\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 55, pp. 1--19.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/55\hfil Homogenization and uniform stabilization]
{Homogenization and uniform stabilization for a nonlinear
hyperbolic equation in domains with holes of small capacity}

\author[M. M. Cavalcanti, V. N. D. Cavalcanti,
J. A. Soriano, \& J. S. Souza\hfil EJDE-2004/55\hfilneg]
{Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti,\\
Juan A. Soriano, \& Joel S. Souza} % in alphabetical order

\address{Marcelo M. Cavalcanti \hfill\break
Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
87020-900 Maring\'a - PR, Brasil} 
\email{mmcavalcanti@uem.br}

\address{Valeria N. Domingos Cavalcanti \hfill\break
Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
87020-900 Maring\'a - PR, Brasil} 
\email{vndcavalcanti@uem.br}

\address{Juan A. Soriano \hfill\break
Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
87020-900 Maring\'a - PR, Brasil} 
\email{jaspalomino@uem.br}

\address{Joel S. Souza \hfill\break
Departamento de Matem\'atica - Universidade
Federal de Santa Catarina\\
80040-900 Florian\'opolis - SC, Brasil}
\email{cido@dme.ufpb.br}

\date{}
\thanks{Submitted September 23, 2003. Published April 9, 2004.}
\subjclass[2000]{35B27, 35B40, 35L05} 
\keywords{Homogenization, asymptotic stability, wave equation}

\begin{abstract}
 In this article we study the homogenization and
 uniform decay of the nonlinear hyperbolic equation
 $$
 \partial_{tt} u_{\varepsilon} -\Delta u_{\varepsilon}
 +F(x,t,\partial_t u_{\varepsilon},\nabla u_{\varepsilon})=0
 \quad\hbox{in }\Omega_{\varepsilon}\times(0,+\infty)
 $$
 where $\Omega_{\varepsilon}$  is a domain containing holes with
 small capacity (i. e. the holes are smaller than a critical size).
 The homogenization's proofs are based on the abstract framework
 introduced by Cioranescu  and Murat \cite {c6} for the study of
 homogenization of elliptic problems.  Moreover,  uniform decay
 rates are obtained by considering the perturbed energy method
 developed by Haraux and Zuazua \cite{h1}.
\end{abstract}

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

\section{Introduction and statement main results}

This paper is devoted to the study of the homogenization and
uniform decay rates of the nonlinear hyperbolic equation
\begin{equation} \label{Pe}
\begin{gathered}
{u_{\varepsilon}''-\Delta u_{\varepsilon}
+F\left(x,t,u_{\varepsilon}',\nabla u_{\varepsilon}\right)=0
\quad\hbox{in } \Omega_{\varepsilon}\times (0,+\infty )}\\
{u_{\varepsilon}=0 \quad\hbox{on } \Gamma_{\varepsilon}\times (0,+\infty )}\\
{u_{\varepsilon}(x,0)=u_{\varepsilon}^0(x);\quad
u_{\varepsilon}'(x,0)=u_{\varepsilon}^1(x);\quad x\in\Omega_{\varepsilon},}
\end{gathered}
\end{equation}
where, for every $\varepsilon >0$,  $\Omega_{\varepsilon}$  is an open domain,
locally located on one side of its smooth boundary $\Gamma_{\varepsilon}$,
obtained by removing, from a given bounded,  connected open set $\Omega$,
a set $S_{\varepsilon}$ of closed subsets (the `holes') of $\Omega$;
i. e., $\Omega_{\varepsilon}=\Omega\backslash S_{\varepsilon}$.  We assume
that the measure of $S_{\varepsilon}$ approaches zero as the parameter
$\varepsilon $ tends to zero.

Now, we state the general hypotheses.
\begin{itemize}
\item[(A1)]  Assumptions on the initial data: Assume that
\begin{equation}
\{u_{\varepsilon}^0,u_{\varepsilon}^1\}\in D(\Omega_{\varepsilon})
\times D(\Omega_{\varepsilon})\label{e1.1}
\end{equation}
and as $\varepsilon\to 0$ we have
\begin{equation}
\left\{\tilde u_{\varepsilon}^0,\tilde u_{\varepsilon}^
1\right\}\rightharpoonup\left\{u^0,u^1\right\}\quad\hbox{weakly in }
 H_0^1(\Omega )\cap H^2(\Omega )\times H_0^1(\Omega), \label{e1.2}
\end{equation}
where the tilde on $\tilde u$ denotes the extension by zero to the
whole domain $\Omega$.

\item[(A2)]  Assumptions on $F(x,t,u',\nabla u)$: Suppose
$F:\Omega\times (0,\infty )\times \mathbb{R}^{n+1}\to \mathbb{R}$
is an element of the space $C^1\left(\Omega\times (0,\infty
)\times \mathbb{R}^{n+1}\right)$ and satisfies
\begin{equation}
|F(x,t,\xi ,\zeta )|\leq C_0\left(1+|\xi|^{
\rho +1}+|\zeta|\right)\label{e1.3}
\end{equation}
where $C_0$ and $\rho$ are positive constants such that $\rho >0$ for
$n=1,2$  and $0<\rho\leq 2/(n-2)$ for $n\geq 3$, and
$\zeta =\left(\zeta_1,\dots ,\zeta_n\right)$.

Assume that there is a non-negative function $\varphi (t)$ in
$W^{1,\infty}(0,\infty )\cap L^1(0,\infty )$ such that for some $\beta >0$,
\begin{equation}
F(x,t,\xi ,\zeta )\eta\geq\beta|\xi|^{\rho}\xi\eta -
\varphi (t)\left(1+|\eta||\zeta|\right),\quad
\hbox{for all }\eta\in \mathbb{R}\,.\label{e1.4}
\end{equation}
Suppose  that there exist positive constants $C_1,\dots ,C_n$ such
that
\begin{gather}
|F_t(x,t,\xi ,\zeta )|
\leq C_0(1+|\xi|^{\rho +1}+|\zeta|),\label{e1.5}\\
F_{\xi}(x,t,\xi ,\zeta )\geq\beta|\xi|^{\rho},\label{e1.6}\\
|F_{\zeta_i}(x,t,\xi ,\zeta )|\leq C_i\quad\hbox{for }
i=1,\dots ,n.\label{e1.7}
\end{gather}
Also assume that  there exists a positive constant $D$ such that
for all $\eta, \hat{\eta}$ in $\mathbb{R}$, one has
\begin{gather}\label{e1.8}
\left(F(x,t,\xi ,\zeta )-F(x,t,\hat\xi ,\hat\zeta\right)\left(\eta
-\hat\eta\right) \\
\geq \beta \left(|\xi|^{\rho}\xi-
|\hat{\xi}|^{\rho}\hat{\xi}\right)(\eta-\hat{\eta})-D|\eta
-\hat\eta||\zeta-\hat\zeta|.\nonumber
\end{gather}
We assume that
\begin{equation}
F(x,t,0,0)=0.\label{e1.9}
\end{equation}
A simple variant of the nonlinear function above is given by
the example
$$
F(x,t,\xi ,\zeta )=\beta|\xi|^{\rho}\xi +\varphi (t)
\sum_{i=1}^n\sin(\zeta_i).
$$
\end{itemize}
Next, we make some remarks about early works concerning homogenization
of distributed systems.

In the framework of homogenization of elliptic problems,
Cioranescu and  Murat  \cite{c6} studied the  problem
\begin{gather*}
{\Delta u_{\varepsilon}=f\quad\hbox{in } \Omega_{\varepsilon}}\\
{u_{\varepsilon}=0\quad\hbox{on } \Gamma_{\varepsilon}}
\end{gather*}
with $f\in H^{-1}(\Omega )$. They showed that for every $\varepsilon>0$ there
exists a unique $u_{\varepsilon}\in H_0^1(\Omega_{\varepsilon})$ such that
$$
\tilde {u}_{\varepsilon}\rightharpoonup u\quad\hbox{weakly in } H_0^1(\Omega ),
\hbox{ as }\varepsilon\to 0
$$
where $\tilde {u}_{\varepsilon}$ is the extension of $u_{\varepsilon}$, by
considering zero, to whole domain $\Omega$,
and  $u$ is the unique solution of the homogenized problem
\begin{gather*}
{-\Delta u+\mu u=f\quad\hbox{in } \Omega}\\
{u=0\quad\hbox{on } \Gamma =\partial\Omega,}
\end{gather*}
where $\mu$ is a non-negative Radon's measure which belongs to $H^{-1}(\Omega )$.
This measure appears in this study and is due to the capacity's
behaviour of the set $S_{\varepsilon}$ when $\varepsilon\to
0$.  For this end it is
necessary that we have small holes, i. e., the diameter of the
holes are smaller than (or equal to) the critical diameter $a_{\varepsilon}$
given by:
$$
a_{\varepsilon}=\begin{cases}
\delta_{\varepsilon}\exp(-C_0/\varepsilon^2) &\hbox{if } n=2 \\
C_0\varepsilon^{n/(n-2)} &\hbox{if }n>2,
\end{cases}
$$
where $C_0$ is a positive constant and $\delta_{\varepsilon}$  is such that
$\varepsilon^2log\delta_{\varepsilon}\to 0$ as $\varepsilon
\to 0$.  From the above condition it is possible to construct an
abstract framework which  plays an essential role to
demonstrate the results. More precisely we have:

There exists a sequence
$\{w_{\varepsilon},\mu_{\varepsilon},\gamma_{\varepsilon}\}$ and $M_0>0$
such that
\begin{equation}
\begin{gathered}
w_{\varepsilon}\in H^1(\Omega )\cap L^{\infty}(\Omega ),\quad
\|w_{\varepsilon}\|_{L^{\infty} (\Omega )}\leq M_0\quad \hbox{for every }
\varepsilon >0,\\
w_{\varepsilon}=0\quad \hbox{on } S_{\varepsilon},\\
w_{\varepsilon}\rightharpoonup 1\quad \hbox{weakly in }
H^1(\Omega)\hbox{ as }\varepsilon\to 0,\\
-\Delta w_{\varepsilon}=\mu_{\varepsilon}-\gamma_{\varepsilon}
\quad \,\quad\hbox{ with }\mu_{\varepsilon},\gamma_{\varepsilon}
\in H^{-1}(\Omega ),\\
\mu_{\varepsilon}\to\mu \,\,\, \hbox{strongly in } H^{-1}(\Omega
)\hbox{ and }
\langle \gamma_{\varepsilon},v_{\varepsilon}\rangle =0\\
\quad\hbox{for every }\{v_{\varepsilon}\} \subset H_0^1(\Omega )
\hbox{ with }v_{\varepsilon}=0\hbox{ on } S_{\varepsilon}\,.
\end{gathered}\label{e1.10}
\end{equation}
In the case above, $\mu$ will be a nonnegative constant when the diameter of the
holes is the critical one. In this case, the additional term of
order zero $\mu u$ (so called  `terme \'etrange') appears in the limit equation.
In  \cite{c6} the authors still showed correctors results; i.e.,
$$
\tilde {u}_{\varepsilon}=w_{\varepsilon}u+R_{\varepsilon},\quad
\hbox{with }R_{\varepsilon}\to 0\hbox{ strongly in } H_0^1(\Omega ).
$$
An example where \eqref{e1.10} is satisfied occurs when
$S_{\varepsilon}$ consists of periodically distributed holes of critical size.
More precisely,
$$
S_{\varepsilon}=\bigcup_{i=1}^{N_{\varepsilon}}T_i^{\varepsilon}
$$
where $T_i^{\varepsilon}$ are spheres of size $r_{\varepsilon}=a_{\varepsilon}$,
periodically distributed (period $2\varepsilon )$ in each axis direction and
$a_{\varepsilon}$ is defined as in \eqref{e1.9}.  In this case \eqref{e1.10} holds
with
\begin{gather*}
\mu =\frac {\pi}2\frac 1{C_0}\quad\hbox{if }n=2,\\
\mu =\frac {\mathcal{S}_n(n-2)}{2n}C_0^{n-2}\quad\hbox{if } n\geq 3
\end{gather*}
where $\mathcal{S}_n$ is the surface of the unit sphere in $\mathbb{R}^n$.

In what concerns evolution equations,  Cioranescu, Donato,
 Murat  and Zuazua  \cite{c4}, studied the homogenization of the
linear wave equation
\begin{equation}
\begin{gathered}
{u_{\varepsilon}''-\Delta u_{\varepsilon}=f_{\varepsilon} \quad
\hbox{in } \Omega_{\varepsilon}\times (0,T)}\\
{u_{\varepsilon}=0\quad\hbox{on } \Gamma_{\varepsilon}\times (0,T )}\\
{u_{\varepsilon}(x,0)=u_{\varepsilon}^0(x);\quad
u_{\varepsilon}'(x,0)=u_{\varepsilon}^1(x);\quad
x\in\Omega_{\varepsilon},}
\end{gathered}\label{e1.11}
\end{equation}
with
$\{u_{\varepsilon}^0,u_{\varepsilon}^1,f_{\varepsilon}\}
\in H_0^1(\Omega_{\varepsilon})\times L^2(\Omega_{\varepsilon})
\times L^1(0,T;L^2(\Omega_{\varepsilon})$
and
\begin{gather*}
\tilde {u}_{\varepsilon}^0\rightharpoonup u^0 \quad\mbox{weakly in }
H_0^1(\Omega ), \\
\tilde {u}_{\varepsilon}^1\rightharpoonup u^1 \quad\mbox{weakly in }L^2(\Omega ),\\
\tilde {f}_{\varepsilon}\rightharpoonup f \quad\mbox{weakly in }
L^1(0,T;L^2(\Omega )).
\end{gather*}
They proved that
$\tilde {u}_{\varepsilon}\rightharpoonup u$ weak-star in
$L^{\infty}(0,T;H_0^1(\Omega ))\cap W^{1,\infty}(0,T;L^2(\Omega ))$,
where $\tilde {u}_{\varepsilon}$ is the unique solution of problem
\eqref{e1.11}, for each $\varepsilon >0$
fixed,  extended by considering zero on the holes, and $u$ is the
unique solution of the homogenized problem
\begin{gather*}
{u''-\Delta u+\mu u=f\quad\hbox{in } \Omega\times (0,T)}\\
{u_{\varepsilon}=0\quad \hbox{on } \Gamma\times (0,T )}\\
{u(x,0)=u^0(x);\quad u'(x,0)=u^1(x);\quad x\in\Omega,}
\end{gather*}
and $\mu$ is a non-negative Radon's measure, which is positive
when one considers holes of critical size.

Now, concerning the exact controllability of the wave equation in
perforated domains, it is important to mention the work of the
authors Cioranescu, Donato and Zuazua  \cite{c5}.  When the size
of the holes is small enough, at the limit, they got the wave
equation with a boundary control and when the holes are of
critical size they obtained at the limit the wave equation with an
additional term of order zero and two controls: the first one on
the boundary and the second one an internal control.

On the other hand it is worth mentioning the papers in connection
with homogenization of attractors for hyperbolic equations of the
authors  Fiedler and Vishik  \cite{f1} as well as Pankratov and
Chueshov \cite{p1}.  Also, we would like to cite some papers where
the damping term is, as in the present paper, in the form
$G(x,t,u_t)$, as, for instance, \cite{c2, pa, pu} and references
therein.

It is important to observe that from the assumption \eqref{e1.2}
one has $\tilde{u}_{\varepsilon}^0 \to u^0$ strongly in
$H_0^1(\Omega)$. Consequently we deduce that: $\mu=0$ or $u^0=0$.
As we are interested in nontrivial initial data, we are forced to
consider $\mu =0 $, which implies that the geometry of the domain
$\Omega$ is such that the holes possess `{\it small capacity}' (i.
e. the holes are smaller than the critical size); see references
\cite{c4,c5} for details.

Since controllability implies stabilization, then we can expect
that we can also stabilize the system \eqref{Pe} by
introducing a suitable dissipative mechanism.  Unfortunately the
controllability is showed basically for linear problems and only
for a few semi-linear problems in a very few class of
nonlinearities. Even if we are dealing with homogenization results
for those domains with `{\it small capacity}', very few is known
for the nonlinear wave equation. For this reason these
homogenization and stabilization results are interesting to be
studied.


In what follows in this work, the geometry of the perforated
domain $\Omega_{\varepsilon}$, will satisfy the conditions given
by \eqref{e1.10} of the abstract framework introduced by Cioranescu
and Murat in  \cite{c6}, having in mind those domains with `{\it small
capacity}'.

Now, we are in a position to state our main result.

\begin{theorem} \label{thm1.1}
Assume that \eqref{e1.1}-\eqref{e1.9} hold. Then, supposing that
\eqref{e1.10} is assumed with $\mu = 0$, the unique solution
$u_{\varepsilon}$ of \eqref{Pe} satisfies
\begin{gather*}
\tilde {u}_{\varepsilon}\to u\quad\mbox{strongly in }
C^0_{\rm loc}([0,\infty );L^2(\Omega )),\\%\label{e1.12} \\
\tilde {u}_{\varepsilon}'\to u'\quad\mbox{strongly in }
C^0_{\rm loc}([0,\infty );L^2(\Omega )),\\% \label{e1.13}\\
\tilde {u}_{\varepsilon}\rightharpoonup u\quad\mbox{weak-star in }
L^{\infty}_{\rm loc}(0,\infty ;H_0^1(\Omega )),\\ %\label{e1.14} \\
\tilde {u}_{\varepsilon}'\rightharpoonup u'\quad \mbox{weak-star in}
L^{\infty}_{\rm loc}(0,\infty ;H_0^1(\Omega )),\\ % \label{e1.15} \\
\tilde {u}_{\varepsilon}''\rightharpoonup u'' \quad \mbox{weak-star in }
L^{\infty}_{\rm loc}(0,\infty ;L^2(\Omega )),%\label{e1.16}
\end{gather*}
where $u$ is the unique solution of the homogenized problem
\begin{equation}
\begin{gathered}
{u''-\Delta u+F\left(x,t,u',\nabla u\right)=0\,\quad\hbox{in }
\Omega\times (0,+\infty)}\\
{u=0\quad\hbox{on }\Gamma\times (0,+\infty )}\\
{u(x,0)=u^0(x);\quad u'(x,0)=u^1(x);\quad x\in\Omega .}
\end{gathered}\label{e1.17}
\end{equation}
Defining the energy related to the homogenized problem as
\begin{equation}
E(t)=\frac 12|u'(t)|_{L^2(\Omega )}^2+\frac 12|\nabla u(t)|_{L^2(\Omega )}^2
\label{e1.18}
\end{equation}
and assuming that $\rho=0$, and
$\varphi (t)\leq C_1e^{-\gamma t}$ for all $t\geq 0$, %\label{e1.19}
where $C_1$ and $\gamma$ are positive constants, we have
\[
E(t)\leq Ce^{-\gamma_0t}, \quad\forall t\geq 0. %\label{e1.20}
\]
Furthermore, supposing that
\[
\varphi (t)\leq\frac {k_1}{(1+t)^{(\rho +2)/\rho}},\quad\forall t\geq 0, %\eqno (1.21}
\]
where $k_1$ is a positive constant, one has
\[
E(t)\leq\frac K{(1+t)^{2/\rho}},\quad\forall t\geq 0 %\label{e1.22}
\]
where $K$ is a positive constant.
\end{theorem}

Our paper is organized as follows: In section 2 we study
the existence and uniqueness of problem \eqref{Pe} for each
$\varepsilon >0$
fixed. In section 3 we obtain the homogenized problem related
to \eqref{Pe} making use of the abstract framework presented in
\eqref{e1.10} and finally in
section 4 we give the proofs of the uniform decay.

\section{Existence and uniqueness of solutions to problem \eqref{Pe}}

In this section,  we prove existence and uniqueness of
solutions to problem \eqref{Pe} for each $\varepsilon >0$ fixed,  assuming that
the initial data belong to the class given by \eqref{e1.1}  and the
nonlinear function $F(x,t,\xi ,\zeta )$ satisfies the hypotheses \eqref{e1.3}-\eqref{e1.8}.

For this end,  let $(\omega_{\nu})_{\nu\in \mathbb{N}}$ be a basis in $
H_0^1(\Omega_{\varepsilon})\cap H^2(\Omega_{\varepsilon})$ which
is an orthornormal system for $L^2(\Omega_{\varepsilon})$.  Let $
V_m$ be the space
generated by $\omega_1,\dots ,\omega_n$  and let
\begin{equation}
u_{\varepsilon m}(t)=\sum_{i=1}^mg_{j\varepsilon m}(t)\omega_j\label{e2.1}
\end{equation}
be the solution to the Cauchy problem
\begin{equation}
\begin{gathered}
{\left(u_{\varepsilon m}''(t),w\right)+\left
(\nabla u_{\varepsilon m}(t),\nabla w\right)+\left(F\left(x,t,u_{
\varepsilon m}'(t),\nabla u_{\varepsilon m}(t)\right),w\right)=0}\\
{\hbox{for all }w\in V_m,}\\
{u_{\varepsilon m}(0)=u_{\varepsilon m}^0\to
u_{\varepsilon}^0\quad\hbox{in } H_0^1(\Omega_{\varepsilon})\cap
H^2(\Omega_{\varepsilon})\quad\hbox{as }m\to\infty,}\\
{u_{\varepsilon m}'(0)=u_{\varepsilon m}^1\to u_{\varepsilon}^1
\quad\hbox{in } H_0^1(\Omega_{\varepsilon})\hbox{ as } m\to\infty ,}
\end{gathered} \label{e2.2}
\end{equation}
where $(\cdot ,\cdot)$ is the inner product in $L^2(\Omega_{\varepsilon})$.
For simplicity we denote
$$
|u|^2=\int_{\Omega_{\varepsilon}}|u(x)|^2dx,
\quad\|u\|_p^p=\int_{\Omega_{\varepsilon}}|u(x)|^pdx.
$$
We observe that the term $\left(F\left(x,t,u_{\varepsilon m}'(t),
\nabla u_{\varepsilon m}(t)\right),w\right)$ is well defined in
view of the assumption \eqref{e1.3}.

By standard methods in differential equations,  we can prove the
existence of a solution to \eqref{e2.2} on some interval
$[0,t_{\varepsilon m})$. Then,  this solution can be extended to
the whole interval $[0,T]$; $T>0$;  by use of the first estimate
below.

\subsection{A Priori estimates}
{\em First Estimate:\/} Taking $w=2u_{\varepsilon m}'(t)$ in \eqref{e2.2}
and considering the assumption \eqref{e1.4}  one has
\begin{equation}
\begin{aligned}
&\frac d{dt}\left\{|u_{\varepsilon m}'(t)|^2
+|\nabla u_{\varepsilon m}(t)|^2\right\}
+2\beta\|u_{\varepsilon m}'(t)\|_{\rho +2}^{\rho +2}\\
&\leq 2\varphi (t)\int_{\Omega_{\varepsilon}}
\left(1+|u_{\varepsilon m}'||\nabla u_{\varepsilon m}|\right)dx\\
&\leq 2\varphi (t)\mathop{\rm meas}(\Omega )+|u_{\varepsilon m}'(t)|^2
+|\nabla u_{\varepsilon m}(t)|^2.
\end{aligned}\label{e2.3}
\end{equation}
Integrating \eqref{e2.3} over $(0,t)$, $t\in [0,t_{\varepsilon m})$,  we obtain
\begin{equation}
\begin{aligned}
&|u_{\varepsilon m}'(t)|^2+|\nabla u_{\varepsilon m}(t)|^2
+2\beta\int_0^t\|u_{\varepsilon m}'(s)\|_{\rho +2}^{\rho +2}ds\\
&\leq|u_{\varepsilon m}^1|^2+|u_{\varepsilon m}^0|^2
+2\|\varphi\|_{L^1(0,\infty )}\mathop{\rm meas}(\Omega)
+\int_0^t\left\{|u_{\varepsilon m}'(s)|^2+|\nabla u_{\varepsilon m}(s)|^2\right\}ds.
\end{aligned} \label{e2.4}
\end{equation}
 From \eqref{e2.4},  considering the convergence in \eqref{e1.2} and \eqref{e2.2}
and employing Gronwall's lemma, we deduce
\begin{equation}
|u_{\varepsilon m}'(t)|^2+|\nabla u_{\varepsilon
m}(t)|^2+2\beta\int_0^t\|u_{\varepsilon m}'(s)\}_{\rho +2}^{\rho +2}ds
\leq L_1\label{e2.5}
\end{equation}
where $L_1$  is a positive constant independent of $m\in \mathbb{N}$,
$t\in [0,T]$ and $\varepsilon >0$. \smallskip

\noindent{\em Second Estimate:\/}
First, we prove that $u_{\varepsilon m}^{
\prime\prime}(0)$ is bounded in $L^2(\Omega_{\varepsilon})$ norm.
Indeed,  taking $w=u_{\varepsilon m}''(0)$ and $t=0$
in \eqref{e2.2}, taking the assumption \eqref{e1.3}  into account;  making use
of Green's formula and  Cauchy-Schwarz inequality and considering
the inequality $ab\leq\frac 12a^2+\frac 12b^2$, we infer
$$
|u_{\varepsilon m}''(0)|^2
\leq\left\{|\Delta u_{\varepsilon m}^0|+C_0[\left(\mathop{\rm meas}
(\Omega\right)^{1/2}+\|u_{\varepsilon m}^1|\|_{2(\rho +1)}^{\rho +1}
+|\nabla u_{\varepsilon m}^0|^2]\right\}|u_{\varepsilon m}''(0)|.
$$
 From the last inequality,  noting that
 $H_0^1(\Omega)\hookrightarrow L^{2(\rho +1)}(\Omega)$ and
considering the convergence in \eqref{e1.2} and \eqref{e2.2} it holds that
\begin{equation}
|u_{\varepsilon m}''(0)|^2\leq L_2 \label{e2.6}
\end{equation}
where $L_2$  is a positive constant independent of $t\in [0,T]$;
$m\in \mathbb{N}$  and $\varepsilon >0$.

Now, taking the derivative of \eqref{e2.2} with respect to $t$  and
substituting $w=2u_{\varepsilon m}''(t)$, it follows that
\begin{align*}
&\frac d{dt}\left\{|u_{\varepsilon m}''(t)|^2
+|\nabla u_{\varepsilon m}'(t)|^2\right\}\\
&=-2\int_{\Omega_{\varepsilon}}F_t\left(x,t,u_{\varepsilon m}',\nabla
u_{\varepsilon m}\right)u_{\varepsilon m}''\,dx
-2\int_{\Omega_{\varepsilon}}F_{u_{\varepsilon m}'}
\left(x,t,u_{\varepsilon m}',\nabla u_{\varepsilon m}\right)
\left(u_{\varepsilon m}''\right)^2dx \\
&\quad -2\sum_{i=1}^n\int_{\Omega_{\varepsilon}}F_{u_{\varepsilon mx_i}}
\left(x,t,u_{\varepsilon m}',\nabla u_{\varepsilon m}\right)
u_{\varepsilon mx_i}'u_{\varepsilon m}''\,dx.
\end{align*}
 From assumptions \eqref{e1.5}-\eqref{e1.7}, taking into account
the above equality and the Cauchy-Schwarz inequality, we obtain
\begin{equation}
\begin{aligned}
&\frac d{dt}\left\{|u_{\varepsilon m}''(t)\|^2
+|\nabla u_{\varepsilon m}'(t)|^2\right\}
+2\beta\int_{\Omega_{\varepsilon}}|u_{\varepsilon m}'|^{\rho}
\left(u_{\varepsilon m}''\right)^2dx\\
&\leq 2C_0\big\{\left(\mathop{\rm meas}(\Omega )\right)^{1/2}|
u_{\varepsilon m}''(t)|+\int_{\Omega_{\varepsilon}}|u_{\varepsilon
m}'|^{\rho /2}|u_{\varepsilon m}'' \|u_{\varepsilon m}'|^{(\rho +2)/2}dx\\
&\quad +|\nabla u_{\varepsilon m}(t)||u_{\varepsilon m}''(t)|\big\}
+2n(M+1)\left\{|u_{\varepsilon m}''(t)|^2+|\nabla u_{\varepsilon m}'(t)\|^2\right\},
\end{aligned} \label{e2.8}
\end{equation}
where $M=\max\{C_i,n\}$; $i=1,\dots ,n$.

Integrating \eqref{e2.8}  over $(0,t)$ and making use of the inequality
$ab\leq\frac 1{4\eta}a^2+\eta b^2$, for an arbitrary $\eta >0$,  we deduce
\begin{equation}
\begin{aligned}
&|u_{\varepsilon m}''(t)|^2+|\nabla u_{\varepsilon m}'(t)|^2
+2C_0(\beta -\eta )\int_0^t\int_{\Omega_{\varepsilon}}|u_{\varepsilon m}'|^{\rho}
\left(u_{\varepsilon m}''\right)^2 dx\,ds\\
&\leq|u_{\varepsilon m}''(0)|^2+|\nabla
u_{\varepsilon m}^1|^2+C_0 \mathop{\rm meas}(\Omega )T\\
&\quad +\frac {C_0}{2\eta}\int_0^t\|u_{\varepsilon m}'(s)\|_{\rho +2}^{\rho +2}ds
+C_1\int_0^t\left\{|u_{\varepsilon m}''(s)|^2+|\nabla u_{\varepsilon m}'(s)|^2
\right\}ds,
\end{aligned}\label{e2.9}
\end{equation}
where $C_1=C_0+2n(M+1)$.
 From \eqref{e2.9},  \eqref{e2.5},  \eqref{e2.6},  considering the convergence 
in \eqref{e1.2}
and \eqref{e2.2}, choosing $\eta >0$ sufficiently small  and employing
Gronwall's lemma,  we obtain the second estimate
\begin{equation}
|u_{\varepsilon m}''(t)|^2+|\nabla
u_{\varepsilon m}'(t)|^2+\int_0^t\int_{\Omega_{
\varepsilon}}|u_{\varepsilon m}'|^{\rho}\left(u_{\varepsilon
m}''\right)^2dx\,ds\leq L_3\label{e2.10}
\end{equation}
where $L_3$ is a positive constant independent of $t\in [0,T]$; $
m\in \mathbb{N}$
and $\varepsilon >0$.

\subsection{Analysis of the nonlinear term $F$}
 From the assumption \eqref{e1.3},  there is a positive constant $N$ such
that
$$
\int_{\Omega_{\varepsilon}}|F\left(x,t,u_{\varepsilon m}',
\nabla u_{\varepsilon m}\right)|^2dx
\leq N\left(1+\|u_{\varepsilon m}'(t)\|_{2(\rho +1)}^{2(\rho +1)}
+|\nabla u_{\varepsilon m}(t)|^2\right).
$$
 Therefore,  from estimates \eqref{e2.5}  and \eqref{e2.10}  and observing
that $H_0^1(\Omega)\hookrightarrow L^{2(\rho +1)}(\Omega)$,  it follows that
\begin{equation}
\left\{F\left(x,t,u_{\varepsilon m}',\nabla u_{\varepsilon m}\right
)\right\}_{m\in \mathbb{N},\varepsilon >0}\quad\hbox{is\,\,\,bounded\,\,\,
in } L^2_{\rm loc}(0,\infty ;L^2(\Omega_{\varepsilon})).\label{e2.11}
\end{equation}
Consequently,  there exists a subsequence $\{u_{\varepsilon\mu}\}$ of $
\{u_{\varepsilon m}\}$ (which  we still denote by the same symbol) and a
function $\chi$ in $L^2_{\rm loc}(0,\infty ;L^2(\Omega_{\varepsilon}))$
such that
\begin{equation}
F\left(x,t,u_{\varepsilon\mu}',\nabla u_{\varepsilon\mu}\right)
\rightharpoonup\chi_{\varepsilon}\quad\hbox{weakly in }
L^2_{\rm loc}(0,\infty ;L^2(\Omega_{\varepsilon}))\mbox{ as }\mu\to \infty .
\label{e2.12}
\end{equation}
 From the above estimates we also deduce that there is a
function $u_{\varepsilon}:\Omega_{\varepsilon}\times (0,\infty )\to \mathbb{R}$
such that
\begin{gather}
u_{\varepsilon\mu}'\rightharpoonup u_{\varepsilon}'\quad\hbox{weak-star in }
 L_{\rm loc}^{\infty}(0,\infty ;L^2(\Omega_{\varepsilon})),\label{e2.13}\\
u_{\varepsilon\mu}\rightharpoonup u_{\varepsilon}\quad\hbox{weak-star in }
 L_{\rm loc}^{\infty}(0,\infty ;H_0^1(\Omega_{\varepsilon})),\label{e2.14}\\
u_{\varepsilon\mu}''\rightharpoonup u_{\varepsilon}''\quad\hbox{weak-star in }
 L_{\rm loc}^{\infty}(0,\infty ;L^2(\Omega_{\varepsilon})),\label{e2.15} \\
u_{\varepsilon\mu}'\rightharpoonup u_{\varepsilon}'\quad\hbox{weak-star in }
L_{\rm loc}^{\infty}(0,\infty ;H_0^1(\Omega_{\varepsilon})).\label{e2.16}
\end{gather}
Moreover,  making use of Aubin-Lions theorem; Lions [13, p. 57], we have
\begin{gather}
u_{\varepsilon\mu}\to u_{\varepsilon}\quad\hbox{strongly in }
L_{\rm loc}^2(0,\infty ;L^2(\Omega_{\varepsilon})),\label{e2.17} \\
u_{\varepsilon\mu}'\to u_{\varepsilon}'\quad\hbox{strongly in }
L_{\rm loc}^2(0,\infty ;L^2(\Omega_{\varepsilon})).\label{e2.18}
\end{gather}
 From the above estimates after passing to the limit, we
conclude that
\begin{equation}
 u_{\varepsilon}''-\Delta u_{\varepsilon}+\chi_{\varepsilon}
=0\quad\hbox{in } D'(\Omega_{\varepsilon}\times (0,T)).\label{e2.19}
\end{equation}
Since
$u_{\varepsilon}'',\chi_{\varepsilon}\in L_{\rm loc}^2(0,\infty
;L^2(\Omega_{\varepsilon}))$ from \eqref{e2.19} we deduce that
$$
\Delta u_{\varepsilon}\in L_{\rm loc}^2(0,\infty ;L^2(\Omega_{\varepsilon}))
$$ %\label{e2.20}
and
\begin{equation}
u_{\varepsilon}''-\Delta u_{\varepsilon}+\chi_{\varepsilon}
=0\quad\hbox{in } L_{\rm loc}^2(0,\infty ;L^2(\Omega_{\varepsilon}
)).\label{e2.21}
\end{equation}
Our goal is to show that
\begin{equation}
\chi_{\varepsilon}=F\left(x,t,u_{\varepsilon}',\nabla u_{\varepsilon}\right).
\label{e2.22}
\end{equation}
Indeed,  integrating \eqref{e2.2} over (0,T) and considering $w=u_{\varepsilon
\mu}(t)$,
we obtain
\begin{equation}
\begin{aligned}
&\int_0^T\left(u_{\varepsilon\mu}''(t),u_{\varepsilon
\mu}(t)\right)dt+\int_0^T|\nabla u_{\varepsilon\mu}(t)|^2dt\\
&+\int_0^T\left(F\left(x,t,u_{\varepsilon\mu}'(t),\nabla u_{\varepsilon
\mu}(t)\right),u_{\varepsilon\mu}(t)\right)dt=0
\end{aligned}\label{e2.23}
\end{equation}
 Then, considering the strong convergence \eqref{e2.17} and the weak
ones \eqref{e2.12} and \eqref{e2.15},  from \eqref{e2.23} we obtain
\begin{equation}
\lim_{\mu\to\infty}\int_0^T|\nabla u_{\varepsilon\mu}(t)|^2dt
=-\int_0^T\left(u_{\varepsilon}''(t),u_{\varepsilon}
(t)\right)dt-\int_0^T\left(\chi (t),u_{\varepsilon}(t)\right)dt.
\label{e2.24}
\end{equation}
Substituting \eqref{e2.21} in \eqref{e2.24} and applying the generalized Green
formula we deduce
\begin{equation}
\lim_{\mu\to\infty}\int_0^T|\nabla u_{\varepsilon\mu}
(t)|^2dt=\int_0^T|\nabla u_{\varepsilon}(t)|^2dt.\label{e2.25}
\end{equation}
 Taking into account that
\begin{align*}
&\int_0^T|\nabla u_{\varepsilon\mu}(t)-\nabla u_{\varepsilon}(t)|^2dt\\
&=\int_0^T|\nabla u_{\varepsilon\mu}(t)|^2dt-2\int_0^
T\left(\nabla u_{\varepsilon\mu}(t),\nabla u_{\varepsilon}\right)
dt+\int_0^T|\nabla u_{\varepsilon}(t)|^2dt,
\end{align*}
 from \eqref{e2.25} and \eqref{e2.14} we deduce that
$\lim_{\mu\to\infty}\int_0^T|\nabla u_{\varepsilon\mu}
(t)-\nabla u_{\varepsilon}(t)|^2dt=0$,
which implies that
\begin{equation}
\nabla u_{\varepsilon\mu}\to\nabla u_{\varepsilon}\quad\hbox{
in } L^2_{\rm loc}(0,\infty ;L^2(\Omega_{\varepsilon}))\quad
\hbox{as }\mu\to\infty .\label{e2.26}
\end{equation}
 Then, from the strong convergence \eqref{e2.17}, \eqref{e2.18} and \eqref{e2.26} we
have
$$
F\left(x,t,u_{\varepsilon\mu}',\nabla u_{\varepsilon\mu}\right)
\to F\left(x,t,u_{\varepsilon}',\nabla u_{\varepsilon}\right
)\quad\hbox{a.e. in } \Omega_{\varepsilon}\times (0,T).
$$
 From the last convergence and considering \eqref{e2.11} we can apply
\cite[Lemma 1.3]{k1}  to obtain
\[
F\left(x,t,u_{\varepsilon\mu}',\nabla u_{\varepsilon\mu}\right)
\rightharpoonup F\left(x,t,u_{\varepsilon}',\nabla u_{\varepsilon}\right
)\quad\hbox{weakly as }\mu\to\infty .%\label{e2.27}
\]
Therefore, \eqref{e2.22} is proved.

\subsection{Uniqueness}
Let $u$ and $\hat {u}$ be two solutions of \eqref{Pe} and put
$z_{\varepsilon}=u_{\varepsilon}-\hat {u}_{\varepsilon}$.
 From assumption \eqref{e1.8}, noting that the map $s \mapsto |s|^{\rho}s$ is increasing and
 taking \eqref{e2.21} and \eqref{e2.22} into account, we deduce
\begin{equation}
\frac d{dt}\big\{|z_{\varepsilon}'(t)|^2+|\nabla
z_{\varepsilon}(t)|^2\big\}\leq D\big\{|z_{\varepsilon}'
(t)|^2+|\nabla z_{\varepsilon}(t)|^2\big\}.\label{e2.28}
\end{equation}
Integrating \eqref{e2.28} over (0,t)  and employing Gronwall's lemma we
conclude that
$|z_{\varepsilon}'(t)|^2=|\nabla z_{\varepsilon}(t)|^2=0$.
Therefore, $u_{\varepsilon}=\hat{u}_{ \varepsilon}$.  This
completes the proofs of section 2.

\section{The homogenized problem}

We begin this section presenting a technical result that will play
an essential role to obtain the homogenized problem.

\subsection{Technical Lemma}

\begin{lemma} \label{lm3.1}
Assume that \eqref{e1.10} is satisfied with $\mu =0$; then
$$
\lim_{\varepsilon\to 0}\int_{ \Omega}\varphi|\nabla
w_{\varepsilon}|^2dx=0;\quad\forall \varphi\in D(\Omega ).
$$
\end{lemma}

\begin{proof}
Let $\varphi\in D(\Omega )$.  Then,  from \eqref{e1.10}, third and fourth equations,
we have
$\langle -\Delta w_{\varepsilon},\varphi w_{\varepsilon}\rangle
=\langle \mu_{\varepsilon}-\gamma_{\varepsilon},\varphi w_{\varepsilon}\rangle
=\langle \mu_{\varepsilon},\varphi w_{\varepsilon}\rangle $
and consequently
\begin{equation}
\lim_{\varepsilon\to 0}\langle -\Delta w_{\varepsilon},\varphi
w_{\varepsilon}\rangle =0 .\label{e3.1}
\end{equation}
On the other hand, we deduce that
\begin{equation}
\langle -\Delta w_{\varepsilon},\varphi w_{\varepsilon}\rangle =\int_{
\Omega}\varphi|\nabla w_{\varepsilon}|^2dx+\int_{\Omega}
w_{\varepsilon}(\nabla w_{\varepsilon}\cdot\nabla\varphi\,)dx.\label{e3.2}
\end{equation}
Now,  from the third equation in \eqref{e1.10}, we have
\begin{equation}
\nabla w_{\varepsilon}\rightharpoonup 0\quad\hbox{weakly in}
L^2(\Omega )\label{e3.3}
\end{equation}
and since the imbedding $H_0^1(\Omega )\hookrightarrow^{}L^2(\Omega)$ is compact,
 we also obtain
\begin{equation}
w_{\varepsilon}\to 1\quad\hbox{strongly in } L^2(\Omega ). \label{e3.4}
\end{equation}
Combining \eqref{e3.1}-\eqref{e3.4}  we conclude that
$$
\lim_{\varepsilon\to 0}\int_{\Omega}\varphi|\nabla
w_{\varepsilon}|^2dx=0 ,
$$
which concludes the proof.
\end{proof}

Next,  we  obtain the homogenized problem when
$\varepsilon\to 0$, making use of the abstract framework given in \eqref{e1.10} and
taking into consideration the estimates obtained in section 2.

\subsection{A priori estimates}
 From the estimates obtained in section 2, there exists a function
$u:\Omega\times (0,\infty )\to \mathbb{R}$  such that
\begin{gather}
\tilde {u}_{\varepsilon}'\rightharpoonup u'\quad\hbox{weak-star in }
L_{\rm loc}^{\infty}(0,\infty ;L^2(\Omega ),\label{e3.9} \\
\tilde {u}_{\varepsilon}\rightharpoonup u\quad\hbox{weak-star in }
L_{\rm loc}^{\infty}(0,\infty ;H_0^1(\Omega ),\label{e3.10} \\
\tilde {u}_{\varepsilon}''\rightharpoonup u'' \quad\hbox{weak-star in }
L_{\rm loc}^{\infty}(0,\infty ;L^2(\Omega ),\label{e3.11} \\
\tilde {u}_{\varepsilon}'\rightharpoonup u'\quad\hbox{weak-star in }
L_{\rm loc}^{\infty}(0,\infty ;H_0^1(\Omega ).\label{e3.12}
\end{gather}
 From Aubin-Lions theorem,  we also deduce
\begin{gather}
\tilde {u}_{\varepsilon}\to u\quad\hbox{strongly in }
L_{\rm loc}^2(0,\infty ;L^2(\Omega )),\label{e3.13} \\
\tilde {u}_{\varepsilon}'\to u'\quad\hbox{strongly in } L_{\rm
loc}^2(0,\infty ;L^2(\Omega )). \label{e3.14}
\end{gather}
On the other hand, let $u_{\varepsilon_1}$ and $u_{\varepsilon_2}$
be two solutions of $ (P_{\varepsilon})$ with initial data
$\{u_{\varepsilon_1}^0,u_{\varepsilon_1}^1\}$ and
$\{u_{\varepsilon_2}^0,u_{\varepsilon_2}^1\}$,
respectively.

Then, considering $z_{\varepsilon}=u_{\varepsilon_1}-u_{\varepsilon_
2}$ and repeating analogous arguments
like those used to prove the uniqueness of solutions in section
2, we obtain
$$
\frac d{dt}\big\{|z_{\varepsilon}'(t)|^2+|\nabla
z_{\varepsilon}(t)|^2\big\}\leq D\big\{|z_{\varepsilon}'
(t)|^2+|\nabla z_{\varepsilon}(t)|^2\big\}.
$$
Integrating the above inequality over (0,t), we infer
\begin{equation}
\begin{aligned}
&|z_{\varepsilon}'(t)|^2+|\nabla z_{\varepsilon}(t)|^2\\
&\leq|u_{\varepsilon_1}^1-u_{\varepsilon_2}^1|^2+|\nabla u_{\varepsilon_1}^0
-\nabla u_{\varepsilon_2}^0|^2 +D\int_0^t\big\{|z_{\varepsilon}'(s)|^2
+|\nabla z_{\varepsilon}(s)|^2\big\}ds.
\end{aligned}\label{e3.15}
\end{equation}
 From \eqref{e3.15} employing Gronwall's inequality, noting that the imbeddings
 $ H^2(\Omega)\hookrightarrow H^1(\Omega)$ and
$H^1(\Omega)\hookrightarrow L^2(\Omega)$ are compact and taking the convergence
in \eqref{e1.2} into account we deduce that
\begin{gather}
\tilde {u}_{\varepsilon}\to u\quad\hbox{strongly in } C^0([0,T];H_0^1(\Omega ));
\quad \forall T>0,\label{e3.16} \\
\tilde {u}_{\varepsilon}'\to u'\quad\hbox{strongly in }
C^0([0,T];L^2(\Omega )); \quad \forall T>0.\label{e3.17}
\end{gather}
\smallskip

\noindent{\bf Remark.} Note that in view of the strong
convergence given in \eqref{e3.16}, it is not necessary to use the equation
 of the abstract framework given in \eqref{e1.10}. However, we
decided to present the passage to the limit making use of the
whole abstract framework in order to facilitate the reader's
comprehension when one has, for instance, a nonlinearity given by
$ F(x,t,u,u')$ where the strong convergence in \eqref{e3.16}
is not required (see convergence in \eqref{e3.13} and \eqref{e3.14}).


\subsection{Passage to the limit}

Multiplying \eqref{e2.21} (taking \eqref{e2.22} into consideration) by
$w_{\varepsilon}\theta\varphi$,
and integrating over $Q_{\varepsilon}=\Omega_{\varepsilon}\times (0,T)$;
where $w_{\varepsilon}$ belongs to the abstract framework \eqref{e1.10},
$\theta\in D(0,T)$ and $\varphi\in D(\Omega )$, we obtain
\begin{equation}
\int_{Q_{\varepsilon}}u_{\varepsilon}''w_{\varepsilon}
\varphi\theta\,dx\,dt-\int_{Q_{\varepsilon}}\Delta u_{\varepsilon}
w_{\varepsilon}\varphi\theta\,dx\,dt+\int_{Q_{\varepsilon}}F\left
(x,t,u_{\varepsilon},\nabla u_{\varepsilon}\right)w_{\varepsilon}
\varphi\theta\,dx\,dt=0.\label{e3.18}
\end{equation}
Employing Green's formula in the second term of \eqref{e3.18}, we
deduce
\begin{equation}
-\int_{Q_{\varepsilon}}\Delta u_{\varepsilon}w_{\varepsilon}\varphi
\theta\,dx\,dt=\int_{Q_{\varepsilon}}\nabla u_{\varepsilon}\cdot\theta
\nabla w_{\varepsilon}\varphi\,dx\,dt+\int_{Q_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\theta w_{\varepsilon}\nabla\varphi dx\,dt.\label{e3.19}
\end{equation}
On the other hand, we also have
\begin{equation}
\int_{Q_{\varepsilon}}\nabla u_{\varepsilon}\cdot\theta\nabla w_{
\varepsilon}\varphi\,dx\,dt=-\langle \Delta w_{\varepsilon},\theta
u_{\varepsilon}\varphi\rangle -\int_{Q_{\varepsilon}}\nabla w_{\varepsilon}
\cdot\theta u_{\varepsilon}\nabla\varphi\,dx\,dt,\label{e3.20}
\end{equation}
where $\langle \cdot ,\cdot\rangle $  means the duality
$L^1(0,T;H^{-1}(\Omega_{\varepsilon}))$ and
$L^{\infty}(0,T;H_0^1(\Omega_{\varepsilon}))$.

Combining \eqref{e3.18}-\eqref{e3.20} we arrive at
\begin{equation}
\begin{aligned}
\int_0^T\int_{\Omega_{\varepsilon}}u_{\varepsilon}''
w_{\varepsilon}\varphi\theta\,dx\,dt-\langle \Delta w_{\varepsilon}
,\theta u_{\varepsilon}\varphi\rangle -\int_0^T\int_{\Omega_{\varepsilon}}
\nabla w_{\varepsilon}\cdot\theta u_{\varepsilon}\nabla\varphi\,d
x\,dt &\\
+\int_0^T\int_{\Omega_{\varepsilon}}\nabla u_{\varepsilon}\cdot
\theta w_{\varepsilon}\nabla\varphi dx\,dt+\int_0^T\int_{\Omega_{
\varepsilon}}F\left(x,t,u_{\varepsilon},\nabla u_{\varepsilon}\right
)w_{\varepsilon}\varphi\theta\,dx\,dt&=0
\end{aligned}\label{e3.21}
\end{equation}
Next,  we  analyze the terms in \eqref{e3.21}. \smallskip

\noindent {\em Estimate for
$I_1:=\int_0^T\int_{\Omega_{\varepsilon}}u_{\varepsilon}^{
\prime\prime}w_{\varepsilon}\varphi\theta\,dx\,dt$.}
Employing Fubini's theorem we deduce
\begin{equation}
I_1=\int_0^T\int_{\Omega}\tilde {u}_{\varepsilon}''
w_{\varepsilon}\varphi\theta\,dx\,dt=\int_{\Omega}w_{\varepsilon}
\varphi\Big(\int_0^T\theta\tilde u_{\varepsilon}''\,
dt\Big)dx.\label{e3.22}
\end{equation}
 From \eqref{e3.4}  and  \eqref{e3.11} we obtain
\begin{equation}
\lim_{\varepsilon\to 0}I_1=\int_{\Omega}\varphi\Big(\int_
0^T\theta u''dt\Big)dx\quad .\label{e3.23}
\end{equation}
{\em Estimate for $I_2:=-\langle \Delta w_{\varepsilon}, \theta
u_{\varepsilon}\varphi\rangle $.} Consider the
$\mathcal{U}_{\varepsilon}\in H_0^1(\Omega )$ defined by
$\mathcal{U}_{\varepsilon}=\int_0^T\theta\tilde
{u}_{\varepsilon}\,dt$. From the convergence \eqref{e3.10} and
since $H_0^1(\Omega )\hookrightarrow L^2(\Omega)$ is compact we
have
\begin{equation}
\begin{gathered}
{\mathcal{U}_{\varepsilon}\rightharpoonup\int_0^T\theta
u\,dt\quad\hbox{weakly in } H_0^1(\Omega )\hbox{ and strongly in } L^2(\Omega ),}\\
{\mathcal{U}_{\varepsilon}=0\quad\hbox{on } S_{\varepsilon}.}
\end{gathered} \label{e3.24}
\end{equation}
In view of \eqref{e1.10}, fourth equation,  $-\Delta
w_{\varepsilon}=\mu_{\varepsilon} -\gamma_{\varepsilon}$.  Then,
applying Fubini's theorem one has
\begin{align*}
I_2&=\langle \mu_{\varepsilon}-\gamma_{\varepsilon},\theta u_{\varepsilon
\varphi}\rangle \\
&=\big\langle \mu_{\varepsilon}-\gamma_{\varepsilon},\big(\int_0^T\theta
\tilde u_{\varepsilon}\,dt\big)\varphi\big\rangle _{H^{-1}(\Omega ),
H_0^1(\Omega )}\\
&=\langle \mu_{\varepsilon},\mathcal{U}_{\varepsilon}\varphi\rangle _{
H^{-1}(\Omega ),H_0^1(\Omega )},
\end{align*}
since $\langle \gamma_{\varepsilon},\varphi \mathcal{U}_{\varepsilon}
\rangle_{H^{-1}(\Omega ),H_0^1(\Omega )}=0$. Consequently from \eqref{e3.24} and
\eqref{e1.10}, fourth equation, we infer
\begin{equation}
\lim_{\varepsilon\to 0}I_2=\big\langle \mu ,\big(\int_0^T\theta
u\,dt\big)\varphi\big\rangle _{H^{-1}(\Omega ),H_0^1(\Omega
)}=0.\label{e3.25}
\end{equation}
{\em Estimate for $I_3:=\int_0^T\int_{\Omega_{\varepsilon}}\nabla
w_{\varepsilon}\cdot\theta u_{\varepsilon}\nabla\varphi\,dx\,dt$.}
>From Fubini's theorem we deduce
\begin{equation}
I_3=\int_{\Omega}\nabla w_{\varepsilon}\cdot\big(\int_0^T\theta
\tilde u_{\varepsilon}dt\big)\nabla\varphi\,dx.\label{e3.26}
\end{equation}
Taking \eqref{e3.3} and \eqref{e3.24} into account,from \eqref{e3.26}  it holds that
\begin{equation}
\lim_{\varepsilon\to 0}I_3=0.\label{e3.27}
\end{equation}
{\em Estimate for $I_4:=\int_0^T\int_{\Omega_{\varepsilon}}\nabla
u_{\varepsilon}\cdot\theta w_{\varepsilon}\nabla\varphi dx\,dt$.}
Analogously, employing Fubini's theorem it follows that
\begin{equation}
I_4=\int_{\Omega}w_{\varepsilon}\nabla\varphi\cdot\nabla\left(\int_
0^T\theta\tilde u_{\varepsilon}dt\right)dx.\label{e3.28}
\end{equation}

Considering \eqref{e3.4} and \eqref{e3.24} from \eqref{e3.28} we conclude
\begin{equation}
\lim_{\varepsilon\to 0}I_4=\int_{\Omega}\nabla\varphi\cdot
\nabla\big(\int_0^T\theta u\,dt\big)dx.\label{e3.29}
\end{equation}
{\em Estimate for $I_5:=\int_0^T\int_{\Omega_{\varepsilon}}
F(x,t,u_{\varepsilon},\nabla u_{\varepsilon})w_{\varepsilon}
\varphi\theta\,dx\,dt$.}
Analogously considering Fubini's theorem and in view of
assumption \eqref{e1.9} we can write
\begin{equation}
I_5=\int_{\Omega}w_{\varepsilon}\varphi\big(\int_0^T\theta
F(x,t,\tilde u_{\varepsilon}',\nabla\tilde u_{\varepsilon})dt\big)dx.
\label{e3.30}
\end{equation}
On the other hand, from the convergence \eqref{e3.16}, \eqref{e3.17} and
making use of Lion's lemma, we deduce that
\begin{equation}
F(x,t,\tilde u_{\varepsilon}',\nabla\tilde u_{\varepsilon})
\rightharpoonup F\left(x,t,u',\nabla u\right)\quad\hbox{weakly
in } L^2(0,T;L^2(\Omega )).\label{e3.31}
\end{equation}
Then, from \eqref{e3.4}, \eqref{e3.30} and \eqref{e3.31} we conclude
\begin{equation}
\lim_{\varepsilon\to 0}I_5=\int_{\Omega}\varphi\big(\theta
\int_0^TF(x,t,u',\nabla u)dt\big)dx.\label{e3.32}
\end{equation}
Combining \eqref{e3.21}, \eqref{e3.23}, \eqref{e3.25},
\eqref{e3.27}, \eqref{e3.29} and \eqref{e3.32} we deduce
\begin{equation}
\langle \int_{\Omega}u''\varphi dx,\theta\rangle +\langle
\int_{\Omega}\nabla u\cdot\nabla\varphi dx,\theta\rangle+ \langle
\int_{\Omega}F\left(x,t,u',\nabla u\right)\varphi dx,\theta\rangle
=0, \label{e3.33}
\end{equation}
where $\langle \cdot ,\cdot\rangle $ means the duality $D'(0,T)$, $D(0,T$),
for all $\varphi\in D(\Omega )$ and for all $\theta\in D(0,T)$.
Then, since $D(\Omega )$ is dense in $H_0^1(\Omega)$ we obtain
\begin{equation}
\left(u''(t),v\right) +(\nabla u(t),v)
+\left(F\left(x,t,u'(t),\nabla
u(t)\right),v\right)=0\quad\hbox{in}\,\,D'(0,T) \label{e3.34}
\end{equation}
for all  $v\in H_0^1(\Omega)$, where $(\cdot ,\cdot)$ is the inner product in
$L^2(\Omega )$. \smallskip

The uniqueness of solutions follows considering analogous
arguments like those ones used to prove \eqref{e2.28}.

\section{Uniform Decay Rates}

In this section we establish uniform  rates of decay (exponential and
algebraic) for the homogenized problem
\begin{equation}
\begin{gathered}
{u''-\Delta u+F\left(x,t,u',\nabla u\right)=0\quad\,\hbox{in } \Omega\times (0,T)}\\
{u=0\quad \hbox{on }\Gamma\times (0,+\infty )}\\
{u(x,0)=u^0(x);\quad u'(x,0)=u^1(x);\quad x\in\Omega .}
\end{gathered} \label{e4.1}
\end{equation}
Since $E(t)=\lim_{\varepsilon \rightarrow 0} E_{\varepsilon}(t)$,
it is sufficient to prove that problem \eqref{Pe} decays
exponentially or polynomially independently of $\varepsilon $.  In
other words, it is enough to prove that there exist positive
constants $C,$ $\gamma_0$, $k_1$, $ k_2$ and $k_3$ independent of
$\varepsilon>0$ and such that
\begin{equation}
E_{\varepsilon}(t)\leq Ce^{-\gamma_0t}\quad\hbox{or}\quad E_{\varepsilon}
(t)\leq k_2\frac {k_3+2k_1}{(1+t)^{2/\rho}}\label{e4.2}
\end{equation}
for all $t\geq 0$ and for all $\varepsilon >0$. \smallskip

\noindent{Remark 2.} It is important to observe that when $\mu >0$
and supposing that one could be able to homogenize the problem
under consideration, an useful alternative to derive uniform decay
rates for the energy
\begin{eqnarray*}
&E^{\mu}(t)= \frac12 \left(|u'(t)|_{L^2(\Omega)^2 + |\nabla
u(t)|_{L^2(\Omega)}^2} + |u(t)|_{L^2(\Omega,\mu)}^2 \right)&
\end{eqnarray*}
would be  making use of the lower semi-continuity of the energy,
or, more precisely, to consider the following estimate:
\begin{eqnarray*}
E^{\mu} \leq \liminf_{\varepsilon \rightarrow 0} E_{\varepsilon}
(t), \quad \hbox{ for all } t\geq 0.
\end{eqnarray*}

\smallskip

In order to obtain \eqref{e4.2}, we consider the following auxiliary lemmas.

\begin{lemma} \label{lm4.1}
Let $E$ be a real $C^1$ positive function satisfying
\begin{equation}
 E'(t)\leq -C_0E(t)+C_1e^{-\gamma t}\label{e4.3}
\end{equation}
 where $C_0$, $C_1$ and $\gamma$  are positive constants.  Then,  there
exists $\gamma_0$  such that
\begin{equation}
E(t)\leq\left(E(0)+(2C_1)/\gamma\right)e^{-\gamma_0t}.\label{e4.4}
\end{equation}
\end{lemma}

\begin{proof} Let $F(t)=E(t)+\frac {2C_1}{\gamma}e^{-\gamma t}$.
Then
$$
F'(t)=E'(t)-2C_1e^{-\gamma t}\leq -C_0E(t)-C_1e^{-\gamma t}\leq-\gamma_0F(t),
$$
where $\gamma_0=\min\{C_0,\frac {\gamma}2\}$.  Integrating the last inequality
over $(0,t)$, we have
$$
F(t)\leq F(0)e^{-\gamma_0t}\quad\hbox{implies}\quad E(t)\leq C_2e^{-\gamma_0t},
$$
where $C_2=E(0)+\frac {2C_1}{\gamma}$.  This completes the proof.
\end{proof}

\begin{lemma} \label{lm4.2}
Let $E$ be a real $C^1$ positive function satisfying
\begin{equation}
E'(t)\leq -k_0[E(t)]^{\frac {\rho +2}2}+\frac {k_1}{\left(1+t\right)
^{\frac {\rho +2}{\rho}}}\label{e4.5}
\end{equation}
where $0<\rho <2$,  and $k_0$ and $k_1$  are positive constants. Then,
 there exists $k_2>0$ such that
\begin{equation}
E(t)\leq k_2\frac {\frac {\rho}2E(0)+2k_1}{\left(1+t\right)^{2/\rho}}.\label{e4.6}
\end{equation}
\end{lemma}

\begin{proof} Consider $h(t)=\frac {2k_1}{\frac {\rho}2(1+t)^{\rho/2}}$ and set
$g(t)=E(t)+h(t)$. We have
\begin{align*}
g'(t)&=E'(t)-\frac {2k_1}{(1+t)^{\frac {\rho +2}{\rho}}}
\leq -k_0\big\{[E(t)]^{\frac {\rho +2}2}+\frac {k_1}{
k_0(1+t)^{\frac {\rho +2}{\rho}}}\big\}\\
&\leq -k_0\big\{[E(t)]^{\frac {\rho +2}2}+(\frac
1{\rho})^{\frac {\rho +2}2}\frac 1{k_0k_1^{\rho /2}}[h
(t)]^{\frac {\rho +2}2}\big\}.
\end{align*}
Let $a_0=\min\{1,(\frac 1{\rho})^{\frac {\rho +2}2}\frac 1{k_0k_1^{\rho /2}}\}$.
Then,
$$ g'(t)\leq -k_0a_0\big\{[E(t)]^{\frac {\rho +2}2}+[h(t)]^{\frac {\rho +2}2}\big\}.
$$
Since there exists a positive constant $a_1$ such that
$$
[E(t)+h(t)]^{\frac {\rho +2}2}
\leq a_1\big\{[E(t)]^{\frac {\rho +2}2}+[h(t)]^{\frac {\rho +2}2}\big\}
$$
we conclude that
$$
g'(t)\leq -\frac {k_0a_0}{a_1}[g(t)]^{\frac {\rho +2}2}.
$$
Integrating the last inequality over (0,t), we deduce
$$
g(t)\leq\frac {(\frac 2{\rho})^{2/\rho}g(0)}{\{\frac
2{\rho}+\frac {k_0a_0}{a_1}[g(0)]^{\rho /2}t\}}
\leq\frac {(\frac 2{\rho})^{\frac {2-\rho}{\rho}}[\frac 2{\rho}E(
0)+2k_1]}{a_2^{2/\rho}(1+t)^{2/\rho}},
$$
where $a_2=\min\{\frac 2{\rho},\frac {k_0a_0}{a_1}[g(0)]^{\rho /2}\}$.
Considering $k_2=\frac 1{a_2}(\frac 2{\rho a_2})^{\frac {2-\rho}{\rho}}$,
it follows the desired result.
\end{proof}

For short notation, we will omit the parameter $\varepsilon$ on the energy
\begin{equation}
E_{\varepsilon}(t)=\frac 12\big(|u_{\varepsilon}'(t)|^2_{L^2(\Omega_{\varepsilon})}
+|\nabla u_{\varepsilon}(t)|_{L^2(\Omega_{\varepsilon})}^2\big)\label{e4.7}
\end{equation}
having in mind that the constants obtained can not depend on
$\varepsilon$. Inspired in the work of Haraux and Zuazua \cite{h1}
let us define the Liapunov functional
\begin{equation}
\psi (t)=[E(t)]^{\rho /2}\left(u'(t),u(t)\right)\label{e4.8}
\end{equation}
where $(\cdot ,\cdot)$ is the inner product in $L^2(\Omega_{\varepsilon})$.

\begin{proposition} \label{prop4.1}
There exists $L>0$, independent of $\varepsilon$, such that
$E(t)\leq L$ for all $t\geq 0$.
\end{proposition}

\begin{proof}  From \eqref{e2.21}, \eqref{e2.22}  and \eqref{e1.4}  we deduce
\begin{equation}
E'(t)\leq -\beta\|u'(t)\|^{\rho +2}_{\rho
+2}+\varphi (t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla u|\right)dx
\leq\varphi (t)\mathop{\rm meas}(\Omega )+\varphi (t)E(t).\label{e4.9}
\end{equation}
Multiplying both sides of the above inequality by $e^{-\int_0^t\varphi(s)ds}$, it
follows that
\begin{equation}
\Big(E(t)e^{-\int_0^t\varphi (s)ds}\Big)'\leq\varphi (t)\mathop{\rm meas}
 (\Omega ).\label{e4.10}
\end{equation}
Integrating \eqref{e4.10}  over $(0,t)$, we obtain
\begin{equation}
E(t)\leq E(0)e^{\int_0^{\infty}\varphi (s)ds}+e^{\int_0^t\varphi
(s)ds}\Big(\int_0^{\infty}\varphi (s)ds\Big)\mathop{\rm meas}(\Omega ).\label{e4.11}
\end{equation}
Considering the convergence in \eqref{e1.2} we deduce that
\begin{equation}
E_{\varepsilon}(0)\leq K;\quad\forall\varepsilon >0\label{e4.12}
\end{equation}
where $K=K(|\nabla u^0|_{L^2(\Omega )},|u^1|_{L^2(\Omega )})$.
Combining \eqref{e4.11} and \eqref{e4.12},  we obtain
$E(t)\leq L$ for all $t\geq 0$, where
\begin{equation}
L=e^{\int_0^{\infty}\varphi (s)ds}\Big(K+\int_0^{\infty}\varphi
(s)ds\,\mathop{\rm meas}(\Omega )\Big),\label{e4.13}
\end{equation}
which concludes the proof
\end{proof}

\begin{proposition} \label{prop4.2}
There exists $\lambda >0$, independent of $\varepsilon$, such that
$$
|\psi (t)|\leq\lambda L^{\rho /2}E(t);\quad\forall t\geq 0.
$$
\end{proposition}

\begin{proof} From \eqref{e4.8} we deduce
$$
|\psi (t)|\leq[E(t)]^{\rho /2}|u'(t)|\lambda|\nabla u(t)|
$$
where $\lambda >0$ comes from the Poincar\'e inequality in $\Omega$; i.e.,
\begin{equation}
|u_{\varepsilon}(t)|_{L^2(\Omega_{\varepsilon})}
=|\tilde u_{\varepsilon}(t)|_{L^2(\Omega )}\leq\lambda|
\nabla\tilde u_{\varepsilon}(t)|_{L^2(\Omega )}
=\lambda|\nabla u_{\varepsilon}(t)|_{L^2(\Omega_{\varepsilon})}.\label{e4.14}
\end{equation}
The above inequalities and Proposition \ref{prop4.1} yield
$$ |\psi (t)|\leq\lambda L^{\rho /2}E(t),
$$
which completes the proof. \end{proof}

\begin{proposition} \label{prop4.3}
Assume that $\rho=0$ and that there exist $C_1$ and $ \gamma$ positive constants
such that
\begin{equation}
\varphi (t)\leq C_1e^{-\gamma t} \quad\forall t\geq 0.\label{e4.15}
\end{equation}
Then, \eqref{e4.3} holds where $C_0$ is a positive constant independent
of $\varepsilon $.  Now, considering that there exists $k_1>0$ such that
\begin{equation}
\varphi (t)\leq\frac {k_1}{\left(1+t\right)^{\frac {\rho +2}{\rho}}}
\quad\forall t\geq 0, \label{e4.16}
\end{equation}
then \eqref{e4.5} holds where $k_0$ is a positive constant independent of
$\varepsilon$.
\end{proposition}

\begin{proof} From \eqref{e4.9}, we have that
\begin{equation}
E'(t)-\varphi (t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla u|\right)dx\leq 0.\label{e4.17}
\end{equation}
Computing the derivative of \eqref{e4.8} with respect to $t$  and
substituting $u''=\Delta u-F\left(x,t,u',\nabla u\right)$, from
\eqref{e1.4} and making
 use of Green formula we deduce
\begin{equation}
\begin{aligned}
\psi'(t)& \leq\frac {\rho}2[E(t)]^{\frac {\rho -2}2}E'(t)\left(u'(t),u(t)\right)\\
&\quad +[E(t)]^{\rho /2}\big\{-|\nabla u(t)|^2-\beta\left(|u'(t)|^{\rho}u'(t),u(t)
\right)\\
&\quad +\varphi (t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla u|\right)dx
+|u'(t)|^2\big\}.
\end{aligned}\label{e4.18}
\end{equation}
Adding and subtracting the term
$$ \frac {\rho}2[E(t)]^{\frac {\rho -2}2}\varphi
(t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla
u|\right)dx\left(u'(t),u(t)\right)
$$
in \eqref{e4.8}, we infer
\begin{equation}
\begin{aligned}
\psi'(t)&\leq-\frac {\rho}2[E(t)]^{\frac {\rho -2}2}\left
(u'(t),u(t)\right)\big[\varphi (t)\int_{\Omega_{\varepsilon}}\left
(1+|u'||\nabla u|\right)dx-E'(t)\big] \\
&\quad +\frac {\rho}2[E(t)]^{\frac {\rho -2}2}\varphi
(t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla
u|\right)dx\left(u'(t),u(t)\right)\\
&\quad +[E(t)]^{\rho /2}\big\{-|\nabla u(t)|^2-
\beta\left(|u'(t)|^{\rho}u'(t),u(t)\right) \\
&\quad +\varphi (t)\int_{\Omega_{\varepsilon}}\left(1+|u'||\nabla u|\right)dx
+|u'(t)|^2\big\}.
\end{aligned}\label{e4.19}
\end{equation}
Observe that from \eqref{e4.14} and taking Proposition \ref{prop4.1} into
account, we can write
\begin{equation}
|\left(u'(t),u(t)\right)|\leq\lambda E(t)\leq\lambda L.\label{e4.20}
\end{equation}
Then, combining \eqref{e4.17}, \eqref{e4.19} and \eqref{e4.20} we deduce
\begin{equation}
\begin{aligned}
\psi'(t)&\leq\frac {\rho\lambda L^{\rho /2}}2\big[\varphi (t)\int_{
\Omega_{\varepsilon}}\left(1+|u'||\nabla u|\right)dx-E'(t)\big]\\
&\quad +\frac {\rho\lambda L^{\rho /2}}2\varphi (t)\int_{\Omega_{\varepsilon}}\left
(1+|u'||\nabla u|\right)dx \\
&\quad +[E(t)]^{\rho /2}\big\{-|\nabla u(t)|^2-
\beta\left(|u'(t)|^{\rho}u'(t),u(t)\right) \\
&\quad +\varphi (t)\int_{\Omega_{\varepsilon}}\left(1+|u'|
|\nabla u|\right)dx+|u'(t)|^2\big\}.
\end{aligned}\label{e4.21}
\end{equation}
Therefore,
\begin{equation}
\begin{aligned}
\psi'(t)
&\leq -\rho\lambda L^{\rho /2}E'(t)+\rho\lambda L^{\rho
/2}\left(\mathop{\rm meas}(\Omega )+L\right)\varphi (t)\\
&\quad +[E(t)]^{\rho /2}\big\{-|\nabla u(t)|^2-
\beta\left(|u'(t)|^{\rho}u'(t),u(t)\right)\\
&\quad + \left(\mathop{\rm meas}(\Omega )+L\right)\varphi
(t)+|u'(t)|^2\big\}.
\end{aligned}\label{e4.22}
\end{equation}
{\em Estimate for $I_1:=\beta\left(|u'(t)|^{\rho}u'(t),u(t)\right)$.}
Making use of H\"older inequality having in mind that
$\frac {\rho +1}{\rho +2}+\frac 1{\rho +2}=1,$ we deduce
\begin{equation}
|I_1|\leq\beta\|u'(t)\|_{\rho +2}^{\rho +1}\|u(t)\|_{\rho +2}.\label{e4.23}
\end{equation}
Now, since $H_0^1(\Omega )\hookrightarrow L^{\rho +2}(\Omega )$, we have
\begin{equation}
\|u_{\varepsilon}(t)\|_{L^{\rho +2}(\Omega_{
\varepsilon})}=\|\tilde u_{\varepsilon}\|_{
L^{\rho +2}(\Omega )}\leq\xi\|\tilde u_{\varepsilon}(t)
\|_{H_0^1(\Omega )}=\xi\|u_{\varepsilon}(t)\|_{H_0^1(\Omega_{\varepsilon})}.\label{e4.24}
\end{equation}
Then, from \eqref{e4.23}, \eqref{e4.24}  and making use of Young's inequality,
 we conclude
\begin{align*}
|I_1|\leq\beta\xi\|u'(t)\|_{\rho +2}^{\rho +1}|\nabla u(t)|
&\leq\frac {(\rho +1)\left(\beta\xi\right)^{\frac {\rho +2}{\rho+1}}}
{\eta\frac 1{\rho +1}}\|u'(t)\|_{\rho+2}^{\rho +2}
+\frac {\eta}{\rho +2}|\nabla u(t)|^{\rho +2},
\end{align*}
where $\eta >0$ is an arbitrary positive constant.
On the other hand, from Proposition \ref{prop4.1} one has
$$
|\nabla u(t)|^{\rho +2}\leq 2^{\rho /2}L^{\rho /2}|\nabla u(t)|^2.
$$
Then,
\begin{equation}
|I_1|\leq\frac {(\rho +1)\left(\beta\xi\right)^{\frac {\rho +2}{
\rho +1}}}{\eta\frac 1{\rho +1}}\|u'(t)\|_{ \rho +2}^{\rho
+2}+\eta\frac {2^{\rho /2}L^{\rho /2}}{\rho +2}|\nabla
u(t)|^2.\label{e4.25}
\end{equation}
Combining \eqref{e4.22} and \eqref{e4.25} choosing $\eta =\frac
{\rho +2}{2^{\frac {\rho +2}2}L^{\rho /2}}$, we infer
\begin{equation}
\begin{aligned}
\psi'(t)&\leq -\rho\lambda L^{\rho /2}E'(t)+\rho\lambda L^{\rho
/2}\left(\mathop{\rm meas}(\Omega )+L\right)\varphi (t) \\
&+\quad [E(t)]^{\rho /2}\big\{-\frac 12|\nabla u(t)|^2
+M\|u'(t)\|_{\rho +2}^{\rho +2}
\left(\mathop{\rm meas}(\Omega )+L\right)\varphi (t)+|u'(t)|^2\big\},
\end{aligned}\label{e4.26}
\end{equation}
where
\begin{equation}
M=\frac {(\rho +1)\left(\beta\xi\right)^{(\rho +2)/(\rho +1)}}{
(\rho +2)\big(\frac {\rho +2}{2^{\frac {\rho +2}2}L^{\rho
/2}}\big)^{1/( \rho +1)}}.\label{e4.27}
\end{equation}
Now, from \eqref{e4.26}, Proposition \ref{prop4.1} and \eqref{e4.9},  we obtain
\begin{equation}
\begin{aligned}
\psi'(t)&\leq -\left(\rho\lambda +M\beta^{-1}L^{\rho /2}\right)E'
(t)+M\beta^{-1}L^{\rho /2}\varphi (t)\int_{\Omega_{\varepsilon}}\left
(1+|u'||\nabla u|\right)dx \\
&\quad -\frac 12[E(t)]^{\rho /2}|\nabla u(t)|^2+
[E(t)]^{\rho /2}|u'(t)|^2+L^{\rho /2}\left(\mathop{\rm meas}(
\Omega )+L\right)\varphi (t).
\end{aligned}\label{e4.28}
\end{equation}
Consequently
\begin{equation}
\begin{aligned}
\psi'(t)&\leq -\left(\rho\lambda +M\beta^{-1}L^{\rho /2}\right)E'(t)
-\frac 12[E(t)]^{\rho /2}|\nabla u(t)|^2 \\
&\quad +[E(t)]^{\rho /2}|u'(t)|^2+N\varphi (t),
\end{aligned}\label{e4.29}
\end{equation}
where
$N=L^{\rho /2}\left(\mathop{\rm meas}(\Omega )+L\right)\left(1+M\beta^{-1}\right)$
%.\label{e4.30}
Defining the perturbed energy by
\begin{equation}
E_{\tau}(t)=(1+\tau R)E(t)+\tau\psi (t);\quad\tau >0,\label{e4.31}
\end{equation}
where
$R=\rho\lambda +M\beta^{-1}L^{\rho /2}$, %\label{e4.32}
from Proposition \ref{prop4.2} we deduce
\begin{equation}
|E_{\tau}(t)-E(t)|\leq\tau\left(R+\lambda L^{\rho /2}\right)E(t).\label{e4.33}
\end{equation}
Setting
$C_1=R+\lambda L^{\rho /2}$, %\label{e4.34}
considering $\tau\in (0,1/2C_1]$, we deduce
\begin{equation}
\frac 12E(t)\leq E_{\tau}(t)\leq 2E(t);\quad\forall t\geq 0,\label{e4.35}
\end{equation}
which implies
\begin{equation}
2^{-\frac {\rho +2}2}[E(t)]^{\frac {\rho +2}2}
\leq [E_{\tau}(t)]^{\frac {\rho +2}2}\leq 2^{\frac {\rho +2}2}
[E(t)]^{\frac {\rho +2}2}.\label{e4.36}
\end{equation}
On the other hand, taking the derivative of \eqref{e4.31} with respect
to $t$ taking \eqref{e4.29} into account, it holds that
$$
E_{\tau}'(t)\leq E'(t)-\frac {\tau}2[E(t)]^{\rho /2}\left
|\nabla u(t)|^2+\tau[E(t)]^{\rho /2}|u'(t)\right
|^2+\tau N\varphi (t).
$$
The last inequality and \eqref{e4.9} yield
\begin{equation}
E_{\tau}'(t)\leq -\beta\|u'(t)\|_{\rho +2}^{\rho +2}
-\frac {\tau}2[E(t)]^{\rho /2}|\nabla u(t)|^2
+\tau[E(t)]^{\rho /2}|u'(t)|^2+\tau N^{*}\varphi (t),
\label{e4.37}
\end{equation}
where
$N^{*}=N+\mathop{\rm meas}(\Omega )+L$. %\label{e4.38}
Having in mind that
\begin{equation}
-\frac 12 |\nabla u(t)|^2=\frac 12 |u'(t)|^2-\frac 12 E(t)
\end{equation}
and noting that $L^{\rho +2}(\Omega )\hookrightarrow L^2(\Omega )$,
from \eqref{e4.37} we deduce
\begin{equation}
E_{\tau}'(t)\leq -\beta\theta^{-(\rho +2)}|u'(t)|^{\rho+2}-\frac {\tau}2[E(t)]^{\frac {\rho +2}2}
+\frac 32\tau[E(t)]^{\rho /2}|u'(t)|^2+\tau N^{*}\varphi (t),
\label{e4.39}
\end{equation}
where $\theta$ comes from the inequality
\begin{equation}
|u_{\varepsilon}'(t)|_{L^2(\Omega_{\varepsilon})}
=|\tilde u_{\varepsilon}'(t)|_{L^2(\Omega )}
\leq\theta \|\tilde u_{\varepsilon}'(t)\|_{L^{\rho +2}(\Omega )}
= \theta\|u_{\varepsilon}'(t)\|_{L^{\rho +2}(\Omega_{\varepsilon})}.\label{e3.40}
\end{equation}
However, since $\frac {\rho}{\rho +2}+\frac 2{\rho +2}=1$,  the H\"older
inequality yields
\begin{equation}
\begin{aligned}
[E(t)]^{\rho /2}|u'(t)|^2
&\leq\frac {\rho}{\rho +2}\left(\eta[E(t)]^{\rho /2}\right)
^{\frac {\rho +2}{\rho}}+\frac 2{\rho +2}\left(\frac1{\eta}|u'( t)|^2\right)
^{\frac {\rho +2}2}\\
&\leq\eta^{\frac {\rho +2}{\rho}}[E(t)]^{\frac {\rho+2}2}
+\frac 1{\eta^{\frac {\rho +2}2}}|u'(t)|^{\rho +2},
\end{aligned}\label{e4.41}
\end{equation}
where $\eta$ is an arbitrary positive constant.
Then,  from \eqref{e4.39}  and \eqref{e4.41}  we obtain
\begin{equation}
E_{\tau}'(t)\leq -\big(\beta\theta^{-(\rho +2)}-\frac {3\tau}2\frac
1{\eta^{\frac {\rho +2}2}}\big)|u'(t)|^{\rho +2}
-\frac {\tau}2\left(1-3\eta^{\frac {\rho +2}{\rho}}\right)
[E(t)]^{\frac {\rho +2}2}+\tau N^{*}\varphi (t).
\label{e4.42}
\end{equation}
Choosing $\eta$ sufficiently small in order to have
$\zeta =1-3\eta^{\frac {\rho +2}{\rho}}>0$
and $\tau$ small enough to have
$$
\beta\theta^{-(\rho +2)}-\frac {3\tau}2\frac 1{\eta^{\frac {\rho
+2}2}}\geq 0,
$$
from \eqref{e4.42} we conclude that
\begin{equation}
E_{\tau}'(t)\leq -\frac {\tau\zeta}2[E(t)]^{\frac {\rho
+2}2}+\tau N^{*}\varphi (t).\label{e4.43}
\end{equation}
At this point, we have to divide our proof into two parts,
namely,

\noindent (A) If $\rho >0$ and $\varphi (t)$ verifies \eqref{e4.16}.
Then, combining \eqref{e4.36} and \eqref{e4.43},  we obtain
$$
E_{\tau}'(t)\leq -k_0^{*}[E_{\tau}(t)]^{\frac {\rho
+2}2}+\frac {k_1^{*}}{(1+t)^{\frac {\rho +2}{\rho}}},
$$
where $k_0^{*}$, $k_1^{*}$  are positive constants independent of
$\varepsilon $.  So,
from Lemma \ref{lm4.2} and considering \eqref{e4.36}, the decay in \eqref{e4.6} holds.

\noindent (B) If $\rho =0$ and $\varphi (t)$  verifies \eqref{e4.15}.
Again, combining \eqref{e4.36}  and \eqref{e4.43}  we obtain
$$E_{\tau}'(t)\leq -C_0^{*}E_{\tau}(t)+C_1^{*}e^{-\gamma t}$$
where $C_0^{*}$ and $C_1^{*}$ are positive constants independent of
$\varepsilon $.

Now, from Lemma \ref{lm4.1}  and taking \eqref{e4.36} into account, \eqref{e4.4}
holds.  This completes the proof of Theorem \ref{thm1.1}.
\end{proof}

\subsection*{Acknowledgements.} The authors would
like to thank the referees for reading the manuscript carefully
and for giving us constructive remarks which resulted in this
final version of our work.

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