{tt}dx
+\alpha \int_{\Omega}(u_{t}-v_{t})u_{tt}dx&=&0
\end{eqnarray*}
and since
\begin{eqnarray*}
\lefteqn{\int_{\Omega}\int^{t}_{0}f_{1}(t-s)\nabla u(s)ds\cdot \nabla u_{tt}dx}\\
&=&\frac{d}{dt}\big\{\int_{\Omega}\int^{t}_{0}f_{1}(t-
s)\nabla u(s)ds\cdot \nabla u_{t}dx\big\} \\
&&-\int_{\Omega}\int^{t}_{0}f_{1}'(t-s)\nabla u(s)ds\cdot \nabla u_{t}dx-f_{1}(0)\int_{\Omega}\nabla u \cdot \nabla u_{t}dx
\end{eqnarray*}
we obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt}&\big\{\int_{\Omega}|u_{tt}|^{2}dx+\int_{\Omega}|\nabla
u_{t}|^{2}dx+2\int_{\Omega}\int^{t}_{0}f_{1}(t-s)\nabla u(s)ds \cdot
\nabla u_{t}dx \big\} \\
=&-g_{1}(0)\int_{\Omega}|u_{tt}|^{2}dx-\int_{\Omega}\int^{t}_{0}f_{1}'(t-s)
[\nabla u(s)-\nabla u(t)]ds \cdot \nabla u_{t}dx \\
&-f_{1}(t)\int_{\Omega}\nabla u \cdot \nabla u_{t}dx+g_{1}(0)\alpha
\int_{\Omega}(u-v)u_{tt}dx -\alpha \int_{\Omega}(u_{t}-v_{t})u_{tt}dx.
\end{align*}
Making use of the inequality
\begin{eqnarray*}
\lefteqn{\Big|\int_{\Omega}\int^{t}_{0}g_{i}(t-s)[\nabla \phi(s)-\nabla \phi(t)]ds
\cdot \nabla \varphi dx \Big|}\\
&\leq&\Big(\int_{\Omega}|\nabla \varphi|^{2}\Big)^{1/2}
\Big(\int^{t}_{0}g_{i}(s)ds\Big)^{1/2}(g_{i}\mathop{\Box}\nabla \phi)
^{1/2}
\end{eqnarray*}
we conclude
\begin{equation} \label{3eq3-6}
\begin{aligned}
\frac{1}{2}&\frac{d}{dt}\left\{\int_{\Omega}|u_{tt}|^{2}dx+\int_{\Omega}
|\nabla u_{t}|^{2}dx+2\int_{\Omega}\int^{t}_{0}f_{1}(t-s)\nabla u(s)ds\cdot
\nabla u_{t}dx\right\} \\
\leq&-g_{1}(0)\int_{\Omega}|u_{tt}|^{2}dx+\frac{g_{1}(0)}{3}\int_{\Omega}|
\nabla u_{t}|^{2}dx+\frac{3}{2}g_{1}(t)(g_{1}(0)+c_{0})^{2}\int_{\Omega}|
\nabla u|^{2}dx \\
&+\frac{3}{2}\frac{(c_{0}g_{1}(0)+c_{2})^{2}}{g_{1}(0)}g_{1}\mathop{\Box}\nabla u
+g_{1}(0)\alpha \int_{\Omega}(u-v)u_{tt}dx
-\alpha \int_{\Omega}(u_{t}-v_{t})u_{tt}dx.
\end{aligned} \end{equation}
Similarly we have
\begin{equation}\label{3eq3-7} \begin{aligned}
\frac{1}{2}&\frac{d}{dt}\left\{\int_{\Omega}|v_{tt}|^{2}dx+\int_{\Omega}|
\nabla v_{t}|^{2}dx+2\int_{\Omega}\int^{t}_{0}f_{2}(t-s)\nabla v(s)ds\cdot
\nabla v_{t}dx\right\} \\
\leq&-g_{2}(0)\int_{\Omega}|v_{tt}|^{2}dx+\frac{g_{2}(0)}{3}
\int_{\Omega}|\nabla v_{t}|^{2}dx+\frac{3}{2}g_{2}(t)(g_{2}(0)
+c_{0})^{2}\int_{\Omega}|\nabla v|^{2}dx \\
&+\frac{3}{2}\frac{(c_{0}g_{2}(0)+c_{2})^{2}}{g_{2}(0)}g_{2}\mathop{\Box}\nabla v
-g_{2}(0)\alpha \int_{\Omega}(u-v)v_{tt}dx
+\alpha \int_{\Omega}(u_{t}-v_{t})v_{tt}dx.
\end{aligned}\end{equation}
Summing the inequalities (\ref{3eq3-6}) and (\ref{3eq3-7}) we obtain
\begin{eqnarray}\label{3eq3-8}
\frac{d}{dt}K(t;u,v)&\leq&-g_{1}(0)\int_{\Omega}|u_{tt}|^{2}dx-
g_{2}(0)\int_{\Omega}|v_{tt}|^{2}dx+\frac{g_{1}(0)}{3}\int_{\Omega}|\nabla u_{t}|^{2}dx \nonumber \\
&&+\frac{g_{2}(0)}{3}\int_{\Omega}|\nabla v_{t}|^{2}dx+\frac{3}{2}g_{1}(t)(g_{1}(0)+c_{0})^{2}\int_{\Omega}|\nabla u|^{2}dx
\nonumber \\
&&+\frac{3}{2}g_{2}(t)(g_{2}(0)+c_{0})^{2}\int_{\Omega}|\nabla
v|^{2}dx+\frac{3}{2}\frac{(c_{0}g_{1}(0)+c_{2})^{2}}{g_{1}(0)}g_{1}\mathop{\Box}\nabla u \nonumber \\
&&+\frac{3}{2}\frac{(c_{0}g_{2}(0)+c_{2})^{2}}{g_{2}(0)}g_{2}\mathop{\Box}\nabla v+g_{1}(0)\alpha \int_{\Omega}(u-v)u_{tt}dx
\nonumber \\
&&-\alpha \int_{\Omega}(u_{t}-v_{t})u_{tt}dx-g_{2}(0)\alpha \int_{\Omega}(u-v)v_{tt}dx \nonumber \\
&&+\alpha \int_{\Omega}(u_{t}-v_{t})v_{tt}dx.
\end{eqnarray}
On the other hand, multiplying equation (\ref{3eq3-4}) by $u_{t}$, integrating over $\Omega$ and using Lemma \ref{Lem2.1} we
get
\begin{eqnarray*}
\frac{d}{dt}I_{1}(t;u)&=&-\int_{\Omega}|\nabla u_{t}|^{2}dx-\frac{1}{2}g_{1}'(t)\int_{\Omega}|\nabla u|^{2}dx \\
&&+\frac{1}{2}g_{1}''\mathop{\Box}\nabla u-\alpha \int_{\Omega}(u_{t}-v_{t})u_{t}dx.
\end{eqnarray*}
>From hypothesis (\ref{1eq1-5}) we obtain
\begin{eqnarray*}
\frac{d}{dt}I_{1}(t;u)&\leq&-\int_{\Omega}|\nabla u_{t}|^{2}dx+\frac{c_{0}}{2}g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx \\
&&+\frac{c_{2}}{2}g_{1}\mathop{\Box} \nabla u - \alpha \int_{\Omega}(u_{t}-v_{t})u_{t}dx.
\end{eqnarray*}
Similarly, we get
\begin{eqnarray*}
\frac{d}{dt}I_{2}(t;v)&\leq&-\int_{\Omega}|\nabla v_{t}|^{2}dx+\frac{c_{0}}{2}g_{2}(t)\int_{\Omega}|\nabla v|^{2}dx \\
&&+\frac{c_{2}}{2}g_{2}\mathop{\Box} \nabla v + \alpha \int_{\Omega}(u_{t}-v_{t})v_{t}dx.
\end{eqnarray*}
Finally, going back to (\ref{3eq3-8}) and from the last two inequalities
our conclusion follows. \hfill$\diamondsuit$ \smallskip

Let us introduce the functional
$$
J(t;\varphi)=\int_{\Omega}\varphi_{t}\varphi dx.
$$

\begin{lem}\label{Lem3.3}
Under hypothesis of the Lemma \ref{Lem3.1} we have
$$
\frac{d}{dt}J(t;u)\leq\alpha_{0}\int_{\Omega}|\nabla u_{t}|^{2}dx-\frac{\beta_{1}}{2}\int_{\Omega}|\nabla
u|^{2}dx+\frac{1}{2\beta_{1}}\left(\int^{t}_{0}g_{1}(s)ds\right)g_{1}\mathop{\Box} \nabla u,
$$
$$
\frac{d}{dt}J(t;v)\leq\alpha_{0}\int_{\Omega}|\nabla v_{t}|^{2}dx-\frac{\beta_{2}}{2}\int_{\Omega}|\nabla
v|^{2}dx+\frac{1}{2\beta_{2}}\left(\int^{t}_{0}g_{2}(s)ds\right)g_{2}\mathop{\Box} \nabla v.
$$
\end{lem}

The proof of this Lemma is similar the proof of the Lemma \ref{Lem3.2},
for this reason we omit it here.

Let us introduce the functionals
\begin{eqnarray*}
{\cal L}(t)&=&N_{1}E(t;u,v)+N_{2}E(t;u_{t},v_{t})+K(t;u,v)+\frac{2g_{1}(0)}{3}I_{1}(t;u) \\
&&+\frac{2g_{2}(0)}{3}I_{2}(t;v)+\frac{g_{1}(0)}{12\alpha_{0}}J(t;u)+\frac{g_{2}(0)}{12\alpha_{0}}J(t;v),
\end{eqnarray*}
\begin{eqnarray*}
{\cal N}(t)&=&\int_{\Omega}|\nabla u|^{2}dx+\int_{\Omega}|\nabla v|^{2}dx+\int_{\Omega}|\nabla
u_{t}|^{2}dx+\int_{\Omega}|\nabla v_{t}|^{2}dx+\int_{\Omega}|u_{tt}|^{2}dx \\
&&+\int_{\Omega}|v_{tt}|^{2}dx+g_{1}\mathop{\Box}\nabla u +g_{2}\mathop{\Box}\nabla v+g_{1}\mathop{\Box}\nabla u_{t} 
+g_{2}\mathop{\Box}\nabla v_{t}.
\end{eqnarray*}
It is not difficult to see that there exist positive constants $q_{0}$
 and $q_{1}$ for which
\begin{eqnarray}\label{3eq3-9}
q_{0}{\cal N}(t)\leq {\cal L}(t) \leq q_{1} {\cal N}(t).
\end{eqnarray}
We will show later that the functional $\cal L$ satisfies the inequality of
the following Lemma.

\begin{lem}\label{Lem3.4}
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{lem}

\paragraph{Proof.}
Suppose that $\gamma_0<\gamma_1$ and define
$$
F(t):=f(t)+\frac{c_0}{\gamma_1-\gamma_0}e^{-\gamma_1 t}.
$$
Then
$$ F'(t)=f'(t)-\frac{\gamma_1c_0}{\gamma_1-\gamma_0}e^{-\gamma_1
t} \leq -\gamma_0F(t).
$$
Integrating from $0$ to $t$ we arrive to
$$
F(t)\leq F(0)e^{-\gamma_0 t}\quad\Rightarrow\quad f(t)\leq
\big(f(0) + \frac{c_0}{\gamma_1-\gamma_0}\big)e^{-\gamma_0 t}.
$$
Now, we shall assume that $\gamma_0\geq \gamma_1$. In this
conditions we get
$$ f'(t)\leq -\gamma_1f(t) +c_0e^{-\gamma_1
t}\quad \Rightarrow\quad [e^{\gamma_1 t}f(t)]'\leq c_0.
$$
Integrating from $0$ to $t$ we obtain
$$ f(t)\leq (f(0) + c_0t)e^{-\gamma_1 t}.
$$
Since $t\leq (\gamma_1-\epsilon)e^{(\gamma_1-\epsilon)t}$
for any $0<\epsilon<\gamma_1$ we conclude that
$$ f(t)\leq [f(0) + c_0(\gamma_1-\epsilon)]e^{-\epsilon t}.
$$
 This completes the proof.  \hfill$\diamondsuit$\smallskip

Now we shall show the main result of this section.

 \begin{theorem}\label{teo3.1}
 Let us suppose that the initial data
 $(u_{0},v_{0}) \in [H^{1}_{0}(\Omega)]^{2}$ and
$(u_{1},v_{1})\in [L^{2}(\Omega)]^{2}$ and that the kernel $g_{i}$
satisfies the conditions
(\ref{1eq1-5}) and (\ref{1eq1-6}). Then there exist positive constants
$k_{1}$ and $k_{2}$ such that
\begin{eqnarray*}
E(t;u,v)+E(t;u_{t},v_{t}) \leq k_{1}(E(0;u,v)+E(0;u_{t},v_{t})) e^{- k_{2}t},
\end{eqnarray*}
for all $t\ge0$.
\end{theorem}

\paragraph{Proof.}
We shall prove this result for strong solutions, that is, for solutions
with initial data $(u_{0},v_{0})\in \left[
H^{2}(\Omega) \cap H^{1}_{0}(\Omega)\right]^{2}$ and
$(u_{1},v_{1})\in [H^{1}_{0}(\Omega)]^{2}$. Our conclusion follows by
standard density arguments. From the Lemmas \ref{Lem3.1}, \ref{Lem3.2} and \ref{Lem3.3} we get
$$
\frac{d}{dt}{\cal L}(t)\leq -c_{1}{\cal N}(t)+c_{2}R^{2}(t)
$$
where $R(t)=g_{1}(t)+g_{2}(t)$. Using the exponential decay of $g_1$ and
$g_2$ and Lemma \ref{Lem3.4} we conclude
$$
{\cal L}(t) \leq \{{\cal L}(0)+c\}e^{-k_{2}t}
$$
for all $t\ge0$. The conclusion of Theorem follows from (\ref{3eq3-9}).
\hfill $\diamondsuit$ \smallskip

As a consequence of Theorem \ref{teo3.1} we have that the first order energy also decays exponentially. We summarize this
result in the following Corollary.

\begin{cor}\label{Cor3.1}
Under the hypotheses of Theorem \ref{teo3.1}, we have that there exist
positive constants $c$ and $k_{1}$ such that
$$
E(t;u,v)\leq c E(0;u,v)e^{-k_{2}t}, \quad \forall t\ge 0.
$$
\end{cor}

\section{Polynomial rate of decay}

In this section we assume that the kernel $g_{i}$ decays polynomially to
zero as time goes to infinity. That is, instead
of hypothesis (\ref{1eq1-5}) we consider
\begin{gather}\label{4eq4-1}
-c_{0}g_{i}^{1+\frac{1}{p}}(t)\leq g_{i}'(t)\leq
-c_{1}g_{i}^{1+\frac{1}{p}}(t), \quad 0 \leq g_{i}''(t)\leq
c_{2}g_{i}^{1+\frac{1}{p}}(t), \\
\label{4eq4-2}
\alpha_{i}:=\int^{\infty}_{0}g_{i}^{1-\frac{1}{p}}(s)ds< \infty
\end{gather}
for some $p>1$ and for $i=1,2$.
The following lemmas will play an important role in the sequel.

\begin{lem}\label{Lem4.1}
Suppose that $g$ and $h$ are continuous functions satisfying $g \in L^{1+\frac{1}{q}}(0,\infty)\cap L^{1}(0,\infty)$ and
$g^{r}\in L^{1}(0,\infty)$ for some $0\leq r<1$. Then
\begin{eqnarray*}
\lefteqn{\int^{t}_{0}|g(t-s)h(s)|ds }\\
&&\leq \big\{\int^{t}_{0}|g(t-s)|^{1+\frac{1-r}{q}}|h(s)|ds\big\}
^{\frac{q}{q+1}}\big\{\int^{t}_{0}|g(t-s)|^{r}|h(s)|ds
\big\}^{\frac{1}{q+1}}.
\end{eqnarray*}
\end{lem}

\paragraph{Proof.}
Without loss of generality we can suppose that $g,h \ge 0$.
Note that for any fixed $t$ we have
$$
\int^{t}_{0}g(t-s)h(s)ds=\lim_{\|\Delta s_{i}\|\rightarrow0}\sum^{m}_{i=1}g(t-s_{i})h(s_{i})\Delta s_{i}.
$$
Letting $I^{r}_{m}:=\sum^{m}_{j=1}g^{r}(t-s_{j})h(s_{j})\Delta s_{j}$,
we may write
$$
\sum^{m}_{i=1}g(t-s_{i})h(s_{i})\Delta s_{i}=\sum^{m}_{i=1}\varphi_{i}\theta_{i},
$$
where
$$
\varphi_{i}=(g^{1-r}(t-s_{i})I^{r}_{m}),\quad\theta_{i}
=\big(\frac{g^{r}(t-s_{i})h(s_{i})\Delta s_{i} }{I^{r}_{m}}\big).
$$
Since the function $F(z):=|z|^{1+\frac{1}{q}}$ is convex, it follows that
$$
F\big(\sum^{m}_{i=1}g(t-s_{i})h(s_{i})\Delta s_{i}\big)
=F\big(\sum^{m}_{i=1}\varphi_{i}\theta_{i}\big)\leq
\sum^{m}_{i=1}\theta_{i}F(\varphi_{i}),
$$
so, we have
\begin{eqnarray}\label{4eq4-3}
\big\{\sum^{m}_{i=1}g(t-s_{i})h(s_{i})\Delta s_{i}\big\}^{1+\frac{1}{q}}\leq
|I^{r}_{m}|^{\frac{1}{q}}\sum^{m}_{i=1}g^{1+\frac{1-r}{q}}(t-s_{i})h(s_{i})\Delta s_{i}.
\end{eqnarray}
In view of
$$
\lim_{\|\Delta s_{i}\|\rightarrow0}I^{r}_{m}=\int^{t}_{0}g^{r}(t-s)h(s)ds,
$$
letting $\|\Delta s_{i}\|\rightarrow 0$ in (\ref{4eq4-3}), we get
$$
\big\{\int^{t}_{0}g(t-s)h(s)ds\big\}^{1+\frac{1}{q}}\leq
 \big\{\int^{t}_{0}g^{r}(t-s)h(s)ds\big\}^{1+q}
\big\{\int^{t}_{0}g^{1+\frac{1-r}{q}}(t-s)h(s)ds \big\},
$$
from which our result follows.\hfill $\diamondsuit$

\begin{lem}\label{Lem4.2}
Let $w\in C(0,T;H^{1}_{0}(\Omega))$ and $g$ be a continuous function
satisfying hypotheses (\ref{4eq4-1})-(\ref{4eq4-2}). Then for $0<r<1$ we have
$$
g\mathop{\Box} \nabla w \leq 2\big\{\int^{t}_{0}g^{r}ds\|w\|_{C(0,T;H^{1}_{0})}\big\}
^{\frac{1}{1+(1-r)p}}\{g^{1+\frac{1}{p}}\mathop{\Box}
\nabla w\}^{\frac{(1-r)p}{1+(1-r)p}},
$$
while for $r=0$ we get
$$
g\mathop{\Box} \nabla w \leq 2 \big\{\int^{t}_{0}\|w(s)\|^{2}_{H^{1}_{0}}ds +
t\|w(t)\|^{2}_{H^{1}_{0}}\big\}^{\frac{1}{p+1}}\{g^{1+\frac{1}{p}}\mathop{\Box} \nabla w\}^{\frac{p}{1+p}}.
$$
\end{lem}

\paragraph{Proof.}
>From the hypotheses on $w$ and Lemma \ref{Lem4.1} we get
\begin{eqnarray}\label{4eq4-4}
g\mathop{\Box} \nabla w &=& \int^{t}_{0}g(t-s)h(s)ds \nonumber \\
&\leq& \Big\{\int^{t}_{0}g^{r}(t-s)h(s)ds\Big\}^{\frac{1}{1+p(1-r)}}
\Big\{\int^{t}_{0}g^{1+\frac{1}{p}}(t-s)h(s)ds\Big\}^{\frac{(1-r)p}{1+p(1-r)}}
\nonumber \\
&\leq& \big\{g^{r}\mathop{\Box} \nabla w\big\}^{\frac{1}{1+p(1-r)}}
\big\{g^{1+\frac{1}{p}}\mathop{\Box}\nabla w\big\}^{\frac{(1-r)p}{1+p(1-r)}}
\end{eqnarray}
where
$$
h(s)=\int_{\Omega}|\nabla w(t)-\nabla w(s)|^{2}ds.
$$
For $0<r<1$ we have
$$
g^{r}\mathop{\Box} \nabla w = \int_{\Omega}\int^{t}_{0}g^{r}(t-s)|\nabla w(t)
-\nabla w(s)|^{2}dsdx
\leq 4 \int^{t}_{0}g^{r}(s)ds\|w\|^{2}_{C(0,T;H^{1}_{0})},
$$
from which the first inequality of Lemma \ref{Lem4.2} follows. To prove the last part, let us take $r=0$ in Lemma
\ref{Lem4.1} to get
\begin{eqnarray*}
1\mathop{\Box} \nabla w &=&\int_{\Omega}\int^{t}_{0}|\nabla w(t)-\nabla w(s)|^{2}dsdx \\
&&\leq 2t\|w(t)\|^{2}_{H^{1}_{0}}+2\int^{t}_{0}\|w(s)\|^{2}_{H^{1}_{0}}ds.
\end{eqnarray*}
Substitution of the above inequality into (\ref{4eq4-4}) yields the second
inequality. The proof is now complete.\hfill $\diamondsuit$
\smallskip

>From the above Lemma, for $0<r<1$, we get
\begin{equation}\label{4eq4-5}
g\mathop{\Box} \nabla w \leq c_{0}(g^{1+\frac{1}{p}}
\mathop{\Box} \nabla w)^{\frac{(1-r)p}{1+p(1-r)}},
\end{equation}

\begin{lem}\label{Lem4.3}
Let $f$ be  a non-negative $C^{1}$ function satisfying
\begin{eqnarray*}
f'(t) \leq - k_{0}[f(t)]^{1+\frac{1}{p}} + \frac{k_1}{(1+t)^{p+1}},
\end{eqnarray*}
for some positive constants $k_{0}$, $k_{1}$ and $p>1$.
Then there exists a positive constant $c_{1}$ such that
\begin{eqnarray*}
f(t) \leq c_{1}\frac{pf(0)+2k_{1}}{(1+t)^{p}}.
\end{eqnarray*}
\end{lem}

\paragraph{Proof.}
Let $h(t):=\frac{2k_{1}}{p(1+t)^{p}}$ and $F(t):=f(t)+h(t)$. Then
\begin{eqnarray*}
F'(t)&=&f'(t)-\frac{2k_{1}}{(1+t)^{p+1}} \\
&\leq&-k_{0}[f(t)]^{1+\frac{1}{p}}-\frac{k_{1}}{(1+t)^{p+1}} \\
&\leq&-k_{0}\Big\{[f(t)]^{1+\frac{1}{p}}+\frac{p^{1+\frac{1}{p}}}
{2k_{0}k_{1}^{\frac{1}{p}}}[h(t)]^{1+\frac{1}{p}}\Big\}.
\end{eqnarray*}
>From which it follows that there exists a positive constant $c_{1}$ such that
$$
F'(t)\leq -c_{1}\{[f(t)]^{1+\frac{1}{p}}+[h(t)]^{1+\frac{1}{p}}\}\leq -c_{1}[F(t)]^{1+\frac{1}{p}},
$$
which gives the required inequality. $\diamondsuit$

\begin{theorem}\label{Teo4.1} Assume  that
$(u_{0},v_{0}) \in [H^{1}_{0}(\Omega)]^{2}$, that
$(u_{1},v_{1})\in [L^{2}(\Omega)]^{2}$, and that
(\ref{1eq1-6}), (\ref{4eq4-1}), (\ref{4eq4-2}) hold.
Then any solution $(u,v)$ of system (\ref{1eq1-1})-(\ref{1eq1-4}) satisfies
\begin{eqnarray*}
E(t;u,v)+E(t;u_{t},v_{t}) \leq c\big\{E(0;u,v)+E(0;u_{t},v_{t})\big\}(1+t)^{-p}
\end{eqnarray*}
for $p>1$.
\end{theorem}

\paragraph{Proof.}
We shall prove this result for strong solutions, that is, for solutions
with initial data $(u_{0},v_{0})\in \left(
H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \right)^{2}$ and $(u_{1},v_{1})\in [H^{1}_{0}(\Omega)]^{2}$. Our conclusion will follow
by standard density arguments. From Lemma \ref{Lem3.1} and hypotheses
(\ref{4eq4-1}) and (\ref{4eq4-2}), we get
\begin{eqnarray}\label{4eq4-6}
\frac{d}{dt}E(t;u,v)&\leq&-\frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u|^{2}dx-g_{2}(t)\int_{\Omega}|\nabla v|^{2}dx \nonumber
\\
&&-\frac{c_{1}}{2}g^{1+\frac{1}{p}}_{1}\mathop{\Box} \nabla u - \frac{c_{1}}{2}g^{1+\frac{1}{p}}_{2}\mathop{\Box} \nabla v, 
\\
\label{4eq4-7}
\frac{d}{dt}E(t;u_{t},v_{t})&\leq&-\frac{1}{2}g_{1}(t)\int_{\Omega}|\nabla u_{t}|^{2}dx-g_{2}(t)\int_{\Omega}|\nabla
v_{t}|^{2}dx \nonumber \\
&&-\frac{c_{1}}{2}g^{1+\frac{1}{p}}_{1}\mathop{\Box} \nabla u_{t} - \frac{c_{1}}{2}g^{1+\frac{1}{p}}_{2}\mathop{\Box} \nabla 
v_{t} \nonumber \\
&&-g_{1}(t)\int_{\Omega}\Delta u_{0}u_{tt}dx-g_{2}(t)\int_{\Omega}\Delta v_{0}v_{tt}dx,
\end{eqnarray}
for some $c_1>0$. Since the inequalities
\begin{multline}\label{4eq4-8}
\Big|\int_{\Omega}\int^{t}_{0}f_{i}(t-s)[\nabla \varphi(s)
-\nabla\varphi(t)]ds\cdot \nabla\varphi_{t}\Big| \\
\leq(c_{0}g_{i}(0)+c_{2})\Big\{ \epsilon_{i}
\int_{\Omega}|\nabla\varphi_{t}|^{2}dx+\frac{(g_{i}(0))^{1/p}}{4\epsilon_i}
\big(\int^{t}_{0}g_{i}(s)ds\big)g_{i}^{
1+\frac{1}{p}}\mathop{\Box} \nabla \varphi\Big\},
\end{multline}
\begin{multline} \label{4eq4-9}
\Big|f_{i}(t)\int_{\Omega}\nabla\varphi\cdot\nabla\varphi_{t}dx\Big| \\
\leq g_{i}(t)\{c_{0}[g_{i}(0)]^{1/p}+g_{i}(0)\}
\Big\{\lambda_{i}\int_{\Omega}|\nabla
\varphi_{t}|^{2}dx+\frac{1}{4\lambda_i}\int_{\Omega}|\nabla \varphi|^{2}dx\Big\}
\end{multline}
hold for any $\epsilon_{i}>0$ and $\lambda_{i}>0$ $(i=1,2)$.
It follows of Lemma \ref{Lem3.2} that
\begin{eqnarray}\label{4eq4-10}
\lefteqn{\frac{d}{dt}\{K(t;u,v)+\frac{2}{3}g_{1}(0)I_{1}(t;u)
+\frac{2}{3}g_{2}(0)I_{2}(t;v)\} }\nonumber \\
&\leq&-\frac{g_{1}(0)}{6}\big\{\int_{\Omega}|u_{tt}|^{2}dx
+\int_{\Omega}|\nabla u_{t}|^{2}dx\big\}-
\frac{g_{2}(0)}{6}\big\{\int_{\Omega}|v_{tt}|^{2}dx
+\int_{\Omega}|\nabla v_{t}|^{2}dx\big\} \nonumber \\
&&+\big\{\frac{3}{2}(g_{1}(0)+[g_{1}(0)]^{\frac{1}{p}}c_{0})^{2}
+\frac{c_{0}}{3}g_{1}^{1+\frac{1}{p}}(0)\big\}g_{1}(t)\int
_{\Omega}|\nabla u|^{2}dx \nonumber \\
&&+\big\{\frac{3}{2}(g_{2}(0)+[g_{2}(0)]^{\frac{1}{p}}c_{0})^{2}
+\frac{c_{0}}{3}g_{2}^{1+\frac{1}{p}}(0)\big\}g_{2}(t)\int
_{\Omega}|\nabla v|^{2}dx \nonumber \\
&&+\Big\{\frac{3(c_{0}g_{1}(0)+c_{2})^{2}}{2g_{1}^{1+\frac{1}{p}}(0)}
+\frac{c_{2}}{3}g_{1}(0)\Big\}g_{1}^{1+\frac{1}{p}}
\mathop{\Box} \nabla u \nonumber \\
&&+\Big\{\frac{3(c_{0}g_{2}(0)+c_{2})^{2}}{2g_{2}^{1+\frac{1}{p}}(0)}
+\frac{c_{2}}{3}g_{2}(0)\Big\}g_{2}^{1+\frac{1}{p}}
\mathop{\Box} \nabla v
\end{eqnarray}
for
$$
\epsilon_{i}=\frac{g_{i}(0)}{6(c_{0}g_{i}(0)+c_{2})};
 \quad \lambda_{i}=\frac{1}{6(g_{i}(0)+[g_{i}(0)]^{\frac{1}{p}}c_{0})}.
$$
Note that
\begin{eqnarray}\label{4eq4-11}
\lefteqn{\int_{\Omega}\int^{t}_{0}g_{i}(t-s)\nabla\varphi(s)ds\cdot \nabla
\varphi dx }\nonumber \\
&=&\int_{\Omega}\int^{t}_{0}g_{i}(t-s)[\nabla \varphi(s)
-\nabla \varphi(t)]ds\cdot \nabla \varphi
dx+\int^{t}_{0}g_{i}(s)ds\int_{\Omega}|\nabla \varphi|^{2}dx \nonumber \\
&\leq&\frac{1}{4\epsilon_{i}}g_{i}^{1+\frac{1}{p}}\mathop{\Box} \nabla
\varphi+(\epsilon_{i}\alpha_{i}+\int^{t}_{0}g(s)ds)\int_{\Omega}|\nabla
\varphi|^{2}dx.
\end{eqnarray}
On taking $\epsilon_{i}=\frac{\beta_{i}}{2\alpha_{i}}$ in (\ref{4eq4-11}) and using Lemma \ref{Lem3.3} it follows that
\begin{eqnarray}\label{4eq4-12}
\frac{d}{dt}J(t;u)&\leq&2\alpha_{0}\int_{\Omega}|\nabla u_{t}|^{2}dx-\frac{\beta_{1}}{2}\int_{\Omega}|\nabla u|^{2}dx
+\frac{\alpha_{1}}{2\beta_{1}}g_{1}^{1+\frac{1}{p}}\mathop{\Box}\nabla u, \\
\label{4eq4-13}
\frac{d}{dt}J(t;v)&\leq&2\alpha_{0}\int_{\Omega}|\nabla v_{t}|^{2}dx-\frac{\beta_{2}}{2}\int_{\Omega}|\nabla v|^{2}dx
+\frac{\alpha_{2}}{2\beta_{2}}g_{2}^{1+\frac{1}{p}}\mathop{\Box}\nabla v.
\end{eqnarray}
>From (\ref{4eq4-6})-(\ref{4eq4-13}) we find that
$$
\frac{d}{dt}{\cal L}(t)\leq c R(t)-k_{0}[M(t)+S(t)]
$$
where
\begin{eqnarray*}
M(t)&=&\int_{\Omega}|u_{tt}|^{2}dx+\int_{\Omega}|v_{tt}|^{2}dx+\int_{\Omega}|\nabla u_{t}|^{2}dx \\
&&+\int_{\Omega}|\nabla v_{t}|^{2}dx+\int_{\Omega}|\nabla u|^{2}dx+\int_{\Omega}|\nabla v|^{2}dx,
\end{eqnarray*}
\begin{eqnarray*}
S(t)=g_{1}^{1+\frac{1}{p}}\mathop{\Box} \nabla u+g_{2}^{1+\frac{1}{p}}\mathop{\Box} \nabla 
v+g_{1}^{1+\frac{1}{p}}\mathop{\Box} \nabla
u_{t}+g_{2}^{1+\frac{1}{p}}\mathop{\Box} \nabla v_{t}.
\end{eqnarray*}
Since the energy is bounded, Lemma \ref{Lem4.2} implies
\begin{gather*}
M(t)\geq c M(t)^{\frac{1+p(1-r)}{p(1-r)}}, \\
S(t)\geq c \big\{g_{1}\mathop{\Box} \nabla u+g_{2}\mathop{\Box} \nabla v+g_{1}\mathop{\Box} \nabla u_{t}
+g_{2}\mathop{\Box} \nabla v_{t}\big\}^{\frac{1+p(1-r)}{p(1-r)}}.
\end{gather*}
It is not difficult to see that we can take $N_{1}$, $N_{2}$ large enough such that $\cal L$ satisfies
\begin{eqnarray}\label{4eq4-14}
c\{E(t;u,v)+E(t;u_{t},v_{t})\}\leq {\cal L}(t)\leq c_{1}\{M(t)+S(t)\}^{\frac{p(1-r)}{1+p(1-r)}}.
\end{eqnarray}
>From which it follows that
$$
\frac{d}{dt}{\cal L}(t)\leq c R(t) - c_{2} {\cal L}(t)^{\frac{1+p(1-r)}{p(1-r)}}.
$$
Using Lemma \ref{Lem4.3} we obtain
$$
{\cal L}(t)\leq c\{ {\cal L}(0)+c_{2}\}\frac{1}{(1+t)^{p(1-r)}}.
$$
>From which it follows that the energies decay to zero uniformly.
Using Lemma \ref{Lem4.2} for $r=0$ we get
\begin{gather*}
M(t)\geq c M(t)^{\frac{1+p}{p}}, \cr
S(t)\geq c \{g_{1}\mathop{\Box} \nabla u+g_{2}\mathop{\Box} \nabla v+g_{1}\mathop{\Box}
\nabla u_{t}+g_{2}\mathop{\Box} \nabla v_{t}\}^{\frac{1+p}{p}}.
\end{gather*}
Repeating the same reasoning as above, we get
$$
{\cal L}(t)\leq c\{ {\cal L}(0)+c_{2}\}\frac{1}{(1+t)^{p}}.
$$
>From which our result follows. The proof is now complete.
\hfil $\diamondsuit$ \smallskip

Finally, as a consequence of Theorem \ref{Teo4.1} we conclude that the
 first order energy also decays polynomially.

\begin{cor}\label{Cor4.1}
Under the hypotheses of Theorem \ref{Teo4.1}, there exists a positive
 constant $c$ such that for $p>1$,
$$
E(t;u,v)\leq c \{E(0;u,v)\}(1+t)^{-p}.
$$
\end{cor}

\paragraph{Remark:} Using the same ideas for proving Theorems \ref{teo3.1}
 and \ref{Teo4.1} it is possible to obtain the
same results for a similar coupled system of the plate equations.


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\noindent
\textsc{Mauro L. Santos}\\
Department of Mathematics, Federal University of Par\'a\\
Campus Universitario do Guam\'a, \\
Rua Augusto Corr\^ea 01, Cep 66075-110, Par\'a, Brazil. \\
e-mail: ls@ufpa.br



\end{document}