\documentclass[twoside]{article} \usepackage{amssymb, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \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 $$\label{P} \begin{gathered} \frac{\partial \beta (u)}{\partial t}-\Delta _{p}u+f(x,t,u)= 0 \quad\mbox{in }\Omega \times (0,\infty)\\ u = 0 \quad \mbox{on } \partial\Omega \times (0,\infty ) \\ \beta (u(.,0)) = \beta (u_{0}) \quad \mbox{in }\Omega , \end{gathered}$$ where $\Delta _{p}u=\mathop{\rm div}\left(|\nabla u|^{p-2}\nabla u\right)$ , $10$, 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 $(x,t)\in \Omega \times \mathbb{R}^{+}$, $$\label{e2.1} \begin{gathered} \mathop{\rm sign}(\xi)f(x,t,\xi ) \geq c_{1}| \beta (\xi )|^{q-1}-c_{2},\\ \lim_{t\to 0^{+}}\sup|f(x,t,\xi )| \leq c_{3}(|\xi|^{q-1}+1) \end{gathered}$$ 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. $(x,t)\in \Omega \times \mathbb{R}^{+}$, $$\label{e2.2} \mathop{\rm sign}(u)f(x,t,u)\geq c_{3}|\beta (u)| -d.$$ 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 $$\label{Pe} \begin{gathered} \frac{\partial \beta _{\varepsilon }(u_{\epsilon })}{\partial t} -\mathop{\rm div}F_{\varepsilon }(\nabla u_{\varepsilon })+f(x,t,u_{\epsilon }) =0 \quad \mbox{in } Q_{T} \\ u_{\epsilon } =0 \quad \mbox{in } S_{T} \\ \beta _{\varepsilon }(u_{\epsilon })_{| t=0} =\beta_{\varepsilon }(u_{0\varepsilon }) \quad \mbox{in }\Omega, \end{gathered}$$ 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\{ $$\label{ped} \begin{gathered} \frac{\partial \beta _{\varepsilon }(u_{\epsilon ,\sigma })}{\partial t} -divF_{\varepsilon }(\nabla u_{\varepsilon ,\sigma })+\sigma f_{m}(x,t,v) =0 \quad \mbox {in } Q_{T},\\ u_{\epsilon ,\sigma } =0 \quad \mbox{in } S_{T}, \\ \beta _{\varepsilon }(\,u_{\epsilon ,\sigma })_{| t=0} =\beta_{\varepsilon }(\sigma u_{0\varepsilon }) \quad \mbox{in } \Omega , \end{gathered}$$ 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 $$\frac{1}{k+2}\frac{d}{dt}{ \int_{\Omega }| \beta _{\varepsilon }(u_{\epsilon })| ^{k+2}dx} +c_{14}\int_{\Omega }| \beta _{\varepsilon }(u_{\epsilon })| ^{k+q}dx\leq c_{15}\int_{\Omega }|\beta _{\varepsilon }(u_{\epsilon })| ^{k+1}dx \label{e3.7}$$ 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 $$y_{\varepsilon ,k}(t)\leq \big( \frac{\alpha _{0}}{\lambda _{0}}\big) ^{ \frac{1}{q-1}}+\frac{1}{\left[ \lambda _{0}(q-2)t\right] ^{\frac{1}{q-2}}} =c_{7}(t)\,,\forall t>0. \label{e3.8}$$ As $k\to+\infty$, and for all $t\geq \tau >0$, we have $$|\beta _{\varepsilon }(u_{\epsilon })(t)| _{L^{\infty }(\Omega )} \leq c_{7}(\tau ); \label{e3.9}$$ which implies $$|u_{\epsilon }(t)| _{L^{\infty }(\Omega )}\leq \max ( \beta _{\varepsilon }^{-1}(c_{7}(\tau )),| \beta _{\varepsilon }^{-1}(-c_{7}(\tau ))|) =\delta _{\varepsilon }\,. \label{e3.10}$$ 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 $$\int_{\Omega }\Psi _{\varepsilon }^{\ast }\,(\beta _{\varepsilon }(u_{\epsilon }))dx+c_{8}\int_{0}^{T}\int_{\Omega }| \,u_{\epsilon }| ^{p}dxds\leq c_{8}(T).\label{e3.12}$$ 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 $$\int_{\Omega }| \nabla u_{\epsilon }| ^{p}dx+\int_{\Omega } \int_{0}^{u_{\varepsilon}(x,t)}f(x,t,y)dy dx \leq \frac{a_{1} +a_{2}}{\tau } +c_{15}(\tau) \quad \forall t\geq \tau>0. \label{e3.17}$$ By using (\ref{e3.10}) and hypothesis (H4), (\ref{e3.17}) leads to $$\int_{\Omega }|\nabla u_{\epsilon }| ^{p}dx\leq c_{16}(\tau ) \forall t\geq \tau >0. \label{e3.18}$$ 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 $$F_{\epsilon }(\nabla u_{\epsilon })\ \mbox{ is bounded in }\ L^{p'}(\tau,T;W^{-1,p'}(\Omega )), \label{e3.21}$$ By Lemma \ref{lm3.5} (i), $$\frac{\partial\beta _{\varepsilon }(u_{\epsilon })}{\partial t} \mbox{ is bounded in }\ L^{2}(\tau ,T;L^{2}(\Omega )), \forall \tau >0. \label{e3.22}$$ 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 $$\| u_{\varepsilon}(t)\| _{L^{\sigma }(\Omega )}\leq c_{\delta }, \quad\mbox{for any }\sigma :1\leq \sigma <\infty ,\label{e4.1}$$ 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 $$\| u(t)\| _{L^{\sigma }(\Omega )}\leq c_{\delta }\mbox{ for any }\sigma :1\leq \sigma <\infty . \label{e4.2}$$ By letting $\sigma$ tends to +$\infty$ in (\ref{e4.2}), we obtain $$\| u(t)\| _{L^{\infty }(\Omega )}\leq c_{\delta }. \label{e4.3}$$ 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 $10}$ 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 10 \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 $$\beta '(w)w^{\prime \prime }+\beta''(w)(w')^{2} -\mathop{\rm div}(( F_{\epsilon }(\nabla w))') +g'(w)w'=0.\label{e5.3}$$ Now multiplying (\ref{e5.3}) by w', integrating over \Omega and using (\ref{e5.2}), gives $$\frac{1}{2}y'(t)+\frac{1}{2}\int_{\Omega }\big[\beta ''(w)(w')^{3} + \frac{4(p-1)}{p ^{2}}| (E(\nabla w)) '| ^{2} +g'(w)(w')^{2}\big]dx\leq 0. \label{e5.4}$$ 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 $$\frac{1}{2}y'(t)+ \frac{2(p-1)}{p ^{2}}\int_{\Omega }| \left( E(\nabla w)\right) '| ^{2}dx \leq c_{33}(y(t)^{\frac{3(1+\alpha )}{2}}+y(t)+1) \leq c_{34}(y(t))^{2}+c_{35} \label{e5.9}$$ for all t\geq \tau >0. By assumption (H6), equation (\ref{e3.16}) can be written as $$\beta '(w)w'-\mathop{\rm div}(F_{\varepsilon }(\nabla w))=h-g(w). \label{e5.10}$$ Taking the scalar product of (\ref{e5.12}) with w', we obtain \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} Hence $$\frac{1}{2}\int_{\Omega }\beta '(w)\left( w'\right) ^{2}dx + \frac{d}{dt}\Big[ \frac{1}{p}\int_{\Omega }\left( | \nabla w| ^{2}+\varepsilon \right) ^{\frac{p}{2}}dx\Big] \leq c_{36}\| g(w)-h\|_{L^{\infty }(\Omega )}^{2}, \label{e5.12}$$ 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 . $$Integrating (5.12) on \left[ t,t+\frac{\tau }{2}\right] yields $$\int_{t}^{t+\frac{\tau}{2}}y(s)ds\leq c_{38}(\tau ),\quad \forall t\geq \frac{\tau}{2}>0. \label{e5.13}$$ Going back to (\ref{e5.9}) and using the uniform Gronwall lemma \cite[p. 89]{tem} with r=\tau/2, g(t)=c_{34}y(t) and h=c_{35} and estimate (\ref{e5.13}) leads to$$y(t+\frac{\tau }{2})\leq c_{39}(\tau )\quad \forall t\geq\frac{\tau }{2}>0.$$Hence y(t)\leq c_{39}(\tau ), for any t\geq\tau >0. The proof of the theorem is now complete. \hfill\Box Using Theorem \ref{thm5.3}, we state the main result of this section. \begin{theorem} \label{thm5.4} Let f,\beta,p satisfies hypotheses (H1)-(H8). Then, for \tau >0, the solution of problem (\ref{P}) satisfies: \begin{gather} \frac{\partial \beta (u)}{\partial t}\in L^{\infty }(\tau ,+\infty ;L^{2}(\Omega )), \label{e5.14} \\ u\in L^{\infty }(\tau ,+\infty ;B_{\infty }^{1+\sigma ,p} \left( \Omega \right) ), \label{e5.15} \end{gather} where B_{\infty }^{1+\sigma ,p}\left( \Omega \right) is a Besov space defined by the real interpolation method \cite{si2}. Moreover, there exists a constant c(\tau )>0, depending on \tau such that $$\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}$$ \end{theorem} \paragraph{Proof.} By Theorem \ref{thm5.3} and hypothesis (H7), :\int_{\Omega }( \frac{\partial \,\beta (u_{\varepsilon })}{\partial t}) ^{2}dx\leq \sigma _{2}y(t)\leq c(\tau ) for t\geq \tau >0. Passing to the limit as \varepsilon goes to 0 then yields (\ref{e5.14}). Now integrating (\ref{e5.9}) on [t,t+1], for any \ t\geq \tau >0, and using Theorem \ref{thm5.4}, yields $$\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}$$ Furthermore, from Lemma \ref{lm5.2}, $$\nabla u_{\varepsilon }\to \nabla u \mbox{ a.e on }Q_{T}. \label{e5.18}$$ By (\ref{e5.17}) and (\ref{e5.18}) we derive the estimate (\ref{e5.16}). On the other hand, by (H8) there is some \sigma ', 0<\sigma '<1, such that :L^{2}(\Omega )\subset W^{-\sigma ',p'}(\Omega ). Now Simon's regularity results \cite{si2}, concerning the equation$$ -\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$. 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Anal. and Applic., {\bf{132}}, 187-212, 1988. \end{thebibliography} \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}