\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small 2004-Fez conference on Differential Equations and Mechanics \newline {\em Electronic Journal of Differential Equations}, Conference 11, 2004, pp. 61--70.\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}} \setcounter{page}{61} \begin{document} \title[\hfilneg EJDE/Conf/11 \hfil Doubly nonlinear parabolic equations] {Doubly nonlinear parabolic equations related to the p-Laplacian operator} \author[F. Benzekri, A. El Hachimi\hfil EJDE/Conf/11 \hfilneg] {Fatiha Benzekri, Abderrahmane El Hachimi} % in alphabetical order \address{Fatiha Benzekri \hfill\break UFR Math\'{e}matiques Appliqu\'{e}es et Industrielles\\ Facult\'{e} des sciences\\ B.P.20, El Jadida, Maroc} \email{benzekri@ucd.ac.ma} \address{ Abderrahmane El Hachimi\hfill\break UFR Math\'{e}matiques Appliqu\'{e}es et Industrielles\\ Facult\'{e} des sciences\\ B.P.20, El Jadida, Maroc} \email{elhachimi@ucd.ac.ma} \date{} \thanks{Published October 15, 2004.} \subjclass[2000]{35K15, 35K60, 35J60} \keywords{p-Laplacian; nonlinear parabolic equations; semi-discretization; \hfill\break\indent discrete dynamical system; attractor} \begin{abstract} This paper concerns the doubly nonlinear parabolic P.D.E. $$ \frac{\partial\beta(u)} {\partial t}-\Delta_p u + f(x,t,u )= 0 \quad \mbox{ in } \Omega\times\mathbb{R}^+, $$ with Dirichlet boundary conditions and initial data. We investigate here a time-discretization of the continuous problem by the Euler forward scheme. In addition to existence, uniqueness and stability questions, we study the long-time behavior of the solution to the discrete problem. We prove the existence of a global attractor, and obtain regularity results under certain restrictions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} We consider problems of the form \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 \big(u(., 0)\big) =\beta(u_0) \quad \mbox{in } \Omega, \end{gathered} \end{equation} where $\Delta_p u =\mathop{\rm div} \big(|\nabla u|^{p-2}\nabla u\big)$, $1 1$, and in the case p = 2 in \cite{emr1}. Here, we shall discretize \eqref{P} and replace it by \begin{equation} \label{Ptaun} \begin{gathered} \beta(U^n) - \tau \Delta_pU^n + \tau f(x, n\tau, U^n) =\beta(U^{n-1}) \quad\mbox{in } \Omega,\\ U^n = 0 \quad \mbox{on } \partial\Omega,\\ \beta(U^0) =\beta(u_0) \quad \mbox{in } \Omega. \end{gathered} \end{equation} The case $p = 2$ is completely studied in \cite{emr2}. Here we shall treat the case $ p > 1$, and obtain existence, uniqueness and stability results for the solutions of $\eqref{Ptaun}$. Then, existence of absorbing sets is given and the global attractor is shown to exist as well. Under restrictive conditions on $f$ and $p$, a supplementary regularity result for the global attractor and, as a consequence, a stabilization result for the solutions of \eqref{Ptaun} are obtained in the case when $\beta(u)=u$. The paper is organized as follows: in section 2, we give some preliminaries and in section 3, we deal with existence and uniqueness of solutions of the problem \eqref{Ptaun}. The question of stability is studied in section 4, while the semi-discrete dynamical system study is done in section 5. finally, section 6 is dedicated to obtain more regularity for the attractor. \section{Preliminaries} \subsection*{Notation} Let $\beta$ a continuous function with $\beta(0) = 0$. we define, for $t\in\mathbb R $, $$\psi(t) =\int_0^t \beta(s)ds . $$ The Legendre transform $\psi^* $ of $\psi$ is defined by $$\psi^{*}(\tau) = \sup_{s\in\mathbb{R}}\{\tau s -\psi(s)\}. $$ Here $\Omega$ stand for a regular open bounded set of $\mathbb{R}^{d}$, $d\geq 1$ and $\partial\Omega$ is it's boundary. The norm in a space X will be denoted as follows: \\ $ \|\cdot \|_r $ if $X = L^r( \Omega )$, $1 \leq r \leq+\infty $; \\ $\|\cdot\|_{1, q}$ if $ X = W^{1, q}( \Omega )$, $1 \leq q \leq+\infty$; \\ $ \|\cdot\|_X $ otherwise; \\ and $\langle.,.\rangle $ denotes the duality between $W_0^{1,p}( \Omega ) $ and $ W^{-1,p'}( \Omega )$. For any $p\geq 1$ we define it's conjugate $p'$ by $\frac{1}{p}+\frac{1}{p'} = 1$. On this paper, $C_i$ and $C$ will denote various positive constants. \subsection*{Assumptions and definition of solution} We consider the following Euler forward scheme for \eqref{P}: \begin{gather*} %( {\mathcal{P}}^n_\tau) \left \{ \beta(U^n) - \tau \Delta_pU^n + \tau f(x, n\tau, U^n) =\beta(U^{n-1})\quad \mbox{in } \Omega,\\ U^n = 0 \quad \mbox{on } \partial\Omega,\\ \beta(U^0) =\beta(u_0) \quad \mbox{in } \Omega, \end{gather*} where $N\tau = T$ is a fixed positive real, and $1\leq n\leq N$. We shall consider the case $u_0 \in L^2( \Omega ) $, and we assume the following hypotheses. \begin{itemize} \item[(H1)] $\beta $ is an increasing continuous function from $\mathbb{R}$ to $\mathbb{R}$, $\beta(0) = 0$, and for some $C_1>0$, $C_2>0$, $\beta(\xi) \leq C_1|\xi| + C_2$ for any $\xi\in \mathbb{R}$ \item[(H2)] For any $\xi$ in $\mathbb{R}$, the map $(x, t)\mapsto f(x, t, \xi)$ is measurable, and $\xi\mapsto f(x, t, \xi)$ is continuous a.e. in $\Omega\times\mathbb{R}^{+}$. Furthermore we assume that there exist $q>\sup(2,p)$ and positives constants $C_3 ,C_4 $ and $ C_5 $ such that \begin{gather*} \mathop{\rm sign} (\xi) f(x, t, \xi)\geq C_3|\xi|^{q-1}-C_4, \\ |f(x, t,\xi)|\leq a(|\xi|) \end{gather*} where $a:\mathbb{R}^+\to\mathbb{R}^+$ is increasing, and $$\limsup_{t\to0^+}|f(x, t,\xi)|\leq C_5(|\xi|^{q-1}+1). $$ \item[(H3)] There is $C_6>0$ such that for almost $(x, t)\in \Omega \times\mathbb{R}^{+} $, $\xi \mapsto f(x, t, \xi)~+C_6\beta(\xi)$ is increasing. \end{itemize} \begin{definition} \label{def2.1}\rm By a weak solution of the discretized problem, we mean a sequence $(U^n)_{0\leq n\leq N}$ such that $\beta(U^0)=\beta(u_0)$, and $U^n$ is defined by induction as a weak solution of the problem \begin{gather*} \beta(U) - \tau \Delta_p U + \tau f(x, n\tau, U) =\beta(U^{n-1}) \quad \mbox{ in } \Omega, \\ U\in W_0^{1,p}(\Omega). \end{gather*} \end{definition} \section{Existence and uniqueness result} \begin{theorem}\label{th33} If $p\geq \frac{2d}{d+2}$, then for each $n =1, \dots, N$ there exists a unique solution $ U^n$ of $\eqref{Ptaun}$ in $ W^{-1,p}(\Omega) $ provided that $0<\tau<\frac{1}{C_6}$. \end{theorem} \begin{proof} We can write \eqref{Ptaun} as \begin{gather*} - \tau \Delta_p U + F(x, U )= h, \\ U\in W_0^{1,p}(\Omega), \end{gather*} where $ U=U^n$, $h = \beta(U^{n-1}) $ and $F(x, \xi) =\tau f(x, n\tau,\xi)+\beta(\xi)$. From (H1) and (H2) we obtain $$ \mathop{rm sign}(\xi) F(x, \xi) \geq -\tau C_4 \quad\text{and}\quad h \in W^{-1,p'}(\Omega) \text{ for } p\geq\frac{2d}{d+2} . $$ Hence the existence follows from a slight modification of a result in \cite{bbm} (there, $C_4 =0$). To obtain uniqueness, we set for simplicity $$w =U^n,\quad \overline{f}(x, w)= f(x, n\tau, U^n),\quad g(x) =\beta (U^{n-1})\,. $$ Then problem \eqref{Ptaun} reads \begin{equation}\label{eq31} \begin{gathered} - \tau \Delta_p w + \tau \overline{f} (x, w)+\beta(w) = g(x),\\ w\in W_0^{1,p}(\Omega) \cap L^\infty(\Omega). \end{gathered} \end{equation} If $w_1$ and $w_2$ are two solutions of (\ref{eq31}), then we have \begin{equation}\label{eq31'} -\tau \Delta_pw_1 + \tau \Delta_pw_2 +\tau(\overline{f}(x,w _1)- \overline{f}(x,w _2)) +\beta(w_1)-\beta(w_2) = 0. \end{equation} Multiplying (\ref{eq31'}) by $w_1-w_2$ and integrating over $\Omega$, gives \begin{equation}\label{eq32} \begin{aligned} &\langle-\tau \Delta_pw_1 + \tau \Delta_pw_2,w_1-w_2\rangle + \tau \int_\Omega\big(\overline{f}(x,w _1)-\overline{f}(x,w _2)\big) (w_1-w_2) dx\\ +& \int_\Omega\big(\beta(w_1)-\beta(w_2)\big) (w_1-w_2) dx = 0. \end{aligned} \end{equation} Applying (H3) yields \begin{equation}\label{eq32'} \int_\Omega \big(\overline{f}(x,w_1)-\overline{f}(x,w _2)\big) (w_1-w_2) dx \geq -C_6\int_\Omega\big(\beta(w_1)-\beta(w_2)\big) (w_1-w_2) dx. \end{equation} Now, (\ref{eq32'}) and the monotonicity condition of the p-Laplacian operator reduce (\ref{eq32}) to $$ (1-\tau C_6)\int_\Omega \big(\beta(w_1)-\beta(w_2)\big) (w_1-w_2)\, dx \leq 0. $$ So by (H1), if $\tau<\frac{1}{C_6}$ , we get $w_1=w_2$. \end{proof} \section{stability} \begin{theorem}\label{th42} Assume $p\geq\frac{2d}{d+2}$. Then there exists a positive constant $C(T, u_0)$ such that, for all $n=1, \dots, N$ \begin{equation}\label{eq46} \int_\Omega \psi^* (\beta(U^n)) dx +\tau \sum_{k=1}^n \|U^k\|_{1, p}^p+ C \tau \sum_{k=1}^n \|U^k\|_q^q \leq C(T, u_0) \end{equation} and \begin{equation}\label{eq47} \max _{1 \leq k \leq n} \|\beta(U^k)\|_2^2 + \sum_{k=1}^n \|\beta(U^k)-\beta(U^{k-1})\|_2^2 \leq C(T, u_0). \end{equation} \end{theorem} \begin{proof} \textbf{(a)}\quad Multiply the first equation of \eqref{Ptaun}, with $k$ instead of $n$, by $U^k$. Then using (H2) and the relation $$ \int_\Omega \psi^* (\beta(U^k))dx - \int_\Omega \psi^* (\beta(U^{k-1})) dx \leq \int_\Omega\left(\beta(U^k)-\beta(U^{k-1})\right)U^k dx , $$ we get, after summing from $k=1$ to $n$, $$ \int_\Omega \psi^* (\beta(U^n)) dx +\tau \sum_{k=1}^n \|U^k\|_{1, p}^p + C \tau \sum_{k=1}^n \|U^k\|_q^q \leq C \tau \sum_{k=1}^n \|U^k\|_1 + \int_\Omega \psi^* (\beta (u_0)) dx. $$ Thanks to Young's inequality, for all $\varepsilon >0$ there exists $C_\varepsilon (T, u_0)$ such that $$ \int_\Omega \psi^* (\beta(U^n)) dx +\tau \sum_{k=1}^n \|U^k\|_{1, p}^p + C \tau \sum_{k=1}^n \|U^k\|_q^q \leq \varepsilon \tau \sum_{k=1}^n \|U^k\|_p^p + C_\varepsilon (T, u_0).$$ Now for a suitable choice of $\varepsilon$, we have $$ \varepsilon\tau \sum_{k=1}^n \|U^k\|_p^p \leq C_\varepsilon (T, u_0). $$ That is, (\ref{eq46}) is satisfied. \\ \textbf{(b)} Multiplying the first equation of \eqref{Ptaun}, with $k$ instead of $n$, by $\beta(U^k)$. Then using (H2), we have \begin{equation}\label{eq401} \int_\Omega \big(\beta (U^k)-\beta (U^{k-1})\big)\beta (U^k) dx + \tau \langle - \Delta_p U^k , \beta (U^k) \rangle \leq C \tau \int_\Omega |\beta (U^k)| dx. \end{equation} With the aid of the identity $2a(a-b) =a^2 -b^2 +(a-b)^2$, we obtain from (\ref{eq401}), \begin{equation}\label{eq402} \| \beta(U^k)\|_2^2 -\|\beta(U^{k-1})\|_2^2 +\| \beta(U^k) - \beta(U^{k-1})\|_2^2 \leq C \tau \| \beta(U^k)\|_1. \end{equation} Summing (\ref{eq402}) from $k=1$ to $n$, yields \begin{equation}\label{eq42} \| \beta(U^n)\|_2^2 + \sum_{k=1}^n \| \beta(U^k) - \beta(U^{k-1})\|_2^2\leq \| \beta (u_0)\|_2^2 + C \tau \sum_{k=1}^n \| \beta(U^k)\|_1 . \end{equation} As in (a), we conclude to (\ref{eq47}). \end{proof} \section{The semi-discrete dynamical system}\label{sect5} We fix $\tau$ such that $0<\tau< \min(1, \frac{1}{C_6})$, and assume that $p> \frac{2d}{d+2}$. Theorem \ref{th33} allows us to define a map $S_\tau$ on $ L^2(\Omega)$ by setting $$S_\tau U^{n-1}=U^n.$$ As $S_\tau$ is continuous, we have $S^n_\tau U^0=U^n$. Our aim is to study the discrete dynamical system associated with $\eqref{Ptaun}$ . We begin by showing the existence of absorbing balls in $L^\infty(\Omega)$. ( We refer to \cite{t} for the definition of absorbing sets and global attractor). \subsection*{Absorbing sets in $L^\infty(\Omega)$} \begin{lemma}\label{l51} If $p>\frac{2d}{d+1}$, then there exists $n(d, p)\in \mathbb{N}^*$ depending on $d$ and $p$, and $C>0$ depending on $d, \Omega$ and the constants in (H1)--(H3) such that \begin{equation}\label{eq51'} U^n \in L^\infty(\Omega) \text{ for all }n \geq n(d, p) \end{equation} and \begin{equation}\label{eq52'} \|U^{n(d, p)}\|_\infty \leq\frac{C}{\tau ^{\alpha+\alpha^2+\dots+\alpha^{n(d, p)}}}\big(\|u_0\|_2^{\alpha^{n(d, p)}}+1 \big), \end{equation} where $\alpha =p'/p$. Moreover, if $d=1, d=2 \text{ or } d<2p$, then $n(d, p) =1$. \end{lemma} The proof of the lemma above follows from a repeated application of the following lemma (cf. \cite{r2}) \begin{lemma} \label{l21} If $u \in W_0^{1,p}(\Omega)$ is a solution to the equation $$-\tau \Delta_p u +F(x, u ) = T, $$ where $T \in W^{-1,r}(\Omega)$ and $F$ satisfies $\xi F(x, \xi) \geq 0$ in $\Omega\times \mathbb{R}$ then we have the following estimates \begin{itemize} \item[(a)] If $r>\frac{d}{p-1} $, then $ u \in L^\infty(\Omega)$ and $$\|u\|_\infty \leq C \big (\frac{{\|T\|}_{-1, r}}{\tau}\big)^{p'/p}$$ \item[(b)] If $p' \leq r < \frac{d}{p-1}$, then $u \in L^{r^*}(\Omega)$ and $$\|u\|_ {r^*} \leq C \big (\frac{{\|T\|}_{-1, r}}{\tau}\big)^{p'/p}, $$ where $\frac{1}{r^*} = \frac{1}{(p-1)r} - \frac{1}{d}$ \item[(c)] If $r = \frac{d}{p-1}$ and $r\geq p' $ then $ u \in L^q(\Omega)$ for any $ q$, $1 \leq q < \infty$ and $$\|u\|_q \leq C \big (\frac{{\|T\|}_{-1, r}}{\tau}\big)^{p'/p}.$$ \end{itemize} \end{lemma} \noindent We can write \eqref{Ptaun} as \begin{gather*} - \tau \Delta_pU^m + F_m(x, U^m) =\beta(U^{m-1})+C_4 sign(U^m) =T_m \quad \mbox{in } \Omega,\\ U^m = 0 \quad \mbox{on } \partial\Omega, \end{gather*} where $$ F_m(x, \xi)=\tau f(x, m\tau, \xi)+\beta(\xi)+C_4sign(\xi). $$ Note that by (H1) and (H2) we have \begin{gather*} \xi F_m(x, \xi) \geq 0 \text{ for all } \xi,\\ T_m \in W^{-1,p'}(\Omega). \end{gather*} Now, applying lemma \ref{l21}, we can find an increasing sequence $\big(\alpha(m)\big)_{m\geq1}$ such that \begin{equation}\label{eq55'} \alpha(m)\geq p', \quad \frac{1}{\alpha (m+1)}=\frac{1}{(p-1)\alpha (m)} -\frac{1}{d} , \end{equation} and \begin{equation}\label{eq56'} \|U^m\|_{\alpha(m)} \leq \frac{C_m}{\tau ^{\alpha+\alpha^2+\dots+\alpha^m }}\big(\|u_0\|_2^{\alpha^ m}+1 \big) \end{equation} We shall stop the iteration on $m$ once we have $\alpha(m-1) > \frac{d}{p}$. Indeed, if $ q > \frac{d}{p}$, then there exists $r> \frac{d}{p-1}$ such that $ L^q(\Omega) \subset W^{-1,r}(\Omega)$. Then we have $T_m \in W^{-1,r}(\Omega)$ and thus $U^m \in L^\infty(\Omega)$. $n(d, p)$ will be the first integer $m$ such that $\alpha( m-1) > \frac{d}{p}$. Finally (\ref{eq52'}) will follow from (\ref{eq56'}) and lemma \ref{l21}. %\end{proof} \begin{remark} \label{rmk5.3}\rm (i) If $d=1$ or $d=2$, then for all $q>1$, we have $ L^2(\Omega) \subset W^{-1,q}(\Omega)$, in particular for $q > \frac{d}{p-1}$. If $d \geq 3$ and $d < 2p$, we can choose $q >1$ to be such that $\frac{d}{p-1} < q < \frac{2d}{d-2}$. In the two cases, $T_1 \in W^{-1,q}(\Omega)$ for some $q > \frac{d}{p-1}$ and, from lemma \ref{l21}, $U^1 \in L^\infty(\Omega)$. We have then $n(d, p)=1$. (ii) If $\alpha (m) \leq \frac{d}{p}$ for all $m$ , then $ l= \lim _{m\to\infty} \alpha(m) $ exists and equals $ \frac{2-p}{p-1}d $. Consequently, for $p>\frac{2d}{d+1}$, we have $l0 , \delta >0$ depending on the data of (H1)--(H3) and $\mu >0$ depending on $n_0, q,\gamma, \delta, k$ such that, for all $n\geq n_0$, we have $$ \|\beta(U^n)\|_\infty \leq{ \big( \frac{\delta}{\gamma} \big)}^{\frac{1}{q-1}} + \frac{C_1 +\mu}{{\big( \tau^\beta (n-n_0+1)\big)} ^{\frac{1}{k-1}}}\quad , $$ where $\beta=\begin{cases} 1 &\text{if } \alpha \leq 1,\\ \alpha^{n_0} &\text{if } \alpha \geq 1. \end{cases} $ \end{lemma} \begin{proof} From lemma \ref{l51}, for $n\geq n_0$ we have $$U^n \in L^\infty(\Omega) \quad\text{and}\quad \|U^{n_0}\|_\infty \leq \frac{C_1}{\tau ^{\alpha+\alpha^2+\dots+\alpha^{n_0}}}. $$ Next, multiplying the first equation of $\eqref{Ptaun}$ by $|\beta(U^n)|^m \beta(U^n)$ for some positive integer $m$, we derive from (H$_1$) and (H$_2$), after dropping some positive terms, that $$ \|\beta(U^n)\|_{m+2}^{m+2} \leq \int_\Omega |\beta(U^n)|^{m+1} \beta(U^{n-1}) dx + C\tau\ |\beta(U^n)\|_{m+1}^{m+1}- C\tau \| \beta(U^n)\|_{m+q}^{m+q}.$$ By setting $$ y_m^n =\|\beta(U^n)\|_{m+2} \text{ and } z_n =\|\beta(U^n)\|_\infty,$$ and using H\"{o}lder's inequality, we deduce the existence of two constants $\gamma >0, \delta > 0$ (not depending on $m$ nor on $U^n$) such that $$ y_m^n + \gamma\tau(y_m^n)^{q-1} \leq \delta \tau + y_m^{n-1}. $$ As $m$ approaches infinity, we then obtain $$ z_n + \gamma\tau z_n^{q-1} \leq \delta \tau + z_{n-1}, $$ with $$ z_{n_0} \leq \frac{C_1}{\tau^{\alpha+ \alpha^2+\dots + \alpha^{n_0}}}. $$ (i) If $\alpha \leq 1$, then $\quad \alpha+ \alpha^2+\dots + \alpha^{n_0} \leq n_0$. So, we have \begin{gather*} z_{n_0} \leq C_1 / \tau^ {n_0},\\ z_n + \gamma\tau z_n^{q-1} \leq \delta \tau + z_{n-1}. \end{gather*} Then we can apply lemma 7.1 in \cite{emr2} to get $$ z_n \leq {\big(\frac{\delta}{\gamma}\big )}^{\frac{1}{q-1}}+ \frac{C_1 +\mu}{{\big( \tau (n-n_0+1)\big)} ^{\frac{1}{k-1}}}\equiv c_\alpha(n). $$ (ii) If $\alpha \geq 1$, then $\alpha+ \alpha^2+\dots + \alpha^{n_0} \leq n_0\alpha^{n_0}$. By setting $\tau_1 = \tau^{\alpha^{n_0}}$, we have \begin{gather*} z_{n_0} \leq C_1/\tau_1^ {n_0}, \\ z_n + \gamma'\tau _1 z_n^{q-1} \leq \delta' \tau _1 + z_{n-1}, \end{gather*} where $\gamma'=\tau ^{1-\alpha^{n_0}}\gamma$ and $\delta'=\tau ^{1-\alpha^{n_0}}\delta $. Then, once again, we can apply lemma 7.1 in \cite{emr2} to get $$ z_n \leq {\big(\frac{\delta}{\gamma}\big )}^{\frac{1}{q-1}}+ \frac{C_1 +\mu}{{\big( \tau _1 (n-n_0+1)\big)} ^{\frac{1}{k-1}}}\equiv c_\alpha(n). $$ \end{proof} \begin{remark} \label{rmk5.5} \rm In the case $\alpha \geq 1$, a slight modification has to be introduced in the proof of lemma 7.1 in \cite{emr2}, since $\mu$ is depending on $\delta'$ and $\gamma'$ and hence on $\tau$. In fact, it suffices to choose in that proof, with the same notation, $\mu$ such that $$\gamma\big(\frac{\delta}{\gamma}\big)^{1-\frac{k}{q-1}}\mu^{k-1} \geq 2^\frac{1}{k-1}/ (k-1).$$ and to remark that $\gamma' \geq \gamma$. \end{remark} Consequently, lemma \ref{l52} implies that there exist absorbing sets in $L^q(\Omega) $ for all $q\in [1, \infty] $. Indeed, this is due to the fact that$$ \|U^n\|_\infty \leq \text{ max } \big(\beta ^{-1}(c_\alpha (n)), |\beta ^{-1}(-c_\alpha(n))|\big), $$ for all $n\geq n_0$, with $c_\alpha(n) \to {\big(\frac{\delta}{\gamma}\big )}^{\frac{1}{q-1}}$ \quad as $n\to\infty$. \subsection*{Absorbing sets in $W_0^{1,p}(\Omega)$, existence of the the global attractor} Multiplying the equation in $\eqref{Ptaun}$ by $\delta_n=U^n-U^{n-1}$, we get \begin{equation}\label{eq51} \begin{aligned} &\langle \frac{\beta(U^n)-\beta(U^{n-1})}{\tau}, \delta_n \rangle+ \int_\Omega |\nabla U^n|^{p-2}\nabla U^n.(\nabla U^n-\nabla U^{n-1})dx \\ &+ \langle f(x, n\tau, U^n), \delta _n \rangle =0. \end{aligned} \end{equation} By setting $$ F_\beta(u) = \int_0^u \big(f(x, n\tau, w) +C_6 \beta(w)\big)dw, $$ we deduce from (H3) that $$ F'_\beta(u)(u-v) \geq F_\beta(u)-F_\beta(v),$$ and then \begin{align*} \langle f(x, n\tau, U^n), \delta_n \rangle &= \langle f(x,n\tau,U^n) +C_6\beta(U^n), \delta_n \rangle -C_6\langle \beta(U^n), \delta_n \rangle\\ &\geq \int_\Omega \left( F_\beta(U^n)-F_\beta(U^{n-1})\right)dx -C_6\langle \beta(U^n), \delta_n \rangle. \end{align*} Now, using (H1), we get $\psi'(v)(u-v) \leq \psi(u)-\psi(v)$. Therefore, we have \begin{align*} &\int_\Omega \beta(U^n)(U^n-U^{n-1})dx\\ &=\int_\Omega \left(\beta(U^n)-\beta(U^{n-1})\right) (U^n-U^{n-1})dx +\int_\Omega \beta(U^{n-1})(U^n-U^{n-1})dx\\ & \leq \int_\Omega \left(\beta(U^n)-\beta(U^{n-1})\right) (U^n-U^{n-1})dx + \int_\Omega \left(\psi(U^n)-\psi( U^{n-1})\right)dx. \end{align*} With the aid of the elementary identity \begin{equation}\label{51'} |a|^{p-2} a.(a-b) \geq \frac{1}{p}|a|^p-\frac{1}{p}|b|^p, \end{equation} we obtain \begin{equation}\label{eq52''} \int_\Omega |\nabla U^n|^{p-2}\nabla U^n.(\nabla U^n-\nabla U^{n-1}) dx \geq \frac{1}{p}\|U^n\|_{1,p}^p-\frac{1}{p}\|U^{n-1}\|_{1, p}^p. \end{equation} % Since $\tau < \frac{1}{C_6}$ , from (\ref{eq51}) we obtain \begin{equation}\label{eq52} \frac{1}{p}\|U^n\|_{1,p}^p + \int_\Omega F_\beta(U^n)dx \leq C_6 \int_\Omega\left(\psi(U^n)-\psi( U^{n-1})\right)dx + \int_\Omega F_\beta(U^{n-1})dx. \end{equation} Now, setting $$ F (u)= \int_0^u f(x, n\tau, w)dw, $$ yields $$ \int_\Omega F_\beta(u)dx= \int_\Omega F (u)dx +C_6 \int_\Omega \psi(u)dx.$$ Hence, from (\ref{eq52}), we get $$ \frac{1}{p}\|U^n\|_{1,p}^p + \int_\Omega F (U^n)dx \leq \frac{1}{p}\|U^{n-1}\|_{1,p}^p + \int_\Omega F (U^{n-1})dx.$$ By setting $$ y_{n}=\frac{1}{p}\|U^n\|_{1,p}^p+ \int_\Omega F (U^n)dx , $$ we get $ y_n \leq y_{n-1}$. And by choosing $N\tau=1$, using the boundedness of $U^n$ and the stability analysis, there exists $n_\tau>0$ such that$$ \tau \sum_{n=n_0}^{n_0+N}y_n \leq a_1, \quad \text{ for all } n \geq n_\tau. $$ Hence we can apply the discrete version of the uniform Gronwall lemma (cf. \cite[Lemma~7.5]{emr2}) with $h_n=0$ to obtain $$ \frac{1}{p}\|U^n\|_{1,p}^p+ \int_\Omega F (U^n)dx \leq C \quad \text{ for all } n \geq n_\tau.$$ % On the other hand, since $U^n$ is bounded, we deduce that $$\|U^n\|_{1,p}\leq C. $$ We have then proved the following result. \begin{proposition}\label{l53} If $\tau <\frac{1}{C_6}$, there exist absorbing sets in $L^\infty(\Omega) \cap W^{-1,p}(\Omega) $. More precisely, for any $u_0 \in L^2(\Omega) $, there exists a positive integer $n_\tau$ such that \begin{equation}\label{eqfin} \|U^n\|_\infty +\|U^n\|_ {1, p}\leq C, \quad \forall n\geq n_\tau, \end{equation} where $C$ does not depend on $\tau$. \end{proposition} In order for the nonlinear map $S_\tau$ to satisfy the properties of a semi-group, namely $S_\tau^{n+p}=S_\tau^noS_\tau^p$, we need $\eqref{Ptaun}$ to be autonomous. So, we assume that $f(x,t,\xi)\equiv f(x,\xi)$. Thus, $S_\tau$ defines a semi-group from $L^2(\Omega)$ into itself and possesses an absorbing ball $B$ in $L^\infty(\Omega)\cap W^{-1,p}(\Omega)$. Setting $$ \mathcal{A}_\tau=\bigcap_{n\in\mathbb{N}} \quad\overline{\bigcup_{m\geq n}S_\tau^m(B)}, $$ we have a compact subset of $L^2(\Omega)$ which attracts all solutions. That implies that for all $u_0\in L^2(\Omega)$, $$ \mathop{\rm dist} \big(\mathcal{A}_\tau, S_\tau^n u_0\big)\mapsto 0 \quad \text{as }n\mapsto\infty. $$ Therefore, we have proved the following result. \begin{theorem}\label{th51} Assuming $u_0\in L^2(\Omega)$ and (H1)--(H3), the discrete problem $\eqref{Ptaun}$ has an associated solution semi-group $S_\tau$ that maps $L^2(\Omega)$ into $L^\infty(\Omega)\cap W^{-1,p}(\Omega)$. This semi-group has a compact attractor which is bounded in $L^\infty(\Omega) \cap W^{-1,p}(\Omega)$. \end{theorem} \section{More regularity for the attractor} In this section, we shall show supplementary regularity estimates on the solutions of problem $\eqref{Ptaun}$ in the particular case where $\beta(\xi)=\xi$. We obtain therefore more regularity for the attractor obtained in section \ref{sect5}. The assumptions are similar to those used for the continuous problem in \cite{ee1}; namely $u_o \in L^2(\Omega)$ and $f$ verifying the following assumption \begin{itemize} \item[(H4)] $f(x,t,\xi)=g(\xi)-h(x)$ where $h \in L^\infty(\Omega)$ and $g$ satisfying the conditions (H1)--(H3). \end{itemize} The problem $\eqref{Ptaun}$ becomes \begin{equation} \label{6n} \delta_n - \Delta_p U^n +g(U^n)= f, \end{equation} where $ \delta _n =\frac{U^n -U^{n-1}}{\tau}$. First, we state the following lemma which we shall use to prove the principal result of this section. % \begin{lemma}\label{l61} There exists a positive real C such that for all $n_0 \geq n_\tau$, and all $N$ in $\mathbb{N} $, we have \begin{equation}\label{601} \tau \sum_{n=n_0} ^{n_0 +N} \|\delta_n\|_2^2 \leq C. \end{equation} \end{lemma} \begin{proof} Multiplying \eqref{6n} by $ \delta _n$, using (\ref{eq52''}), (\ref{eqfin}), (H4) and Young's inequality, we get after some simple calculations \begin{equation}\label{eq602} \frac{1}{4} \tau \|\delta_n\|_2^2 +\frac{1}{p}\|U^n\|_{1,p}^p-\frac{1}{p}\|U^{n-1}\|_{1, p}^p \leq C\tau . \end{equation} Summing (\ref{eq602}) from $ n = n_0$ to $n = n_0 +N$ , yields \begin{equation}\label{603} \frac{1}{4} \tau \sum_{n=n_0}^{n_0+N} \|\delta_n\|_2^2 +\frac{1}{p}\|U^{n_0+N}\|_{1,p}^p \leq \frac{1}{p}\|U^{n_0}\|_{1, p}^p + CN\tau. \end{equation} Now choosing $n_0\geq n_\tau$ shows that $U^{n_0}$ is in an $W^{-1,p}(\Omega)$--absorbing ball. As $\tau N =1$, we therefore obtain (\ref{601}) from (\ref{603}). \end{proof} \begin{theorem}\label{t61} For all $ n \geq n_\tau$, we have $\|\delta_n\|_2 \leq C$, where $C$ is a positive constant. \end{theorem} \begin{proof} By subtracting equation \eqref{6n}, with $n-1$ instead of $n$, from \eqref{6n} and multiplying the difference by $\delta_n$, we deduce from the monotonicity of the p-Laplacian operator, Young's inequality and (H3) that $$ \frac{1}{2}\|\delta_n\|_2^2 \leq \frac{1}{2} \|\delta_{n-1}\|_2^2+ C\tau\|\delta_n\|_2^2. $$ Setting $$ y_n = \frac{1}{2}\|\delta_n\|_2^2\quad \text{and}\quad h_n = C\|\delta_n\|_2^2. $$ and using \cite[Lemma 7.5]{emr2} and Lemma \ref{l61}, we deduce that $$ y_{ n+N} \leq \frac{C}{N\tau} +C. $$ If $n \geq n_\tau$ and $N\tau =1$, then we get the desired estimate. \end{proof} Using this theorem, we have the following regularizing estimates. \begin{corollary}\label{c61} If $p>\frac{2d}{d+2}$ and $ p\neq 2$, then there exists some $\sigma,$ $0< \sigma <1$, such that$$ \|U^n\|_{B_\infty^{1+\sigma, p}(\Omega)} \leq C \text{ for all } n \geq n_\tau,$$ where $B_\infty^{\alpha, p}(\Omega)$ denotes a Besov space defined by real interpolation method. \\ If $p=2$, then $$\|U^n\|_{W^{2, 2}(\Omega)} \leq C \text{ for all } n \geq n_\tau.$$ \end{corollary} \begin{proof} (i) If $\frac{2d}{d+2}< p<2$ then there exists some $ \sigma', 0<\sigma'<1$ such that \begin{equation}\label{eq71} L^2(\Omega)\hookrightarrow W^{-\sigma', p'}(\Omega) \end{equation} By $(6)_n$, (\ref{eq71}), (H4) and theorem \ref{t61} we get $$ \|-\Delta_p U^n\|_{B_\infty^{-\sigma', p'}(\Omega)} \leq C \quad \text{ for all } n \geq n_\tau. $$ Therefore, Simon's regularity result in \cite{s} yields $$ \|U^n\|_{B_\infty^{1+(1-\sigma')(p-1)^2, p}(\Omega)} \leq C \quad \text{ for all } n \geq n_\tau. $$ (ii) If $p>2$, then, by \eqref{6n}, (H4) and theorem \ref{t61}, we get $$ \|-\Delta_p U^n\|_{p'} \leq C \quad \text{ for all } n \geq n_\tau. $$ Therefore, Simon's regularity result in \cite{s} yields $$ \|U^n\|_{B_\infty^{1+\frac{1}{(p-1)^2}, p}(\Omega)} \leq C \quad \text{ for all } n \geq n_\tau. $$ (iii) For $p=2$, see \cite{emr2}. \end{proof} \begin{thebibliography}{00} \bibitem{bbm} A. 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