\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 113, pp. 1--14.\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 2003 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/113\hfil Doubly nonlinear parabolic equations]
{Doubly nonlinear parabolic equations related to the
$p$-Laplacian  operator:  Semi-discretization}

\author[F. Benzekri \& A. El Hachimi\hfil EJDE--2003/113\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{Submitted April 16, 2003. Published November 11, 2003.}
\subjclass[2000]{35K15, 35K60, 35J60} 
\keywords{P-Laplacian, nonlinear parabolic equations,
 semi-discretization, \hfill\break\indent discrete dynamical system, attractor}

\begin{abstract}
 We study the doubly nonlinear parabolic equation
 $$
 \frac{\partial\beta(u)} {\partial t}-\triangle_p u + f(x,t,u )= 0
 \quad\mbox{in } \Omega\times\mathbb{R}^+,
  $$
 with Dirichlet boundary conditions and initial data.
 We investigate a time-discretization of the continuous
 problem by the Euler forward scheme.  In addition to proving
 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 its
 regularity under additional conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{remark}[thm]{Remark}


\section{Introduction}

In this paper we study a  doubly nonlinear parabolic partial
differential equation related to the $p$-Laplacian operator
estudied in \cite{ee1}. We examine the validity of numerical solutions as
approximations to solutions for long times.
This work  is inspired, on one hand by the results of El Hachimi
and  El Ouardi  \cite{ee1},  and, on the other hand, by the
work of Eden, Michaux  and Rakotoson  \cite{emr2}. It is a generalization
in different directions of several results.

The problem under consideration has the form
\begin{equation} \label{P}
 \begin{gathered}
  \frac{\partial \beta(u)}{\partial t} -\triangle_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 $\Delta_p u =\mathop{\rm div} \big(|\nabla u|^{p-2}\nabla u\big)$,
$1<p<+ \infty$, $\beta$ is a nonlinearity of porous
medium type, and $f$ is a nonlinearity of reaction-diffusion type.
 The continuous problem  \eqref{P} has been extensively treated in \cite{ee1}
for $p > 1$, and for the case $p = 2$ in \cite{emr1}.
Here, we shall discretize  \eqref{P} and replace it by
\begin{gather*}
 \beta(U^n) - \tau \triangle_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*}
The case $p = 2$ of this equation is studied in  \cite{emr2}.  Here, we
study the case $ p > 1$ to obtain   existence, uniqueness
and stability results.
Furthermore, we obtain existence of absorbing sets and of a global attractor.
Under some conditions on  $f$ and $p$, additional regularity result
for the global attractor and, as a consequence, a stabilization result
are obtained when $\beta(u)=u$.

 This paper is organized as follows: In section 2, we give some preliminaries.
In section 3, we show the existence and uniqueness of solutions of problem
\eqref{Pn}. 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 obtaining some
regularity for the attractor.

\section{Preliminaries}

\subsection{Notations and useful lemmas}

Let $\beta$ be a continuous function with $\beta(0) = 0$. For $t\in\mathbb{R}$,
define
$$ \psi(t) =\int_0^t \beta(s)ds . $$
The Legendre transform  is defined as
$\psi^{*}(\tau) = \sup_{s\in\mathbb{R}}\{\tau s -\psi(s)\}$.
Let $\Omega$ stand for a regular open bounded set of $\mathbb{R}^{d}$,
$d\geq 1$ and $\partial\Omega$ be it's boundary.

The norm in a space $X$ will be denoted by
\begin{itemize}
\item $\|.\|_r $ if  $X = L^r( \Omega )$,  $1 \leq r \leq+\infty$;
\item $\|.\|_{1, q}$    if  $X = W^{1, q}( \Omega )$,
 $1 \leq q \leq+\infty $;
\item $ \|.\|_X $ otherwise
\end{itemize}
and $\langle.,.\rangle $ denotes  the duality between
$W_0^{1,p}(  \Omega ) $ and $ W^{-1,p'}( \Omega )$.

For $p\geq 1$ we define it's conjugate $p'$  by
$\frac{1}{p}+\frac{1}{p'} = 1$.
In this paper, $C_i$ and $C$ will denote various
positive constants.
We shall use the following results.

 \begin{lemma}[\cite{r2}]  \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}


\begin{lemma} \label{l22}
 Let $g(x, s, \xi)$ be a Caratheodory function such that
 $\mathop{\rm sign}\xi\, g(x, s, \xi) \geq -C_1$
 and
$|g(x, s, \xi)| \leq b(|s|)(|\xi|^p + c(x))$,
 where $b$ is a continuous and increasing function with (finite)
 values on  $\mathbb{R}^+$, $c \in\ L^1~(\Omega)$, $ c \geq 0 $
 and $C_1$ is  a nonnegative real.
 Also let $h \in\ W^{-1,p'}(\Omega )$.
Then the problem
 \begin{gather*}
-\Delta_p u +g(x, u, \nabla u) = h \quad \text{in } \mathcal{D}'(\Omega),\\
    u \in  W_0^{1,p}(\Omega),
   \end{gather*}
has at least one  solution.
\end{lemma}

\begin{remark} \rm
Since in  \cite{bbm},  $C_1 = 0$, a slight modification has to be
introduced in the  proof therein. Indeed, we consider
 $u_\varepsilon \in  W_0^{1,p}
(\Omega)$ such that
 $$-\Delta_p  u_\varepsilon +g_\varepsilon(x,
u_\varepsilon,\nabla u_\varepsilon) = h, $$
 where $ g_\varepsilon =g/(1+\varepsilon |g|)$.
 Thanks to  the  \emph{sign condition}, it is easy to obtain
  a $W_0^{1,p}(\Omega)$-estimate on $u_\varepsilon$.  By  extracting a subsequence,
 $u_\varepsilon$ tends to u in $W_0^{1,p}(\Omega)$ weak.
The problem will be solved whenever the convergence is proved to
be strong in $W_0^{1,p}(\Omega)$, and this follows the same lines
as in \cite{bbm} provided we replace $h$ by $h +C_1$.
 \end{remark}

\subsection{Assumptions and definition of solution}

For \eqref{P}, we consider the  Euler forward scheme
\begin{equation} \label{Pn}
\begin{gathered}
 \beta(U^n) - \tau \triangle_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}
where $N\tau = T$, $T$ a fixed positive real, and $1\leq n\leq N$.
We shall be concerned with one of the following two cases:

\noindent\textbf{case 1} \quad $u_0 \in\ L^\infty( \Omega )$, and
      we assume the following hypotheses:
\begin{itemize}
\item[(H1)] The function $\beta$ is an increasing and continuous
    from $\mathbb{R}$  to  $\mathbb{R}$, and $\beta(0)=0 $.

\item[(H2)] For $\xi \in\ \mathbb{R} $, the map   $(x,t)  \mapsto f (x, t, \xi)$
is measurable and, a.e. in  $\Omega\times\mathbb{R}^+$, $ \xi \mapsto f(x, t, \xi)$
is continuous.  Furthermore we assume that there exists
    $C_1>0$, such that for a.e. $(x, t)\in\Omega\times\mathbb{R}^+$
    $\mathop{\rm sign} \xi f(x, t, \xi) \geq-C_1$.

\item[(H3)] There is $C_2>0$, such that  for almost
    $(x, t)\in\Omega\times\mathbb{R}^+$,
$ \xi \mapsto f(x, t, \xi) +  C_2\beta(\xi) $    is
    increasing.
\end{itemize}



\noindent \textbf{Case 2} $u_0 \in\ L^2( \Omega )$, and
      we assume the following hypotheses:
\begin{itemize}
\item[(H1')] The function $\beta $ is increasing and continuous
    from $\mathbb{R}$  to  $\mathbb{R}$, $\beta(0) = 0$, and for
    some $C_3>0$, $C_4>0$,
    $\beta(\xi) \leq C_3|\xi| + C_4$  for all $\xi\in \mathbb{R}$.

\item[(H2')]  For any $\xi$ in $\mathbb{R}$, the map $(x,t)\mapsto f(x, t, \xi)$
is measurable and, a.e, in $\Omega\times\mathbb{R}^+$,  $\xi\mapsto f(x, t,\xi)$ is
continuous.  Furthermore we assume that there exist $q>sup(2,p)$ and
positives constants   $C_5  ,C_6 $ and $ C_7  $   such that
$$
\mathop{\rm sign} \xi f(x, t, \xi)\geq C_5|\xi|^{q-1}-C_6.
$$
Also assume that $|f(x, t,\xi)|\leq a(|\xi|)$ where
$a:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is increasing and
$$
\limsup_{t\rightarrow0^+}|f(x, t,\xi)|\leq C_7(|\xi|^{q-1}+1).
$$

 \item[(H3')] There is $C_2>0$ such that for almost all
    $(x, t)\in \Omega \times\mathbb{R}^+$,   $\xi \mapsto f(x, t, \xi)+C_2\beta(\xi)$
 is  increasing.
\end{itemize}

\begin{remark} \label{rmk2.4} \rm
    In the hypothesis  (H2'), the monotonicity condition on  $a$
      is not restrictive since we can replace $a$ by
    the increasing function
    $\widetilde{a}(s)=\sup_{0\leq t \leq s}a(t)$.
\end{remark}

\begin{defn} \label{dfn2.5} \rm
 By a weak solution to 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 \triangle_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{defn}

\section{Existence and uniqueness result}

\subsection*{Case 1:} $u_0\in\ L^\infty(\Omega)$.
 Assume (H1)--(H3), we derive an a priori  estimate.

\begin{lemma} \label{l31}
The function $U^n$ is in $L^\infty(\Omega)$ for  $n=0,\ldots,N$.
\end{lemma}

\begin{proof}
In this case  $U^0\in\ L^\infty(\Omega)$. To show that $U^1\in\
L^\infty(\Omega)$, we can write \eqref{Pn} as
\begin{gather*}
 - \tau \triangle_p U^1  + F_1(x, U^1)=\beta(u_0)+C_1\mathop{\rm sign}(U^1)
=\varphi _1  \\
U^1\in\ W_0^{1,p}(\Omega)\,,
\end{gather*}
 where $ F_1(x, \xi) =\tau f(x, \tau, \xi)+\beta(\xi) + C_1\mathop{\rm sign}(\xi) $,
and  $ \varphi_1 \in\ L^\infty(\Omega) $.
 According to (H1) and (H2),
 $\xi F_1(x, \xi) \geq 0$ for all $\xi \in\mathbb{R}$.
By lemma \ref{l21} we can conclude that $U^1\in\ L^\infty(\Omega) $.
By a simple  induction, we deduce  that $U^n\in\ L^\infty(\Omega)$ for all
$ n = 0, \ldots, N$.
 \end{proof}

\begin{thm} \label{th32}
For $n=1, \ldots, N $, there exists a unique solution $U^n$
of \eqref{Pn}  in $W_0^{1,p}(\Omega) \cap  L^\infty(\Omega)$ provided
that $0<\tau<\frac{1}{C_2}$.
\end{thm}

\begin{proof}
We can write \eqref{Pn} as
\begin{gather*}
 - \tau \triangle_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
$F(x, \xi) =\tau f(x, n\tau,\xi)+\beta(\xi)$.
According to (H1) and (H2),
 $$
 \mathop{\rm sign} \xi\, F(x, \xi) \geq -\tau C_1\quad \text{and}\quad
  h \in\ W^{-1,p'}(\Omega).
$$
 Hence the existence follows from lemma \ref{l22}.

Next, we obtain uniqueness. For simplicity, we set
$$ w =U^n,\quad
 \overline{f}(x, w)= f(x, n\tau, U^n),\quad
 \text{and}\quad  g(x) =\beta (U^{n-1})
$$
Then  problem \eqref{Pn} reads
\begin{equation}\label{eq31}
  \begin{gathered}
- \tau \triangle_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
 \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{multline}\label{eq32}
\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{multline}
 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_2\int_\Omega\big(\beta(w_1)-\beta(w_2)\big) (w_1-w_2)  dx.
\end{equation}
Using this equation and the monotonicity condition  of the $p$-Laplacian
 operator, (\ref{eq32}) reduces to
$$
(1-\tau C_2)\int_\Omega \big(\beta(w_1)-\beta(w_2)\big)
(w_1-w_2)\, dx \leq 0.
$$
Then by (H1), if $\tau<1/C_2$, we get $w_1=w_2$.
\end{proof}

Uniqueness can be also obtained under the following assumption:
\begin{itemize}
\item[(H3'')] For all $M>0$, there exists $C_M>0$ such
that, if $|\xi|+|\xi'|\leq M$ then
$$
|f(t, x, \xi)- f(t, x, \xi')|^\alpha \leq C_M
\big(\beta(\xi)-\beta(\xi')\big)(\xi-\xi')
$$
where
$\alpha=\begin{cases} 2 & \text{if }  1<p\leq2,\\
p'&  \text{if }  p\geq 2.
\end{cases}$
\end{itemize}

\begin{prop} \label{prop3.3}
Assume in the case 1 that   (H1), (H2) and (H3'')
hold, and that $p\geq 2d/(d+2)$.  Then the solution of \eqref{Pn}
is unique provided that $0<\tau<\tau_1$,  where $\tau_1$ is a
prescribed constant.
\end{prop}

\begin{proof}
 Let $w_1$ and $w_2$ be two solutions of (\ref{eq31}). Using the stability
result which we will establish below (see theorem \ref{th41}), we have
\begin{equation}\label{eq33}
     \|w_1\|_\infty + \|w_2\|_\infty \leq M \quad\text{and}\quad
\tau^{1/p}(\|w_1\|_{1, p} + \|w_2\|_{1, p}) \leq K,
\end{equation}
where $M$ and $K$ are positive constants which do not depend on
$N$.
Now, let us  recall the relations verified by the  $p$-Laplacian
(see \cite{gm} or \cite{s} for example).
For every $u $ and $v$ in $W_0^{1,p}(\Omega)$, we have
\begin{gather}\label{eq34}
     \langle -\Delta_p u + \Delta_p v, u-v \rangle \geq C_p \|u-v\|_{1,
     p}^p\quad\text{if } p\geq2 \\
\label{eq35}
\langle -\Delta_p u + \Delta_p v, u-v \rangle \geq C_p \frac{\|u-v\|_{1,
     p}^2}{({\|u\|_{1,p}+\|v\|_{1,p}})^{2-p}} \quad\text{if } 1< p \leq 2
\end{gather}
(i) If $p\geq 2$ , then from (\ref{eq32}), (\ref{eq34}),
(H3''), Young's  and Poincare's inequalities, we get
\begin{align*}
&\lambda_1C_p\tau \|w_1-w_2\|_p^p+ \int_\Omega \left(\beta(w_1)- \beta(w_2)\right)
(w_1-w_2) dx  \\
& \leq\frac{1}{p'C_M}\|\overline{f}(x, w_1)-\overline{f}(x, w_2)\|_{p'}^{p'}+
  \frac{\tau^pC_M^{p/p'}}{p}\|w_1-w_2\|_p^p \\
& \leq \frac{1}{p'}\int_\Omega \left(\beta(w_1)- \beta(w_2)\right)(w_1-w_2) dx +
  \frac{\tau^pC_M^ {p/p'}}{p}\|w_1-w_2\|_p^p ,
\end{align*}
where $\lambda_1$ is the first eigenvalue of $-\Delta_p$.
Then, from (H1), we obtain
$$
\Big(\lambda_1C_p\tau - \frac{\tau^pC_M^
{p/p'}}{p}\Big)\|w_1-w_2\|_p^p \leq 0.
$$
Therefore, when  $0 <\tau <\big(\frac{p \lambda_1C_p}{C_M^{p/p'}}\big)^ {1/(p-1)}$,
we get $w_1=w_2$.

\noindent (ii)  If $\frac{2d}{d+2}\leq p\leq 2$, then from (\ref{eq32}),
(\ref{eq35}), (H3'') and Young's  inequality, we obtain
\begin{align*}
&\tau^{2/p} \frac{C_p}{K^{2-p}} \|w_1-w_2\|_{1, p}^2+
   \int_\Omega \left(\beta(w_1)- \beta(w_2)\right)(w_1-w_2) dx\\
& \leq\frac{1}{2C_M}\|\overline{f}(x, w_1)-\overline{f}(x, w_2)\|_{2}^{2}+
  \frac{\tau^2 C_M }{2}\|w_1-w_2\|_2^2 \\
& \leq \frac{1}{2}\int_\Omega \left(\beta(w_1)-
\beta(w_2)\right)(w_1-w_2) dx +  \frac{\tau^2C_M}{2}\|w_1-w_2\|_2^2.
\end{align*}
Since $p\geq\frac{2d}{d+2}$, we have
$\|w_1-w_2\|_2 \leq C'_p \|w_1-w_2\|_{1, p}$.
Then, from (H1), we obtain
$$
 \tau^{2/p}\left( \frac{C_p}{K^{2-p}} - \frac{1}{2}\tau ^{2/p'}
 C_M {C'_p}^2\right) \|w_1-w_2\|_{1, p}^2 \leq 0.
$$
 Therefore, when $0< \tau < (\frac{2C_p}{C_M{C'_p}^2K^{2-p}})^ {p'/2}$,
  we obtain $w_1 =w_2$.
 \end{proof}

\subsection*{Case 2} The function $u_0$ is in $L^2(\Omega)$.


\begin{thm}\label{th33}
We assume the hypotheses (H1')--(H3') and $p\geq \frac{2d}{d+2}$, then
for each $n =1, \cdots, N$ there exists a unique solution $ U^n$
of \eqref{Pn} in $ W_0^{1,p}(\Omega) $ provided that $0<\tau<1/C_2$.
\end{thm}

The proofs of existence and  uniqueness are the same as  those of
Theorem  \ref{th32}, with $h = \beta(U^{n-1})$ in
$L^2(\Omega) \subset W^{-1,p'}(\Omega)$ for $p\geq 2d/(d+2)$. Therefore,
we omit it.

\section{stability}
\subsection*{Case 1} The function $u_0$ in $L^\infty(\Omega)$.

\begin{thm}\label{th41}
Assume (H1)--(H3). Then there exists $C(T,u_0) > 0$ depending
on $T$, $u_0$, $\beta$ , $g$ and $\Omega$, but not on $N$, such that
for all $n = 1, \cdots, N$,
 \begin{gather}\label{eq41'}
 \|U^n\|_\infty \leq C(T,u_0) , \\
\label{eq42'}\int_\Omega \psi^* (\beta(U^n))  dx
 +\tau \sum _{k=1}^n \|U^k\|_{1, p}^p  \leq C(T,u_0) , \\
\label{eq43'}\sum_{k=1}^n  \|\beta(U^k)-\beta(U^{k-1})\|_2^2 \leq  C(T,u_0) .
 \end{gather}
 \end{thm}

\begin{proof}
\textbf{(a)} From lemma \ref{l31}, $U^n \in{L^\infty(\Omega)}$.  Then,
multiplying the first equation of \eqref{Pn} by
$|\beta(U^n)|^k\beta(U^n)$, using H\"{o}lder's inequality and the
hypotheses on $f$,  we obtain
$$
\| \beta(U^n)\|_{k+2}^{k+2}
 \leq \|\beta(U^n)\|_{k+2}^{k+1}\|\beta(U^{n-1})\|_{k+2}
 +C\tau\|\beta(U^n)\|_{k+1}^{k+1}.
$$
Since $\|\beta(U^n)\|_{k+1}\leq C \|\beta(U^n)\|_{k+2}$,
it follows that
$$
|\beta(U^n)\|_{k+2}  \leq \|\beta(U^{n-1})\|_{k+2}+C\tau,
$$
 and, by induction, we deduce that
$$
\|\beta(U^n)\|_{k+2}\leq  \|\beta (u_0)\|_{k+2} +NC\tau.
$$
Finally, as $k\rightarrow \infty$ , we obtain
$\|U^n\|_\infty \leq C(T,u_0)$.
Thus (\ref{eq41'}) is satisfied.

\noindent\textbf{(b)}  Multiplying the first equation of
\eqref{Pn} (with $k$ instead of $n$) by $U^k$, and  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 obtain
\begin{equation}\label{eq40}
\int_\Omega \psi^* (\beta(U^k))   dx - \int_\Omega \psi^*( \beta(U^{k-1}) ) dx
   +\tau\|U^k\|_{1,p}^p
 \leq C_1 \tau\|U^k\|_1 .
\end{equation}
 Now, summing (\ref{eq40}) from $k=1$ to $n$, gives
 \begin{equation}\label{eq41}
\int_\Omega \psi^*( \beta(U^n)) \, dx +\tau \sum_{k=1}^n \|U^k\|_{1,p}^p
 \leq C \tau \sum_{k=1}^n \|U^k\|_1 +\int_\Omega \psi^* (\beta (u_0)  )   dx.
\end{equation}
>From (\ref{eq41}) and Lemma \ref{l31}, we deduce (\ref{eq42'}).

\noindent\textbf{(c)} Multiplying the first equation of
\eqref{Pn} (with $k$ instead of $n$) by $\beta(U^k)$ and 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_1 \tau \int_\Omega
|\beta (U^k)| dx.
\end{equation}
 With the aid of the identity
$2a(a-b) =a^2 -b^2 +(a-b)^2$,
 from (\ref{eq401}) we   obtain
 \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}
 Now 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}
Hence, by (\ref{eq42}) and Lemma \ref{l31}, we conclude (\ref{eq43'}).
\end{proof}

\subsection*{Case 2:} The function $u_0$ is in $L^2(\Omega)$.

\begin{thm}\label{th42}
We assume hypotheses (H1')--(H3') and $p\geq 2d/(d+2)$.  Then there
exists a positive constant $C(T,u_0)$ such that, for all $n=1, \cdots,
N$,
 \begin{gather}\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) \\
\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{gather}
\end{thm}

\begin{proof}
Since the proof is nearly the same as that of theorem \ref{th41},
we just sketch it.

\noindent\textbf{(a)}\quad As for (\ref{eq41}) , we obtain
$$
\int_\Omega \psi^*(\beta(U^n)) dx
 +\tau \sum_{k=1}^n \|U^k\|_{1, p}^p + C \tau \sum_{k=1}^n \|U^k\|_q^q \leq
C \tau \sum_{k=1}^n \|U^k\|_1 + \int_\Omega \psi^* (\beta (u_0)) dx.
$$
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
 $$
\varepsilon\tau \sum_{k=1}^n \|U^k\|_p^p \leq C_\varepsilon  (T, u_0).
$$
Therefore, (\ref{eq46}) is satisfied.

\noindent\textbf{(b)}\quad From (\ref{eq42}), (H1') and (H2'), we
obtain
$$
\|\beta(U^n)\|_2^2 + \sum_{k=1}^n  \|\beta(U^k)-\beta(U^{k-1})\|_2^2 \leq
 \|\beta (u_0)\|_2^2 + C \tau \sum_{k=1}^n  \|\beta(U^k)\|_1
$$
 As in (a), we conclude (\ref{eq47}).
\end{proof}

\section{The semi-discrete dynamical system}\label{sect5}

In the remainder sections, we assume $u_0 \in L^2(\Omega)$ and the
hypotheses (H1')--(H3'), we fix $\tau$   such that
$0<\tau< \min(1,\frac{1}{C_2})$, and   assume that $p>2d/(d+2)$.
Theorem \ref{th33}  allows us to define a map $S_\tau$ on $ L^2(\Omega)$
by setting
$$ S_\tau   U^{n-1}=U^n. $$
Since $S_\tau$ is continuous, we have $S^n_\tau   U^0=U^n$.

Our aim is to study the discrete dynamical system associated with
\eqref{Pn}. 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 $p>2d/(d+1)$, 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{gather}\label{eq51'}
U^n \in L^\infty(\Omega) \quad \text{for all }n \geq n(d, p), \\
\label{eq52'}
\|U^{n(d, p)}\|_\infty \leq\frac{C}{\tau
^{\alpha+\alpha^2+\cdots+\alpha^{n(d,p)}}}\big(\|u_0\|_2^{\alpha^{n(d, p)}}
+1 \big),
\end{gather}
where $\alpha =p'/p$.  Moreover, if $d=1, d=2$  or
 $d<2p$ then $n(d, p) =1$.
\end{lemma}

\begin{proof}
The proof follows from a repeated application of lemma \ref{l21}.
We can write \eqref{Pn} as
\begin{gather*}
   - \tau \triangle_p(U^m) +   F_m(x, U^m)
  =\beta(U^{m-1})+C_6 sign(U^m) =T_m \quad  \mbox{in } \Omega,\\
 U^m = 0 \quad \mbox{on } \partial\Omega,
 \end{gather*}
where  $F_m(x, \xi)=\tau f(x, m\tau, \xi)+\beta(\xi)+C_6sign(\xi)$.
Note that by (H1') and (H2') we have
$\xi F_m(x, \xi) \geq 0  \text{ for all } \xi$
and   $T_m \in W^{-1,p'}(\Omega)$.

 Now, applying lemma \ref{l21}, we can find an increasing
 sequence $\big(\alpha(m)\big)_{m\geq1}$
 such that
\begin{gather}\label{eq55'}
\alpha(m)\geq p', \quad
\frac{1}{\alpha (m+1)}=\frac{1}{(p-1)\alpha (m)} -\frac{1}{d} , \\
\label{eq56'}
\|U^m\|_{\alpha(m)} \leq \frac{C_m}{\tau^{\alpha+\alpha^2+\cdots
+\alpha^m}}\big(\|u_0\|_2^{\alpha^ m}+1 \big)\,.
 \end{gather}
We shall stop the iteration on $m$ once we have $\alpha(m-1) >d/p$.
Indeed, if $ q > d/p$, then there exists
$r> d/(p-1)$ 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 $\alpha(m-1) > d/p$.
Then (\ref{eq52'}) follows  from  (\ref{eq56'}) and lemma \ref{l21}.
\end{proof}

\begin{remark} \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  $n(d, p)=1$.

\noindent(ii) If   $\alpha (m) \leq \frac{d}{p}$ for all $m$, then
$ l=\lim _{m\rightarrow\infty} \alpha(m)$ exists  and equals
$\frac{2-p}{p-1}d$.
 Consequently, for $p>2d/(d+1)$, we have  $l<p'$ ,
which contradicts the fact that  $\alpha (m)\geq p'$.
Hence, the existence of $n(d, p)$ is justified.
\end{remark}

In the remaining of this article, we set $n_0 =n(d, p)$ and
$ C_1 =C\big(\|u_0\|_2^{\alpha^{n_0}}+1 \big)$.

\begin{lemma} \label{l52}
Let $k$ be such that $1<k< q-1$ and $k\leq 1+\frac{1}{n_0}$.
Then, there exist $\gamma>0 , \delta >0$ 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
$$
\|\beta(U^n)\|_\infty \leq \big( \frac{\delta}{\gamma}\big)^{1/(q-1)}
+ \frac{C_1 +\mu}{\big( \tau^\beta (n-n_0+1)\big) ^{1/(k-1)}},
$$
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
 $$
U^n \in L^\infty(\Omega)\text{ and } \|U^{n_0}\|_\infty
\leq C_1/\tau^{\alpha+\alpha^2+\cdots+\alpha^{n_0}}.
$$
Multiplying the first equation of \eqref{Pn} by $|\beta(U^n)|^m \beta(U^n)$
for some positive integer $m$,
we derive from (H1') and (H2'), 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
$z_{n_0} \leq C_1/\tau^{\alpha+ \alpha^2+\cdots +\alpha^{n_0}}$.

\noindent (i) If $\alpha \leq 1$, then
$\alpha+ \alpha^2+\cdots +\alpha^{n_0} \leq n_0$.  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 \cite[Lemma 7.1]{emr2} to obtain
$$
 z_n \leq {\big(\frac{\delta}{\gamma}\big )}^{1/(q-1)}+
\frac{C_1 +\mu}{{\big( \tau   (n-n_0+1)\big)}
^{\frac{1}{k-1}}}\equiv c_\alpha(n).
$$
  (ii) If $\alpha \geq 1$, then
$\alpha+ \alpha^2+\cdots + \alpha^{n_0} \leq n_0\alpha^{n_0}$.
 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  \cite[lemma 7.1]{emr2} to obtain
$$
 z_n \leq {\big(\frac{\delta}{\gamma}\big )}^{1/(q-1)}+
\frac{C_1 +\mu}{{\big( \tau _1 (n-n_0+1)\big)}
^{\frac{1}{k-1}}}\equiv c_\alpha(n).
$$
\end{proof}

\begin{remark} \rm
In the case $\alpha \geq 1$, a slight modification has to be
introduced in the proof of  \cite[lemma 7.1]{emr2}, since  $\mu$
depends on $\delta'$ and $\gamma'$ and hence on $\tau$.  In
fact, with the same notation, it suffices to choose $\mu$ such that
$$
\gamma\left(\frac{\delta}{\gamma}\right)^{1-\frac{k}{q-1}}\mu^{k-1}
\geq
  2^\frac{1}{k-1}/ (k-1).$$
  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
$$
 \|U^n\|_\infty \leq \max  \big(\beta ^{-1}(c_\alpha (n)),
 |\beta ^{-1}(-c_\alpha(n))|\big),
 $$
 for all $n\geq n_0$, with
   $c_\alpha(n) \rightarrow
 {\big(\frac{\delta}{\gamma}\big )}^{1/(q-1)}$  as  $n\rightarrow\infty$.

\subsection{Absorbing sets in $W_0^{1,p}(\Omega )$, existence of the the global
attractor}

 Multiplying equation \eqref{Pn} by $\delta_n=U^n-U^{n-1}$, we obtain
\begin{multline}\label{eq51}
\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{multline}
By setting
$$
F_\beta(u) = \int_0^u \big(f(x, n\tau, w) +C_2 \beta(w)\big)dw,
$$
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_2\beta(U^n), \delta_n \rangle -C_2\langle
\beta(U^n), \delta_n \rangle\\
&\geq  \int_\Omega \left( F_\beta(U^n)-F_\beta(U^{n-1})\right)dx
-C_2\langle \beta(U^n), \delta_n \rangle.
\end{align*}
Now, using (H1'), we get $\psi'(v)(u-v) \leq \psi(u)-\psi(v)$.
Therefore,
\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 inequality
\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 $\tau < 1/C_2$, 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_2
\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
$$
 \int_\Omega F_\beta(u)dx= \int_\Omega F (u)dx +C_2 \int_\Omega \psi(u)dx.$$
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
$$
 \tau \sum_{n=n_0}^{n_0+N}y^n \leq a_1,  \quad \text{for all } n \geq n_\tau.
$$
Then we apply the discrete version of
\emph{the uniform Gronwall lemma} \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{prop}\label{l53}
If $\tau <1/C_2$, there exist absorbing sets   in $L^\infty(\Omega) \cap W_0^{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{prop}

 For the nonlinear map $S_\tau$ to satisfy the properties
of the semi-group, namely $S_\tau^{n+p}=S_\tau^noS_\tau^p$, we
need \eqref{Pn} 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
$L^\infty(\Omega)\cap W_0^{1,p}(\Omega)$. Setting
$$
\mathcal{A}_\tau=\bigcap_{n\in\mathbb{N}}
\overline{\cup_{m\geq n}S_\tau^m(B)},
$$
$\mathcal{A}_\tau$ is a compact subset of $L^2(\Omega)$ which attracts
all solutions.  That means 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{thm}\label{th51}
Assuming that $u_0\in L^2(\Omega)$ and (H1')--(H3'), the discrete
problem \eqref{Pn} has an associated solution semi-group $S_\tau$
that maps $L^2(\Omega)$ into $L^\infty(\Omega)\cap
W_0^{1,p}(\Omega)$. This semi-group has a compact attractor which
is bounded in $L^\infty(\Omega) \cap W_0^{1,p}(\Omega)$.
\end{thm}

\section{Additional regularity for the attractor}

In this section, we shall show supplementary regularity estimates
on the solutions of problem \eqref{Pn} 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{Pn} becomes
\begin{equation} \label{6n}
\delta_n -\triangle_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 main result of this section.

 \begin{lemma}\label{l61}
 There exists a positive constant $C$ 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, if  $n_0\geq n_\tau$,  $U^{n_0}$ is in an
$W_0^{1,p}(\Omega)$-absorbing ball, and  Choosing $N\tau  =1$, we
therefore obtain (\ref{601}) from (\ref{603}).
\end{proof}

\begin{thm}\label{t61}
 For all $ n \geq n_\tau$, we have
$\|\delta_n\|_2  \leq C$,
where $C$ is a positive constant.
\end{thm}

\begin{proof}
By subtracting equation  \eqref{6n} with $n-1$ instead of $n$,
from equation  \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 $y_n =  \frac{1}{2}\|\delta_n\|_2^2$ and
$h_n= C\|\delta_n\|_2^2$,
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{cor}\label{c61}
If $p>2d/(d+2)$ 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
$\|U^n\|_{W^{2, 2}(\Omega)} \leq C$ for all $n \geq n_\tau$.
\end{cor}

\begin{proof}
(i) If $2d/(d+2)< p<2$ 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 \eqref{6n}, (\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
$\|-\Delta_p U^n\|_{p'} \leq C$ 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}
\section*{Acknowledgement}
The authors would like to thank the referee for his interest on
this work and his helpful remarks.

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