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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 128, pp. 1--18.\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-2007/128\hfil wave equations with memory]
{On the wave equations with memory in noncylindrical domains}

\author[M. L. Santos\hfil EJDE-2007/128\hfilneg]
{Mauro L. Santos} 

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

\thanks{Submitted March 8, 2007. Published October 2, 2007.}
\subjclass[2000]{35K55, 35F30, 34B15}
\keywords{Wave equation; noncylindrical domain; memory dissipation}

\begin{abstract}
 In this paper we prove the exponential and polynomial decays rates
 in the case $n > 2$, as time approaches infinity of regular solutions
 of the wave equations with memory
 $$
 u_{tt}-\Delta u+\int^{t}_{0}g(t-s)\Delta u(s)ds=0 \quad \mbox{in } \widehat{Q}
 $$
 where $\widehat{Q}$ is a non cylindrical domains of $\mathbb{R}^{n+1}$, 
 $(n\ge1)$.  We show that the dissipation produced by memory effect is
 strong enough to produce exponential decay of solution provided
 the relaxation function $g$ also decays exponentially. When the
 relaxation function decay polynomially, we show that the solution
 decays polynomially with the same rate. For this we introduced a
 new multiplier that makes an important role in the obtaining of
 the exponential and polynomial decays of the energy of the system.
 Existence, uniqueness and regularity of solutions for any
 $n \ge 1$ are investigated. The obtained result extends known
 results from cylindrical to  non-cylindrical domains.
\end{abstract}

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

\section{Introduction}

 Let $\Omega $ be an open bounded domain of $\mathbb{R}^{n}$ containing the origin
and having $C^{2}$ boundary. Let $\gamma :[0,\infty[ \to \mathbb{R}$ be a
continuously differentiable function. See hypothesis
\eqref{1eq1-11}--\eqref{1eq1-13} on $\gamma$. Consider the family
of subdomains $\{ \Omega _{t}\} _{0\leq t<\infty }$ of $\mathbb{R}^n$
given by
\begin{equation*}
\Omega _{t}=T(\Omega),\quad  T:y\in \Omega \mapsto x=\gamma
 (t) y
\end{equation*}
whose boundaries are denoted by $\Gamma _{t}$, and let
 the noncylindrical domain of $\mathbb{R}^{n+1}$ be
\begin{equation*}
\widehat{Q}=\cup_{0\leq t<\infty } \Omega _{t}\times \{ t\}
\end{equation*}
with lateral boundary
\[
\widehat{\Sigma}=\cup_{0\leq t<\infty }\Gamma _{t}\times \{ t\} .
\]
Let us consider the Hilbert space $L^2(\Omega)$ endowed with the
inner product
$$
(u,v)=\int_{\Omega}u(x)v(x)dx
$$
and corresponding norm $\|u\|^2_{L^2(\Omega)}=(u,u)$. We also
consider the Sobolev space $H^1(\Omega)$ endowed with the scalar
product
$$
(u,v)_{H^1(\Omega)}=(u,v)+(\nabla u, \nabla v).
$$
We define the subspace of $H^1(\Omega)$, denoted by
$H^1_0(\Omega)$, as the closure of $C^\infty_0(\Omega)$ in the
strong topology of $H^1(\Omega)$. By $H^{-1}(\Omega)$ we denote
the dual space of $H^1_0(\Omega)$. This space endowed with the
norm induced by the scalar product
$$
((u,v))_{H^1_0(\Omega)}=(\nabla u, \nabla v)
$$
is a Hilbert space; due to the Poincar\'e inequality
$$
\|u\|^2_{L^2(\Omega)} \leq C \|\nabla u\|^2_{L^2(\Omega)}.
$$
 For  $1 \le p < \infty$, we define
$$
\|u\|^p_{L^p(\Omega)}= \int_{\Omega}|u(x)|^pdx,
$$
and for $p=\infty$,
$$
\|u\|_{L^\infty(\Omega)}= \mathop{\rm esssup}_{x \in \Omega}|u(x)|.
$$
In this work we study the existence and uniqueness of strong
global solutions, as well the exponential and polynomial decays of
the energy for the wave equation
\begin{gather}\label{1eq1-1}
u_{tt} - \Delta u + \int^{t}_{0}g(t-s)\Delta u(s)ds  = 0
\quad\mbox{in }\widehat{Q},\\
\label{1eq1-2}
u=0 \quad \mbox{on } \widehat{\sum},\\
\label{1eq1-3} u(x,0)=u_{0}(x),\quad
u_{t}(x,0)= u_1(x)\quad\mbox{in }\quad{\Omega_{0}},
\end{gather}
where $u$ is the transverse displacement. The method used for
proving  existence and uniqueness is based on transforming
our problem into another initial boundary value problem defined
over a cylindrical domain, whose sections are not time-dependent.
This is done using a suitable change of variable. Then we show the
existence and uniqueness for this new problem. Our existence
result on non-cylindrical domains will follows using the inverse
transformation. That is, using the diffeomorphism
$\tau:\widehat{Q}\to Q$ defined by
\begin{equation} \label{1eq1-4}
 \tau:\widehat{Q}\to Q,
\quad (x,t) \in \Omega_{t}\mapsto(y,t)=(\frac{x}{\gamma(t)},t)
\end{equation}
and $\tau^{-1}:Q\to\widehat{Q}$ defined by
\begin{equation}\label{1eq1-5}
\tau^{-1}(y,t)=(x,t)=(\gamma(t)y,t).
\end{equation}
Denoting by $v$ the function
\begin{equation}\label{1eq1-6}
v(y,t)=u\circ \tau^{-1}(y,t)=u(\gamma(t)y,t)
\end{equation}
the initial boundary value problem \eqref{1eq1-1}--\eqref{1eq1-3}
becomes
\begin{gather}\label{1eq1-7}
v_{tt}-\gamma^{-2}\Delta
v+\int^{t}_{0}g(t-s)\gamma^{-2}(s)\Delta v(s)ds
-A(t)v+a_{1}\cdot \nabla \partial_{t}v+a_{2}\cdot\nabla v = 0
\quad \mbox{in }  Q, \\
\label{1eq1-8}
 v|_{\Gamma} = 0,\\
\label{1eq1-9}  v|_{t=0}=v_{0}, \quad v_{t}|_{t=0}=v_{1} \quad
\mbox{in }  \Omega,
\end{gather}
where
$$
A(t)v=\sum_{i,j=1}^{n}\partial_{y_{i}}(a_{ij}\partial_{y_{j}}v)
$$
and
\begin{equation}\label{1eq1-10}
\begin{gathered}
a_{ij}(y,t)=-(\gamma'\gamma^{-1})^{2}y_{i}y_{j} \quad
(i,j=1,\dots,n),\\
a_{1}(y,t)=-2 \gamma' \gamma^{-1}y,\\
a_{2}(y,t)=-\gamma^{-2}y(\gamma''\gamma+(\gamma')^2(n-1)).
\end{gathered}
\end{equation}
To show the existence of strong solution we will use the following
hypotheses:
\begin{gather}\label{1eq1-11}
\gamma'\leq 0 \quad \mbox{if } n>2, \quad
\gamma'\ge 0 \quad \mbox{if }  n \leq 2,\\
\label{1eq1-12}
 \gamma(\cdot)\in L^{\infty}(0,\infty), \quad
\inf_{0\leq t<\infty}\gamma(t)=\gamma_{0}>0,\\
\label{1eq1-13} \gamma' \in W^{2,\infty}(0,\infty) \cap
W^{2,1}(0,\infty).
\end{gather}

Note that the assumption \eqref{1eq1-11} means that $\widehat{Q}$ is
decreasing if $n>2$ and increasing if $n \leq 2$ in the sense that
when $t>t'$ and $n>2$ then the projection of $\Omega_{t'}$ on the
subspace $t=0$ contain the projection of $\Omega_{t}$ on the same
subspace. The opposite holds in the case $n \leq 2$.

The above method was introduced by Dal Passo and Ughi
\cite{Dal-Ughi} to study certain class of parabolic equations in
non cylindrical domains.

We only obtained the exponential and polynomial
decays of solution for our problem for the case  $n>2$. The main
difficulty to prove the exponential and polynomial decays for the
case $n\leq 2$ are in the Lemma \ref{Lem3.3}, \ref{Lem3.4} and
\ref{Lem3.5}, where appears the boundary terms, since we worked
directly in $\widehat{Q}$. To control those terms we used the
hypothesis (\ref{1eq1-11}). Therefore the case $n \leq 2$ is an
important open problem.

 The equation (\ref{1eq1-1}) can be seen as a model
of propagation of seismic waves, where the function g represents
the medium of propagation of waves. In the considered case, the
medium is elastic.

The wave equations with dissipation was studied by several
authors. All of them consider essentially two types of dissipative
mechanisms (or a combination of them):

 (a) The frictional dissipation, obtained by introducing a frictional
damping that can be acting either on the boundary or in a neighborhood of the
boundary;

 (b)The viscoelastic dissipation given by the memory effects as
in \cite{Rivera, Santos2, Santos, Santos1}.

The frictional damping is the simplest dissipative mechanism when
one is working either in the whole domain $\Omega$ or over a
strategic part of the domain (locally). It was proved by
\cite{Assila, Cavalcanti2, Cavalcanti3, Conrad, Lasieka, Nakao1,
Nakao2, Vasco, Zuazua} that the first-order energy decays
exponentially to zero as time goes to infinity.

Finally, the memory effect produces a suitable dissipative
mechanism which depends on the ralaxation function (see
\cite{Santos2, Santos, Santos1}). They proved that the energy
decays uniformly exponentially or algebraically with the same rate
of decay as the relaxation function.

In a non cylindrical domain, the problem of existence, uniqueness
and exponential decay of the solutions for the wave equations with
memory and weak damping was studied by Ferreira and Santos
\cite{Ferreira1}. They proved that the energy decays uniformly
exponentially to zero as time goes to infinity.

The main result of this paper is to extend the result obtained by
Ferreira and Santos \cite{Ferreira1}. That is, to remove the term
$u_t$ of the equation
$$
u_{tt}-\Delta u+\int^t_0 g(t-s)\Delta u(s)ds+u_t=0
$$
in \cite{Ferreira1}.

The main technical difficulty it is to control the term
$\int_{\Omega_t}|u_t|^2dx$ of the total energy of the system
\eqref{1eq1-1}--\eqref{1eq1-3}. To solve this problem we introduced
a new multiplier $(g\ast u)_t$, that makes key role to obtain the
exponential and polynomial decays.

The present paper  extends the results from cylindrical to non
cylindrical domains.

To see the dissipative properties of the system we have to
construct a suitable functional whose derivative is negative and
is equivalent to the first order energy. This functional is
obtained using the multiplicative technique following Komornik
\cite{Kormo}, Lions \cite{Lions1} and Rivera \cite{MRivera}.

The notation we use in this paper are standard and can be found in
Lion's book \cite{Lions1,Lions2}. In the sequel by $C$ (sometimes
$C_1,C_2,\dots$) we denote various positive constants which do not
depend on $t$ or on the initial data. This paper is organized as
follows. In section 2 we prove the existence, regularity and
uniqueness of solutions. We use Galerkin approximation,
Aubin-Lions theorem, energy method introduced by Lions
\cite{Lions1} and some technical ideas to show existence
regularity and uniqueness of regular solution for the problem
\eqref{1eq1-1}--\eqref{1eq1-3}. Finally, in the section 3 and 4, we
establish the results on the exponential and polynomial decays of
the regular solution to the problem \eqref{1eq1-1}--\eqref{1eq1-3}.
We use the technique of the multipliers introduced by Kormornik
\cite{Kormo}, Lions \cite{Lions1} and Rivera \cite{Rivera} coupled
with some technical lemmas and some technical ideas.

\section{Existence and Regularity}
In this section we shall study the existence and regularity of
solutions for the system \eqref{1eq1-1}--\eqref{1eq1-3}. For this
we assume that the kernel $g:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is in
$W^{2,1}(0,\infty)$ and satisfy
\begin{gather}\label{2eq2-1}
g,-g' \ge 0, \\
\label{2eq2-0}
 \gamma_{1}^{-2}-\int^{\infty}_{0}g(s) \gamma^{-2}(s)ds=\beta>0,
\end{gather}
where $\gamma_{1}= \sup_{0< t < \infty}\gamma(t)$. Note that
\eqref{2eq2-0} implies
$$
\beta \leq \gamma(t)^{-2}-\int^t_0 g(s) \gamma^{-2}(s)ds \leq
\frac{1}{\gamma_0^{2}}.
$$
On the other hand, since all the dissipation of the system is
contained only in the memory term, we also have to require that
$g\neq 0$, and this explains \eqref{2eq2-0}.

 Typical examples of functions $\gamma$ and $g$ satisfying
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0}
are
$$
\gamma(t)=e^{-\sigma_0 t}+\sigma_1,\quad g(t)=\sigma_2e^{-\sigma_3
t}
$$
where $\sigma_i$, $i=0,1,2,3$, are positive constants. To simplify
our analysis, we define the binary operator
$$
g\Box \frac{\nabla
u(t)}{\gamma(t)}=\int^{t}_{0}g(t-s)\gamma^{-2}(s)\int_{\Omega}|\nabla
u(t)-\nabla u(s)|^{2}dx\,ds.
$$
With this notation we have the following statement.

\begin{lemma}\label{Lem2.1}
For $v \in C^{1}(0,T:H^{1}(\Omega))$ and $g\in C^1(0,\infty)$ we have
\begin{align*}
\int_{\Omega}\int^t_0\frac{g(t-s)}{\gamma^2(s)}\nabla v ds \cdot
\nabla v_{t}dx
&=-\frac{1}{2} \frac{g(t)}{\gamma^{2}(0)}\int_{\Omega}|\nabla v|^{2}dx +
 \frac{1}{2}g'\Box \frac{\nabla v}{\gamma} \\
&\quad -\frac{1}{2}\frac{d}{dt} \big[g \Box \frac{\nabla
v}{\gamma}-(\int^{t}_{0}\frac{g(t-s)}{\gamma^{2}(s)}ds
)\int_{\Omega}|\nabla v|^{2}dx \big]\\
&\quad +\int^t_0g(t-s)\frac{\gamma'(s)}{\gamma^3(s)}\int_{\Omega}|\nabla
u|^2dx\, ds.
\end{align*}
\end{lemma}

The proof of this lemma follows by differentiating the term
$$
g\Box \frac{\nabla
u(t)}{\gamma(t)}-\int^t_0\frac{g(t-s)}{\gamma^2(s)}\int_{\Omega}|\nabla
u|^2dx ds.
$$

The well-posedness of system (\ref{1eq1-7})-(\ref{1eq1-9}) is
given by the following theorem.

\begin{theorem}\label{teo2.1}
Let us take $v_{0}\in H_{0}^{1}(\Omega)\cap H^{2}(\Omega)$,
$v_{1}\in H_{0}^{1}(\Omega)$ and let us suppose that assumptions
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0} hold. Then
there exists a unique solution $v$ of the problem
(\ref{1eq1-7})-(\ref{1eq1-9}) satisfying
\begin{gather*}
v \in L^{\infty}(0,\infty:H_{0}^{1}(\Omega)\cap H^{2}(\Omega)), \\
v_{t} \in L^{\infty}(0,\infty:H_{0}^{1}(\Omega)), \\
v_{tt} \in L^{\infty}(0,\infty:L^{2}(\Omega)).
\end{gather*}
\end{theorem}

\begin{proof}
 The main idea is to use the Galerkin method. To do this let us take a basis
$\{w_j\}_{j \in \mathbb{N}}$ to $H^1_0(\Omega) \cap H^2(\Omega)$ which is orthonormal
in $L^2(\Omega)$ and we represent by $V_m$ the space generated by $w_1, w_2, \dots, w_m$.
 Let us denote by
$$
v^{m}_{0}=\sum^{m}_{j=1}(v_0,w_j)w_j, \quad
v^{m}_{1}=\sum^{m}_{j=1}(v_1,w_j)w_j.
$$
Note that for any $(v_0,v_1)\in (H^1_0(\Omega) \cap
H^2(\Omega))\times H^{1}_{0}(\Omega)$, we have $v^{m}_{0} \to v_0$
strong in $H^1_0(\Omega) \cap H^2(\Omega)$ and $v^{m}_{1} \to v_1$
strong in $H^{1}_{0}(\Omega)$.

Standard results on ordinary differential equations imply the
existence of a local solution $v^{m}$ of the form
$$
v^{m}(t)=\sum^{m}_{j=1}g_{jm}(t)w_{j},
$$
to the system
\begin{gather}\label{2eq2-2}
\begin{aligned}
&\int_{\Omega}v^{m}_{tt} w_{j}dy - \gamma^{-2} \int_{\Omega}
\Delta v^{m} w_{j}dy + \int_{\Omega} \int^{t}_{0}g(t-s)
\gamma^{-2}(s)\nabla v^{m}(s) \cdot \nabla w_{j}ds dy  \\
&\quad +\int_{\Omega}A(t)v^{m} w_{j}dy + \int_{\Omega}a_{1} \cdot
\nabla v^{m}_{t} w_{j}dy + \int_{\Omega}a_{2} \cdot \nabla
v^{m} w_{j} dy = 0, \quad (j=1,\dots,m),
\end{aligned}\\
\label{2eq2-3}
v^{m}(x,0)=v^{m}_{0}, \quad v^{m}_{t}(x,0)=v^{m}_{1}.
\end{gather}
The extension of this solution to the whole interval $[0, \infty)$
is a consequence of the a priori estimate below.

\subsection*{First estimate}
Multiplying the equation (\ref{2eq2-2}) by $g'_{jm}(t)$, summing
up the product result in $j=1, 2,\dots, m$, and using the
Lemma \ref{Lem2.1} we get
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\pounds^{m}_{1}(t,v^{m})+
\int_{\Omega}A(t)v^{m} v^{m}_{t}dy+\int_{\Omega}a_{1} \cdot \nabla
v^{m}_{t} v^{m}_{t}dy +\int_{\Omega}a_{2} \cdot \nabla v^{m} v^{m}_{t} dy\\
&= -\frac{1}{2}\frac{g(t)}{\gamma^{2}(0)}\|\nabla
v^{m}\|^{2}_{L^{2}(\Omega)}+\frac{1}{2}g'\Box \frac{\nabla
v^{m}}{\gamma}\\
&\quad -\frac{\gamma'}{\gamma^{3}}\|\nabla
v^{m}\|^{2}_{L^{2}(\Omega)}+\int^t_0g(t-s)\gamma'(s)\gamma^{-3}(s)ds
\int_{\Omega}|\nabla v^m|^2dx,
\end{align*}
where
$$
\pounds^{m}_{1}(t,v^{m})=\|v^{m}_{t}\|^{2}_{L^{2}(\Omega)}+
\big(\frac{1}{\gamma^{2}(t)}-\int^{t}_{0}\frac{g(t-s)}{\gamma^2(s)}ds\big)
\|\nabla v^{m}\|^{2}_{L^{2}(\Omega)}+g \Box \frac{\nabla v^{m}}{\gamma}.
$$
Taking into account (\ref{1eq1-11}), \eqref{1eq1-13},
\eqref{2eq2-1} and \eqref{2eq2-0} we obtain
\begin{equation}\label{2eq2-4}
\frac{1}{2}\frac{d}{dt}\pounds^{m}_{1}(t,v^{m}) \leq
C(|\gamma'|+|\gamma''|)\pounds^{m}_{1}(t).
\end{equation}
Integrating the inequality (\ref{2eq2-4}), taking account \eqref{1eq1-13} and
using Gronwall's Lemma we get
\begin{equation}\label{2eq2-5}
\pounds^{m}_{1}(t,v^{m}) \leq C, \quad \forall m \in \mathbb{N}, \;
\forall t \in [0,T].
\end{equation}

\subsection*{Second estimate}
 From equation (\ref{2eq2-2}) we get
\begin{equation}\label{2eq2-6}
\|v^{m}_{tt}(0)\|^{2}_{L^{2}(\Omega)} \leq C, \quad \forall m \in
\mathbb{N}.
\end{equation}
Differentiating the equation (\ref{2eq2-2}) with respect to the
time, we obtain
\begin{equation}\label{2eq2-7}
\begin{aligned}
&\int_{\Omega}v^{m}_{ttt}w_{j}dy-\gamma^{-2}\int_{\Omega}\Delta
v^{m}_{t} w_{j}dy +2\frac{\gamma'}{\gamma^{3}}\int_{\Omega}\Delta
v^{m}w_{j}dy-\frac{g(0)}{\gamma^{2}(0)}\int_{\Omega}\Delta
v^{m}_{0}w_{j}dy  \\
&+ \int_{\Omega}\int^{t}_{0}g'(t-s) \gamma^{-2}(s)\nabla
v^{m}(s) \cdot \nabla w_{j}ds dy
+\int_{\Omega}\frac{d}{dt}(A(t)v^{m})w_{j}dy \\
&+\int_{\Omega}\frac{d}{dt}(a_{1}\cdot \nabla v^{m}_{t})w_{j}dy +
\int_{\Omega}\frac{d}{dt}(a_{2}\cdot \nabla v^{m})w_{j}dy=0.
\end{aligned}
\end{equation}
Multiplying (\ref{2eq2-7}) by $g''_{jm}(t)$, summing up the
product result in $j=1, 2, \dots, m$ and using similar arguments
as (\ref{2eq2-5}) we obtain
\begin{equation}\label{2eq2-8}
\pounds^{m}_{1}(t,v^{m}_{t})+\int^{t}_{0}\|v^{m}_{ss}(s)\|^{2}_{L^{2}(\Omega)}ds
\leq C, \quad \forall t \in [0,T], \; \forall m\in \mathbb{N}.
\end{equation}

The first and second a priori estimates allow us to obtain a
subsequence of $(v_m)$ which from now on will be also denoted by
$(v_m)$ and a function $v:\Omega \times (0,\infty) \to \mathbb{R}$
satisfying:
\begin{gather*}
v^m \to v \quad \mbox{weak star in }  L^{\infty}(0,\infty;H^1_0(\Omega)) \\
v^m_t \to v \quad \mbox{weak star in }  L^{\infty}(0,\infty;H^1_0(\Omega)) \\
v^m_{tt} \to v_{tt} \quad \mbox{weak star in }
L^{\infty}(0,\infty;L^2(\Omega)).
\end{gather*}
The above convergence allows us to pass to the limit in the
problem (\ref{2eq2-2})-(\ref{2eq2-3}).

Letting $m\to \infty$ in the equation (\ref{2eq2-2}) we conclude
that
\[
v_{tt}-\gamma^{-2}\Delta
v+\int^{t}_{0}g(t-s)\gamma^{-2}(s)\Delta v(s)ds
-A(t)v+a_{1}\cdot \nabla \partial_{t}v+a_{2}\cdot\nabla v = 0
\]
in $L^{\infty}(0,\infty:L^{2}(\Omega))$.
Therefore, using the elliptic regularity, we have that
$$
v\in L^{\infty}(0,\infty:H^{1}_{0}(\Omega)\cap
H^{2}(\Omega)).
$$

\subsection*{Uniqueness}
 Suppose we have two solutions $v$ and
$\widehat{v}$ in the conditions of Theorem \ref{teo2.1}.
Then $\phi=v-\widehat{v}$ satisfies the same conditions and $\phi(0)=0$,
 $\phi_t(0)=0$.
Let us prove that $\phi=0$ on $\Omega \times [0,\infty[$.

Multiplying the equations (\ref{1eq1-7}) by
$\phi_t$, summing up the product
result and using the Lemma \ref{Lem2.1} we get
\[
\frac{1}{2}\frac{d}{dt}\pounds_1(t,\phi) \leq
C(|\gamma'|+|\gamma''|)\pounds_1(t),
\]
where
\[
\pounds_{1}(t,\phi)=\|\phi_{t}\|^{2}_{L^{2}(\Omega)}+
(\frac{1}{\gamma^{2}(t)}-\int^{t}_{0}\frac{g(t-s)}{\gamma^2(s)}ds)\|\nabla
\phi\|^{2}_{L^{2}(\Omega)}
+g \Box \frac{\nabla \phi}{\gamma}.
\]
Integrating with respect to the time the above inequality and applying
 Gronwall's inequality we conclude that $\phi=0$ on
$\Omega \times [0,\infty[$.
\end{proof}

To show the existence in non cylindrical domains, we return to our
original problem in the non cylindrical domains by using the
change variable given in (\ref{1eq1-4}) by $(y,t)=\tau(x,t)$,
$(x,t)\in \widehat{Q}$. Let $v$ be the solution obtained from Theorem
\ref{teo2.1} and $u$ defined by (\ref{1eq1-6}), then $u$ belongs to
the class
\begin{gather}\label{2eq2-9}
u \in L^{\infty}(0,\infty:H^{1}_{0}(\Omega_t)), \\
\label{2eq2-10}
 u_{t} \in L^{\infty}(0,\infty:H^{1}_{0}(\Omega_t)),\\
 \label{2eq2-11}
 u_{tt} \in L^{\infty}(0,\infty:L^{2}(\Omega_t)).
 \end{gather}
 Denoting by
 $$
 u(x,t)=v(y,t)=(v\circ \tau)(x,t),
 $$
 from (\ref{1eq1-6}) it is easy to see that $u$ satisfies
\begin{equation}\label{2eq2-12}
u_{tt} - \Delta u + \int^{t}_{0}g(t-s)\Delta u(s)ds  = 0 \quad
\mbox{in} \quad L^{\infty}(0,\infty:L^{2}(\Omega_{t})).
\end{equation}
 Using regularity elliptic, we obtain
\begin{equation}\label{2eq2-13}
u\in L^{\infty}(0,\infty:H^1_0(\Omega_t)\cap H^2(\Omega_t)).
\end{equation}
Let $u_1$, $u_2$ be two solutions to (\ref{1eq1-1}), and $v_1$,
$v_2$ be the functions obtained through the diffeomorphism $\tau$
given by (\ref{1eq1-4}). Then $v_1$, $v_2$ are the solutions to
(\ref{1eq1-7}). By the uniqueness result Theorem \ref{teo2.1}, we
have $v_1=v_2$, so $u_1=u_2$. Therefore, we have the
 following result.

 \begin{theorem}\label{teo2.2}
Let us take $u_{0}\in H_{0}^{1}(\Omega_0)\cap H^{2}(\Omega_0)$,
$u_{1}\in H_{0}^{1}(\Omega_0)$ and let us suppose that assumptions
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0} hold. Then
there exists a unique solution $u$ of the problem
\eqref{1eq1-1}--\eqref{1eq1-3} satisfying
(\ref{2eq2-9})-(\ref{2eq2-13}).
\end{theorem}

\section{Exponential Rate of Decay}

In this section we show that the solution of system
\eqref{1eq1-1}--\eqref{1eq1-3}
decays exponentially. To this end we
will assume that the memory $g$ satisfies:
\begin{equation}\label{3eq3-1}
g'(t) \leq -C_1g(t)
\end{equation}
for all $t \ge 0$, with positive constant $C_1$. Additionally, we
assume that the function $\gamma(\cdot)$ satisfies the conditions
\begin{gather}\label{3eq3-2}
\gamma' \leq 0, \quad t \ge 0, \quad n>2,\\
 \label{3eq3-3}
 0<\max_{0\leq t<\infty}
|\gamma'(t)|\leq \frac{1}{d},
\end{gather}
where $d=\mathop{\rm diam}(\Omega)$. The condition \eqref{3eq3-3} implies that
our domains is ``time like'' in the sense that
$$
|\underline{\nu}| < |\overline{\nu}|
$$
where $\underline{\nu}$ and $\overline{\nu}$ denote the
$t$-component and $x$-component of the outer unit normal of
$\widehat{\Sigma}$. To facilitate our calculations we introduce the
notation
\begin{align*}
(g \Box \nabla u)(t) = \int_{\Omega_{t}}\int^{t}_{0}g(t-s)|\nabla
u(t) - \nabla u(s)|^{2} ds\,dx.
\end{align*}
First, we  prove the following two lemmas that will be
used in the sequel.

\begin{lemma}\label{Lem3.1}
Let $F(\cdot,\cdot)$ be the smooth function defined in
$\Omega_{t}\times [0,\infty[$, $(t \in [0,\infty[$. Then
\begin{equation}\label{3eq3-4}
\frac{d}{dt}\int_{\Omega_{t}}F(x,t)dx=\int_{\Omega_{t}}
\frac{d}{dt}F(x,t)dx+\frac{\gamma'}{\gamma}\int_{\Gamma_{t}}F(x,t)(x\cdot
\overline{\nu})d \Gamma_{t},
\end{equation}
where $\overline{\nu}$ is the $x$-component of the unit normal
exterior $\nu$.
\end{lemma}

\begin{proof}
By a change variable $x=\gamma(t)y$, $y\in \Omega$, we have
\begin{align*}
\frac{d}{dt}\int_{\Omega_{t}}F(x,t)dx
&=\frac{d}{dt}\int_{\Omega}F(\gamma (t)y,t)\gamma^{n}(t)dy \\
&=\int_{\Omega} (\frac{\partial F}
{\partial{t}})\gamma^{n}(t)dy+\sum^{n}_{i=1}\int_{\Omega}
\frac{\gamma'}{\gamma}x_{i}(\frac{\partial F}{\partial t}) \gamma^{n}(t)dy\\
&\quad +n \int_{\Omega}\gamma'(t)\gamma^{n-1}(t)F(\gamma(t)y,t)dy.
\end{align*}
If we return at the variable $x$, we get
\begin{align*}
\frac{d}{dt}\int_{\Omega_{t}}F(x,t)dx= \int_{\Omega_t}
\frac{\partial F}{\partial t}dx+
\frac{\gamma'}{\gamma}\int_{\Omega_t}x \cdot \nabla F(x,t) dx + n
\frac{\gamma'}{\gamma}\int_{\Omega_t}F(x,t)dx.
\end{align*}
Integrating by part in the last equality we obtain the formula
(\ref{3eq3-4}).
\end{proof}

\begin{lemma}\label{Lem3.2}
For any functions $g \in C^{1}(\mathbb{R}_+)$ and
$u \in C^{1}((0,T):H^{2}(\Omega_t))$, we have
\begin{align*}
&\int_{\Omega_t}\int^{t}_{0} g(t-s) \nabla u(s) \cdot \nabla
u_{t} ds\,dx \\
& = -\frac{1}{2} g(t) \int_{\Omega_t}|\nabla u(t)|^{2}dx
 + \frac{1}{2} g' \Box \nabla u
 - \frac{1}{2} \frac{d}{dt}\Big[g \Box \nabla u -
(\int^{t}_{0}g(s) ds ) \int_{\Omega_t}|\nabla
u|^{2} \Big] \\
&\quad + \frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^{t}_{0}g(t-s)|\nabla
u(t)-\nabla u(s)|^{2}(\overline{\nu}\cdot x)d \Gamma_{t}\\
&\quad -\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^{t}_{0}g(t-s)|\nabla
u(t)|^{2}(\overline{\nu}\cdot x)d \Gamma_{t}.
\end{align*}
\end{lemma}

\begin{proof}
Differentiating the term $g \Box \nabla u$ and applying the
lemma \ref{Lem3.1} we obtain
\begin{align*}
\frac{d}{dt}g \Box \nabla u
&=\int_{\Omega_t}\int^t_0g'(t-s)|\nabla u(t)-\nabla u(s)|^2 ds\,dx
\\
&\quad -2\int_{\Omega_t}\int^t_0g(t-s) \nabla u(s)\cdot \nabla u_t ds\,dx\\
&\quad + \Big( \int^t_0 g(t-s)ds \Big) \int_{\Omega_t}
\frac{d}{dt}|\nabla u(t)|^2 dx\\
&\quad +\frac{\gamma'}{\gamma}\int_{\Gamma_t}\int^t_0g(t-s)|\nabla
u(t)-\nabla u(s)|^2 (x\cdot\overline{\nu})ds d \Gamma_t.
\end{align*}
 From where it follows that
\begin{align*}
&2\int_{\Omega_t}\int^t_0g(t-s) \nabla u(s)\cdot \nabla u_t ds\,dx\\
&=-\frac{d}{dt}\{g\Box \nabla u - \int^t_0 g(t-s) ds
 \int_{\Omega_t} |\nabla u(t)|^2 dx \} \\
&\quad + \int_{\Omega_t} \int^t_0 g'(t-s)|\nabla u(t)- \nabla u(s)|^2
 ds\,dx - g(t) \int_{\Omega_t}|\nabla u(t)|^2 dx\\
&\quad +\frac{\gamma'}{\gamma}\int_{\Gamma_t}\int^t_0g(t-s)|\nabla
 u(t)-\nabla u(s)|^2 (x\cdot\overline{\nu})ds d \Gamma_t\\
&\quad -\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^{t}_{0}g(t-s)|\nabla
u(t)|^{2}(\overline{\nu}\cdot x)d \Gamma_{t}.
\end{align*}
The proof is  complete.
\end{proof}

Let us introduce the functional
\begin{align*}
E(t)=\|u_t\|^{2}_{L^{2}(\Omega_t)}+\Big(1-\int^{t}_{0}g(s)ds\Big)
\| \nabla u\|^{2}_{L^{2}(\Omega_t)}+g \Box \nabla u.
\end{align*}

\begin{lemma}\label{Lem3.3}
  Let us take $u_{0} \in
H^{1}_{0}(\Omega_0)\cap H^{2}(\Omega_0)$, $u_{1} \in
H^{1}_{0}(\Omega_0)$ and let us suppose that assumptions
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0} hold.
Then any regular solution of system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
 \begin{align*}
&\frac{d}{dt}E(t)-\int_{\Gamma_t}\frac{\gamma'}{\gamma}(\overline{\nu}\cdot
x)(|u_t|^{2}+|\nabla u|^{2})d \Gamma_t \\
&-\int_{\Gamma_t}\frac{\gamma'}{\gamma}(\overline{\nu}\cdot x)
\int^{t}_{0}g(t-s)|\nabla u(t)-\nabla u(s)|^{2}ds d \Gamma_t \\
&= -\frac{1}{2}\int_{\Omega_t}g(t)|\nabla u|^{2}dx+\frac{1}{2}g'
\Box \nabla u.
\end{align*}
\end{lemma}

\begin{proof}
 Multiplying the equation (\ref{1eq1-1}) by $u_t$ and integrating over
 $\Omega_t$ we get
\[
\frac{1}{2}\int_{\Omega_t}\frac{d}{dt}|u_t|^2dx
+\frac{1}{2}\int_{\Omega_t}\frac{d}{dt}|\nabla
u|^2dx-\int_{\Omega_t}\int^t_0g(t-s) \nabla u(s) \cdot \nabla u_t
ds\,dx=0.
\]
Using Lemmas \ref{Lem3.1} and \ref{Lem3.2} we obtain
 \begin{align*}
&\frac{d}{dt}E(t)-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu}\cdot
 x)|u_t|^{2}d \Gamma_t
 -\frac{\gamma'}{2\gamma}\Big(1-\int^t_0g(s)ds\Big)
 \int_{\Gamma_t}(\overline{\nu}\cdot x)|\nabla u|^2d \Gamma_t\\
&-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^t_0(\overline{\nu}\cdot
x)g(t-s) |\nabla u(\cdot,t)-\nabla u(\cdot,s)|^2ds d \Gamma_t \\
&= -\frac{1}{2}g(t)\int_{\Omega}|\nabla u|^2dx+\frac{1}{2}g' \Box
\nabla u .
\end{align*}
The proof is complete.
\end{proof}

For the estimate of the term $\int_{\Omega_t}|u_t|^2dx$ we
introduced the  functional
\[
\varphi(t)=-\int_{\Omega_t}u_t(g\ast u)_t
dx+\frac{1}{2}\int_{\Omega}|g\ast \nabla u|^2dx
\]
where
$(g\ast u)_t=g(0)u+g' \ast u$.

\begin{lemma}\label{Lem3.4}
Let us take $u_{0} \in H^{1}_{0}(\Omega_0)\cap H^{2}(\Omega_0)$,
$u_{1} \in H^{1}_{0}(\Omega_0)$ and let us suppose that
assumptions \eqref{1eq1-11}--\eqref{1eq1-13} and
\eqref{2eq2-1}--\eqref{2eq2-0} hold. Then any regular solution of
system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\varphi(t)-\frac{\gamma'}{2
\gamma}\int_{\Gamma_t}(\overline{\nu} \cdot
x)|g \ast \nabla u|^2 d \Gamma_t \\
& \leq
-\frac{g(0)}{2}\int_{\Omega_t}|u_t|^2dx+\frac{3g(0)}{2}\int_{\Omega_t}|\nabla
u|^2dx+g(t)\int_{\Omega_t}|\nabla u|^2dx \\
&\quad +\frac{(\int^t_0|g'(s)|ds)}{2g(0)}|g'|\Box
\nabla u  + \frac{|g'(t)|^2}{g(0)}\int_{\Omega_t}|u_0|^2 dx \\
&\quad -g(t)\int_{\Omega_t}|u_t|^2dx+\frac{(\int^t_0|g'(s)|ds)}{g(0)}|g'|\Box
 u_t .
\end{align*}
\end{lemma}

\begin{proof}
 From the equation (\ref{1eq1-1}) and using the fact that
$u=0$ on the boundary we get
\begin{equation}\label{3eq3-5}
\begin{aligned}
&-\frac{d}{dt}\int_{\Omega_t}u_t(g\ast u)_t dx\\
&=\int_{\Omega_t}(-\Delta u+g\ast \Delta u)(g \ast u)_tdx
 -g(0)\int_{\Omega_t}|u_t|^2dx-\int_{\Omega_t}u_t(g'\ast u) dx\\
&=g(0)\int_{\Omega_t}|\nabla u|^2dx+\int_{\Omega_t}\nabla u \cdot
 (g' \ast \nabla u) dx -
\frac{1}{2}\int_{\Omega_t}\frac{d}{dt}|g \ast \nabla u|^2dx  \\
&\quad -g(0)\int_{\Omega_t}|u_t|^2dx-\int_{\Omega_t}u_t(g'\ast u)_tdx.
\end{aligned}
\end{equation}
Using Lemma \ref{Lem3.1} we have
\begin{equation}\label{3eq3-6}
\frac{1}{2}\int_{\Omega_t}\frac{d}{dt}|g \ast \nabla u|^2dx =
\frac{1}{2}\frac{d}{dt}\int_{\Omega_t}|g \ast \nabla
u|^2dx-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu} \cdot
x)|g \ast \nabla u|^2 d \Gamma_t.
\end{equation}
Define
$$
I_1:=\int_{\Omega_t}\nabla u \cdot \int^t_0g'(t-s)\nabla u(s)ds\,dx;
$$
then we have
\begin{equation}\label{3eq3-7}
\begin{aligned}
I_1&=\int_{\Omega_t}g(t)|\nabla u|^2dx+\int_{\Omega_t}\nabla u
\cdot \int^t_0g'(t-s)(\nabla u(\cdot,t)-\nabla u(\cdot,s))ds\,dx \\
&\leq (\int_{\Omega_t}|\nabla
u|^2dx)^{1/2}(\int^t_0|g'(s)|ds)^{1/2}(|g'|\Box\nabla u)^{1/2}
 +g(t)\int_{\Omega_t}|\nabla u|^2dx.
\end{aligned}
\end{equation}
Define
\begin{align*}
I_2&:=-\int_{\Omega}u_t(g \ast u)_tdx\\
&=-\int_{\Omega_t}u_t(g'(t)u_0+g' \ast u_t)dx \\
&=-\int_{\Omega_t}g'(t)u_tu_0dx-\int_{\Omega_t}g(t)|u_t|^2dx \\
&\quad -\int_{\Omega_t}u_t\int^t_0g'(t-s)(u_t(\cdot,s)- u_t(\cdot,t))ds\, dx;
\end{align*}
then thanks to the Young inequality, we obtain
\begin{equation}\label{3eq3-8}
|I_2| \leq
\frac{g(0)}{2}\int_{\Omega_t}|u_t|^2dx+\frac{|g'|^2}{g(0)}\int_{\Omega_t}|u_0|^2dx
-\int_{\Omega_t}|u_t|^2dx+\frac{|g(t)-g(0)|}{g(0)}|g'|\Box u_t.
\end{equation}
Substituting the inequalities (\ref{3eq3-6}), (\ref{3eq3-7}) and
(\ref{3eq3-8}) into (\ref{3eq3-5}) we obtain the conclusion of
lemma.
\end{proof}

For the estimate of the term $\int_{\Omega_t}|g\ast\nabla u|^2dx$
we introduced the following functional
$$
\eta(t):=\frac{1}{2}\int_{\Omega_t}g \Box u_t
dx-\int_{\Omega_t}\Big(\int^t_0g(s)ds\Big)|u_t|^2dx
-\frac{1}{2}\int_{\Omega_t}|g\ast\nabla u|^2dx.
$$

\begin{lemma}\label{Lem3.5}  Let us take
$u_{0} \in H^{1}_{0}(\Omega_0)\cap H^{2}(\Omega_0)$, $u_{1} \in
H^{1}_{0}(\Omega_0)$ and let us suppose that assumptions
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0} hold.
Then any regular solution of system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
\begin{align*}
&\frac{d}{dt}\eta(t)
+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu}
 \cdot x)|g \ast \nabla u|^2 d \Gamma_t\\
&-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^t_0(\overline{\nu}
\cdot x)g(t-s) |u_t(\cdot,t)-u_t(\cdot,s)|^2ds d \Gamma_t
\\
&+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu} \cdot
x)(\int^t_0g(s)ds)|u_t|^2 d \Gamma_t \\
& \leq  -g(t)\int_{\Omega_t}|u_t|^2dx+g' \Box u_t
-\frac{g(0)}{2}\int_{\Omega_t}|\nabla u|^2dx
\\
&\quad +g(t)\int_{\Omega_t}|\nabla u|^2dx+\frac{(\int^t_0|g'(s)|ds
)}{2g(0)}|g'| \Box \nabla u.
\end{align*}
\end{lemma}

\begin{proof}
Multiplying the equation (\ref{1eq1-1}) by $g \ast u_t$ and
using similar argument as in the lemma \ref{Lem3.4} we obtain
\begin{equation}\label{3eq3-9}
\begin{aligned}
&\frac{d}{dt}\eta(t)+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu}
\cdot x)|g \ast \nabla u|^2 d \Gamma_t  \\
&-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^t_0(\overline{\nu}
\cdot x)g(t-s) |u_t(\cdot,t)-u_t(\cdot,s)|^2ds d \Gamma_t\\
&+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu}
\cdot x)(\int^t_0g(s)ds)|u_t|^2 d \Gamma_t   \\
&=-\frac{1}{2}g(t)\int_{\Omega_t}|u_t|^2dx+\frac{1}{2}g'
\Box u_t -g(0)\int_{\Omega_t}|\nabla u|^2dx  \\
&\quad +g(t)\int_{\Omega_t}|\nabla u|^2dx+\int_{\Omega_t}\nabla
u\int^t_0g'(t-s)(\nabla u(\cdot,s)-\nabla u(\cdot,t))ds\,dx
\end{aligned}
\end{equation}
Noting that
\begin{align*}
&\int_{\Omega_t}\nabla u \int^t_0 g'(t-s)(\nabla
u(\cdot,s)-\nabla u(\cdot,t))ds\,dx \\
&\leq \Big(\int_{\Omega_t}|\nabla
u|^2dx\Big)^{1/2} \Big(\int^t_0|g'(s)|ds\Big)^{1/2}(|g'|\Box\nabla
u)^{1/2},
\end{align*}
considering the inequality $ab \leq \frac{a^2}{2}+\frac{b^2}{2}$
and  using the Cauchy-Schwarz inequality we deduce that
\begin{align*}
&\frac{d}{dt}\eta(t)+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu}
\cdot x)|g \ast \nabla u|^2 d \Gamma_t \\
&-\frac{\gamma'}{2\gamma}\int_{\Gamma_t}\int^t_0(\overline{\nu}
\cdot x)g(t-s) |u_t(\cdot,t)-u_t(\cdot,s)|^2ds d \Gamma_t
\\
&+\frac{\gamma'}{2\gamma}\int_{\Gamma_t}(\overline{\nu} \cdot
x)(\int^t_0g(s)ds)|u_t|^2 d \Gamma_t \\
& \leq -\frac{1}{2}g(t)\int_{\Omega_t}|u_t|^2dx+\frac{1}{2}g'
\Box u_t -\frac{g(0)}{2}\int_{\Omega_t}|\nabla u|^2dx \\
&\quad +g(t) \int_{\Omega_t}|\nabla
u|^2dx+\frac{(\int^t_0|g'(s)|ds)}{2g(0)}|g'| \Box \nabla u .
\end{align*}
The proof is  complete.
\end{proof}

 To estimate the term $(1-\int^t_0 g(s)
ds)\int_{\Omega_t}|\nabla u|^2 dx$, we introduced the
functional
\begin{align*}
\psi(t)= \int_{\Omega_t}u_tu dx .
\end{align*}

\begin{lemma}\label{Lem3.6}
  Let us take $u_{0} \in H^{1}_{0}(\Omega_0)\cap H^{2}(\Omega_0)$,
$u_{1} \in H^{1}_{0}(\Omega_0)$ and let us suppose that assumptions
\eqref{1eq1-11}--\eqref{1eq1-13} and \eqref{2eq2-1}--\eqref{2eq2-0} hold.
Then any regular solution of system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
\begin{align*}
\frac{d}{dt}\psi(t)
& \leq  -(1-\int^t_0 g(s) ds)\int_{\Omega_t}|\nabla u|^2 dx
+g(0)\int_{\Omega_t}|\nabla u|^2 dx \\
&\quad +\frac{(\int^t_0g(s)ds )}{4g(0)}g \Box \nabla u  +
\int_{\Omega_t}|u_t|^2 dx.
\end{align*}
\end{lemma}

\begin{proof}
 From  (\ref{1eq1-1}) we get
\begin{align*}
\frac{d}{dt}\psi (t)=-\int_{\Omega_t}|\nabla u|^2
dx+\int_{\Omega_t}\nabla u \int^t_0g(t-s)\nabla u(\cdot,s)ds\,dx
+\int_{\Omega_t}|u_t|^2 dx.
\end{align*}
Considering the inequality $ab \leq \frac{a^2}{2}+\frac{b^2}{2}$,
making use of the Cauchy-Schwarz inequality and using similar
arguments as in the lemmas \ref{Lem3.4} and \ref{Lem3.5}, follows
the conclusion of lemma.
\end{proof}

The following lemma is the key to obtain  exponential decay.

\begin{lemma}  \label{lem-tec-exp}
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}

\begin{proof}
 First,  suppose that $\gamma_0<\gamma_1$. 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 at
$$
F(t)\leq F(0)e^{-\gamma_0 t}\quad\Rightarrow\quad
f(t)\leq \Big(f(0) + \frac{C_0}{\gamma_1-\gamma_0}\Big)e^{-\gamma_0 t}.
$$
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
$$
f(t)\leq \big(f(0) + C_0t\big)e^{-\gamma_1 t}.
$$
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.
\end{proof}

  Let us introduce the functional
\begin{equation}\label{3eq3-11}
\mathcal{L} (t) = N_1 E(t)+N_2 \eta(t)+\epsilon \psi(t)+\varphi (t),
\end{equation}
with $N_1>N_2>0$ and $\epsilon>0$ small enough. It is not
difficult to see that $\mathcal{L}(t)$ verifies
\begin{equation}\label{3eq3-12}
k_{0}E(t) \leq \mathcal{L} (t) \leq k_{1}E(t),
\end{equation}
for $k_{0}$ and $k_{1}$ positive constants.
Now we are in a position to show the main result of this paper.

\begin{theorem}\label{teo3.1}
Let us take $u_{0} \in H^{1}_{0}(\Omega_0)$, $u_{1} \in
L^2(\Omega_0)$ and let us suppose that assumptions
\eqref{1eq1-12}, \eqref{1eq1-13}, \eqref{2eq2-1}, \eqref{2eq2-0},
\eqref{3eq3-2} and \eqref{3eq3-3} hold. Then any regular solution
of system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
\[
E(t) \leq C e^{- \xi t} E(0),\quad \forall t\ge 0
\]
where $C$ and $\xi$ are positive constants.
\end{theorem}

\begin{proof}
 We shall prove this result for strong solutions, that is,
for solutions with initial data $u_{0} \in H^{1}_{0}(\Omega_0)\cap
H^{2}(\Omega_0)$, $u_{1} \in H^{1}_{0}(\Omega_0)$. Our conclusion
will follows by standard density arguments. Taking $N_1$, $N_2$
large enough, with $N_1>N_2$, $\epsilon>0$ small enough and using
the lemmas (\ref{Lem3.3}), (\ref{Lem3.4}), (\ref{Lem3.5}) and
(\ref{Lem3.6}), we conclude that there exist positive constants
$\alpha_0$ and $C_0$ such that
\[
\frac{d}{dt} \mathcal{L} (t) \leq - \alpha_0 \mathcal{L}(t)+C_0g^2(t)E(0).
\]
Using the lemma \ref{lem-tec-exp} we obtain
\[
\mathcal{L}(t) \leq \{\mathcal{L}(0)+C\} e^{-\alpha_1t}
\]
where $C$ and $\alpha_1$ are positive constants. From equivalence
relation (\ref{3eq3-12}) our conclusion follows.
\end{proof}


\section{Polynomial Rate of Decay}

In this section we assume that the memory $g$ satisfies:
\begin{gather}\label{4eq4-1}
g'(t) \leq -C_1g^{1+\frac{1}{p}}(t)\\
 \label{4eq4-2}
\alpha:=\int^{\infty}_{0}g^{1-\frac{1}{p}}(s)ds<\infty
\end{gather}
for some $p>1$ and $t \ge 0$, with positive constant $C_1$. The
following lemmas will play an important role in the sequel.

\begin{lemma}\label{Lem4.1}
Suppose that $g$ and $h$ are continuous functions,
$g\in L^{1+\frac{1}{q}}(0,\infty) \cap L^1(0,\infty)$ and
$g^r \in L^1(0,\infty)$ for some $0\leq r<0$. Then
\begin{align*}
&\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{align*}
\end{lemma}

\begin{proof}
Without loss of generality we can suppose that $g,h \ge 0$.
Note that for any fixed $t$ we have
\[
\int^t_0g(t-s)h(s)ds=\lim_{\|\Delta s_i\| \to 0}
\sum^m_{i=1}g(t-s_i)h(s_i)\Delta s_i.
\]
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
\[
 F(\sum^m_{i=1}g(t-s_i)h(s_i)\Delta s_i)=F(\sum^m_{i=1}\varphi_i
 \theta_i) \leq \sum^m_{i=1}\theta_iF(\varphi_i),
\]
 so, we have
\begin{equation}\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{equation}
In view of
\[
 \lim_{\|\Delta s_i\|\to 0}I^r_m=\int^t_0g^r(t-s)h(s)ds,
\]
letting $\|\Delta s_i\| \to 0$ in (\ref{4eq4-3}), we get
 $$
\big\{\int^t_0g(t-s)h(s)ds\big\}^{1+\frac{1}{q}}\leq
\big\{\int^t_0g^r(t-s)h(s)ds\big\}^{1+q}
\big\{\int^t_0g^{1+\frac{1-r}{q}}\big\},
$$
 from which our result follows.
\end{proof}

\begin{lemma}\label{Lem4.2}
 Let $w \in C(0,T;H^1_0(\Omega_t))$ and $g$ be a continuous
 function satisfying hypothesis \eqref{4eq4-1}--\eqref{4eq4-2}.
 Then for $0<r<1$ we have
 $$
 g \Box \nabla w \leq
 2\big\{\int^t_0g^rds\|w\|_{C(0,T;H^1_0(\Omega_t))}\big\}^{\frac{1}{1+(1-r)p}}
\big\{g^{1+\frac{1}{p}}\Box
 \nabla w\big\}^{\frac{(1-r)p}{1+(1-r)P}},
 $$
 while for $r=0$ we have
 $$
 g \Box \nabla w \leq
 2\big\{\int^t_0\|w(s)\|^2_{H^1_0(\Omega_t)}ds+t\|w(t)\|^2_{H^1_0(\Omega_t)}\big\}
\big\{g^{1+\frac{1}{p}}\big\}^{\frac{p}{1+p}}.
 $$
 \end{lemma}

\begin{proof}
>From hypotheses on $w$ and Lemma \ref{Lem4.1} we get
\begin{equation}\label{4eq4-4}
\begin{aligned}
g \Box \nabla w & =  \int^t_0g(t-s)h(s)ds  \\
& \leq  \big\{\int^t_0g^r(t-s)h(s)ds \big\}^{\frac{1}{1+p(1-r)}}
\big\{\int^t_0g^{1+\frac{1}{p}}(t-s)h(s)ds\big\}^{\frac{(1-r)p}{1+p(1-r)}}  \\
& \leq  \{g^r \Box \nabla w \}^{\frac{1}{1+p(1-r)}}
\{g^{1+\frac{1}{p}} \Box \nabla w\}^{\frac{(1-r)p}{1+p(1-r)}}
\end{aligned}
\end{equation}
where
$$
h(s)=\int_{\Omega_t}|\nabla w(t)-\nabla w(s)|^2ds.
$$
For $0<r<1$, we have
$$
g^r \Box \nabla w=\int_{\Omega_t}\int^t_0g^r(t-s)|\nabla
w(t)-\nabla w(s)|^2 ds\,dx \leq
4\int^t_0g^r(s)\|w\|^2_{C(0,T:H^1_(\Omega_t))},
$$
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{align*}
1 \Box \nabla w
&= \int_{\Omega_t}\int^t_0|\nabla w(t)-\nabla
w(s)|^2ds\,dx \\
& \leq  2t \|w(t)\|^2_{H^1_0(\Omega_t)}+2\int^t_0\|w(s)\|^2_{H^1_0(\Omega_t)}.
\end{align*}
Substitution of the above inequality into (\ref{4eq4-4}) yields
the second inequality. The proof is complete.
\end{proof}

\begin{lemma}\label{Lem4.3}
Let $f$ be a non-negative $C^1$ function satisfying
$$
f'(t) \leq -k_0[f(t)]^{1+\frac{1}{p}}+\frac{k_1}{(1+t)^{p+1}},
$$
for some positive constants $k_0$, $k_1$ and $p>1$. There exists a
positive constant $C_1$ such that
$$
f(t) \leq C_1\frac{pf(0)+2k_1}{(1+t)^p}
$$
\end{lemma}

\begin{proof}
Let $h(t):=\frac{2k_1}{p(1+t)^{p+1}}$ and $F(t):=f(t)+h(t)$. Then
\begin{align*}
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_0k^{\frac{1}{p}}_1}[h(t)]^{1+\frac{1}{p}}
\Big\}.
\end{align*}
 From which it follows that there exists a positive constant $C_1$
such that
$$
F'(t) \leq -C_1 \{[f(t)]^{1+\frac{1}{p}}+[h(t)]^{1+\frac{1}{p}}
\},
$$
which gives the required inequality.
\end{proof}


\begin{theorem}\label{teo4.1}
Let us take $u_{0} \in H^{1}_{0}(\Omega_0)$, $u_{1} \in L^2(\Omega_0)$
and let us suppose that assumptions \eqref{1eq1-12}, \eqref{1eq1-13}, \eqref{2eq2-1} \eqref{2eq2-0},
\eqref{4eq4-1} and \eqref{4eq4-2} hold. Then any regular solution
of system \eqref{1eq1-1}--\eqref{1eq1-3} satisfies
\begin{align*}
E(t) \leq C E(0)(1+t)^{-p},
\end{align*}
where $C$ is a positive constant and $p>1$.
\end{theorem}

\begin{proof}
We shall prove this result for strong solutions, that is,
for solutions with initial data $u_{0} \in H^{1}_{0}(\Omega_0)\cap
H^{2}(\Omega_0)$, $u_{1} \in H^{1}_{0}(\Omega_0)$. Our conclusion
will follows by standard density arguments. From the Lemmas
\ref{Lem3.3}, \ref{Lem3.4}, \ref{Lem3.5} and \ref{Lem3.6} we get
\begin{align*}
\frac{d}{dt}\mathcal{L}(t)
&\leq -C_0\Big\{ \int_{\Omega_t}|u_t|^2
dx+\big(1-\int^t_0g(s)ds \big)\int_{\Omega_t}|\nabla
u|^2dx- g' \Box \nabla u  \Big\} \\
&\quad + C_1g^2(t) \int_{\Omega_0}|u_0|^2dx.
\end{align*}
Using hypothesis \eqref{4eq4-1} we have
\begin{align*}
\frac{d}{dt}\mathcal{L}(t)
&\leq -C_0\Big\{ \int_{\Omega_t}|u_t|^2
dx+\big(1-\int^t_0g(s)ds \big)\int_{\Omega_t}|\nabla
u|^2dx\Big\} \\
&\quad -C_1 g^{1+\frac{1}{p}}(t) \Box \nabla u  + C_2g^2(t)
\int_{\Omega_0}|u_0|^2dx.
\end{align*}
for some positive constants $C_0$, $C_1$ and $C_2$. Let us define
 the  functional
$$
\mathcal{N}(t)=\int_{\Omega_t}|u_t|^2dx+\int_{\Omega_t}|\nabla u|^2dx.
$$
Since the total energy is bounded, Lemma \ref{Lem4.2} implies
\begin{gather*}
\mathcal{N}(t) \ge C_2 \mathcal{N}(t)^{\frac{(1+(1-r)p)}{(1-r)p}},\\
g^{1+\frac{1}{p}}(t) \Box \nabla u  \ge C_2 \left \{\ g \Box
\nabla u  \right \}^ {\frac{(1+(1-r)p)}{(1-r)p}}.
\end{gather*}
It is not difficult to see that for $N_1$, $N_2$ large enough,
with $N_1>N_2$, and $\epsilon$ small enough the inequality
$$
CE(t) \leq \mathcal{L}(t) \leq C_3 \{\mathcal{N}(t)+g \Box \nabla u  \}
\leq C_4E(t)
$$
holds. From this follows that
$$
\frac{d}{dt}\mathcal{L}(t) \leq -C_5\mathcal{L}(t)^{\frac{(1+(1-r)p)}{(1-r)p}}+C_2g^2(t) \int_{\Omega_0}|u_0|^2
dx.
$$
Using  Lemma \ref{Lem4.3}, we obtain
$$
\mathcal{L}(t) \leq C\{ \mathcal{L}(0)+C_6\} \frac{1}{(1+t)^{p(1-r)}}
$$
where $C$ and $C_6$ are positive constants independent on the
initial data. From which it follows that the energy decay to zero
uniformly.

Using Lemma \ref{Lem4.2} for $r=0$ we get
\begin{gather*}
\mathcal{N}(t) \ge C_2 \mathcal{N}(t)^{\frac{(1+p)}{p}}, \\
g^{1+\frac{1}{p}}(t) \Box \nabla u  \ge C_2 \big\{ g \Box \nabla
u  \big\}^{\frac{1}{p}}.
\end{gather*}
Repeating the same reasoning as above, we obtain
$$
\mathcal{L}(t) \leq C\{ \mathcal{L}(0)+C_6\} \frac{1}{(1+t)^p}
$$
 From which our result follows. The proof is now complete.
\end{proof}


\subsection*{Acknowledgements} The author is thankful to the
anonymous referee of this paper for his/her valuable suggestions which
improved this paper.

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\end{document}