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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 288, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/288\hfil Asymptotic behaviour of nonlinear wave equations]
{Asymptotic behaviour of nonlinear wave equations in a noncylindrical domain \\
becoming unbounded}

\author[A. Aibeche, S. Hadi, A. Sengouga  \hfil EJDE-2017/288\hfilneg]
{Aissa Aibeche, Sara Hadi, Abdelmouhcene Sengouga}

\address{Aissa Aibeche \newline
Department of Mathematics,
University Setif 1,
Route de Scipion, 19000 Setif, Algeria}
\email{aibeche@univ-setif.dz}

\address{Sara Hadi \newline
Department of Mathematics,
University Setif 1,
Route de Scipion, 19000 Setif, Algeria}
\email{sarra\_math@yahoo.fr}

\address{Abdelmouhcene Sengouga (corresponding author)\newline
Laboratory of Functional Analysis and Geometry of Spaces,
University of M'sila, 28000 M'sila, Algeria}
\email{amsengouga@gmail.com}

\dedicatory{Communicated by Goong Chen}

\thanks{Submitted August 7, 2017. Published November 21, 2017.}
\subjclass[2010]{35B35, 35B40, 35L70}
\keywords{Nonlinear wave equation; asymptotic behaviour in time;
\hfill\break\indent noncylindrical domains}

\begin{abstract}
 We study the asymptotic behaviour for the solution of nonlinear wave
 equations in a noncylindrical domain, becoming unbounded in some directions,
 as the time $t$ goes to infinity. If the limit of the source term is
 independent of these directions and $t$, the wave converges to the solution
 of an elliptic problem defined on a lower dimensional domain. The rate of
 convergence depends on the limit behaviour of the source term and on the
 coefficient of the nonlinear term.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In recent years, there is much interest in evolution problems set in
time-dependent domains. These problems arise in many real world applications
when the spatial domain of the considered phenomena depends strongly on
time, see for instance the survey paper \cite{KnK:15} and the references
cited therein.

Let us denote the points in $\mathbb{R}^{n_1}\times
\mathbb{R}^{n_2}$ as
\begin{equation*}
x=(X_1,X_2)=( x_1,\dots ,x_{n_1},x_1',\dots ,x_{n_2}') ,
\end{equation*}
where $n_1$ and $n_2$ are positive integers. Then we consider a
time-dependent family of bounded subsets in
 $\mathbb{\mathbb{R}}^{n_1}\times\mathbb{R}^{n_2}$ defined as
\begin{equation*}
\Omega _{t}:=( -\ell _0-\ell t,\ell _0+\ell t) ^{n_1}\times
\omega ,\quad t\geq 0,
\end{equation*}
where $\omega $ is a bounded open subset of $\mathbb{R}^{n_2}$
with sufficiently smooth boundary, $\ell _0>0$ and the speed of
expansion $\ell $ is constant. In $\mathbb{R}^{+}\times\mathbb{R}^{n_1+n_2}$,
we obtain the  noncylindrical domain and its
lateral boundary
\begin{equation*}
Q_{t}:=\cup_{0<s<t}\{ s\} \times \Omega _{s},\quad
\Sigma_{t}:=\cup_{0<s<t}\{ s\} \times \partial \Omega _{s},\quad t>0.
\end{equation*}

We are interested in the asymptotic behaviour, as $t\to +\infty $,
of the solution of the following nonlinear wave equation set in $Q_{t}$,
\begin{equation}
\begin{gathered}
u''-\Delta u+\beta u'+\gamma ( t)
| u| ^{\rho }u=f( t,x) , \quad\text{in }Q_{t}, \\
u( t,x) =0,\quad  \text{on }\Sigma _{t}, \\
u( 0,x) =u^{0}( x) ,\quad u'(0,x) =u^1( x) , \quad \text{in }\Omega _0,
\end{gathered}  \label{p1}
\end{equation}
where the prime stands for the time derivative, $\Delta $ is the Laplace
operator, $\beta$ is a positive constant and $\gamma $ is a nonnegative
function.

This study is motivated by some recent works on the asymptotic behaviour of
the solutions of boundary value problems in a domain $\Omega _{\ell }$, when
the size of $\Omega _{\ell }$ becomes unbounded in some directions, as the
parameter $\ell \to +\infty $ (independently of the time). See for
instance \cite{Chip:02,ChR:01,ChY:08,Gue:09} for elliptic and parabolic
problems and \cite{BrG:07,Gue:08a} for hyperbolic problems. In the paper at
hand, we give to $\ell t$ the same role of the parameter $\ell $ in these
papers.

The existence and uniqueness of solutions for wave problems in
noncylindrical domains was considered by several authors, see
 \cite{Lion:69,Med:72,CoB:73,CoM:72,Fer:95,FeL:96,NaN:78} and related works. To
focus on the asymptotic behaviour, we considered Problem \eqref{p1} whose
existence and uniqueness can be established by arguing as in \cite{FeL:96}.

Many works dealt with the asymptotic behaviour in time for the solutions of
evolution problems in noncylindrical domains. Using the multiplier method,
Bardos and Chen \cite{BaC:81} proved that the energy of the linear wave
equation decays when the domain is timelike and expanding. Nakao and
Narazaki \cite{NaN:78} and Rabello \cite{Rab:94} studied the decay of the energy
for weak solutions of nonlinear wave problems in expanding domains. There
idea relays on the penalization method, introduced by Lions \cite{Lion:69}.
Another method consists in considering a suitable change of variables that
transforms the noncylindrical domain to a cylindrical one, establish energy
estimates for the new problem, then derive the desired energy estimates for
the noncylindrical problem, see for instance \cite{HaP:11, LaS:02}. The
drawback of this method is that the differential operator of the transformed
problem is, in general, more complicated.

In this work, we study the problem directly in the noncylindrical domain,
without any change of variables. The idea is based on the use of some
special cut-off functions,\ depending on $( t,X_1) $, to obtain
local estimates of the difference between the wave and its limit. This
technique was recently introduced by Guesmia \cite{Gue:13} for a parabolic
problem in a noncylindrical domain, see also \cite{ChY:08}. Roughly
speaking, if $f( t,x) $ converges to some $f_{\infty }(
X_2) $ and $\gamma ( t) $ converges to $0$, faster enough
in a sense to be made precise later, we obtain the convergence $u(
t) \to u_{\infty }$ in interior regions of the domain $Q_{t}$.
Here $u_{\infty }$ is the solution of an elliptic problem defined on $\omega
$. Then, the rate convergence $u( t) \to u_{\infty }$ is
analysed and improved under some assumptions.

The main features of this work can be summarized as follows:

$\bullet $ In \cite{HaP:11, NaN:78,Rab:94}, the size of the domain is assumed
to remain bounded as $t\to +\infty $ and the limit of the solution
of the considered problem is zero. This situation arises when the decay in
the energy of the solution, due to the expansion of the domain and
damping terms, overtakes the contribution of the source term. In this work,
$\Omega _{t}$ becomes unbounded in $n_1$ directions and the limit of the solution,
in interior regions of the domain, is not
necessarily zero, as $t\to +\infty $. To the best of
our knowledge, the asymptotic behaviour of such problems has not been considered
before.

$\bullet $ In contrast with \cite{Gue:13}, the source term $f$ in this work
depends on all the variables $( t,x) \in \mathbb{R}^{+}\times
( -\ell _0-\ell t,\ell _0+\ell t) ^{n_1}\times \omega $ and
not only on $X_2\in \omega $.

The rest of this article is organized as follows: In the next section, we
state an existence and uniqueness result for $u( t) $, solution
of Problem \eqref{p1}. Then we define $u_{\infty }$, the candidate limit
$u( t)$ as $t\to +\infty $, and the cut-off functions
needed in the sequel. In section 3, we give an energy estimate for
$u(t) $ as well as a local energy estimate for the difference
$u(t) -u_{\infty }$. In the last section, we give the convergence results
and discuss some particular cases where the rate of convergence is
exponential.

\section{Preliminaries}

\subsection{Existence and uniqueness of solutions}

First, let us state our assumptions:

$\bullet $ Concerning the speed of expansion, in the $n_1$ first
directions, it satisfies
\begin{equation}
0\leq \ell \leq 1.  \label{tlike}
\end{equation}
This ensures that $\Sigma _{t}$ satisfies the so-called timelikness condition
\begin{equation*}
| \nu _{t}| \leq | \nu _{x}| \quad\text{on  }\Sigma _{t},\text{ for }t>0,
\end{equation*}
where $\nu _1=(\nu _{t},\nu _{x})$ is the unit outward normal to
$\Sigma_{t}$ and $|\cdot | $ denotes the usual Euclidian norm.

$\bullet $ The nonlinear term in Problem \eqref{p1} is subject to the
following assumptions (Recall that $x\in \mathbb{R}^{n_1+n_2}$)
\begin{gather}
0<\rho \leq \frac{2}{( n_1+n_2) -2}, \text{ if }
n_1+n_2>2,\quad
 0<\rho \leq \infty \text{ if }n_1=n_2=1, \label{4} \\
\gamma \geq 0,\quad \gamma '\leq 0,\quad \gamma ,\gamma '\in L^{\infty }( 0,t ) .
 \label{5}
\end{gather}

$\bullet $ The initial data and the source term satisfy
\begin{equation}
u^{0}\in H_0^2( \Omega _0) ,\quad
u^1\in H_0^1(\Omega _0),\quad
f\in H^1( 0,t ;L^2(\Omega _{s}) ) .  \label{6}
\end{equation}
Then we have the following existence and uniqueness result.

\begin{theorem}\label{thm0}
Let $t>0$. Under the assumptions eqref{tlike}--\eqref{6}
there exists a unique solution for Problem \eqref{p1}, in the sense
that
\begin{equation*}
u\in L^{\infty }( 0,t;H_0^1( \Omega _{s}) \cap H^2( \Omega _{s}) ) ,\quad
u'\in L^{\infty}( 0,t;H^1( \Omega _{s}) ) ,\quad
u''\in L^2( 0,t;L^2( \Omega _{s}) )
\end{equation*}
and we can take $u'$ as a test function, i.e. the following
identity holds
\begin{equation*}
\int_{\Omega _{s}}( u''-\Delta u+\beta u'
+\gamma ( s) | u| ^{\rho }u) u'( s) dx
=\int_{\Omega _{s}}f( s) u'(s) dx,
\end{equation*}
 for a.e. $s\in (0, t)$.
\end{theorem}

\begin{proof}
To express $\Omega _{s}$ using the notation of \cite{FeL:96}, we consider
$K( s) =1+\frac{\ell }{\ell _0}s$. Then $\Omega _{s}$ can also
defined as
\begin{equation*}
\Omega _{s}=\{ ( X_1,X_2) \in \mathbb{R}
^{n_1}\times \omega \text{ }|\text{ }X_1=K( s) Y_1,Y_1\in
( -\ell _0,\ell _0) ^{n_1}\}, \quad s\in (0,t).
\end{equation*}
The rest of the proof becomes similar to the proof of \cite[Theorem 3.1]{FeL:96},
hence it is omitted.
\end{proof}

\subsection{Limit problem}

We set
\begin{gather*}
\nabla _{X_1}u=( \partial _{x_1}u,\dots ,\partial _{x_{n_1}}u) ^{T},\quad
\nabla _{X_2}u=( \partial_{x_1'}u,\dots ,\partial _{x_{n_2}'}u) ^{T}, \\
\nabla u=\begin{pmatrix}\nabla _{X_1}u \\
\nabla _{X_2}u
\end{pmatrix}) ,\quad \nabla _{x,t}u=\begin{pmatrix}
u' \\
\nabla u
\end{pmatrix}
\end{gather*}
and we assume that the source term becomes independent of the variables
$(t,X_1)$, i.e.
\begin{equation*}
f( t,X_1,X_2)  \to f_{\infty }(X_2) ,\quad \text{as }t\to +\infty ,
\end{equation*}
for some
\begin{equation}
f_{\infty }\in L^2( \omega ) .  \label{f00}
\end{equation}
To handle the nonlinear term, in the estimations below, we need to assume that
\begin{equation*}
\gamma ( t) \to 0\quad \text{as }t\to +\infty .
\end{equation*}
The sense of these two convergences will be made precise below.

Passing formally to the limit in \eqref{p1}, one expects the limit problem
to become independent of $( t,X_1) $, as $t\to +\infty $.
More precisely, the candidate limit of $u( t) $, as
$t\to +\infty $, is the solution of the  elliptic problem
defined on $\omega $,
\begin{equation}
\begin{gathered}
-\Delta _{X_2}u_{\infty }=f_{\infty } \quad \text{in }\omega , \\
u_{\infty }=0 \quad \text{on }\partial \omega ,
\end{gathered} \label{p2}
\end{equation}
where $\Delta _{X_2}:=\partial _{x_1'}^2+\dots +\partial
_{x_{n_2}'}^2$. It is well known that Problem \eqref{p2} has a
unique solution $u\in H_0^1( \omega ) $ and one can check
easily that
\begin{equation}
| \nabla _{X_2}u_{\infty }| _{L^2( \omega
) }\leq | f_{\infty }| _{L^2( \omega )}.  \label{u00bound}
\end{equation}

\begin{remark} \label{rmk1} \rm
By the Sobolev embedding theorem (Recall that $\omega \subset
\mathbb{R}^{n_2}$), we have:

$\bullet $ if $n_2\in \{ 1,2\} $, then $H^1( \omega
) \subset L^{\rho +2}( \omega ) $ for $0<\rho \leq \infty $.

$\bullet $ if $n_2\geq 3$, then due to \eqref{4} we have
$0<\rho \leq \frac{2}{( n_1+n_2) -2}$ which implies that
$0<\rho \leq \frac{2}{n_2-2}$, hence
$H^1( \omega ) \subset L^{\rho +2}( \omega ) $.

Therefore, under assumption \eqref{4}, it holds that
\begin{equation*}
| u_{\infty }| _{L^{\rho +2}( \omega ) }\leq
C_{S}| \nabla u_{\infty }| _{L^2( \omega ) },
\end{equation*}
for $n_2\geq 1$ and some constant $C_{S}$ depending only on $\omega $.
Combining this inequality with \eqref{u00bound} we have
\begin{equation}
| u_{\infty }| _{L^{\rho +2}( \omega ) }\leq
C_{S}| f_{\infty }| _{L^2( \omega ) }.
\label{limbound2}
\end{equation}
\end{remark}

\subsection{Special cut-off functions}

To estimate the converge of $u( t) $ towards $u_{\infty} $, we consider the functions
\begin{gather*}
w( t,X_1,X_2)  :=u( t,X_1,X_2) -u_{\infty}( X_2) , \\
F( t,X_1,X_2)  :=f( t,X_1,X_2) -f_{\infty}( X_2) ,
\end{gather*}
for $( X_1,X_2) \in \Omega _{t}$ and $t\geq 0$. Since
$u_{\infty }$ depends only on $X_2$, then the function $w$
satisfies the equation
\begin{equation}
w''-\Delta w+\beta w'+\gamma | u|^{\rho }u=F\quad \text{in }Q_{t},  \label{p3}
\end{equation}
with the initial conditions
\begin{equation*}
w( 0,x) =u^{0}( x) -u_{\infty }( X_2) ,\quad
w'( 0,x) =u^1( x) .
\end{equation*}
Observe that if $u_{\infty }\neq 0$ on $\Sigma _{t}$, then $w\neq 0$ on
$\Sigma _{t}$. As a consequence $w( t) \notin H_0^1(\omega ) $,
hence it is not a valid test function for equation \eqref{p3}.
This motivates the consideration of the next cut-off functions.

For a fixed $t>1$, let $m$ be a integer such that $0\leq m\leq t-1$. On
one hand, we consider the sequence of sets
\begin{equation*}
S_{m}^{t}:=\{ (s,X_1): t-m<s<t,\; |x_i| <\ell _0+\ell ( m-t+s) ,  \text{ for }
 i=1,\dots ,n_1\} .
\end{equation*}
This sequence is increasing in $m$, i.e. $S_{m}^{t}\subset S_{m+1}^{t}$, and
satisfies
\begin{equation*}
\quad S_{m}^{t}\subset \cup_{t-m<s<t}\{ s\} \times
( -\ell _0-\ell s,\ell _0+\ell s) ^{n_1} \subset (t-m,t) \times
\mathbb{R}^{n_1}.
\end{equation*}
On the other hand, we consider a sequence of smooth cut-off functions,
depending on $( s,X_1) $,
\begin{equation*}
\varrho _{m}=\varrho _{m}( s,X_1) :( 0,t) \times
\mathbb{R}^{n_1}\mathbb{\to R}
\end{equation*}
and satisfying
\begin{gather*}
\varrho _{m}=\begin{cases}
1 & \text{in }S_{m}^{t}, \\
0 & \text{in }\{ ( 0,t) \times \mathbb{R}^{n_1}\}\setminus S_{m+1}^{t},
\end{cases}\\
0\leq \varrho _{m}\leq 1,\quad | \nabla _{X_1}\varrho_{m}| ,
| \varrho _{m}'| \leq \theta ,
\end{gather*}
where $\theta $ is a constant independent of $t$ and $m$. We have in
particular $\varrho _{m}( 0,X_1) =0$ and $\varrho _{m}=0$ near
the lateral boundary $\Sigma _{t}$.
The supports of $\nabla _{X_1}\varrho_{m}\ $and $\varrho _{m}'$
are included in $S_{m+1}^{t}\backslash S_{m}^{t}$.

\section{Energy Estimates}

In this section, we establish some useful lemmas needed in the sequel. The
first one gives an estimation for $u$ and its derivatives.

\begin{lemma} \label{lem1} Under the assumptions \emph{\eqref{tlike}--\eqref{6},}
the solution of Problem \emph{\eqref{p1}} satisfies,
\begin{align*}
&\int_{\Omega _{t}}| u'( t) |
^2+| \nabla u( t) | ^2+\frac{\gamma (
t) }{\rho +2}| u( t) | ^{\rho
+2}dx+\int_{Q_{t}}\beta | u'| ^2+
\frac{2| \gamma '| }{\rho +2}|
u| ^{\rho +2}\,dx\,ds \\
&\leq C_0\Big( 1+| f| _{L^2( Q_{t})
}^2\Big) ,\quad \text{for }t> 0 ,
\end{align*}
where $C_0$ is a positive constant independent of $t$.
\end{lemma}

\begin{proof}
Since the solutions $u$ satisfies $u=0$ on $\Sigma _{t}$, then all the
tangential derivatives of $u$ are also vanishing on $\Sigma _{t}$, so
$\nabla _{x,t}u=\frac{\partial u}{\partial \nu }\ \nu ,$\ on $\Sigma _{t}$,
which implies that
\begin{equation*}
u'=\frac{\partial u}{\partial \nu }\nu _{t},\quad
\nabla u=\frac{\partial u}{\partial \nu }\nu _{x},\quad \text{on }\Sigma _{t}.
\end{equation*}
Thanks to Theorem \ref{thm0}, we can take $u'$ as a test function
and arguing as in \cite{BaC:81}, we obtain
\begin{align*}
&\frac{1}{2}\int_{\Omega _{t}} | u'(t) | ^2+| \nabla u( t) |
^2+\frac{\gamma ( t) }{\rho +2}| u( t)
| ^{\rho +2}dx+\int_{Q_{t}}\beta | u'| ^2-\frac{\gamma '}{\rho +2}| u|
^{\rho +2}\,dx\,ds \\
&= \frac{1}{2}\int_{\Omega _0}| u^1|
^2+| \nabla u^{0}| ^2+\frac{\gamma ( 0) }{
\rho +2}| u^{0}| ^{\rho+2}dx
+\int_{Q_{t}}fu'\,dx\,ds \\
&\quad +\frac{1}{2}\int_{\Sigma _{t}}( \frac{\partial u}{\partial
\nu }) ^2\text{ }\nu _{t}\text{ }( | \nu
_{x}| ^2-\nu _{t}^2) \text{ }d\sigma ,
\end{align*}
for $t>0$. Using the fact that $| \nu _{t}| \leq | \nu _{x}| $ on
$\Sigma _{t}$ and noting that $\nu _{t}\leq 0$ for expanding domains, we
infer that the
boundary integral term in the right-hand side is nonpositive.
Then applying Young's inequality
$fu'\leq \frac{\beta }{2}( u') ^2+\frac{1}{2\beta }f^2$, we obtain
\begin{align*}
&\int_{\Omega _{t}}| u'( t) |^2+| \nabla u( t) | ^2
 +\frac{\gamma( t) }{\rho +2}| u( t) | ^{\rho+2}dx
 +\int_{Q_{t}}\beta | u'| ^2+\frac{2| \gamma '| }{\rho +2}| u| ^{\rho +2}\,dx\,ds \\
& \leq \int_{\Omega _0}| u^1|^2+| \nabla u^{0}| ^2+\frac{\gamma (0)}{\rho +2}
| u^{0}| ^{\rho +2}dx
+\frac{1}{\beta }\int_{Q_{t}}f^2\,dx\,ds.
\end{align*}
This completes the proof.
\end{proof}

The second lemma, gives an estimation for the difference
$u( t)-u_{\infty }$ in interior regions of $\Omega_{t}$ and $Q_{t}$.
For simplicity, we set
\begin{equation}
D( t,x) :=| w'( t,x) |^2+| \nabla w( t,x) | ^2+\gamma (
t) | u( t,x) | ^{\rho +2},\quad \text{for }x\in \Omega _{t},\; t\geq 0.  \label{dtx}
\end{equation}
Then we have the following energy inequality.

\begin{lemma} \label{lem2}
Under  assumptions \eqref{tlike}--\eqref{f00}, the
solutions of Problem \eqref{p1} and Problem \eqref{p2} satisfy
\begin{align*}
&\int_{\Omega _{t}}D( t) \varrho _{m}^2( t)
dx+\int_{S_{m}^{t}\times \omega }D\ \,dx\,ds\\
& \leq C_1\int_{(S_{m+1}^{t}\backslash S_{m}^{t}) \times \omega }
D\,dx\,ds
 +C_1\int_{S_{m+1}^{t}\times \omega }F^2+\gamma |
u_{\infty }| ^{\rho +2}\,dx\,ds,\quad \text{for a.e. } t> 0,
\end{align*}
for some positive constant $C_1$ independent of $t$.
\end{lemma}

\begin{proof}
To derive local energy estimates, we  use  $\varrho _{m}$
and its proprieties.

$\bullet $ \emph{A local energy identity.} Let us multiply \eqref{p3} by
 $2w\varrho _{m}^2$, it yields
\begin{align*}
&\frac{\partial }{\partial s}( \beta \varrho _{m}^2w^2+2\varrho
_{m}^2ww')  -2\beta \varrho _{m}'\varrho
_{m}w^2-2\varrho _{m}^2| w'| ^2-4\varrho
_{m}'\varrho _{m}ww'+2\gamma | u|
^{\rho }uw\varrho _{m}^2 \\
& +2\varrho _{m}^2| \nabla w| ^2-2\nabla \cdot (
\varrho _{m}^2w\nabla w) +4\varrho _{m}w( \nabla \varrho
_{m}\cdot \nabla w) =2w\varrho _{m}^2F.
\end{align*}
Then, multiplying \eqref{p3} by $2\alpha w'\varrho _{m}^2$,
for some constant $\alpha >0$, yields
\begin{align*}
&\frac{\partial }{\partial s} \Big( \alpha \varrho _{m}^2|
w'| ^2+\alpha \varrho _{m}^2| \nabla
w| ^2+\frac{2\alpha \gamma }{\rho +2}| u|
^{\rho +2}\varrho _{m}^2\Big) \\
& -2\alpha \varrho _{m}'\varrho _{m}| w'| ^2+2\alpha \beta \varrho _{m}^2| w'| ^2
-\frac{2\alpha \gamma '}{\rho +2}|
u| ^{\rho +2}\varrho _{m}^2-\frac{4\alpha \gamma }{\rho +2}
| u| ^{\rho +2}\varrho _{m}'\varrho _{m} \\
& -2\alpha \varrho _{m}'\varrho _{m}| \nabla w|
^2-2\alpha \nabla \cdot ( \varrho _{m}^2w'\nabla w)
+4\alpha \varrho _{m}w'( \nabla \varrho _{m}\cdot \nabla
w) =2\alpha w'\varrho _{m}^2F.
\end{align*}
Summing  the above identities, we obtain
\begin{align*}
&\frac{\partial }{\partial s} \Big( \beta \varrho _{m}^2w^2+2\varrho
_{m}^2ww'+\alpha \varrho _{m}^2| w'| ^2+\alpha \varrho _{m}^2| \nabla w| ^2+
\frac{2\alpha \gamma }{\rho +2}| u| ^{\rho +2}\varrho
_{m}^2\Big) \\
& -2\varrho _{m}^2| w'| ^2+2\alpha \beta
\varrho _{m}^2| w'| ^2+2\varrho
_{m}^2| \nabla w| ^2-2\alpha \varrho _{m}'\varrho _{m}| \nabla w| ^2 \\
& +2\gamma | u| ^{\rho +2}\varrho _{m}^2-2\gamma
| u| ^{\rho }uu_{\infty }\varrho _{m}^2-\frac{2\alpha
\gamma '}{\rho +2}| u| ^{\rho +2}\varrho
_{m}^2-\frac{4\alpha \gamma }{\rho +2}| u| ^{\rho
+2}\varrho _{m}'\varrho _{m} \\
& -2\beta \varrho _{m}'\varrho _{m}w^2-4\varrho _{m}'\varrho _{m}ww'-2\alpha \varrho _{m}'\varrho
_{m}| w'| ^2-2\nabla \cdot ( \varrho
_{m}^2w\nabla w) +4\varrho _{m}w( \nabla \varrho _{m}\cdot
\nabla w) \\
& -2\alpha \nabla \cdot ( \varrho _{m}^2w'\nabla w)
+4\alpha \varrho _{m}w'( \nabla \varrho _{m}\cdot \nabla
w) =2w\varrho _{m}^2F+2\alpha w'\varrho _{m}^2F.
\end{align*}
Collecting the terms with derivatives of $\varrho $ in the right-hand
side of the above identity, we obtain
\begin{align*}
&\frac{\partial }{\partial s} \Big( \beta \varrho _{m}^2w^2+2\varrho
_{m}^2ww'+\alpha \varrho _{m}^2| w'| ^2+\alpha \varrho _{m}^2| \nabla w| ^2+
\frac{2\alpha \gamma }{\rho +2}| u| ^{\rho +2}\varrho
_{m}^2\Big) \\
& 2( \alpha \beta -1) \varrho _{m}^2| w'| ^2+2\varrho _{m}^2| \nabla w|
^2+2\big( \gamma -\frac{\alpha \gamma '}{\rho +2}\big)
| u| ^{\rho +2}\varrho _{m}^2 \\
&= 2\beta \varrho _{m}'\varrho _{m}w^2+4\varrho _{m}'\varrho _{m}ww'
 +2\alpha \varrho _{m}'\varrho
_{m}| w'| ^2+2\alpha \varrho _{m}'\varrho _{m}| \nabla w| ^2 \\
&\quad -4\varrho _{m}w( \nabla \varrho _{m}\cdot \nabla w) -4\alpha
\varrho _{m}w'( \nabla \varrho _{m}\cdot \nabla w)
+2\alpha \nabla \cdot ( \varrho _{m}^2w'\nabla w) \\
&\quad +2\nabla \cdot ( \varrho _{m}^2w\nabla w) +\frac{4\alpha }{
\rho +2}| u| ^{\rho +2}\gamma \varrho _{m}'\varrho _{m}+2\gamma ( | u| ^{\rho }u)
u_{\infty }\varrho _{m}^2\\
&\quad +2w\varrho _{m}^2F+2\alpha w'\varrho_{m}^2F.
\end{align*}
Integrating on $Q_{t}$ and taking into account the fact that $\varrho _{m}=0$
for $t=0$ and on $\Sigma _{t}$, we end up with the  identity
\begin{equation}
\begin{aligned}
&\int_{\Omega _{t}} \Big( \beta w^2( t) +2ww'(
t) +\alpha | w'( t) |
^2+| \nabla w( t) | ^2+\frac{2\alpha
\gamma ( t) }{\rho +2}| u( t) |
^{\rho +2}\Big) \varrho _{m}^2(t)dx  \notag \\
& +\int_{Q_{t}}2( \alpha \beta -1) \varrho
_{m}^2| w'| ^2+2\varrho _{m}^2|
\nabla w| ^2+2( \gamma -\frac{\alpha \gamma '}{
\rho +2}) | u| ^{\rho +2}\varrho _{m}^2\,dx\,ds
\notag \\
&= \int_{Q_{t}}2\beta \varrho _{m}'\varrho
_{m}w^2+4\varrho _{m}'\varrho _{m}ww'+2\alpha \varrho
_{m}'\varrho _{m}| w'| ^2+2\alpha
\varrho _{m}'\varrho _{m}| \nabla w| ^2 \notag \\
&\quad +\frac{4\alpha \gamma }{\rho +2}| u| ^{\rho +2}\varrho
_{m}'\varrho _{m}\,dx\,ds
 -\int_{Q_{t}}4\varrho _{m}w( \nabla \varrho _{m}\cdot \nabla
w) -4\alpha \varrho _{m}w'( \nabla \varrho _{m}\cdot
\nabla w) \,dx\,ds  \notag \\
&\quad +\int_{Q_{t}}2\gamma ( | u| ^{\rho
}u) u_{\infty }\varrho _{m}^2\,dx\,ds+\int_{Q_{t}}2w\varrho
_{m}^2F+2\alpha w'\varrho _{m}^2F\,dx\,ds.
\end{aligned}  \label{local=}
\end{equation}

$\bullet $ \emph{Estimate for the left-hand side of \eqref{local=}.} Using the
inequality
\begin{equation*}
2ww'\geq -\Big( \beta w^2+\frac{1}{\beta }| w'| ^2\Big) ,
\end{equation*}
then choosing $\alpha >1/\beta $, we obtain
\[
\beta \varrho _{m}^2w^2+2\varrho _{m}^2ww'+\alpha \varrho
_{m}^2| w'| ^2+\alpha \varrho
_{m}^2| \nabla w| ^2\geq \delta _0\varrho
_{m}^2| w'| ^2+\alpha \varrho
_{m}^2| \nabla w| ^2,
\]
where $\delta _0=( \alpha -\frac{1}{\beta }) >0$. Integrating
on $Q_{t}$ , and taking into account that $\gamma '\leq 0$, we
deduce that the left-hand side of \eqref{local=} is bounded below by
\begin{align*}
&\int_{\Omega _{t}}\Big( \delta _0| w'( t)
| ^2+\alpha | \nabla w( t) | ^2+
\frac{2\alpha \gamma ( t) }{\rho +2}| u( t)
| ^{\rho +2}\Big) \varrho _{m}^2( t) dx \\
&+2\int_{Q_{t}}\Big( \beta \delta _0| w'| ^2+| \nabla w| ^2+\big( \gamma +\frac{
\alpha | \gamma '| }{\rho +2}\big) |
u| ^{\rho +2}\Big) \varrho _{m}^2\,dx\,ds.
\end{align*}

$\bullet $ \emph{Estimate for the right-hand side of \eqref{local=}.}
Given that the supports of $\varrho _{m}'$ and
$| \nabla \varrho _{m}| $ are included in the set
$S_{m+1}^{t}\backslash S_{m}^{t}$, the right-hand side of \eqref{local=} can
be estimated above by
\begin{align*}
&c_0\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times \omega }
| w'| ^2+| w|
^2+| \nabla w| ^2+\gamma | u|
^{\rho +2}\,dx\,ds \\
& +\int_{Q_{t}}2\gamma ( | u| ^{\rho
}u) u_{\infty }\varrho _{m}^2\,dx\,ds
+\int_{Q_{t}}2w\varrho _{m}^2F+2\alpha w'\varrho _{m}^2F\,dx\,ds.
\end{align*}
Here and in the sequel, $c_i$ denotes positive constants depending (at
most) on $\theta ,\alpha $ and $\omega $, but not on $t$. To estimate the
second integral, containing $( | u| ^{\rho }u)
u_{\infty }$, we apply Young's inequality $ab\leq \frac{\varepsilon a^{p}}{p}
+\frac{1}{\varepsilon ^{q/p}}\frac{b^{q}}{q}$ for $p=$ $\frac{\rho +2}{\rho
+1}$, $q=\rho +2$ and $\varepsilon \in ( 0,1) $. We obtain
\begin{equation*}
( | u| ^{\rho }u) u_{\infty }\leq \frac{(
\rho +1) \varepsilon }{\rho +2}| u| ^{\rho +2}+
\frac{1}{( \rho +2) \varepsilon ^{( \rho +1) }}
| u_{\infty }| ^{\rho +2}.
\end{equation*}
The same inequality, for $p=q=2$, yields
\begin{gather*}
2w\varrho _{m}^2F+2\alpha w'F \leq \varepsilon (
w^2+| w'| ^2) +\frac{1+\alpha ^2}{
\varepsilon }F^2, \\
2ww' \leq w^2+| w'| ^2, \\
2w| \nabla w| \leq w^2+| \nabla w|^2.
\end{gather*}
Then, the right-hand side of \eqref{local=} is bounded above by
\begin{align*}
&c_0\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times \omega }
| w'| ^2+| w|
^2+| \nabla w| ^2+\gamma | u|
^{\rho +2}\,dx\,ds \\
& +c_1\varepsilon \int_{Q_{t}}( | w'|
^2+| w| ^2+\gamma | u| ^{\rho
+2}) \varrho _{m}^2\,dx\,ds
+\frac{c_1}{\varepsilon ^{( \rho
+1) }}\int_{Q_{t}}( F^2+\gamma | u_{\infty
}| ^{\rho+2}) \varrho _{m}^2\,dx\,ds.
\end{align*}
Since $\omega $ is bounded, then Poincar\'{e}'s inequality in the
$X_2$-direction yields
\begin{equation*}
\int_{\Omega _{t}}| w( t) |^2\varrho _{m}^2(t)dx
\leq c_{\omega }^2\int_{\Omega_{t}}| \nabla _{X_2}w( t) | ^2\varrho
_{m}^2(t)dx\leq c_{\omega }^2\int_{\Omega _{t}}| \nabla
w( t) | ^2\varrho _{m}^2(t)dx,
\end{equation*}
where $c_{\omega }\ $is the Poincar\'{e} constant. Thus the right-hand side
of \eqref{local=} is bounded above by
\begin{align*}
&c_2\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times \omega}| w'| ^2+| \nabla w|
^2+\gamma | u| ^{\rho +2}\,dx\,ds \\
&+c_2\varepsilon \int_{Q_{t}}( | w'|^2+| \nabla w| ^2+\gamma | u|
^{\rho +2}) \varrho _{m}^2\,dx\,ds+\frac{c_2}{\varepsilon ^{(
\rho +1) }}\int_{Q_{t}}( F^2+\gamma |
u_{\infty }| ^{\rho +2}) \varrho _{m}^2\,dx\,ds.
\end{align*}

$\bullet $ \emph{End of proof.}
The estimations of the two sides of \eqref{local=} yields
\begin{align*}
& \int_{\Omega _{t}}\Big( \delta _0| w'( t)
| ^2+\alpha | \nabla w( t) | ^2+
\frac{2\alpha \gamma ( t) }{\rho +2}| u( t)
| ^{\rho +2}\Big) \varrho _{m}^2( t) dx \\
&  +2\int_{Q_{t}}\Big( \beta \delta _0| w'| ^2
 +| \nabla w| ^2+\big( \gamma +\frac{\alpha | \gamma '| }{\rho +2}\big) |
u| ^{\rho +2}\Big) \varrho _{m}^2\,dx\,ds \\
& \leq c_2\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times
\omega }| w'| ^2+| \nabla
w| ^2+\gamma | u| ^{\rho +2}\,dx\,ds \\
& \quad +c_2\varepsilon \int_{Q_{t}}\big( | w'|
^2+| \nabla w| ^2+\gamma | u|
^{\rho +2}\big) \varrho _{m}^2\,dx\,ds \\
&\quad +\frac{c_2}{\varepsilon ^{(\rho +1) }}\int_{Q_{t}}\big( F^2+\gamma |
u_{\infty }| ^{\rho +2}\big) \varrho _{m}^2\,dx\,ds.
\end{align*}
For $\varepsilon $ small enough, we end up with
\begin{align*}
&\int_{\Omega _{t}} \big( | w'( t) |
^2+| \nabla w( t) | ^2+\gamma (
t) | u( t) | ^{\rho +2}\big) \varrho_{m}^2( t) dx \\
&+\int_{Q_{t}}( | w'| ^2+| \nabla w| ^2+\gamma |
u| ^{\rho +2}) \varrho _{m}^2\,dx\,ds \\
& \leq c_{3}\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times
\omega }| w'| ^2+| \nabla w| ^2+\gamma | u| ^{\rho +2}\,dx\,ds \\
&\quad +c_{3}\int_{Q_{t}}( F^2+\gamma | u_{\infty
}| ^{\rho +2}) \varrho _{m}^2\,dx\,ds.
\end{align*}
This completes the proof.
\end{proof}

\begin{remark} \label{rmk2} \rm
Thanks to Inequality \eqref{limbound2}, we obtain
\begin{align*}
&\int_{S_{m+1}^{t}\times \omega }\gamma | u_{\infty }|
^{\rho +2}\varrho _{m}^2\,dx\,ds=| u_{\infty }| _{L^{\rho
+2}( \omega ) }^{\rho +2}\int_{S_{m+1}^{t}}\gamma \varrho
_{m}^2dX_1ds \\
&\leq C_{S}^{\rho +2}| f_{\infty }|
_{L^2( \omega ) }^{\rho +2}\int_{S_{m+1}^{t}}\gamma \varrho
_{m}^2dX_1ds
\end{align*}
and since $0\leq \varrho _{m}\leq 1$, we obtain
\begin{equation*}
\int_{Q_{t}}\gamma | u_{\infty }| ^{\rho +2}\varrho
_{m}^2\ \,dx\,ds\leq C_{S}^{\rho +2}| f_{\infty }|
_{L^2( \omega ) }^{\rho +2}2^{n_1}( \ell _0+\ell
t) ^{n_1}\int_{t-m-1}^{t}\gamma ( s) ds.
\end{equation*}
Thus
\begin{equation}
\int_{S_{m+1}^{t}\times \omega }\gamma | u_{\infty }|
^{\rho +2}\varrho _{m}^2\,dx\,ds\leq C_2( \ell _0+\ell t)
^{n_1}\int_{t-m-1}^{t}\gamma ( s) ds
\end{equation}
where $C_2$ is a constant independent of $t$ and $m$.
\end{remark}

\section{Main Results}

In this section, we establish the convergence $u(t)\to u_{\infty }$, in
bounded interior region of $\Omega _{t}$ and $Q_{t}$, under some
assumptions involving the asymptotic behaviour of $f$ and $\gamma $ as
$t\to +\infty $.

\subsection{Convergence theorems}

Let us consider the nonnegative real function
\begin{equation}
g_0( t) :=\sum_{j=1}^{[ t] -1}(
k^{j}\int_{S_{j+1}^{t}\times \omega }| f-f_{\infty }|
^2+\gamma | u_{\infty }| ^{\rho +2}\,dx\,ds),\quad t\geq 2,  \label{g0}
\end{equation}
where $[ \cdot ] $ denotes the integer part and
$k:=C_1/(1+C_1)$, ($C_1>0$is the
constant considered in Lemma \ref{lem2}). Then, we have the
following convergence on $S_1^{t}\times \omega $.

\begin{theorem}\label{thm1}
Assume  \eqref{tlike}--\eqref{f00} and
\begin{gather}
g_0( t) \to 0,\quad \text{as }t\to +\infty ,  \label{g} \\
t| f| _{L^2( Q_{t}) }^2 =o( e^{\mu _0t}) ,\quad \text{as }t\to +\infty   \label{f}
\end{gather}
where $\mu _0:=\ln ( 1+\frac{1}{C_1}) $. Then we have
\begin{gather}
u'\to 0,\quad \nabla _{X_1}u\to 0,\quad
\nabla _{X_2}u\to \nabla _{X_2}u_{\infty }\quad \text{in }
L^2( S_1^{t}\times \omega ) , \label{conv1}\\
\gamma ^{\frac{1}{\rho +2}}u\to 0\quad \text{in }L^{\rho +2}(
S_1^{t}\times \omega ) , \label{conv2}
\end{gather}
as $t\to +\infty$. Moreover, if $f=f_{\infty }$ and $\gamma =0$,
the above convergences are exponential.
\end{theorem}

\begin{proof}
The main idea is an iteration technique on the increasing sequence of sets
$S_{m}^{t}\times \omega $. First, we observe that
\begin{equation*}
\int_{( S_{m+1}^{t}\backslash S_{m}^{t}) \times \omega }D
\,dx\,ds
=\int_{S_{m+1}^{t}\times \omega }D\ \,dx\,ds
-\int_{S_{m}^{t}\times \omega}D\ \,dx\,ds
\end{equation*}
and therefore Lemma \ref{lem2} yields in particular
\begin{equation*}
( 1+C_1) \int_{S_{m}^{t}\times \omega }D\ \,dx\,ds\leq
C_1\int_{S_{m+1}^{t}\times \omega }D\,dx\,ds
+C_1\int_{S_{m+1}^{t}\times \omega }F^2+\gamma |u_{\infty }| ^{\rho +2}\,dx\,ds.
\end{equation*}
Since $k=\dfrac{C_1}{1+C_1}$, then $0<k<1$ and we can rewrite the precedent
inequality as
\begin{equation}
\int_{S_{m}^{t}\times \omega }D\ \,dx\,ds\leq
k\int_{S_{m+1}^{t}\times \omega }D\,dx\,ds
+k\int_{S_{m+1}^{t}\times \omega }F^2+\gamma |
u_{\infty }| ^{\rho +2}\,dx\,ds.  \label{k}
\end{equation}
This is an inequality that we can iterate for $m=1,\dots ,[t]-1$. It follows that
\begin{align*}
\int_{S_1^{t}\times \omega }D\ \,dx\,ds
& \leq k\int_{S_2^{t}\times \omega}D\ \,dx\,ds
+k\int_{S_2^{t}\times \omega }\big( F^2+\gamma |
u_{\infty }| ^{\rho +2}\Big) \,dx\,ds \\
& \leq k^2\int_{S_{3}^{t}\times \omega }D\ \,dx\,ds
+\sum_{j=1}^2(k^{j}\int_{S_{1+j}^{t}\times \omega }F^2+\gamma | u_{\infty
}| ^{\rho +2}\,dx\,ds) \\
& \dots \\
& \leq k^{[ t] -1}\int_{S_{[ t] }^{t}\times \omega }D\
\,dx\,ds+\sum_{j=1}^{^{[ t] -1}}( k^{j}\int_{S_{1+j}^{t}\times
\omega }F^2+\gamma | u_{\infty }| ^{\rho +2}\,dx\,ds).
\end{align*}
Note that $t-1<[ t] \leq t$ and $\mu _0=-\ln k>0$. Then
$k^{[ t] -1}=e^{( [ t] -1) \ln k}=e^{-\mu
_0( [ t] -1) }$ and it follows that
\begin{equation}
\begin{aligned}
&\int_{S_1^{t}\times \omega }D\ \,dx\,ds\\
&\leq c_{5}e^{-\mu _0t}\int_{S_{[t] }^{t}\times \omega }D\,dx\,ds
+\sum_{j=1}^{[ t] -1}\Big(k^{j}\int_{S_{1+j}^{t}\times \omega }F^2+\gamma | u_{\infty
}| ^{\rho +2}\,dx\,ds\Big) .
\end{aligned} \label{c5}
\end{equation}
To estimate the first integral term in the right-hand side of \eqref{c5},
we write
\begin{align*}
\int_{S_{[ t] }^{t}\times \omega }D \,dx\,ds
&\leq \int_{Q_{t}}D\,dx\,ds \\
& \leq \int_{Q_{t}}| u'| ^2+| \nabla
u| ^2+| \nabla _{X_2}u_{\infty }|
^2+\gamma | u| ^{\rho +2}\,dx\,ds \\
& \leq \int_{Q_{t}}| u'| ^2+| \nabla
u| ^2+\gamma | u| ^{\rho +2}\,dx\,ds\\
&\quad +|\nabla _{X_2}u_{\infty }| _{L^2( \omega )
}^2\int_0^{t}( \int_{( -\ell _0-\ell s,\ell _0+\ell s) ^{n_1}}dX_1) ds.
\end{align*}
Taking into account Lemma \ref{lem1} and \eqref{u00bound}, it follows that
\begin{align*}
\int_{S_{[ t] }^{t}\times \omega }D\ \,dx\,ds
&\leq c_{6}t( 1+| f| _{L^2( Q_{t})}^2) +\frac{2^{n_1}}{\ell ( n_1+1) }|
f_{\infty }| _{L^2( \omega )
}^2( \ell _0 +\ell t) ^{n_1+1} \\
&\leq c_{7}( t^{n_1+1}|f_{\infty }| _{L^2( \omega )
}^2+t| f| _{L^2(Q_{t}) }^2)
\end{align*}
for large $t$. Substituting this in \eqref{c5} and expending the expression
of $D( t,x) $, we obtain
\begin{equation}
\begin{aligned}
&\int_{S_1^{t}\times \omega }| u'|
^2+| \nabla _{X_1}u| ^2 +| \nabla
_{X_2}( u-u_{\infty }) | ^2+\gamma |
u| ^{\rho +2}\,dx\,ds   \\
& \leq c_{8}\big( t^{n_1+1}|
f_{\infty }| _{L^2( \omega )
}^2+t| f| _{L^2(
Q_{t}) }^2\big) e^{-\mu _0t}+g_0( t)
\end{aligned}   \label{c7}
\end{equation}
where $g_0$ is the function given by \eqref{g0}. Since \eqref{g} and
\eqref{f} ensure that the left-hand side of \eqref{c7} tends to zero, as
$t\to +\infty $, then the convergences \eqref{conv1} and ( \ref{conv2}) follow.

If $f=f_{\infty }$ and $\gamma =0$ then $g_0=0$ and
$|f| _{L^2( Q_{t}) }^2$ grows polynomially in time, hence the
claimed exponential convergences are a consequence of \eqref{c7}. This completes
the proof.
\end{proof}

\begin{remark} \label{rmk3} \rm
(i) The source term $f$ satisfies \eqref{f} for example when
$|f| _{L^2( \Omega _{t}) }$ is bounded or grows
polynomially in time.

(ii) The function $g_0$ satisfies \eqref{g} if the convergences
 $f( t) \to f_{\infty }$, $\gamma ( t) \to 0$, as $t\to +\infty $,
are strong enough. Some examples
are given below.

(iii) If $f_{\infty }=0$, and by consequence $u_{\infty }=0$, then $g_0$
does not depend on $\gamma $. In this case, Theorem \ref{thm1} holds
without any convergence assumption of $\gamma ( t) $ towards $0$.
\end{remark}

The next corollary gives the convergence on the domain $\Omega _1$.

\begin{corollary} \label{cor1}
Under  assumptions \eqref{tlike}--\eqref{f00}, \eqref{g} and \eqref{f}, we have
\begin{gather*}
u'( t) \to 0,\quad \nabla _{X_1}u(t) \to 0,\quad \nabla _{X_2}u( t)
\to \nabla _{X_2}u_{\infty }\quad \text{in }L^2( \Omega_1) , \\
\gamma ( t) ^{\frac{1}{\rho +2}}u( t) \to 0 \quad
\text{in }L^{\rho +2}( \Omega _1) ,
\end{gather*}
as $t\to +\infty $. Moreover, if $f=f_{\infty }$ and $\gamma =0$,
the above convergences are exponential.
\end{corollary}

\begin{proof}
Using Lemma \ref{lem2}, we have in particular for $m=1$,
\begin{align*}
\int_{\Omega _1}D( t) dx
&\leq \int_{\Omega _{t}}D( t) \varrho _1^2( t) dx \\
&\leq C_1\int_{S_2^{t}\times \omega }D\,dx\,ds
+C_1\int_{S_2^{t}\times \omega }F^2+\gamma
| u_{\infty }| ^{\rho +2}\,dx\,ds.
\end{align*}
Then we can estimate the integral
$\int_{S_2^{t}\times \omega }D \,dx\,ds$ by using the above iteration
technique for $m=2,\dots ,[ t] -1$.
Arguing as in the proof of Theorem \ref{thm1}, we end up with
\begin{equation*}
\int_{\Omega _1}D( t) dx\leq c_{9}(t^{n_1+1}|
f_{\infty }| _{L^2( \omega )}^2+t| f| _{L^2( Q_{t}) }^2)
e^{-\mu _0t}+g_0( t) .
\end{equation*}
Hence the corollary follows.
\end{proof}

\subsection{Convergence in arbitrary interior regions}

The assumptions \eqref{g} and \eqref{f} can be considerably weakened to
involve only the asymptotic behaviours of $f$ and $\gamma $ for large $t$.
Moreover, we show that the above convergences hold for an arbitrary interior
bounded region of $\Omega _{t}$ and $Q_{t}$.

Let $O$ be a bounded subset of $\mathbb{R}^{n_1}\times \omega $ and $a$
 be a positive constant. Since $\Omega _{t}$
is increasing in time and becomes unbounded in the $X_1$ direction, as
$t\to +\infty $, then there exists $m_0>a $ such that
\begin{equation}
( t-a,t) \times O\Subset ( t-m_0,t) \times \Omega _{m_0},  \label{m0}
\end{equation}
and we can check that
\begin{equation*}
( t-m_0,t) \times \Omega _{m_0}\Subset
S_{2m_0}^{t}\times \omega ,\text{ \ for }t>2m_0.
\end{equation*}
Let us consider the function
\begin{equation}
g_{m_0}( t) :=\sum_{j=2m_0+1}^{[t/2]} \Big( k^{j}
\int_{S_{1+j}^{t}\times \omega }| f-f_{\infty
}| ^2+\gamma | u_{\infty }| ^{\rho
+2}\,dx\,ds\Big) .  \label{gm0}
\end{equation}
Then, we have the following convergences on $( t-a,t) \times O$.

\begin{theorem}\label{thm1b}
Under the assumptions \eqref{tlike}--\eqref{f00} and
\begin{equation}
g_{m_0}( t) \to 0\text{ and  }t|f| _{L^2( Q_{t}) }^2=o( e^{\frac{\mu _0}{2}
t}) ,\quad \text{as }t\to +\infty ,  \label{gm}
\end{equation}
we have
\begin{gather*}
u'\to 0,\quad \nabla _{X_1}u\to 0,\quad
\nabla _{X_2}u\to \nabla _{X_2}u_{\infty }\quad \text{in }
L^2( ( t-a,t) \times O) , \\
\gamma ^{\frac{1}{\rho +2}}u\to 0\quad \text{in }
L^{\rho +2}(( t-a,t) \times O) ,
\end{gather*}
as $t\to +\infty $. Moreover, if $f=f_{\infty }$ and $\gamma =0$,
the above convergences are exponential.
\end{theorem}

\begin{proof}
Let us take $t>4m_0+2$, i.e., $[t/2] >2m_0$. Since
$( t-a,t) \times O\subset \subset S_{2m_0}^{t}\times \omega $,
then iterating Inequality \eqref{k} for $m=2m_0,\dots $,
$[t/2]-1$, we obtain
\begin{align*}
&\int_{( t-a,t) \times O}D\ \,dx\,ds \\
&\leq \int_{S_{2m_0}^{t}\times \omega }D\ \,dx\,ds   \\
&\leq k^{[t/2] -2m_0}\int_{S_{[ \frac{t}{2}
] }^{t}\times \omega }D\ \,dx\,ds+\sum_{j=2m_0+1}^{^{[ \frac{t}{2}
] }}( k^{j-2m_0}\int_{S_{j}^{t}\times \omega }F^2+\gamma
| u_{\infty }| ^{\rho +2}\,dx\,ds)
\end{align*}
hence
\begin{equation}
\int_{( t-a,t) \times O}D\,dx\,ds
\leq c_{10}\Big( (
t^{n_1+1}|
f_{\infty }| _{L^2( \omega )
}^2+t| f| _{L^2( Q_{t}) }^2)
e^{-\frac{\mu _0}{2}t}+g_{m_0}( t) \Big)  \label{c8}
\end{equation}
where $c_{10}>0$ and $g_{m_0}$ is defined by \eqref{gm0}. Under the
assumption \eqref{gm}, the right-hand side tends to zero, as
$t\to +\infty $, and the theorem follows.
\end{proof}

\begin{remark} \label{rmk4} \rm
In contrast with $g_0$ defined in $\eqref{g0}$, by a sum that involves the
values of $f-f_{\infty }$ and $\gamma $ on
 $S_{[ t] }^{t}\times \omega $ (which is identical to $Q_{t}$ if $t$
is an integer), the
function $g_{m_0}$ involves only the values of $f-f_{\infty }$ and
$\gamma $ on $S_{[t/2] +1}^{t}\times \omega $, included in the
strip $( \frac{t}{2}-1,t) \times \mathbb{R}^{n_1}\times \omega $.
\end{remark}

\begin{corollary} \label{cor2}
Under the assumptions \eqref{tlike}--\eqref{f00} and
\eqref{gm}, we have
\begin{gather*}
u'( t) \to 0,\quad \nabla _{X_1}u(t) \to 0,\quad
\nabla _{X_2}u( t)\to \nabla _{X_2}u_{\infty }\quad \text{in }L^2( O) ,\\
\gamma ^{\frac{1}{\rho +2}}u( t) \to 0\quad \text{in }
L^{\rho +2}( O) ,
\end{gather*}
as $t\to +\infty $. Moreover, if $f=f_{\infty }$ and $\gamma =0$,
the above convergences are exponential.
\end{corollary}

\begin{proof}
Using Lemma \ref{lem2}, we have for $m=2m_0$ and $t>2m_0+1$
\begin{align*}
\int_{O}D( t) dx
&\leq \int_{\Omega _{t}}D( t) \varrho_{2m_0}( t) dx\\
&\leq C_1\int_{S_{2m_0+1}^{t}\times \omega }D\,dx\,ds
 +C_1\int_{S_{2m_0+1}^{t}\times \omega}F^2+\gamma | u_{\infty }| ^{\rho +2}\,dx\,ds.
\end{align*}
The integral $\int_{S_{2m_0+1}^{t}\times \omega }D \,dx\,ds$ in the
right-hand side can be estimated as above by iteration for
$m=2m_0+1,\dots ,[t/2] -1$. The rest of the proof is
similar to the proof of Theorem \ref{thm1} and hence is omitted.
\end{proof}

\subsection{Exponential convergence}

We give now some assumptions on the asymptotic behaviour of $\gamma $ and $f$
for large $t$, other than the trivial case $f=f_{\infty }$ and $\gamma =0$, that
ensure an exponential rate of convergences.

\begin{theorem}\label{thm2}
Assume \eqref{tlike}--\eqref{f00}, and that
\begin{equation}
\gamma ( t) ,\; | f( t) -f_{\infty
}| _{L^2( \Omega _{t}) }^2\leq K_2e^{-\mu _1t},
\label{gf}
\end{equation}
for large $t$ and some positive constants $K_2$ and $\mu _1$. Then we
have
\begin{gather*}
| u'| _{L^2( ( t-a,t) \times O) },\;
| \nabla _{X_1}u| _{L^2(( t-a,t) \times O) },\;
| \nabla _{X_2}( u-u_{\infty }) | _{L^2( (t-a,t) \times O) }
\leq M e^{-\mu 't}, \\
| \gamma ^{\frac{1}{\rho +2}}u| _{L^{\rho +2}(
( t-a,t) \times O) }\leq M\ e^{-\frac{2\mu '}{\rho +2}t},
\end{gather*}
for some positive constants $M$ and $\mu '$, such that
$0<\mu'<\min \{\mu _0/2,\mu _1\}/2$.
\end{theorem}

\begin{proof}
On one hand, $| f| _{L^2( Q_{t}) }^2$
grows polynomially since \eqref{gf} yields
\begin{equation}
\begin{aligned}
| f| _{L^2( Q_{t}) }^2
&\leq
2\int_0^{t}| f_{\infty }| _{L^2( \omega )
}^2\Big(\int_{( -\ell _0-\ell s,\ell _0+\ell s)
^{n_1}}dX_1+2K_2e^{-\mu _1s}\Big)ds \\
&\leq c_{11}t^{n_1+1} 
\end{aligned} \label{f2}
\end{equation}
for large $t$. On the other hand, by Remark \ref{rmk2} we derive
\begin{align*}
&\int_{S_{1+j}^{t}\times \omega }F^2+\gamma | u_{\infty
}| ^{\rho +2}\,dx\,ds \\
&\leq \int_{t-(1+j)}^{t}\int_{\Omega
_{s}}F^2\,dx\,ds+C_2( \ell t+\ell _0)
^{n_1}\int_{t-(1+j)}^{t}\gamma (s) ds \\
&\leq K_2( 1+C_2( \ell t+\ell _0) ^{n_1})
\int_{t-(1+j)}^{t}e^{-\mu _1s}ds \\
&\leq K_2( 1+C_2( \ell t+\ell _0) ^{n_1})
(1+j)e^{-\mu _1t}\times e^{\mu _1(1+j)} \\
&\leq c_{12}t^{n_1+1}e^{-\mu _1t}\times e^{\mu _1j},
\end{align*}
for large $t$. Since $k^{j}=e^{-\mu _0j}$ then we have
\[
k^{j}\int_{S_{1+j}^{t}\times \omega }F^2+\gamma | u_{\infty
}| ^2\,dx\,ds\leq c_{12}t^{n_1+1}e^{-\mu _1t}\times e^{(
\mu _1-\mu _0) j},
\]
for $2m_0+1\leq j\leq [t/2]$.
Summing  the above inequalities from $2m_0+1$ to $[t/2] $, we obtain
\begin{equation*}
g_{m_0}( t) \leq c_{12}t^{n_1+1}e^{-\mu _1t}
\sum_{j=2m_0+1}^{[t/2]}e^{( \mu _1-\mu_0) j}.
\end{equation*}
If $\mu _1<\mu _0$, then the sum term in the right-hand is bounded
independently of $t$. If $\mu _1\geq \mu _0$, then
\begin{equation*}
\sum_{j=2m_0+1}^{[t/2]}e^{( \mu _1-\mu
_0) j}\leq c_{13}te^{( \mu _1-\mu _0) \frac{t}{2}}.
\end{equation*}
Therefore, in both cases it holds that
\begin{equation}
g_{m_0}( t) \leq c_{14}t^{n_1+2}e^{-\min \{\frac{\mu _0+\mu
_1}{2},\mu _1\}t},  \label{g3}
\end{equation}
for large $t$. The estimations \eqref{f2} and \eqref{g3} means
that Assumption \eqref{gm} is satisfied.

Going back to \eqref{c8} we derive that
\begin{align*}
&\int_{( t-a,t) \times O}D( t,x) \,dx\,ds\\
&\leq c_{10}( t^{n_1+1}|f_{\infty }| _{L^2( \omega )
}^2+c_{11}t^{n_1+2}) e^{-\frac{\mu _0}{2}
t}+c_{14}t^{n_1+2}e^{-\min \{\frac{\mu _0+\mu _1}{2},\mu _1\}t}.
\end{align*}
Expending the expression of $D( t,x) $, we end up with
\begin{align*}
&\int_{( t-a,t) \times O}| u'| ^2+| \nabla _{X_1}u| ^2+|
\nabla _{X_2}( u-u_{\infty }) | ^2+\gamma
| u| ^{\rho +2}\,dx\,ds \\
&\leq c_{15}t^{n_1+2}\ e^{-\min \{\frac{\mu _0}{2},\mu _1\}t}.
\end{align*}
This completes the proof.
\end{proof}

\begin{remark} \label{rmk5} \rm
(i) Under  assumption \emph{\eqref{gf}}, the convergences in Corollary
\ref{cor2} are also exponential.

(ii) Theorem \ref{thm2} also holds if we replace the assumption
\eqref{gf} by the following one
\begin{equation*}
\int_{t-1}^{t}\gamma ( s) ds,\; \int_{t-1}^{t}\int_{
\Omega _{s}}| f-f_{\infty }| ^2\,dx\,ds\leq K_{3}e^{-\mu
_2t},
\end{equation*}
for large $t$ and some positive constants $K_{3}$ and $\mu _2$.
\end{remark}

\begin{remark} \label{rmk6} \rm
As long as the existence result of Theorem \ref{thm0} holds, we can obtain
the same results as in this article for more general domains, e.g.
\begin{equation*}
\Omega _{t}=\Big( \prod_{i=1}^{n_1}( -\alpha _i( t)
,\beta _i( t) ) \Big) \times \omega ,\quad t\geq 0,
\end{equation*}
where $\alpha _i( t)$ and $\beta _i( t) $ are smooth functions satisfying
\begin{equation*}
\beta _i( 0) +\alpha _i( 0) >0\text{  and }
\alpha_i( t) ,\beta _i( t) \to +\infty ,\quad \text{as }t\to +\infty
\end{equation*}
and their derivatives satisfy
\begin{equation*}
0<\alpha _i'( t) ,\; \beta _i'( t)<1,\quad  \text{for } i=1,\dots ,n_1.
\end{equation*}
Of course, the definitions of $S_{m}^{t}$ and $\varrho _{m}$ must be adapted
to this case.
\end{remark}

\subsection*{Acknowledgments}
The authors would like to thank Dr. Senoussi Guesmia, Qassim University (KSA),
 for his useful suggestions and comments on the manuscript.

\begin{thebibliography}{99}

\bibitem{BaC:81} C.~Bardos, G.~Chen;
\newblock {Control and stabilization for the wave equation. III: Domain with
  moving boundary.} \newblock {\em {SIAM J. Control Optim.}}, 19:123--138,
1981.

\bibitem{BrG:07} B.~Brighi, S.~Guesmia;
 \newblock Asymptotic behaviour of
solutions of hyperbolic problems on a cylindrical domain.
\newblock {\em Discrete Contin. Dyn. Syst. suppl}, (Special):160--169, 2007.

\bibitem{Chip:02} M.~Chipot;
 \newblock {\em $\ell $ goes to plus infinity}.
\newblock Birkh\"{a}user, 2002.

\bibitem{ChR:01} M.~Chipot, A.~Rougirel;
\newblock On the asymptotic behaviour of the solution of parabolic problems
in cylindrical domains of large size in some directions.
\newblock {\em Discrete Contin. Dyn. Syst. Ser. B}, 1(3):319--338, 2001.

\bibitem{ChY:08} M.~Chipot, K.~Yeressian;
 \newblock Exponential rates of convergence by an iteration technique.
\newblock {\em C. R. Math. Acad. Sci. Paris, S\'{e}r. I}, 346:21--26, 2008.

\bibitem{CoB:73} J.~Cooper,  C.~Bardos;
\newblock{A nonlinear wave
equation in a dependent domain.} \newblock {\em {J. Math. Anal. Appl.}},
42:29--60, 1973.

\bibitem{CoM:72} J.~Cooper, L.~A. Medeiros;
\newblock {The Cauchy problem for non linear wave equations in domains with
  moving boundary.}
\newblock {\em {Ann. Sc. Norm. Super. Pisa, Sci. Fis.
Mat., III. Ser.}}, 26:829--838, 1972.

\bibitem{Fer:95} J.~Ferreira;
\newblock Nonlinear hyperbolic-parabolic
partial differential equation in noncylindrical domain.
\newblock {\em
Rend. Circ. Mat. Palermo (2)}, 44(1):135--146, 1995.

\bibitem{FeL:96} J.~Ferriera, N. A. Lar'kin;
\newblock Global solvability of a mixed problem for a nonlinear
hyperbolic-parabolic equation in noncylindrical domains. \newblock
 {\em Portugal. Math.}, 53(4):381--395, 1996.

\bibitem{Gue:08a} S.~Guesmia;
\newblock Some results on the asymptotic
behaviour for hyperbolic problems in cylindrical domains becoming unbounded.
\newblock {\em J. Math. Anal. Appl.}, 341(2):1190--1212, 2008.

\bibitem{Gue:09} S.~Guesmia;
 \newblock Some convergence results for quasilinear parabolic boundary value
problems in cylindrical domain of large size. \newblock {\em Nonlinear Anal.},
 70(9):3320--3331, 2009.

\bibitem{Gue:13} S.~{Guesmia};
\newblock {Large time and space size behaviour of the heat equation in
  noncylindrical domains.} \newblock {\em {Archiv der Mathematik}},
101(3):293--299, 2013.

\bibitem{HaP:11} T.~G. Ha, J.~Y. Park;
\newblock {Global existence and uniform decay of a damped Klein-Gordon equation
  in a noncylindrical domain.}
\newblock {\em {Nonlinear Anal., Theory Methods Appl., Ser. A, Theory
  Methods}}, 74(2):577--584, 2011.

\bibitem{KnK:15} E.~Knobloch, R.~Kerchetnikov;
\newblock Problems on time-varying domains: Formulation, dynamics, and challenges.
\newblock {\em
Acta Applicandae Math.}, 137(1):123--157, Dec. 2015.

\bibitem{LaS:02} N.~A. Lar'kin, M.~H. Sim{\~o}es;
\newblock Nonlinear wave equation with a nonlinear boundary damping in a
 noncylindrical domain.
\newblock {\em Matematica Contemporânea}, 23:19--34, 2002.

\bibitem{Lion:69} J.~L. Lions;
\newblock {\em Quelques m\'{e}thodes de r\'{e}solution des probl\`{e}mes aux
  limites non lin\'{e}aires}. \newblock Dunod-Gautier Villars, 1969.

\bibitem{Med:72} L.~A. Medeiros;
\newblock {Nonlinear wave equations in
domains with variable boundary.} \newblock {\em {Arch. Ration. Mech. Anal.}},
 47:47--58, 1972.

\bibitem{NaN:78} M.~L. Nakao, T.~Narazaki;
 \newblock Existence and decay of solutions of some nonlinear wave equations
in noncylindrical domains.
\newblock {\em Math. Reports, Kyushu Univ.}, 11(2):117--125, 1978.

\bibitem{Rab:94} T.~N. Rabello;
\newblock Decay of solutions of a nonlinear
hyperbolic system in noncylindrical domain.
\newblock {\em {Int. J. Math. Math. Sci.}}, 17(3): 561--570, 1994.

\end{thebibliography}

\end{document}