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\markboth{\hfil Existence and regularity of a global attractor
\hfil EJDE--2002/45} {EJDE--2002/45\hfil 
Abderrahmane El Hachimi \& Hamid El Ouardi \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 45, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Existence and regularity of a global attractor for
  doubly nonlinear parabolic equations
 %
\thanks{ {\em Mathematics Subject Classifications:} 35K15, 35K60, 35K65.
\hfil\break\indent 
{\em Key words:} p-Laplacian, a-priori
estimate, long time behaviour, dynamical system,
\hfil\break\indent 
absorbing set, global attractor.
\hfil\break\indent 
\copyright 2002 Southwest Texas State University. \hfil\break\indent 
Submitted January 15, 2001. Published May 24, 2002.} }

\date{}
%
\author{Abderrahmane El Hachimi \& Hamid El Ouardi}
\maketitle

\begin{abstract}
   In this paper we consider a doubly nonlinear parabolic partial
   differential equation
   $$
   \frac{\partial \beta (u)}{\partial t}-\Delta _{p}u+f(x,t,u)=0
   \quad \mbox{in }\Omega \times\mathbb{R}^{+},
   $$
   with Dirichlet boundary condition and initial data given.
   We prove the existence of a global compact attractor
   by using a dynamical system approach. Under additional
   conditions on the nonlinearities $\beta$, $f$, and on $p$,
   we prove more regularity for the global attractor
   and obtain stabilization results for the solutions.
\end{abstract}

\numberwithin{equation}{section}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}


\section{Introduction}

This paper is devoted to the study of a doubly nonlinear parabolic
P.D.E. related to the p-Laplacian operator. More precisely, we are
interested in the existence, uniqueness and long time behaviour of
the solutions of  problem
\begin{equation} \label{P}
\begin{gathered}
\frac{\partial \beta (u)}{\partial t}-\Delta _{p}u+f(x,t,u)=  0
\quad\mbox{in }\Omega \times (0,\infty)\\
 u  =  0 \quad \mbox{on } \partial\Omega \times (0,\infty ) \\
 \beta (u(.,0)) =  \beta (u_{0}) \quad \mbox{in }\Omega ,
\end{gathered}
\end{equation}
 where $\Delta _{p}u=\mathop{\rm div}\left(|\nabla u|^{p-2}\nabla u\right)$ ,
$1<p<+\infty $ and $\Omega $ is a regular
bounded open subset of $\mathbb{R}^{N}$, $N\geq 1$.

These problems arise in many applications in the fields of mechanics,
physics and biology (non Newtonian fluids, gas flow in porous
media, spread of biological populations, etc.). There are a lot of
works dedicated to the existence of solutions
\cite{al,ba,bf,dp,ra} and to the large time behaviour of these equations
\cite{dt, emr1,fct,lp,sch,tsu}.

Our work is inspired by the results of El Hachimi and
de Th\'elin \cite{ht1,ht2} and of Eden,  Michaux and Rakotoson \cite{emr1}.
The  aim here is to study the long time behaviour of solutions of (\ref{P})
via a dynamical systems approach (in the framework of Foias and Temam
\cite{ft}). As is well known, the presence of a dissipative term,
in many infinete dimensional systems, implies  the
existence of a compact set $\mathcal{A}$ which attracts all the
trajectories. This set, called the global attractor, has usually
finite Haussdorf and fractal dimensions, and it is studied by reducing it
to a finite dimensional system.

For $p=2$, problem (\ref{P}) has  been studied in
\cite{emr1,emr2}. Here, we shall consider general $p$ under the
same assumptions on $\beta $ and $f$ as in these references, and
extend some of the results therein.

This paper is organized as follows: After some preliminaries in Section 2,
we give, in section 3, an existence result for solutions of problem (\ref{P}).
Then section 4 is devoted to the existence of the global attractor
$\mathcal{A}$. Finally in section 5 we give, under restrictive
conditions on $\beta,f, p$,
a supplementary regularity result for $\mathcal{A}$ and a stabilization
result for the solutions of (\ref{P}).

\section{Preliminaries}
\paragraph{Notation}
Let $\beta $ be a continuous function with $\beta
(0)=0$. For $t\in \mathbb{R}$, we define
$\Psi (t)=\int_{0}^{t}\beta (\tau)d\tau $. Then the Legendre transform of
$\Psi $ is defined as
$\Psi ^{\ast }(\tau )=\sup_{s\in \mathbb{R}} \{{\tau s-\Psi(s)}\}$.
Let $\Omega $ be a regular open bounded subset of $\mathbb{R}^{N}$
and $\partial \Omega $ its boundary.
For $T>0$, we set $Q_{T}=\Omega \times (0,T)$ and
$S_{T}=\partial \Omega \times (0,T)$.
 The norm in a space $X$ will be denoted by
$\|\cdot\|_X$. However, $\|\cdot\|_{r}$ is the norm when $X=L^{r}(\Omega )$
with $1\leq r\leq +\infty $, and
$\|\cdot\|_{1,q}$ when $X=W^{1,q}(\Omega )$ with $1\leq q\leq +\infty$.
Let $\langle\cdot,\cdot \rangle _{X,X'}$ denote the duality product
between $X$ and its dual $X'$.
For  $l>1$ we denote by $\ell'$ the conjugate of $\ell$; that is the real
number $l'$ satisfying $\frac{1}{l}+\frac{1}{l'}=1$.
For $1\leq r<+\infty $, we shall denote by $W_{r}^{2,1}((0,T)\times \Omega )$
the set of all functions $v$ such that
$$
\int_{0}^{T}\int_{\Omega }\Big( | v|
^{r}+| Dv| ^{r}+| D^{2}v| ^{r}+\big|\frac{\partial v}{\partial T}\big| ^{r}
\Big) dx\, dt<\infty.
$$

We shall consider the following hypotheses.
\begin{enumerate}
\item[(H1)]
$u_{0}$ and $\beta (u_{0})$ are in $L^{2}(\Omega )$.
\item[(H2)]
$\beta $ is an increasing locally Lipschitzian function
from $\mathbb{R}$ to $\mathbb{R}$, with $\beta (0)=0$.
\item[(H3)]
For each $\zeta \in \mathbb{R}$, the map
$(x,t)\to f(x,t,\zeta )$ is measurable and $\zeta \to
f(x,t,\zeta )$ is continuous  almost
everywhere in $\Omega \times \mathbb{R}^{+}$.
Furthermore, we assume that there exist positive constants
$c_{1}, c_{2},c_{3}$ such that, for a.e $(x,t)\in
\Omega \times \mathbb{R}^{+}$,
\begin{equation} \label{e2.1}
\begin{gathered}
\mathop{\rm sign}(\xi)f(x,t,\xi )  \geq c_{1}| \beta (\xi )|^{q-1}-c_{2},\\
\lim_{t\to 0^{+}}\sup|f(x,t,\xi )|  \leq c_{3}(|\xi|^{q-1}+1)
\end{gathered} \end{equation}
with $q>\sup (2,p)$. Also assume that $| f(x,t,\xi )| \leq a(|\xi | )$
almost everywhere in $\Omega \times \mathbb{R}^{+}$, where
$a:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is an increasing function.
\item[(H4)]
For each $M>0$ and $|\zeta)| \leq M$,
 $\frac{\partial f}{\partial t}(x,t,\zeta )$ exists,
there exists a positive constant $C_{M}$ such that
$| \frac{\partial f}{\partial t}(x,t,\zeta )| \leq C_M$
for almost every $(x,t)\in \Omega \times \mathbb{R}^{+}$.
\item[(H5)]
There exist $c_{4}>0$ such that
$\zeta \to f(x,t,\zeta )+c_{4}\beta (\zeta)$, is increasing
for almost $(x,t)\in \Omega \times \mathbb{R}$.
\end{enumerate}

\paragraph{Remarks} (i) By hypothesis (H5) and properties of $\beta $,
if the function
 $f_{0}:(x,t)\to |f(x,t,0)|$ is bounded by a positive constant $d$,
for  a.e. $(x,t)\in \Omega \times \mathbb{R}^{+}$,
\begin{equation} \label{e2.2}
\mathop{\rm sign}(u)f(x,t,u)\geq c_{3}|\beta (u)| -d.
\end{equation}
When this condition is satisfied, Condition (\ref{e2.1}) is also satisfied.
\\
(ii) From (H1), it follows that
$\Psi ^{\ast }(\beta(u_{0}))\in L^{1}(\Omega )$.
\\
(iii) When $\beta $ satisfies the condition $|\beta (s)| \leq d_{1}|s| +d_{2}$,
for any $ s\in \mathbb{R}$, with positive constants $d_{1}$ and $d_{2}$, as in
\cite{emr1}, we have the implications:
 $$
 u_{0}\in L^{2}(\Omega)\Rightarrow \beta (u_{0})\in L^{2}(\Omega )
 \Rightarrow \Psi ^{\ast }(\beta(u_{0}))\in L^{2}(\Omega ).
 $$

\paragraph{Definition}
By a weak solution to (\ref{P}), we mean a function $u$ such that:
\begin{gather*}
u\in L^{p}(0,T;W_{0}^{1,p}(\Omega ))\cap L^{q}(0,T;L^{q}(\Omega ))\cap L^{\infty
}(\tau ,T;L^{\infty }(\Omega )) \quad \forall \tau >0,\\
\frac{\partial \beta(u)}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\Omega
))+L^{q'}(0,T;L^{q'}( \Omega ) ),
\end{gather*}
 for all
 $\phi \in L^{p}(0,T;W_{0}^{1,p}(\Omega ))\cap L^{\infty }(0,T;L^{\infty }
 (\Omega))$ it holds
  $$ \int_{0}^{T}\big\langle \frac{\partial \beta (u)}{\partial t},\phi
\big\rangle _{X,X'}dt+\int_{0}^{T}\int_{\Omega }F(\nabla u)\nabla
\phi dx dt=-\int_{0}^{T}\int_{\Omega }f(x,t,u)\phi dx dt;
$$
and if $\frac{\partial \phi }{\partial t}\in L^{2}(0,T;L^{2}
( \Omega ))$, with $\phi (T)=0$, then
$$ \int_{0}^{T}\big\langle \frac{\partial \beta
(u)}{\partial t},\phi \big\rangle _{X,X'}dt
=-\int_{0}^{T}\int_{\Omega }( \beta (u)-\beta (u_{0}))
\frac{\partial \phi }{\partial t}dx dt,
$$
where $X=L^{\infty }(\Omega )\cap W_{0}^{1,p}(\Omega )$,
$X'=L^{1}(\Omega )+W^{-1,p'}(\Omega )$
and $ F(\xi )=| \xi| ^{p-2}\xi $ for any $\xi \in \mathbb{R}^{N}$.

\section{Existence and uniqueness}
Our main result reads as follows.
\begin{theorem} \label{thm3.1}
Under Hypotheses (H1)-(H5), Problem (\ref{P}) has a weak
solution $u$ such that
 $u\in L^{p}(0,T;W_{0}^{1,p}(\Omega ))\cap L^{\infty
}(\tau,T;W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega ))$, for all $\tau
>0$ and
$\beta (u)\in L^{q}(Q_{T})\cap L^{\infty }(0,T;L^{2}(\Omega ))$.
\end{theorem}

\paragraph{Remark}  % 3.1
 For a solution $u$  of (\ref{P}), by the first equation in
 (\ref{Pe}), we have
$$
\frac{\partial \beta (u)}{\partial t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))
+L^{q'}(0,T;L^{q'}( \Omega ) ).
$$
 Since $q>\sup (2,p)$,
 we get $\beta (u)\in L^{q}(Q_{T})\cap L^{\infty }
 (0,T;L^{2}(\Omega ))$ wich is a subset of
 $L^{q'}(0,T;L^{q'}(\Omega )+W_{0}^{-1,p'}(\Omega ))$.
 Thus, from Lion's lemma of compactness \cite[p.23]{lio},
 we deduce that at least $\beta (u)$ is in $ C(0,T;L^{q'}(\Omega ))$;
 so that the third condition (\ref{P}) makes sense.

\subsection*{Proof of the main result}
\textbf{a) Existence.} The proof of Theorem \ref{thm3.1} is based on a priori
estimates. From $\beta $, we construct a sequence
$\beta _{\varepsilon }\in C^{1}(\mathbb{R})$ such that:
 $\varepsilon \leq \beta _{\varepsilon }'$,
$\beta _{\varepsilon }(0)=0$, $\beta _{\varepsilon }\to \beta $ in
$C_{{\rm loc}}(\mathbb{R})$ and
$|\beta _{\varepsilon }| \leq |\beta |$.

 Let $(u_{0\varepsilon })_{\varepsilon >0}$ be a sequence in
$D(\Omega )$ such that $u_{0\varepsilon }\to u_{0}$ almost everywhere
in $\Omega $ and $\| u_{0\varepsilon }\| _{L^{2}(\Omega )},\|
\beta _{\varepsilon }(u_{0\varepsilon })\| _{L^{2}(\Omega )}\leq c$, with
a constant $c>0$. Consider the problem
\begin{equation} \label{Pe}
\begin{gathered}
\frac{\partial \beta _{\varepsilon }(u_{\epsilon })}{\partial t}
-\mathop{\rm div}F_{\varepsilon }(\nabla u_{\varepsilon })+f(x,t,u_{\epsilon }) =0
\quad \mbox{in } Q_{T} \\
 u_{\epsilon }  =0 \quad \mbox{in } S_{T} \\
\beta _{\varepsilon }(u_{\epsilon })_{| t=0}
 =\beta_{\varepsilon }(u_{0\varepsilon }) \quad  \mbox{in }\Omega,
\end{gathered}
\end{equation}
where $F_{\varepsilon }(\xi )=( | \xi |^{2}+\varepsilon )^{(p-2)/2}\xi$,
 for $\xi \in \mathbb{R}^{N}$.

 \paragraph{Remark} %3.2
  In this paper, we shall denote by
$c_{i}$ different constants, depending on $p$ and $\Omega $, but not on
$\varepsilon$, or $T$. Sometimes we shall refer to a constant depending
on specific parameters: $c(\tau )$, $c(T)$, $c(\tau ,T)$, etc.

\begin{lemma} \label{lm3.2}
There exists a unique solution of (\ref{Pe}), such that
 $u_{\varepsilon }\in L^{\infty }(Q_{T})\cap L^{\infty }
 (0,T;W_{0}^{1,p}(\Omega ))$. Moreover,
 $u_{\varepsilon }\in W_{r}^{2,1}((0,T)\times \Omega )$ for
 $1\leq r<\infty$,
\end{lemma}

\paragraph{Proof.}
 The proof is similar to that in \cite[lemma 5]{emr1} and we
shall give here only a sketch. For a fixed positive integer $m$, consider
the function
$$ f_{m}(x,t,u)= \begin{cases}
f(x,t,u) &\mbox{if }| \beta (u)| \leq m \\[2pt]
c_{1}(| \beta(u)| ^{q-1}-m^{q-1})\mathop{\rm sign}(u) \\
+f(x,t,\beta ^{-1}(u)\mathop{\rm sign}(u))) &\mbox{otherwise}.
\end{cases}
$$
Then
$$
\mathop{\rm sign}(u)f_{m}(x,t,u)\geq c_{1}| \beta _{\varepsilon }(u)|
^{q-1}-c_{2}.
$$
Indeed, if $|\beta (u)| \leq m$, by properties of $\beta_{\varepsilon }$,
we get
$$
\mathop{\rm sign}(u)f_{m}(x,t,u)=\mathop{\rm sign}(u)f(x,t,u)\geq c_{1}| \beta (u)|
^{q-1}-c_{2}\geq c_{1}| \beta _{\varepsilon }(u)| ^{q-1}-c_{2},
$$
and if $|\beta (u)| \geq m$ then, as
$\mathop{\rm sign}(u)/\mathop{\rm sign}(\beta^{-1}(m \mathop{\rm sign}(u)))=1$,
 we deduce by properties of $\beta _{\varepsilon }$ that
\begin{align*}
\mathop{\rm sign}(u)f_{m}(x,t,u) \geq &
 c_{1}(| \beta (u)|^{q-1}-m^{q-1})+c_{1}|
 \beta (\beta ^{-1}(m \mathop{\rm sign}(u)))|^{q-1}-c_{2} \\
\geq & c_{1}| \beta (u)| ^{q-1}-c_{2}\geq c_{1}|
\beta _{\varepsilon }(u)| ^{q-1}-c_{2}.
\end{align*}
For $\sigma \in [0,1]$, define the map $K(\sigma ,.)$ by
$K(\sigma,v)=u_{\varepsilon ,\sigma }$ which is the solution to
%(P_{\varepsilon ,\sigma }) \left\{
\begin{equation} \label{ped}
\begin{gathered}
\frac{\partial \beta _{\varepsilon }(u_{\epsilon ,\sigma })}{\partial t}
-divF_{\varepsilon }(\nabla u_{\varepsilon ,\sigma })+\sigma f_{m}(x,t,v) =0
\quad \mbox {in } Q_{T},\\
 u_{\epsilon ,\sigma }  =0 \quad \mbox{in } S_{T}, \\
\beta _{\varepsilon }(\,u_{\epsilon ,\sigma })_{| t=0}
 =\beta_{\varepsilon }(\sigma u_{0\varepsilon }) \quad \mbox{in } \Omega ,
\end{gathered}
\end{equation}
 For each $\sigma \in [0,1]$, the operator $K(\sigma ,.)$ is compact
from $L^{p}(0,T;W_{0}^{1,p}(\Omega ))$ into itself. Indeed, for a
fixed $v\in L^{p}(0,T;W_{0}^{1,p}(\Omega ))$, one has a unique
solution $u_{\varepsilon,\sigma }\in L^{p}(0,T;W_{0}^{1,p}(\Omega
))\cap W_{r}^{2,1}((0,T)\times \Omega )$ by using the theory of
Ladyzenskaya et al  \cite[chap. V]{lso}. Therefore, arguing
exactly as in \cite[Lemma5]{emr1},
 we deduce that, for each $\sigma \in [0,1]$,
 $K(\sigma,.)$ is a compact operator from $L^{p}(0,T;W_{0}^{1,p}(\Omega ))$
 into itself
and that the map $\sigma \to K(\sigma ,.)$ is continuous and
$K(0,v)=u_{\varepsilon ,0}=0$. Thus, from Leray-Schauder fixed-point theorem,
there exists a fixed point
$u_{\varepsilon }\equiv u_{\varepsilon ,1}=K(1,v)$.
 Moreover, arguing also as in \cite[Lemma 5]{emr1} and
using (\ref{e3.7}), we obtain
$|\beta _{\varepsilon }(u_{\varepsilon })|_{L^{\infty }
(0,T;L^{\infty }(\Omega ))}\leq c(u_{0\varepsilon})$, where
$c(u_{0\varepsilon })$ is a positive constant depending only on
$u_{0\varepsilon }$. Thus,
$f_{m}(x,t,u_{\varepsilon })=f(x,t,u_{\varepsilon })$
for $m\geq c(u_{0\varepsilon })$ and then $u_{\varepsilon }$ is a solution of
(\ref{Pe}).

The uniqueness property of a solutions can be derived
from \cite[Theorem 3, p. 1095]{dt}. If we show that $\frac{\partial
\beta _{\varepsilon }(u_{\varepsilon })}{\partial t}\in L^{2}(0,T;L^{2}(\Omega
))$. To avoid repetition, we claim that it is a consequence of
Lemma \ref{lm3.4} below.

 Now we give the a priori estimates needed for the remainder of the proof.

\begin{lemma} \label{lm3.3}
Under the hypothesis (H1)-(H3), there exists
constants $c_{i}$ such that for any $\varepsilon \in ]0,1[ $ and any
$\tau >0$, the following estimates hold
\begin{gather}
\| u_{\epsilon }\| _{_{L^{\infty }(\tau ,T;L^{\infty
}(\Omega ))}} \leq   c_{4}(\tau ,T),   \label{e3.4} \\
\| \beta _{\varepsilon }(u_{\epsilon })\|_{{L^{\infty
}(0,T;L^{2}(\Omega ))\cap L^{q}(Q_{T})}} \leq  c_{5}(T) \label{e3.5}\\
|u| _{L^{p}(0,T;W_{0}^{1,p}(\Omega ))}\leq c_{6}(T).\label{e3.6}
\end{gather}
\end{lemma}

\paragraph{Proof}
(i) Multiplying the first equation in (\ref{Pe}) by $| \beta
_{\varepsilon }(u_{\epsilon })| ^{k}\beta _{\varepsilon
}(u_{\epsilon })$ and using the growth condition on $f$ and the
properties of $\beta _{\varepsilon }$, we deduce that
\begin{equation}
\frac{1}{k+2}\frac{d}{dt}{ \int_{\Omega }| \beta _{\varepsilon
}(u_{\epsilon })| ^{k+2}dx} +c_{14}\int_{\Omega }| \beta _{\varepsilon
}(u_{\epsilon })| ^{k+q}dx\leq c_{15}\int_{\Omega }|\beta _{\varepsilon
}(u_{\epsilon })| ^{k+1}dx \label{e3.7}
\end{equation}
Setting
$y_{\varepsilon,k}(t)=\| \beta _{\varepsilon }(u_{\epsilon })\| _{L^{k+2}(\Omega
)}$ and using H\"{o}lder's inequality on both sides of (\ref{e3.7}),
there exist two constants $\alpha _{0}>0$ and $\lambda _{0}>0 $ such that
$$
\frac{dy_{\varepsilon ,k}(t)}{dt}+\lambda _{0}y_{\varepsilon ,k}^{q-1}(t)
\leq \alpha _{0};
$$
which implies from Ghidaglia's lemma \cite{tem} that
\begin{equation}
y_{\varepsilon ,k}(t)\leq \big( \frac{\alpha _{0}}{\lambda
_{0}}\big) ^{ \frac{1}{q-1}}+\frac{1}{\left[ \lambda _{0}(q-2)t\right]
^{\frac{1}{q-2}}} =c_{7}(t)\,,\forall t>0.  \label{e3.8}
\end{equation}
As $k\to+\infty $, and for all $t\geq \tau >0$, we have
\begin{equation}
|\beta _{\varepsilon }(u_{\epsilon })(t)| _{L^{\infty }(\Omega )}
\leq c_{7}(\tau );  \label{e3.9}
\end{equation}
which implies
\begin{equation}
|u_{\epsilon }(t)| _{L^{\infty }(\Omega )}\leq \max ( \beta
_{\varepsilon }^{-1}(c_{7}(\tau )),| \beta _{\varepsilon }^{-1}(-c_{7}(\tau
))|) =\delta _{\varepsilon }\,.  \label{e3.10}
\end{equation}
Since $\beta _{\varepsilon }$ converges to $\beta $ in
$C_{{\rm loc}}(\mathbb{R})$, then the sequence $\delta
_{\varepsilon }$ is bounded in $\mathbb{R}$ as $ \varepsilon \to
+\infty $.  Thus $\delta _{\varepsilon }$ is bounded by $\max ( \beta
^{-1}(c_{7}(\tau )),| \beta ^{-1}(-c_{7}(\tau ))|) $, which is finite.
Whence (\ref{e3.4}) is satisfied.
 On the other hand,  taking $k=0$ in (\ref{e3.7}), using H\"older
inequality and integrating on $[0,T] $ yields (\ref{e3.5}).

(ii) Multiplying the first equation in (\ref{Pe}) by $u_{\epsilon
}$, integrating on $\Omega $ and using (\ref{e2.1}) and the
properties of $\beta _{\varepsilon }$, gives
\begin{multline}
\frac{d}{dt}\Big(\int_{\Omega }\Psi _{\varepsilon }^{\ast }
(\beta _{\varepsilon }(u_{\epsilon }))dx\Big)
+\int_{\Omega }( |\nabla u_{\epsilon }|
^{2}+\epsilon ) ^{(p-2)/2}| \nabla u_{\epsilon }|
^{2}dx+c_{1}\int_{\Omega }| \beta _{\varepsilon }(u_{\epsilon
})| ^{q-1}dx \\
\leq c_{2},     \label{e3.11}
\end{multline}
where $\Psi _{\varepsilon }^{\ast }$ is the Legendre transform of $\Psi
_{\varepsilon }$ and $\Psi _{\varepsilon }(t)=\int_{0}^{t}\beta _{\varepsilon
}(s)ds$. By  hypotheses (H1) and (H2), and the remark (ii) in Chapter 2,
we can assume that
$ \int_{\Omega }\Psi _{\varepsilon }^{\ast }(\beta _{\varepsilon }(u_{0\epsilon
}))dx$ converges to $\int_{\Omega }\Psi ^{\ast }(\beta (u_{\epsilon }))dx\leq
c$, where $c$ is some positive constant. So, integrating (\ref{e3.10})
from $0$ to $T$ yields
\begin{equation}
\int_{\Omega }\Psi _{\varepsilon }^{\ast }\,(\beta _{\varepsilon
}(u_{\epsilon }))dx+c_{8}\int_{0}^{T}\int_{\Omega }| \,u_{\epsilon }|
^{p}dxds\leq c_{8}(T).\label{e3.12}
\end{equation}
Hence (\ref{e3.6}) follows.  \hfill$\Box$

\begin{lemma} \label{lm3.4}
Assume (H1)-(H4). Then there exist constants $c_{11}(\tau )$ and
$c_{i}(\tau ,T)$ $(i=9,10)$ such that for $\varepsilon \in ]0,1[ $ the
following estimates hold
\begin{gather}
 \| u_{\varepsilon }\| _{L^{\infty }(\tau
,T;W_{0}^{1,p}(\Omega ))} \leq c_{9}(\tau ,T), \label{e3.13}\\
\int_{\tau}^{T}\int_{\Omega }\beta _{\varepsilon }'(u_{\varepsilon
})(\frac{\partial u_{\varepsilon }}{\partial t})^{2} dx ds
\leq c_{10}(\tau,T)\label{e3.14} \\
\int_{t}^{t+\tau }\int_{\Omega }\beta _{\varepsilon
}'(u_{\varepsilon })(\frac{\partial u_{\varepsilon }}{\partial t}
)^{2}dxds\leq c_{11}(\tau ),\mbox{ for any }t\geq \tau >0. \label{e3.15}
\end{gather}
\end{lemma}

\paragraph{Proof.}
 Multiplying the first equation in (\ref{Pe}) by $\frac{\partial u_{\epsilon
}}{\partial t}$ , integrating on $\Omega $ and using (\ref{e3.10})
and (H4),  it follows that for any $t\geq \tau >0$,
\begin{multline}
\int_{\Omega }\beta_{\varepsilon }'(u_{\epsilon })(\frac{
\partial u_{\epsilon }}{\partial t})^{2}dx+\frac{d}{dt}\Big[\frac{1}{p}
\int_{\Omega }(|\nabla u_{\epsilon }| ^{2}+\epsilon )
^{\frac{p}{2}}dx+\int_{\Omega }\int_{0}^{u_{\epsilon }}f(x,t,y)dy\,
dx\Big] \\
\leq|\int_{\Omega }\int_{0}^{u_{\epsilon}}\frac{\partial f}{\partial t}(x,t,y)
dy dx| \leq c_{12}(\tau )\,, \label{e3.16}
\end{multline}
where $c_{12}(\tau )$ is some positive constant.
 Now integrating (\ref{e3.11}) on $[t,t+\frac{\tau }{2}]$ and
observing that $\varepsilon \in ]0,1[$, yields
$$
\int_{t}^{t+\frac{\tau }{2}}\int_{\Omega }(|\nabla u_{\epsilon }|
^{2}+\epsilon ) ^{\frac{p}{2}}dxdt\leq c_{13}(\tau )\quad \forall
t\geq \frac{\tau }{2}.
$$
Furthermore, by (\ref{e3.10}) we have:
$|\int_{\Omega }\int_{0}^{u_{\varepsilon }(x,t)}f(x,t,y)dy\, dx|
\leq c_{13}(\tau )$. Then, applying the uniform Gronwall's lemma
\cite[p.89]{tem} with $a_{1}=c_{13}(\tau)$, $a_{2}=c_{14}(\tau )$,
$h=c_{12}(\tau )$ and
$$
y(t)=\int_{\Omega}( | \nabla u_{\epsilon }| ^{2}+\epsilon )
^{p/2}dx+\int_{\Omega }\int_{0}^{u_{\varepsilon}(x,t)}f(x,t,y)dy dx,
$$
 gives
\begin{equation}
 \int_{\Omega }| \nabla u_{\epsilon }| ^{p}dx+\int_{\Omega }
\int_{0}^{u_{\varepsilon}(x,t)}f(x,t,y)dy dx \leq \frac{a_{1}
+a_{2}}{\tau } +c_{15}(\tau) \quad \forall t\geq \tau>0.
\label{e3.17}
\end{equation}
By using (\ref{e3.10}) and hypothesis (H4), (\ref{e3.17}) leads  to
\begin{equation}
\int_{\Omega }|\nabla u_{\epsilon }| ^{p}dx\leq c_{16}(\tau )
\forall t\geq \tau >0.  \label{e3.18}
\end{equation}
Hence (\ref{e3.13}) is satisfied.
On the other hand, by the mean value theorem and (\ref{e3.6}),
we conclude that for any $\tau >0$, there exists
$\tau _{\varepsilon }\in ]\frac{\tau }{ 4},\frac{\tau }{2}[ $
such that
$$
\int_{\Omega }| \nabla u_{\epsilon }(\tau _{\varepsilon })|
^{p}dx=\frac{2}{\tau }\int_{\frac{\tau }{4}}^{\frac{\tau }{2}}\int_{\Omega }|
\nabla u_{\epsilon }| ^{p}dxdt\leq c_{17}(\tau ).
$$
Now, integrating (\ref{e3.16}) on $[\tau _{\varepsilon },T] $ and using
(\ref{e3.10}), (\ref{e3.18}) and (H4), we easily deduce (\ref{e3.14}).
To conclude (\ref{e3.15}), it suffices to integrate (\ref{e3.16}) on
$[ t,t+\tau ]$ and to use once again (\ref{e3.10}), (\ref{e3.18}) and
hypothesis (H4). Whence the lemma is proved. \hfill$\Box$

As a consequence of Lemma \ref{lm3.4}, we get the following lemma.

\begin{lemma} \label{lm3.5}
(i) The following estimates hold:
\begin{gather*}
\int_{\tau }^{T}\int_{\Omega }\big( \frac{\partial \beta _{\varepsilon
}(u_{\varepsilon })}{\partial t}\big) ^{2}dx\,ds\leq c_{18}(\tau ,T),
\quad\mbox{for }T\geq \tau >0, \\
\int_{t}^{t+\tau }\int_{\Omega }\big( \frac{\partial \beta _{\varepsilon
}(u_{\varepsilon })}{\partial t}\big) ^{2}dx\,ds\leq c_{19}(\tau ),
\quad\mbox{for } \tau >0.
\end{gather*}
(ii) When $f$ does not depend on $t$,
$$
\int_{\tau }^{T}\int_{\Omega }\beta _{\varepsilon }'(u_{\varepsilon
})\big( \frac{\partial u_{\varepsilon }}{\partial t}\big) ^{2}dxds\leq
c_{22}(\tau ),\quad\mbox{for }T\geq \tau >0.
$$
\end{lemma}

\paragraph{Proof.} (i) Let $L$ be the Lipschitz constant of $\beta $
on $[-\delta ,\delta ] $, where $\delta $ is the bound in the
proof of lemma 3.3 (i). It is possible to choose $\beta
_{\varepsilon }$ so that $\beta _{\varepsilon }'\leq L$
on $\left[ -\delta ,\delta \right]$. Then (\ref{e3.12}) implies
$$
\frac{1}{L}\int_{\tau }^{T}\int_{\Omega }\big( \frac{\partial
\beta _{\varepsilon }(u_{\varepsilon })}{\partial t}\big)
^{2}dxds\leq c_{23}(\tau ,T),\mbox{ for any }T \geq \tau >0.
$$
{\bf{(ii)}} From (\ref{e3.15}), and using the notation on the equation
preceding (\ref{e3.17}) now we have
$$
\int_{\Omega }\beta _{\varepsilon }'(u_{\epsilon })(
(u_{\epsilon })_t)^{2} dx + \frac{d}{dt}\Big[
\int_{\Omega}\frac{1 - p}{p} \left( | \nabla u_{\epsilon
}| ^{2}+\epsilon \right) ^{\frac{p}{2}}dx+ y(t) \Big] \leq
0.
$$
Integrating this expression on $[ \tau_{\varepsilon},T]$
and using (\ref{e3.18}), it follows (\ref{e3.14}). \hfill$\Box$

\paragraph{Passage to the limit in (\ref{Pe}) as
$\varepsilon \to +\infty $.}
 By estimates (\ref{e3.6}) and
(\ref{e3.13}), $F_{\epsilon }(\nabla u_{\epsilon })$ is bounded in
$L^{p'}(0,T;L^{p'}(\Omega ))$. Hence
\begin{equation}
F_{\epsilon }(\nabla u_{\epsilon })\ \mbox{ is bounded in }\
L^{p'}(\tau,T;W^{-1,p'}(\Omega )),
\label{e3.21}
\end{equation}
By Lemma \ref{lm3.5} (i),
\begin{equation}
 \frac{\partial\beta _{\varepsilon }(u_{\epsilon })}{\partial t}
 \mbox{ is bounded in }\ L^{2}(\tau ,T;L^{2}(\Omega )), \forall \tau
>0. \label{e3.22}
\end{equation}
Therefore, by estimates (\ref{e3.4}), (\ref{e3.5}), (\ref{e3.6}),
(\ref{e3.9}), (\ref{e3.13}) and (\ref{e3.21}), there
exists a subsequence (denoted again by $u_{\varepsilon }$) such that as
$\varepsilon \to 0$, we have
\begin{gather}
u_{\epsilon } \to u \quad \mbox{weak in } L^{p}(0,T;W_{0}^{1,p}(\Omega )),
\label{e3.23} \\
u_{\epsilon } \to u \quad \mbox{weak star in }L^{\infty }
(\tau ,T;W_{0}^{1,p}(\Omega )),\quad \forall \tau >0,
\label{e3.24}  \\
\mathop{\rm div}F_{\epsilon }(\nabla u_{\epsilon }) \to
\chi \quad\mbox{weak in } L^{p'}(0,T;W^{-1,p'}(\Omega )),
\label{e3.25} \\
 \beta _{\varepsilon }(u_{\varepsilon }) \to
\xi \quad\mbox{weak in }L^{q}(Q_{T}),
\label{e3.26}\\
\beta_{\varepsilon }(u_{\varepsilon }) \to \xi
\quad\mbox{weak star in }L^{\infty }(\tau ,T;L^{\infty }(\Omega )).
\label{e3.27}
\end{gather}
Now according to (\ref{e3.10}), (\ref{e3.22}), (\ref{e3.26}), (\ref{e3.27}),
 and Aubin's lemma \cite[Corol. 4]{si1}, we derive that
 $\beta _{\varepsilon }(u_{\varepsilon})\to \xi$ strongly in
 $C([0,T] ,L^{2}(\Omega ))$ and by a similar way
as that in (\cite{bf}, page 1048), we consequently obtain $\beta (u)=\xi$.
Moreover standard monotonicity argument \cite{bf,lio} gives
$\chi = \mathop{\rm div} F(\nabla u)$.

To conclude that $u$ is a weak solution of (\ref{P}) it
suffices to observe, as in \cite[p. 108]{emr1}, that $f(x,t,u_{\varepsilon
})\to f(x,t,u)$ strongly in $L^{1}(Q_{T})$ and in
$L^{s}(\tau ,T;L^{s}(\Omega ))$ for all $\tau >0$ and for all $s\geq 1$,
as $\varepsilon \to 0$. (One should use the growth condition on
$f_{\varepsilon }$ and Vitali's theorem).

\paragraph{b) Uniqueness.}
 By Lemma \ref{lm3.4}, the solutions of (\ref{P}) satisfy
$$
\frac{\partial \beta (u)}{\partial t}\in L^{2}(\tau ,T;L^{2}(\Omega )) \quad
\forall \tau >0.
$$
Therefore, by \cite[Theorem 3, p.\ 1095]{dt}, we
deduce that the solution is unique. \hfill $\Box$

\begin{corollary} \label{coro3.6}
Under the hypotheses of Theorem \ref{thm3.1} with $f$ independent of time,
Problem (\ref{P}) generates a continuous semi-group
$S(t: L^{2}(\Omega )\to L^{2}(\Omega ) $ defined by
$S(t)u_{0}=\beta (u(t,.))$. Moreover the solution of
problem (\ref{P}) satisfies
$\frac{\partial \beta (u)}{\partial t}\in
 L^{2}(\tau ,+\infty ;L^{2}(\Omega ))$ for all $\tau>0$.
\end{corollary}

\section{Existence and regularity of the attractor}

For the concepts of absorbing sets and global attractors used here, we refer
the reader to \cite{tem}. Using estimates in Lemma \ref{lm3.3},
we deduce the following statement.
\begin{proposition} \label{prop4.1}
Under hypotheses (H1)-(H5), the semi-group S(t) associated with problem
(\ref{P}) is such that
\begin{enumerate}
\item[(i)] There exist absorbing sets in $L^{\sigma }(\Omega )$,
for $1\leq \sigma \leq +\infty $.
\item[(ii)] There exist absorbing sets in $W_{0}^{1,p}(\Omega )$.
\end{enumerate}
\end{proposition}

\paragraph{Proof.}
Let $u$ be solution of (\ref{P}) and $u_{\varepsilon }$ solution of
(\ref{Pe}) approximating $u$,
then for fixed $t\geq \tau >0$,
(\ref{e3.10})  and Sobolev's injection theorem imply
\begin{equation}
\| u_{\varepsilon}(t)\| _{L^{\sigma }(\Omega )}\leq c_{\delta },
\quad\mbox{for any }\sigma :1\leq \sigma <\infty ,\label{e4.1}
\end{equation}
where $c_{\sigma }$ is some positive
constant depending on $\mathop{\rm meas}(\Omega )$ and $\delta $,
with $\delta =\max ( \beta^{-1}(c(\tau )),| \beta ^{-1}(-c(\tau ))|)$
as in the proof of Lemma \ref{lm3.3} (i).
From (\ref{e4.1}), we then obtain
\begin{equation}
\| u(t)\| _{L^{\sigma }(\Omega
)}\leq c_{\delta }\mbox{ for any }\sigma :1\leq \sigma <\infty .
\label{e4.2}
\end{equation}
 By letting $\sigma $ tends to +$\infty $ in (\ref{e4.2}), we  obtain
\begin{equation}
\| u(t)\| _{L^{\infty }(\Omega )}\leq c_{\delta }. \label{e4.3}
\end{equation}
Thus, by (\ref{e4.2}) and (\ref{e4.3}), the open ball $B(0,c_{\delta })$
centered at 0 and with radius $c_{\delta }$ is an absorbing set
in $L^{\sigma }(\Omega )$, $1\leq\sigma \leq +\infty$.
On the other hand, by (\ref{e3.17}), (\ref{e3.23}) and
the lower semi-continuity of the norm, we get
$$ \int_{\Omega }| \nabla
u| ^{p}(t)dx\leq c_{16}(\tau ), \mbox{ for any }t\geq \tau .
$$
Therefore the open ball $B(0,c_{16}(\tau ))$ is an absorbing set in
$ W_{0}^{1,p}(\Omega)$. Whence part (ii) is verified.
\hfill$Box$

Assuming that the nonlinear function $f$ does not depend on time,
Proposition \ref{prop4.1} then gives
assumptions (1.1), (1.4) and (1.12) of \cite[Theorem 1.1, p. 23]{tem},
with $U=L^{2}(\Omega )$. So, by means of the uniform compactness lemma in
\cite[p. 111]{emr1}, we get the following result.

\begin{theorem} \label{thm4.2}
Assume that (H1)-(H5) are satisfied and that $f$ does not depend on time.
Then the semi-group $S(t)$ associated with the boundary value problem (\ref{P})
possesses a maximal attractor $A$ which is bounded in $W_{0}^{1,p}(\Omega )\cap
L^{\infty }(\Omega )$, compact and connected in $L^{2}(\Omega )$.
Its domain of attraction is the whole space $L^{2}(\Omega )$.
\end{theorem}

\section{More regularity for the attractor}

In this section we shall show supplementary regularity estimates on the
solution of problem (\ref{P}) and by use of them, we shall obtain more
regularity on the attractor obtained in Section 4.
To this end, we consider the following hypotheses on the data.
\begin{enumerate}
\item[(H6)] $f(x,t,u)=g(u)-h(x)$, where $h\in L^{\infty
}(\Omega )$ and $g\in C^{1}(\mathbb{R})$ are such that $f$ satisfies the
conditions already prescribed in (H3), (H4) and (H5).
\item[(H7)] $\beta \in C^{2}(\mathbb{R})$ is such that there exist
$\sigma _{1}, \sigma _{2}>0$ with
$\sigma _{1}\leq \beta '(s)\leq \sigma _{2}$ for all $s\in \mathbb{R}$.
\end{enumerate}
Let $u_{\varepsilon }$ be solution of (\ref{Pe}) with
$f=g-h$. For simplicity, we shall denote
$$w:=u_{\varepsilon },\quad
w'=\frac{\partial u_{\varepsilon }}{\partial t}, \quad
w''=\frac{\partial ^{2}u_{\varepsilon }}{\partial t^{2}}, \quad
( E(\nabla w)) '=\frac{\partial }{\partial t}( E(\nabla w)),
$$
with $E(\xi )=| \xi | ^{(p-2)/2}\xi$, for all $\xi \in \mathbb{R} ^{N}$
and $( F_{\varepsilon }(\nabla w)) '
=\frac{\partial }{\partial t}( F_{\varepsilon }(\nabla w))$.

The following two lemmas are used in the proof of the main results of this
section.

\begin{lemma} \label{lm5.1}
For  $1<p<2$, there exists a positive constant $c_{24}$ such that
\begin{equation}
\int_{\Omega }|\nabla w '| ^{p}dx \leq c
_{24}\int_{\Omega }| \nabla w| ^{p}dx + \frac{2(p-1)}{p ^{2}}
\int_{\Omega }| \left( E(\nabla w)\right) '| ^{2}dx,
\label{e5.1}
\end{equation}
\end{lemma}

\paragraph{Proof.} Straightforward calculations, \cite{ht1},
give
\begin{equation}
 \int_{\Omega }\left( F_{\varepsilon }(\nabla w)\right)
'.\nabla w'dx\geq \frac{4(p-1)}{p ^{2}}
\int_{\Omega }| \left( E(\nabla w)\right) '|^{2}dx. \label{e5.2}
\end{equation}
Since $\nabla w=| E(\nabla w)| ^{\frac{2-p}{p}}\ E(\nabla w)$,
it follows that
$\nabla w'=\frac{2}{p}| E(\nabla w)| ^{\frac{2-p}{p}}(E(\nabla w))'$.
So, as $1<p<2$, the H\"older and Young
inequalities lead to
\begin{align*}
\int_{\Omega }| \nabla w'| ^{p}dx &=c_{25}\int_{\Omega
}| E(\nabla w)| ^{2-p}| (E(\nabla w))'| ^{p}dx \\
 &\leq \frac{c_{26}}{2}\int_{\Omega }| \left( E(\nabla
w)\right) | ^{2}dx\ + \frac{2(p-1)}{p ^{2}}\int_{\Omega
}| \left( E(\nabla w)\right) '| ^{2}dx,
\end{align*}
 where $c_{25}=(2/p)^{p}$ and $c_{26}$ is a positive constant. Hence
estimate \ref{e5.1} follows. \hfill $\Box$

\begin{lemma} \label{lm5.2}
Assuming (H1)-(H8), the sequence $(u_{\varepsilon })_{\varepsilon
>0}$ converges strongly to the solution $u$ of (\ref{P}) in
$L^{p}(0,T;W^{1,p}(\Omega))$.
\end{lemma}

The proof of this lemma is similar to that of \cite[Lemma 2]{ht2}
and is omitted here. For stating the next theorem we introduce the hypothesis
\begin{enumerate}
\item[(H8)] $N=1$ and $1<p<2$ or $N\geq 2$ and $\frac{3N}{N+2}\leq p<2$.
\end{enumerate}

\begin{theorem} \label{thm5.3}
Let $f$ and $\beta $ satisfy hypotheses (H1)-(H7), and (H8) be satisfied.
Let $y(t)=\int_{\Omega }\beta '(w)(w')^{2}dx$. Then
$$
y(t)\leq c_{27}(\tau ),\quad \forall t,\tau ,
\varepsilon \mbox{ with } t\geq \tau >0  \mbox{ and } 0<\varepsilon <1.
$$
\end{theorem}

\paragraph{Proof.}
Differentiating equation (\ref{e3.15}) (with $f=g-h$) with respect to
$t$ (the justification can be done by passing to finite dimension as in
\cite{ht2}), we get
\begin{equation}
\beta '(w)w^{\prime \prime }+\beta''(w)(w')^{2}
-\mathop{\rm div}(( F_{\epsilon }(\nabla w))') +g'(w)w'=0.\label{e5.3}
\end{equation}
 Now multiplying (\ref{e5.3}) by $w'$, integrating over $\Omega $ and using
 (\ref{e5.2}),
gives
\begin{equation}
 \frac{1}{2}y'(t)+\frac{1}{2}\int_{\Omega }\big[\beta ''(w)(w')^{3}
 + \frac{4(p-1)}{p ^{2}}| (E(\nabla w)) '| ^{2}
 +g'(w)(w')^{2}\big]dx\leq 0. \label{e5.4}
\end{equation}
On the other hand, by using hypotheses (H7) and (H8) and relation
(\ref{e3.4}) and applying successively Gagliardo-Nirenberg's
inequality (see for example \cite{lso}), Young's inequality and
Lemma \ref{lm5.1}, it follows that
\begin{multline}
 -\frac{1}{2}\int_{\Omega }\beta ^{\prime \prime}(w)(w')^{3}dx \\
 \leq c_{31}| | w'| | _{2}^{3(1+\alpha )}
c_{32}| | \nabla w| | _{p}^{p}\ +
\frac{4(p-1)}{p ^{2}} \int_{\Omega }| ( E(\nabla w)) '| ^{2}dx, \label{e5.5}
\end{multline}
where $\theta =\frac{1}{3}( \frac{Np}{Np+2p-2N}) $
and $\alpha = \frac{N(3-p)}{3Np+6p-9N}$.
Estimate (\ref{e3.4}) and hypothesis (H6) and (H7) imply
\begin{gather}
\int_{\Omega }g'(w)(w')^{2}dx \leq \|
g'(w)\| _{L^{\infty }(\Omega )}\int_{\Omega
}(w')^{2}dx\leq M_{1}\| w'\| _{2}^{2},  \label{e5.6} \\
  \sigma _{1}\| w'\| _{2}^{2} \leq y(t), \label{e5.7}
\end{gather}
where $M_{1}$ is a positive constant. Therefore, using (\ref{e5.5}) and
(\ref{e5.6}), (\ref{e5.4}) becomes
\begin{multline}
\frac{1}{2}y'(t)+ \frac{2(p-1)}{p ^{2}}\int_{\Omega
}| \left( E(\nabla w)\right) '| ^{2}dx \\
 \leq c_{31}\| w'\| _{2}^{3(1+\alpha )}
+ c_{32}\| \nabla w\| _{p}^{p}+M_{1}\| w'\| _{2}^{2}. \label{e5.8)}
\end{multline}
Now (\ref{e5.7}) and estimate (\ref{e3.5}) give
\begin{equation}
\frac{1}{2}y'(t)+ \frac{2(p-1)}{p ^{2}}\int_{\Omega
}| \left( E(\nabla w)\right) '| ^{2}dx
\leq c_{33}(y(t)^{\frac{3(1+\alpha )}{2}}+y(t)+1)
\leq c_{34}(y(t))^{2}+c_{35} \label{e5.9}
\end{equation}
for all $t\geq \tau >0$.
By assumption (H6), equation (\ref{e3.16}) can be written as
\begin{equation}
 \beta '(w)w'-\mathop{\rm div}(F_{\varepsilon }(\nabla w))=h-g(w).
 \label{e5.10}
\end{equation}
Taking the scalar product of (\ref{e5.12}) with
$w'$, we obtain
\begin{equation} \label{e5.11}
\begin{aligned}
\int_{\Omega }&\beta '(w)\left( w'\right) ^{2}dx
+\frac{d}{dt }\Big[ \frac{1}{p}\int_{\Omega }\left( | \nabla
w| ^{2}+\varepsilon \right) ^{\frac{p}{2}}dx\Big] \\
&=\int_{\Omega }(g(w)-h)w'dx \\
&\leq \int_{\Omega }\frac{(g(w)-h)}{\sqrt{\beta '(w)}}.\sqrt{\beta '(w)}w'dx   \\
&\leq \frac{1}{2\sigma _{2}}\| g(w)-h\| _{2}^{2}
+\frac{1}{2} \int_{\Omega }\beta '(w)\left( w'\right) ^{2}dx.
\end{aligned}
\end{equation}
Hence
\begin{equation}
 \frac{1}{2}\int_{\Omega }\beta '(w)\left( w'\right) ^{2}dx
 + \frac{d}{dt}\Big[ \frac{1}{p}\int_{\Omega
}\left( | \nabla w| ^{2}+\varepsilon \right) ^{\frac{p}{2}}dx\Big]
\leq c_{36}\| g(w)-h\|_{L^{\infty }(\Omega )}^{2},
 \label{e5.12}
\end{equation}
where $c_{36}$ depends on $\sigma _{2}$ and $\mathop{\rm meas}(\Omega )$.
Estimate (\ref{e3.13}) of Lemma \ref{lm3.4}  gives
$$
\frac{1}{p}\int_{\Omega}\left( | \nabla w| ^{2}+\varepsilon \right)
^{\frac{p}{2}}(t)dx\leq c_{37}(\tau ),\quad \forall t\geq \frac{\tau }{2}>0 .
$$
 Integrating (5.12) on $\left[ t,t+\frac{\tau
}{2}\right] $ yields
\begin{equation}
 \int_{t}^{t+\frac{\tau}{2}}y(s)ds\leq c_{38}(\tau ),\quad
 \forall t\geq \frac{\tau}{2}>0. \label{e5.13}
\end{equation}
Going back to (\ref{e5.9}) and using the
uniform Gronwall lemma \cite[p. 89]{tem} with
$r=\tau/2$, $g(t)=c_{34}y(t)$ and $h=c_{35}$ and estimate (\ref{e5.13})
leads to
$$y(t+\frac{\tau }{2})\leq c_{39}(\tau )\quad \forall t\geq\frac{\tau }{2}>0.$$
Hence  $y(t)\leq c_{39}(\tau )$, for any $t\geq\tau >0.$
The proof of the theorem is now complete. \hfill$\Box$

 Using Theorem \ref{thm5.3}, we state the main result of this
section.
\begin{theorem} \label{thm5.4}
Let $ f,\beta,p$ satisfies hypotheses (H1)-(H8). Then, for $\tau >0$,
 the solution of problem (\ref{P}) satisfies:
\begin{gather}
 \frac{\partial \beta (u)}{\partial t}\in L^{\infty }(\tau ,+\infty
;L^{2}(\Omega )), \label{e5.14} \\
u\in L^{\infty }(\tau ,+\infty ;B_{\infty }^{1+\sigma ,p}
\left( \Omega \right) ),  \label{e5.15}
\end{gather}
where $B_{\infty }^{1+\sigma ,p}\left( \Omega \right)$ is a Besov space
defined by the real interpolation method \cite{si2}.
Moreover, there exists a constant $c(\tau )>0$, depending on $\tau$ such that
\begin{equation}
 \lim_{t\to +\infty }\| \nabla u| ^{(p-2)/2}
 \frac{\partial \nabla u}{\partial t}
 \| _{L^{2}(t,t+1;L^{2}(\Omega ))}\ \leq c(\tau ).\label{e5.16}
\end{equation}
\end{theorem}

\paragraph{Proof.} By Theorem \ref{thm5.3} and hypothesis (H7),
:$\int_{\Omega }( \frac{\partial \,\beta (u_{\varepsilon })}{\partial t})
^{2}dx\leq \sigma _{2}y(t)\leq c(\tau )$
for $t\geq \tau >0$. Passing to
the limit as $\varepsilon $ goes to 0 then yields (\ref{e5.14}).
Now integrating (\ref{e5.9})
on $[t,t+1]$, for any $\ t\geq \tau >0$, and using Theorem \ref{thm5.4},
 yields
\begin{equation}
\int_{t}^{t+1}\int_{\Omega }| \left( E(\nabla u_{\varepsilon })\right)
'| ^{2}dx\,ds\leq c(\tau ),\quad \forall \tau >0. \label{e5.17}
\end{equation}
Furthermore, from Lemma \ref{lm5.2},
\begin{equation}
\nabla u_{\varepsilon }\to \nabla u \mbox{ a.e on }Q_{T}. \label{e5.18}
\end{equation}
By (\ref{e5.17}) and (\ref{e5.18}) we derive the estimate (\ref{e5.16}).
On the other hand, by (H8) there is some $\sigma '$, $0<\sigma '<1$,
such that :$L^{2}(\Omega )\subset W^{-\sigma ',p'}(\Omega )$.
Now Simon's regularity results \cite{si2}, concerning the equation
$$
-\Delta _{p}u=h(x)-g(u)-\beta (u)_{t}\in L^{\infty }(\tau ,+\infty
;B_{\infty }^{-\sigma ',p'}( \Omega) ),
$$
implies that for any $t\geq \tau $,
$$
\|u(.,t)\| _{B_{\infty }^{1+(1-\sigma ')(1-p)^{2},p}\left( \Omega \right) }
\leq c_{41}(\tau )\| g(u)-h(.)\| _{B_{\infty }^{-\sigma ',p'}
\left( \Omega \right) }+c_{42}(\tau ).
$$
Hence estimate (5.15) follows. \hfill$\Box$

\paragraph{Remark} %\label{rm5.5}
Integrating (\ref{e5.9}) on $[t,t+h]$ and letting $h$ tends to $0$ leads
to the estimate
$$
\lim_{h\to 0}\frac{1}{h}\int_{t}^{t+h}\int_{\Omega }| \nabla u|
^{p-2}| \frac{\partial }{\partial t}\nabla u| ^{2}dx\,ds\leq c(\tau
),\quad \forall t\geq \tau >0\,.
$$
Let
$$\omega (u_{0})=\big\{ w\in W_{0}^{1,p}(\Omega )\cap L^{\infty }(\Omega
): \exists t_{n}\to +\infty :u(.,t_{n})\to w\mbox{ in }
W_{0}^{1,p}(\Omega )\big\}.
$$
\begin{corollary} \label{coro5.6}
Under the hypotheses of Theorem \ref{thm5.3},
$\omega (u_{0})$ is not empty and $\omega (u_{0})\subset E$, where $E$
is the set of solutions of the associated elliptic problem
\begin{gather*}
-\Delta _{p} w  =g(w)-h(x)  \quad \mbox{in }\Omega , \\
 w =0 \quad \mbox{on }\partial \Omega .
\end{gather*}
\end{corollary}

\paragraph{Proof.}
 Note that  $\omega (u_{0})$ is not empty because
 $B_{\infty}^{1+r,p}( \Omega) $ is compactly imbedded in $ W^{1,p}(\Omega )$.
Let $w=\lim_{n\to \to +\infty } u(.,t_{n})\in
\omega (u_{0})$. By the regularity estimate $\frac{\partial u}{\partial t}\in
L^{2}(\tau ,+\infty ;L^{2}(\Omega ))$, we can conclude as in \cite{ht2} that
$w\in \mathcal{E}$. \hfill$\Box$

\paragraph{Concluding remarks.}
1) In the case $\beta (u)=u$, a regularity property stronger than
(\ref{e5.16}) is obtained in \cite{ht2}; namely,
$$
| \nabla u| ^{(p-2)/2}\frac{\partial \nabla u}{\partial t}\in L^{2}
(\tau ,+\infty ;L^{2}(\Omega )) \quad \forall \tau >0.
$$
 2) In \cite{emr1}, the authors obtained
that the attractor $\mathcal{A}$ satisfies $\mathcal{A}\subset W^{2,6}(\Omega
)$ if $ p=2$, and $ N\leq 3$. In fact, their result still holds for
 $N=4$ and the proof follows the same lines as in Theorem \ref{thm5.3}
 with $p=2$.\\
3) In \cite{ht1} and \cite{ht2}, it is obtained that ~ $A\subset B_{\infty
}^{1+\frac{1}{(p-2)^{2}},p}(\Omega )\,~$if $\ p>2$ and $\beta (u)=u$.
Unfortunately for general $\beta $\ and $p>2$, Lemma \ref{lm5.1}
no longer applies.\\
4) In a forthcoming paper, we shall study a time semi-discretization scheme
associated to problem (\ref{P}) and related questions.

\paragraph{Acknowledgement}
The authors would like to thank professor F. de
Th\'{e}lin for reading a preliminary version of this work.

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\noindent\textsc{Abderrahmane El Hachimi } \\ 
UFR Math\'ematiques Appliqu\'ees et Industrielles\\ 
Facult\'{e} des Sciences \\ 
B.P. 20, El Jadida - Maroc\\ 
e-mail adress: elhachimi@ucd.ac.ma
\smallskip

\noindent\textsc{Hamid El Ouardi} \\ 
Ecole Nationale Sup\'erieure d'Electricit\'e et de M\'ecanique\\ 
B.P. 8118 -Casablanca-Oasis, Maroc\\ 
and\\
UFR Math\'ematiques Appliqu\'ees et Industrielles \\
Facult\'e des Sciences, El Jadida -  Maroc\\ 
e-mail adress: elouardi@ensem-uh2c.ac.ma
\end{document}