\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. 117--128.\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}{117}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Thermistor problem: A nonlocal parabolic problem]
{Thermistor problem: A nonlocal parabolic problem}

\author[A. El Hachimi,  M. R. Sidi Ammi\hfil EJDE/Conf/11 \hfilneg]
{Abderrahmane El Hachimi,  Moulay Rchid Sidi Ammi}  % in alphabetical order


\address{Abderrahmane El Hachimi \hfill\break
 UFR Math\'ematiques Appliqu\'ees et Industrielles\\
 Facult\'{e} des Sciences \\
 B.P. 20, El Jadida - Maroc}
\email{elhachimi@ucd.ac.ma}

\address{Moulay Rchid Sidi Ammi \hfill\break
 UFR Math\'ematiques Appliqu\'ees et Industrielles\\
 Facult\'{e} des Sciences \\
 B. P. 20, El Jadida - Maroc}
\email{rachidsidiammi@yahoo.fr}

\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{35K15, 35K60, 35J60} 
\keywords{Semi-discretization;
thermistor; a nonlocal; existence;  attractor; \hfill\break\indent
 discrete dynamical system}


\begin{abstract}
 In this paper, we study a nonlocal parabolic problem arising in
 Ohmic heating. Firstly, some existence and uniqueness results
 for the continuous problem are proposed. secondly, a time
 discretization technique by Euler forward scheme is proposed and
 a study of the discrete associated dynamical system is presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

In this work, we shall deal with the following nonlocal parabolic
problem
\begin{equation} \label{11}
\begin{gathered}
  \frac{\partial u}{\partial t}-\triangle u = \lambda
  \frac{f(u)}{( \int_{\Omega} f(u)\, dx )^2} , \quad
   \mbox{in } \Omega \times ]0;T[, \\
   u = 0 \quad \mbox{on }  \partial \Omega \times ]0;T[,  \\
   u(0)= u_0  \quad \mbox{in }  \Omega,
  \end{gathered}
\end{equation}
where $\Omega \subset \mathbb{R}^{d}$ $(d\geq 2)$ is a bounded
regular domain,
  $\lambda$ is a positive parameter and $f$ is a function with
  prescribed conditions.
 Let us recall first that \eqref{11} arises by reducing the
following system of two equations which model a thermistor problem
\begin{equation} \label{12}
\begin{gathered}
     u_{t}= \nabla .(k(u)\nabla u) + \sigma (u)|\nabla \varphi |^2, \\
     \nabla ( \sigma (u) \nabla \varphi )= 0,
\end{gathered}
\end{equation}
where, $u$ represents the temperature generated by the electric current
flowing through a conductor,
$\varphi$ the electric potential, $\sigma (u)$ and $k(u)$ are respectively
the electric and thermal conductivities. For more information, we refer
the reader to \cite{es1, lac1, lac2, tza}.\\
In section 2, our gaol concerns the existence and uniqueness of
weak solutions to \eqref{11}. Some results
   have been obtained by many authors in the case where $N=1$
  and $f$ taking particular forms: Montesinos and Gallego \cite{mg} proved
  the existence of weak solution under
   \begin{equation} \label{13}
   0< \sigma _{1}\leq \sigma (s) \leq \sigma _{2}, \forall s\in \mathbb{R}.
   \end{equation}
  In  \cite{lac1, lac2, tza}, major emphasis is placed on
    cases where the spatial dimension $N$ is $1$ or $2$ and $f$ is of the form
     $f(u)= \exp (u) or \exp (-u)$. In these works, additional regularity
      assumptions are made on $u_{0}$ and a combination of  usual Lyapounov
      functional and a comparison method is the main ingredient. Our purpose
       is to extend some of the results therein to problem \eqref{11},
        where here, the condition \eqref{13} is weakened to (H2) below.

We recall also that the Euler forward method has been used by several authors
in the semi-discretization of non linear parabolic problems, see for example
\cite{ra,eb}. Concerning the existence and uniqueness of solutions to
\eqref{11} under particular forms of $f$, we refer the reader to \cite{bl}
and the references therein.
On the other hand, little is known about the solutions to
the following discrete problem:
\begin{equation} \label{13a}
  \begin{gathered}
  U^{n}-\tau \triangle U^{n}= U^{n-1}+  \lambda \tau
  \frac{f(U^{n})}{ \big ( \int_{\Omega} f(U^{n})\, dx \big )^2} ,
   \mbox{ in } \Omega , \\
   U^{n} = 0 \quad \mbox{on }  \partial \Omega,  \\
   U^{0}= u_0  \quad \mbox{in }  \Omega.
\end{gathered}
\end{equation}
Whereas, semi-discretization has been used  for  equations of the
thermistor problem in \cite{psx,al}. Our aim here is to continue
the study of problem \eqref{11} initiated in section 2, where an a
priori
  $L^{\infty}-$estimate is derived. In addition to the usual
existence and uniqueness questions concerning the solutions of
\eqref{13}, we shall prove some results of stability and proceed
to error estimates analysis. In \cite{al}, the authors derived an
$L^2$ and $H^{1}$ norm error by requiring regularity on the
solution $u$, for instance $u, u_{t}$ in $H^2(\Omega)\cap
W^{1,\infty}(\Omega)$.
 Unfortunately, such smoothness is not always possible since the
function $f$ is non linear. We end this paper by studying the asymptotic
behaviour of the solutions to the discrete dynamical system associated with
\eqref{13}.

\section{Existence and uniqueness for the continuous problem} %2

We assume the following hypotheses:
\begin{itemize}
\item[(H1)] $f: \mathbb{R} \rightarrow \mathbb{R} $ is a locally
Lipschitzian function.
  \item[(H2)] There exist positive constants $\sigma ,c_{1}, c_{2}$ and
$\alpha$ such
  that $\alpha < \frac{4}{d-2}$ and for all $\xi \in \mathbb{R}$
  $$
  \sigma \leq f(\xi )\leq c_{1}| \xi|^{\alpha +1}+c_{2}.
  $$
  \end{itemize}
  We adopt the following weak formulation for \eqref{11}: $u$ is a solution
  of \eqref{11} if and only if
  \begin{gather*}
 u \in L^{\infty}(\tau ,+\infty ,H_{0}^{1}(\Omega)\cap L^{\infty}(\Omega))
\mbox{ with} \frac{\partial u}{\partial t}\in L^{2}(\tau ,+\infty
,L^{2}(\Omega) ) \\ \mbox{ for any } \tau >0,  \mbox{ and satisfying }\\
 \int_{0}^{T}\int_{\Omega} u \frac{\partial }{\partial t}\phi -\nabla u
\nabla \phi \, dx\,dt=
 \int_{0}^{T}(  \frac{\lambda}{ \big ( \int_{\Omega} f(u)\, dx \big )^{2}}
  \int_{\Omega} f(u) \phi dx)dt, \\ \mbox{for any } \phi \in
  C^{\infty}((0,\infty),\Omega ).
\end{gather*}
 Now, we state our main result.

  \begin{theorem}
   Let hypotheses (H1)-(H2) be satisfied. Assume that
   $u_{0}\in L^{k_{0}+2}(\Omega)$ with $ k_{0}$ such that
  \begin{equation} \label{21}
     k_{0} \geq \max \big ( 0,\frac{\alpha N}{2}-2 \big ).
   \end{equation}
   Then, there exists $d_{0}>0$ such that if $\| u_{0}\|_{k_{0}+2}<d_{0},$
   the problem \eqref{11} admits a solution $u$ verifying for all $\tau >0$
   \begin{gather*}
   u \in L^{\infty}(\tau ,+\infty ,L^{k_{0}+2}(\Omega)),  \quad
   |u|^{\gamma}u \in L^{\infty}(\tau ,+\infty ,H_{0}^{1}(\Omega)),
    \mbox{ with } \gamma = \frac{k_{0}}{2}.
    \end{gather*}
    Moreover, if $u_{0}\in L^{\infty}(\Omega ),$ then
    $u \in L^{\infty}(\tau ,+\infty ,L^{\infty }(\Omega))$ and is unique.
   \end{theorem}

   \noindent\textbf{Remark.}\quad The value of $d_{0}$ will be given in
   the course of the proof. 

\begin{proof}
    We use a Faedo-Galerkin method see \cite{jll}. Let $u_{m}\subseteq
    D(\Omega)$ be such that $u_{0m}\rightarrow u_{0}$ in $H_{0}^{1}(\Omega)$
     and let $(w_{j})_{j}\subseteq H_{0}^{1}(\Omega)$ a special basis. We seek
     $u$ to be the limit of a sequence $(u_{m})_{m}$ such that
$$  u_{m}(t)= \Sigma _{j=1}^{m}g_{jm}(t)w_{j},      
$$
 where $g_{jm}$ is the solution of the following ordinary differential
     system
\begin{equation} \label{22a}
 \begin{gathered}
 \langle u_{m}',w_{j}\rangle +(u_{m},w_{j})= \frac{\lambda }
 { \big ( \int_{\Omega} f(u_{m})\, dx \big )^{2} } \,
 \langle f(u_{m}),w_{j} \rangle  , \, 1\leq j \leq m , \\
 u_{m}(0)=u_{om}.
 \end{gathered}
 \end{equation}
 It is easy to see that \eqref{22a} has a unique solution $u_{m}$ according to
 hypotheses (H1)--(H2) and Cartan's existence theorem concerning
 ordinary differential equation (see \cite{car}). This solution is shown to
 exist on a maximal interval $[0;t_{m}[$. The following estimates  enable
 us to assert that it can be continued on the hole interval $[0;T]$. We shall
 denote by $C_{i}$ different positive constants, depending on data, but not on
 $m$.
\end{proof}

  \begin{lemma}
For any $ \tau >0$, there exists a constant $c_{3}(\tau ), c_{4}(\tau )$
  such that
  \begin{gather}\label{23a}
  \| u_{m}(t)\|_{k_{0}+2} \leq  c_{3}(\tau ), \forall t\geq \tau , \\
\label{24a}
  \| u_{m}(t)\|_{\infty } \leq  c_{4}(\tau ), \forall t\geq \tau .
 \end{gather}
 \end{lemma}
 
\begin{proof}
 (i) Multiplying the first equation of \eqref{22} by $|u_{m}|^{k} g_{jm},$
integrating on $\Omega $, adding from $j=1$ to $m$ and using
(H1)-(H2), yields
 \begin{equation}\label{25}
    \frac{1}{k+2}\frac{d}{dt}\| u_{m}\|_{k+2}^{k+2} +\frac{4}{(k+2)^{2}}
     \| \nabla |u_{m}|^{\frac{k}{2}} u_{m}\|_{2}^{2} \leq
     c_{5} \| u_{m}\|_{k+\alpha +2}^{k+\alpha +2}+ c_{6}.
 \end{equation}
 By using well-known Sobolev's and Gagliardo-Nirenberg's inequalities, we have
 \begin{equation}\label{26}
     \| u_{m}\|_{k_{0}+\alpha +2}^{k_{0}+\alpha +2}\leq
     c_{7} \| u_{m}\|_{k_{0}+2}^{\alpha }
     \| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2},
 \end{equation}
 Thus, from \eqref{25} and \eqref{26}, we obtain
 \begin{equation}\label{27}
    \frac{1}{k_{0}+2}\frac{d}{dt}\| u_{m}\|_{k_{0}+2}^{k_{0}+2} \leq
     ( c_{8} \| u_{m}\|_{k_{0}+2}^{\alpha }-
     \frac{4}{(k_{0}+2)^{2}})\| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2}
     + c_{6}.
     \end{equation}
  We shall make the following compatibility condition on $u_{0}$
      \begin{equation} \label{28}
        \| u_{0}\|_{k_{0}+2} <
         \Big( \frac{4}{c_{8}(k_{0}+2)^{2}} \Big )^{1/\alpha}=d_{0}.
 \end{equation}
    Then, there exists a small $\tau >0$ such that
   \begin{equation} \label{29}
   \| u_{m}(t)\|_{k_{0}+2} <   d_{0} \mbox{ for } t\in ]0,\tau [.
    \end{equation}
  Hence
    \begin{equation} \label{210}
  \frac{1}{k_{0}+2}\frac{d}{dt}\| u_{m}\|_{k_{0}+2}^{k_{0}+2}
         + c_{9}\| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2}
          \leq c_{6}\quad  \forall  \quad 0<t<\tau  .
    \end{equation}
    By Poincar\'e's inequality and after integrating, it follows that
    $$ 
\| u_{m}(t)\|_{k_{0}+2}\leq c_{10}, \quad  \forall  \quad 0<t<\tau , 
$$
    Therefore, relation \eqref{23} is achieved by iterating successively the
    same process with initial condition calculated at the last one. \\
 (ii)  By using H\^older's inequality, we get
   \begin{equation} \label{211}
           \| u_{m}\|_{k+\alpha +2}^{k+\alpha +2}\leq c_{11}
            \| u_{m}\|_{k+2}^{\theta _{1}}   \| u_{m}\|_{k_{0}+2}^{\theta _{2}}
             \| u_{m}\|_{q}^{\theta _{3}} ,
      \end{equation}
 with $\theta _{1}, \theta _{2}$ and $\theta _{3}$ satisfying
 $$
  \frac{\theta _{1}}{k+2}+ \frac{\theta _{2}}{k_{0}+2}+
  \frac{\theta _{3}}{q} =1 \quad \mbox{ and } \quad
   \theta _{1}+\theta _{2}+\theta _{3}=k+\alpha +2.
 $$
  We require moreover
   $$
  \frac{\theta _{1}}{k+2}+ \frac{\theta _{3}}{2(\gamma +1)}=1 .
  $$
   Using the boundedness of $\| u_{m}\|_{k_{0}+2}$, the choice of $q$,
   Sobolev's inequality and young's inequality, we have from \eqref{211} that
    \begin{align*}
  c_{5} \| u_{m}\|_{k+\alpha +2}^{k+\alpha +2}
  &\leq c_{12}
      \| u_{m}\|_{k+2}^{\theta _{1}} \| \nabla |u_{m}|^{\gamma}u_{m}\|
      _{2}^{\frac{\theta _{3}}{\gamma +1}} \\
  &\leq c_{13}(k+2)^{\theta _{4}} \| u_{m}\|_{k+2}^{k+2}
     + \frac{2}{(k+2)^{2}}
     \| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2} ,
 \end{align*}
      where $\theta _{4}$ is some positive constant.
      Hence  \eqref{25} becomes
      $$
        \frac{1}{k+2}\frac{d}{dt}\|u_{m}\|_{k+2}^{k+2} +\frac{c_{14}}{(k+2)^{2}}
        \| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2} \leq
         c_{15}(k+2)^{\theta _{4}} \| u_{m}\|_{k+2}^{k+2} +c_{5}.
      $$
      Therefore, by applying   \cite[lemma 4]{jf} we conclude to \eqref{24}.

       \noindent\textbf{Passage to the limit in \eqref{22} as
       $m \rightarrow \infty$.} \quad
       Multiplying the jth equation of system \eqref{22} by $g_{jm}(t)$,
       adding these equations for $j=1,\dots, m$ and integrating with respect to
       the time variable, we deduce the existence of a subsequence of $u_{m}$
       such that
       \begin{gather*}
      u_{m} \to u  \quad\mbox{ weak star in } L^{\infty }(0,T;L^{2}(\Omega)), \\
       u_{m} \to u  \quad\mbox{ weak in } L^{2 }(0,T;H^{1}_{0}(\Omega)), \\
        u_{mt} \to u_{t} \quad \mbox{ weak in } L^{2 }(0,T;H^{-1}(\Omega)), \\
         u_{m} \to u  \quad\mbox{ strongly in } L^{2 }(0,T;L^{2}(\Omega))
          \\mbox{ and a.e in } Q_{T}.
        \end{gather*}
   Straightforward standard compactness arguments allow us to assert that
        $u$ is a solution of problem \eqref{11}   
	
 \noindent\textbf{Uniqueness.} \quad
       Consider $u_{1}$ and $u_{2}$ two weak solutions of the problem
        \eqref{11} and define $w=u_{1}-u_{2}$. Substracting the equations
        verified by $ u_{1}$ and $u_{2}$, we obtain 
 \begin{equation} \label{212}
  \begin{aligned}
 \frac{dw}{dt}-\triangle w &=
         \frac{\lambda }{ \big ( \int_{\Omega}f(u_{1})\, dx \big )^{2}}
	 \Big( f(u_{1})-f(u_{2}) \Big)\\
 &\quad+\lambda \frac{\Big( \int_{\Omega } f(u_{2})-f(u_{1})\,dx \Big)
   \Big( \int_{\Omega }f(u_{2})+f(u_{1})\,dx \Big)}
  {\big ( \int_{\Omega} f(u_{1})\, dx \big )^{2}
  \big ( \int_{\Omega} f(u_{2})\, dx \big )^{2}} f(u_{2}).
  \end{aligned}
\end{equation}
  Taking the inner product of  \eqref{212} by $w$ and using (H1) and
 \eqref{24a}, we get
  $$
  \frac{1}{2}\frac{d}{dt}\| w(t)\|^{2}_{2}\leq c_{16} \| w(t)\|^{2}_{2},
  $$
  which implies that $w=0$. Hence the solution is unique.  
\end{proof} 

\section{The semi-discrete problem}

\subsection*{Existence and uniqueness} 

We consider the Euler scheme \eqref{13}, with $N\tau = T$, $T>0$
fixed and $1\leq n \leq N$. In the sequel, $(\cdot,\cdot)$ will
denote the associated inner product in $L^2(\Omega)$ or the
duality product between $H_{0}^{1}(\Omega)$ and its dual
$H^{-1}(\Omega)$.

\begin{theorem}
Assume (H1)-(H2). Then, for each $n$, there exists a
unique solution $U^{n}$ of \eqref{13} in
$H_{0}^{1}(\Omega) \cap L^{\infty}(\Omega)$
provided that $\tau$ is small enough.
  \end{theorem}

\begin{proof}
  For simplicity, we write $U=U^{n}$, $h(x)=U^{n-1}$. Then \eqref{13} becomes
\begin{equation} \label{21a}
\begin{gathered}
  U-\tau \triangle U= h(x) +  \lambda
  \frac{f(U)}{( \int_{\Omega} f(U)\, dx)^2} ,
   \quad \mbox{in } \Omega , \\
   U  = 0 \quad \mbox{on }  \partial \Omega,
\end{gathered}
\end{equation}
\textbf{Existence.}
Define the map $S(\mu , .)$ by $U=S(\mu ,v), \mu \in [0, 1]$ if
\begin{equation} \label{22}
  \begin{gathered}
  U - \tau \triangle U= \mu g(x, v) \quad\mbox{in } \Omega , \\
   U  = 0 \quad \mbox{on }  \partial \Omega , \\
   U^{0}=\mu u_{0},
\end{gathered}
\end{equation}
where $g(x, v)= h(x)+ \lambda f(v)/\big ( \int_{\Omega} f(v)\, dx
\big )^2$.

  For a fixed $v\in H_{0}^{1}(\Omega)$, \eqref{22} has a unique solution $U
\in H_{0}^{1}(\Omega)  $. Then, for each $\mu \in [0, 1]$, the operator
$S(\mu , .)$ is well defined.
  Moreover, $S(\mu , .)$ is compact from $H_{0}^{1}(\Omega)$ into it self.
  Indeed, using (H2), we have the  estimate
  $$
   |U|_{2}^2+\tau  |\nabla U|_{2}^2\leq c_{17}.
  $$
  We can easily see that $\mu \to S(\mu , v)$ is continuous and that
  $S(0 , v)=U$, for any $v$, if and only if $U=0$. From the
  Leray-Schauder fixed point
  theorem, there exists therefore a fixed point $U$ of $S(\mu , .)$.
\end{proof}

  Now, we derive an a priori estimate.

  \begin{lemma} \label{lm22}
  If $u_{0} \in L^{\infty}(\Omega)$, then for all $n \in \{ 1,\dots,N \}$,
$U^{n}  \in L^{\infty}(\Omega)$.
  \end{lemma}

The proof of the above lemma is similar to the one used by de
Thelin in \cite{th} in a different problem; we shall give here
only a sketch.
  Suppose $d\geq 2$ and define
   $$
\delta =\begin{cases} \frac{2d}{d-2} & \mbox{if } 2<d , \\
2(\alpha +2) & \mbox{if } d=2.
\end{cases}
$$
Let $q_{1}=\delta $ and let
$$
q_{k}=\{ (\frac{\delta}{2})^{k-1}(\delta -\gamma )-(2-\gamma)\}
\frac{\delta}{\delta -2}, \quad k\geq 2\,.
$$
Then we have
$$
q_{k+1}=(q_{k}+2-\gamma)\frac{\delta}{2} \mbox{ with } \gamma =\alpha +2,
  \quad \mbox{ for all } k  \in \mathbb{N^{*}}.
$$

\begin{lemma} \label{lm23}
For $k$ in $\mathbb{N}^{*}$, $U^{n} \in L^{q_{k}}(\Omega)$ and
\begin{equation} \label{23}
| U^{n}|_{\infty}=\limsup  | U^{n}|_{q_{k}}<+ \infty.
\end{equation}
  \end{lemma}

\begin{proof}  We prove by recurrence that $U \in L^{q_{k}}$.
  This property is true for $k=1$, since
  $H_{0}^{1}(\Omega)\subset L^{\delta}(\Omega)$.
Now we show that $U \in L^{q_{k+1}}$. Let $m \in \mathbb{N}$,
$1\leq m\leq k$.
  Multiplying \eqref{21} by $|U|^{q_{m}-\gamma}U$, using (H2) and
Young's inequality, we get
$$
  (q_{m}-\gamma +1)\int_{\Omega}|\nabla U|^2 |U|^{q_{m}-\gamma}\, dx\leq
  c_{18}|U|^{q_{m}}_{q_{m}} +  c_{19}.
$$
  On the other hand,
  $$
  |U|^{q_{m}+2-\gamma}_{q_{m+1}}\leq c_{20}(1+\frac{q_{m}-\gamma}{2})^2
  \int_{\Omega}|\nabla U|^2 |U|^{q_{m}-\gamma}\, dx .
  $$
  Therefore,
  $$
    |U|^{q_{m}+2-\gamma}_{q_{m+1}}\leq (  c_{21}+ c_{22}|U|^{q_{m}}_{q_{m}} )
    (q_{m}+2-\gamma).
    $$
Thus,
$$
(|U|^{q_{k+1}}_{q_{k+1}})^{2/\delta}\leq (  c_{21}+
c_{22}|U|^{q_{k}}_{q_{k}} ) (q_{k}+2-\gamma).
$$
The rest of the proof follows the same lines as in \cite[p. 383-384]{th}.
\end{proof}

\noindent\textbf{Uniqueness.}\;
    Consider $U$ and $V$ two different solutions of \eqref{21} and define
    $w=U-V$. Then, we have
\begin{equation} \label{24}
\begin{aligned}
w- \tau \triangle w
&= \frac{\lambda \tau }{( \int_{\Omega}f(U)\, dx )^2} \big( f(U)-f(V) \big)\\
&\quad +\lambda \tau \frac{\big( \int_{\Omega } f(U)-f(V)\,dx \big)
   \big( \int_{\Omega }f(V)+f(U)\,dx \big)}
  {( \int_{\Omega} f(U)\, dx )^2
  ( \int_{\Omega} f(V)\, dx )^2} f(V).
\end{aligned}
\end{equation}
  Multiplying \eqref{24} by $w$, integrating on $\Omega$ and using the
  $L^{\infty}-$estimate obtained in lemma \ref{lm22}, we obtain
  $$
  |w|_{2}^2 +\tau  |\nabla w|_{2}^2 \leq c_{30}\tau  |w|_{2}^2 .$$
  Therefore, $w=0$ when $\tau \leq 1/c_{30}$.


\section{Stability}

  \begin{theorem}
   Assume  (H1)-(H2). Then, there exists $c(T, u_{0})>0$ depending on
   the data but not on $N$ such that for any $n\in \{1, \dots, N\}$
   \begin{gather*}
   |U^{n}|_{L^{\infty}(\Omega)} \leq c(T,u_{0}),  \label{ea}\\
   |U^{n}|_{2}^2+\tau \sum _{k=1}^{n} |\nabla U^{k}|_{2}^2
\leq c(T,u_{0}), \label{eb}\\
   \sum _{k=1}^{n}|U^{k}-U^{k-1}|_{2}^2 \leq c(T,u_{0}). \label{ec}
   \end{gather*}
  \end{theorem}

\begin{proof}
   (i) Multiplying \eqref{13} by $|U^{k}|^{m}U^{k}$ for some integer $m\geq 1$,
using lemma \ref{22} and H\^older's inequality, we obtain after
   simplification
    \begin{equation}\label{31}
        |U^{k}|_{m+2} \leq |U^{k-1}|_{m+2}+c_{31}\tau .
    \end{equation}
By induction and taking the limit in the resulting inequality as $m \to
    +\infty$, we get
    $$
     |U^{k}|_{L^{\infty}(\Omega)} \leq c(T,u_{0}).
$$
(ii) Multiplying the first equation of \eqref{13} by  $U^{k}$ and using
     the hypotheses on $f$, one easily has
$$
     (U^{k}-U^{k-1}, U^{k})+ \tau |\nabla U^{k}|_{2}^2 \leq c_{32}\, \tau
     |U^{k}|_{1}.
$$
Using the elementary identity $2a(a-b)=a^2-b^2+(a-b)^2$ and
summing  from $k=1$ to $n$, we obtain
$$
|U^{n}|_{2}^2+ \sum _{k=1}^{n}|U^{k}-U^{k-1}|_{2}^2+
 \tau \sum _{k=1}^{n} |\nabla U^{k}|_{2}^2 \leq |u_{0}|_{2}^2
 + \tau \, c_{33} \sum _{k=1}^{n}|U^{k}|_{1}.
$$
Then, the inequalities(b) and (c) of the lemma hold by using
relation \eqref{23} and (a).
\end{proof}

 \section{Error estimates for solutions}

 We shall adopt the following notation concerning the time discretization
for problem \eqref{11}. Let us denote the time step by $\tau =\frac{T}{N}$,
$t^{n}=n\tau$ and $I_{n}=( t^{n},  t^{n-1})$ for $n=1,\dots, N$. If $z$ is
a  continuous function (respectively summable), defined in $(0, T)$ with
    values in $H^{-1}(\Omega)$ or $L^2(\Omega)$ or $H^{1}_{0}(\Omega)$, we
define $z^{n}= z(t^{n}, .)$,
$\overline{z}^{n}=\frac{1}{\tau}\int_{I_{n}}z(t,.)dt$,
$\overline{z}^{0}=z^{0}=z(0, .)$; the error $e_{n}=u(t)-U^{n}$ for all
$t \in I_{n}$ and the local errors $e_{u}^{n}$ and $e^{n}$ defined by
     $e_{u}^{n}=\overline{u}^{n}(t)-U^{n}, e^{n}=u^{n}-U^{n}$.

    \begin{theorem}
Let (H1)-(H2) hold. Then, the following error bounds are
     satisfied
\begin{gather*}
\| e_{n}\|^2_{L^{\infty}(0, T,H^{-1}(\Omega))}+\int_{0}^{T}|
   e_{n}|^2dt \leq c_{34}\, \tau , \\
 \| e^{m}\|_{ H^{-1}(\Omega)} \leq   c_{35} \, \tau ^{1/2} , \\
|\nabla \int_{0}^{T}e_{n}(t)\,dt|_{2} \leq   c_{36} \,\tau ^{1/4}.
\end{gather*}
  \end{theorem}

\begin{proof} We consider the following variational formulation of
discrete   problem \eqref{13}:
   \begin{equation} \label{41}
  (U^{n}-U^{n-1}, \varphi )
  +\tau (\nabla U^{n}, \nabla \varphi ) = \frac{\lambda \tau}
  { \big ( \int_{\Omega} f(U^{n})\, dx \big )^2} (f(U^{n}), \varphi ),
\end{equation}
 for all $\varphi \in H^{1}_{0}(\Omega)$.
  Integrating the continuous problem \eqref{11} over $I_{n}$, we get
  \begin{equation} \label{42}
   (u^{n}-u^{n-1}, \varphi )
  +\tau (\nabla \overline{u}^{n}, \nabla \varphi ) = \lambda \tau
  \int_{I_{n}}\frac{(f(u^{n}), \varphi )}
  {\big ( \int_{\Omega} f(u^{n})\, dx \big )^2},
  \quad \forall \, \varphi \in H^{1}_{0}(\Omega)
   \end{equation}
  Subtracting \eqref{42} from \eqref{41} and adding from $n=1$ to $m$
with  $m\leq N$, we obtain
\begin{equation} \label{43}
\begin{aligned}
&\sum_{n=1}^{m}(e^{n}-e^{n-1}, \varphi)+\tau \sum_{n=1}^{m}(\nabla
e_{u}^{n}, \nabla   \varphi ) \\
&\leq c_{37} \, \tau |\sum_{n=1}^{m}(\overline{f(u)}^{n}-f(U^{n}),
\varphi)|
  +c_{38} \, \tau |\sum_{n=1}^{m}(f(U^{n}), \varphi )| .
\end{aligned}
\end{equation}
Let $(-\triangle)^{-1}$ the green operator satisfying
  $$
    (\nabla (-\triangle)^{-1}\psi , \nabla \varphi)=
    (\psi, \varphi)_{H^{-1}(\Omega), H^{1}_{0}(\Omega)}
  $$
for all $\psi \in H^{-1}(\Omega), \varphi \in H^{1}_{0}(\Omega)$. Choosing
$ \varphi = (-\triangle)^{-1}(e^{n})$ as test function, we then obtain
\begin{equation} \label{44}
     I_{1}+I_{2} \leq I_{3} + I_{4}  ,
\end{equation}
where
\begin{gather*}
 I_{1}=  \sum_{n=1}^{m}(e^{n}-e^{n-1}, (-\triangle)^{-1}(e^{n})), \quad
 I_{2}=  \tau \sum_{n=1}^{m}( e_{u}^{n}, e^{n}), \\
 I_{3}  \leq c_{37}  \tau |\sum_{n=1}^{m}(\overline{f(u)}^{n}-f(U^{n}),
    (-\triangle)^{-1}(e^{n}) )| ,\\
 I_{4}= c_{38} \tau |\sum_{n=1}^{m}(f(U^{n}),(-\triangle)^{-1}(e^{n}) )| .
\end{gather*}
    With the aid of the elementary identity $2a(a-b)=a^2-b^2+(a-b)^2$
and  the property of $(-\triangle)^{-1}$, $I_{1}$ reduces after straightforward
  calculations to
  $$
  I_{1}=\frac{1}{2}\| e^{m}\|_{ H^{-1}(\Omega)}^2
  +\frac{1}{2}\sum _{n=1}^{m}\|e^{n}-e^{n-1}\|_{H^{-1}(\Omega)}^2 .
  $$
On the other hand
  \begin{align*}
   I_{2}&=  \tau \sum_{n=1}^{m}( e_{u}^{n}, e^{n})  \\
  & = \sum_{n=1}^{m}\int_{I_{n}}(u(t)-U^{n}, u(t)-U^{n})\,dt
  +\sum_{n=1}^{m}\int_{I_{n}}(u(t)-U^{n}, u^{n}-u(t))\, dt \\
& =I_{21}+ I_{22}. \\
I_{22}&= \sum_{n=1}^{m}\int_{I_{n}}(u(t), u^{n}-u(t))\, dt
- \sum_{n=1}^{m}\int_{I_{n}}(U^{n}, u^{n}-u(t))\, dt  \\
&= I_{22}^{1}+I_{22}^2.
\end{align*}
We now estimate $I_{22}^{1}$.
   \begin{align*}
|I_{22}^{1}|&=|\sum_{n=1}^{m}\int_{I_{n}}(u(t), \int_{t}^{t^{n}}
  \frac{\partial u}{\partial s}\,ds) \,dt|  \\
& \leq  \sum_{n=1}^{m}\int_{I_{n}}(\int_{t}^{t^{n}}
\| \frac{\partial u}{\partial s}\|_{H^{-1}(\Omega)}\,ds )
\|u(t)\|_{H^{1}_{0}(\Omega)}\, dt  \\
&  \leq \tau  \| \frac{\partial u}{\partial s}\|_{L^2(0, t^{m},
H^{-1}(\Omega))}
\, \|u\|_{L^2(0, t^{m}, H^{1}_{0}(\Omega))} \\
& \leq c_{39}\, \tau .
   \end{align*}
In the same manner,
\[
 |I_{22}^2|\leq \tau \| \frac{\partial u}{\partial s}\|_{L^2(0, t^{m}, H^{-1}
 (\Omega))}
     (\tau \sum_{n=1}^{m}\|U^{n}\|^2_{H^{1}_{0}(\Omega))})^{1/2}
\leq c_{40}\, \tau .
\]
   Next, we estimate the first term on the right-hand side of \eqref{44} by
   using H\^older's and Young's inequalities and (H1)
  \begin{align*}
   |I_{3}|&\leq |\sum_{n=1}^{m}( \int_{I_{n}}[f(u)-f(U^{n})] \, dt,
   (-\triangle)^{-1}(e^{n}))|\\
   &\leq
c_{41}\,\tau^{1/2}\sum_{n=1}^{m}(\int_{I_{n}}|f(u)-f(U^{n})|_{2}^2
   \,dt)^{1/2} \|e^{n}\|_{H^{-1}(\Omega)} \\
   & \leq \eta \sum_{n=1}^{m}(\int_{I_{n}}|f(u)-f(U^{n})|_{2}^2\, dt)
   + \frac{c_{42}}{\eta} \, \tau
\sum_{n=1}^{m}\|e^{n}\|_{H^{-1}(\Omega)}^2
   \\
   &\leq c_{43} \, \eta  \sum_{n=1}^{m}(\int_{I_{n}}|e_{n}|_{2}^2 \, dt )
   +  \frac{c_{42}}{\eta} \, \tau
\sum_{n=1}^{m}\|e^{n}\|_{H^{-1}(\Omega)}^2.
\end{align*}
Moreover, we have
$$
|I_{4}|\leq  c_{44}\,\tau + c_{45}\, \tau
\sum_{n=1}^{m}\|e^{n}\|_{H^{-1}(\Omega)}^2.
$$
Choosing suitably $\eta$, we conclude that
\begin{equation} \label{45}
\begin{aligned}
&\| e^{m}\|_{ H^{-1}(\Omega)}^2
  +\sum _{n=1}^{m}\|e^{n}-e^{n-1}\|_{H^{-1}(\Omega)}^2
+  \sum _{n=1}^{m} \int_{I_{n}}|e_{n}|_{2}^2 \, dt\\
&\leq c_{46}\,\tau + c_{47}\, \tau
\sum_{n=1}^{m}\|e^{n}\|_{H^{-1}(\Omega)}^2.
\end{aligned}
\end{equation}
On the other hand, setting
$y^{m}=\sum_{n=1}^{m}\|e^{n}\|_{H^{-1}(\Omega)}^2$, from \eqref{45},
  we get
$$
   y^{m}-y^{m-1} \leq c_{46}\,\tau +   c_{47}\,\tau  y^{m}.
$$
By applying the discrete Gronwall inequality, we deduce that $y^{m}\leq
c(T)$.Therefore,
$$
  \| e^{m}\|_{ H^{-1}(\Omega)}\leq  c_{48}\,\tau^{1/2}.
$$
On the other hand, we have
$$ \sup_{t\in (0, t_{m})} \| e_{n}(t)\|_{
H^{-1}(\Omega)}-c_{48}\tau^{1/2}
  \leq \max_{1\leq n\leq m}\| e_{n}(t^{n})\|_{ H^{-1}(\Omega)}=
  \max_{1\leq n\leq m}\| e^{n}\|_{ H^{-1}(\Omega)} .
$$
Thus,
$$
\|e_{n}\|_{L^{\infty}(0, T, H^{-1}(\Omega))}- c_{48}\,\tau^{1/2}
\leq \max_{1\leq n\leq m}  \| e^{n}\|_{ H^{-1}(\Omega)} .
$$
   From the last inequality, we obtain
\begin{gather*}
  \|e_{n}\|_{L^{\infty}(0, T,
  H^{-1}(\Omega))}^2+\int_{0}^{T}|e_{n}|_{2}^2\, dt
  \leq c_{49}\,\tau ,\\
\sum _{n=1}^{m}\|e^{n}-e^{n-1}\|_{H^{-1}(\Omega)}^2 \leq
c_{49}\,\tau .
\end{gather*}
Choosing $\varphi =\tau  \sum _{n=1}^{m}(\overline{u}^{n}-U^{n}) $ in
   \eqref{43} , we obtain
\begin{align*}
   &\tau \int_{\Omega}(u^{m}-U^{m})(\sum_{n=1}^{m}(\overline{u}^{n}-U^{n})\, dx)
+\tau^2|\sum _{n=1}^{m}\nabla (\overline{u}^{n}-U^{n})|_{2}^2  \\
& \leq
c_{50}\tau^2|\int_{\Omega}\sum_{n=1}^{m}(\overline{f(u)}^{n}-f(U^{n}))
   (\sum _{n=1}^{m}(\overline{u}^{n}-U^{n}))dx|\\
&\quad  +c_{51} \, \tau^2 |\sum_{n=1}^{m}(f(U^{n}), \sum
_{n=1}^{m}(\overline{u}^{n}-U^{n}) )| .
\end{align*}
  This implies
\begin{align*}
  &\tau^2|\sum _{n=1}^{m}\nabla (\overline{u}^{n}-U^{n})|_{2}^2
  = |\nabla \int_{0}^{t^{m}}e_{n}\, dt|_{2}^2
  \leq \tau |\int_{\Omega}(u^{m}-U^{m})(\sum_{n=1}^{m}(\overline{u}^{n}-U^{n})\, dx)|\\
&\quad +
c_{50}\tau^2|\int_{\Omega}\sum_{n=1}^{m}(\overline{f(u)}^{n}-f(U^{n}))
   (\sum _{n=1}^{m}(\overline{u}^{n}-U^{n})dx| \\
 &\quad +c_{51} \, \tau^2 |\sum_{n=1}^{m}(f(U^{n}), \sum_{n=1}^{m}(\overline{u}^{n}-U^{n}) )|  .\\
& \leq I+II+III .
\end{align*}
Clearly
\begin{align*}
I &\leq \| e^{m}\|_{ H^{-1}(\Omega)}(\sum_{n=1}^{m}\int_{I_{n}}\|u(t)\|
  _{H^{1}_{0}(\Omega)}\,dt+\tau
\sum_{n=1}^{m}\|U^{n}\|_{H^{1}_{0}(\Omega)})\\
  & \leq c_{52} \| e^{m}\|_{ H^{-1}(\Omega)}
   \leq c_{53} \tau^{1/2}.
\end{align*}
We  get also
\begin{align*}
II&\leq
(\int_{\Omega}(\sum_{n=1}^{m}\int_{I_{n}}(f(u)-f(U^{n}))\,dt)^2\,dx)^{1/2}
\times
(\int_{\Omega}(\sum_{n=1}^{m}\int_{I_{n}}(u(t)-U^{n})\,dt)^2\,dx)^{1/2}
\\
&\leq
T^2(\sum_{n=1}^{m}\int_{I_{n}}|f(u)-f(U^{n}|_{2}^2\,dt)^{1/2}
\times (\sum_{n=1}^{m}\int_{I_{n}}|u(t)-U^{n}|_{2}^2\,dt)^{1/2}
\\ &
\leq
T^2(\sum_{n=1}^{m}\int_{I_{n}}|f(u)-f(U^{n}|_{2}^2\,dt)^{1/2}
\times (2\|u\|^2_{L^2(0, T, H^{1}_{0}(\Omega))}
+2\tau \sum_{n=1}^{m} |U^{n}|_{2}^2)^{1/2} \\
& \leq c_{54}\, \tau^{1/2}.
\end{align*}
The last inequality follows by using simultaneously the
$L^{\infty}-$estimate of $u(t)$ , $U^{n}$ and the error bound
given in \eqref{ea}. Arguing as in the previous estimate, we get
$$
III \leq
T^2(\sum_{n=1}^{m}\int_{I_{n}}|f(U^{n}|_{2}^2\,dt)^{1/2}
\times (2\|u\|^2_{L^2(0, T, H^{1}_{0}(\Omega))}
+2\tau \sum_{n=1}^{m} |U^{n}|_{2}^2)^{1/2} .
  $$
Using again the hypothesis (H1) and the estimates above, we obtain
$$
III \leq  c_{55}\, \tau^{1/2}.
$$
Finally collecting these results, it follows that
$$
  |\nabla \int_{0}^{T}e_{n}\, dt|_{2}^2 \leq c_{56}\, \tau^{1/2}.
$$
This completes the proof.
\end{proof}

\begin{corollary} \label{coro4.3}
   Under hypotheses (H1)-(H2), problem \eqref{13} generates a
   continuous semi-group $S_{\tau}$ defined by $S_{\tau}U^{n-1}=U^{n}$.
\end{corollary}

\section{The semi-discrete dynamical system}

The aim here is to study the discrete dynamical system \eqref{13} via the
concepts of absorbing sets and global attractors (see Temam \cite{tem}).

\begin{theorem} \label{thm5.1}
The semi-group associated with \eqref{13} possesses a compact attractor
$\mathbb{A_{\tau}}$ which is bounded in $H_{0}^{1}(\Omega)\cap
L^{\infty}(\Omega)$ for $\tau$ small enough.
\end{theorem}

\begin{proof}
We begin by showing the existence of an
absorbing set in  $H_{0}^{1}(\Omega)\cap L^{\infty}(\Omega)$.

\noindent (i) Denoting $y_{m}^{n}=|U^{n}|_{m+2}$ and
$y^{n}=|U^{n}|_{L^{\infty}(\Omega)}$, then from
\eqref{31}, we have
$$
    y_{m}^{n} \leq c_{57}\,y_{m}^{n-1} +  c_{58}\tau .
$$
Letting $m$ approach infinity, we deduce that
$$
    y^{n} \leq c_{57}\,y^{n-1} +  c_{58}\tau .
$$
  On the other hand, we have
$$ \tau \sum_{n=n_{0}}^{n_{0}+N}y^{n} \leq a_{1}, \quad \forall n_{0}\geq
n_{\tau},
$$
  for some positive real number $a_{1}$ which do not depend on $n_{0}$. \\
  Applying the discrete uniform Gronwall's lemma (\cite{tem}), we get
  $$
   |U^{n}|_{L^{\infty}(\Omega)}\leq c_{59},  \quad \forall \, n\geq
n_{\tau},
  $$
which implies the existence of absorbing sets in $L^{\infty}(\Omega)$.

\noindent (ii) To obtain existence of absorbing sets in $H_{0}^{1}(\Omega)$,
multiply \eqref{13} by $U^{n}-U^{n-1}$. By using H\^older's and Poincar\'e's
inequalities,
we have
$$
|\nabla U^{n}|_{2}^2 \leq |\nabla U^{n-1}|_{2}^2+ c_{60}\tau
,\quad \forall \,n\geq n_{\tau}.
$$
Using again the relation $(b)$ and the discrete uniform Gronwall's lemma,
we get
$$
   \|U^{n}\|_{H_{0}^{1}(\Omega)}\leq  c_{61}, \quad \forall n\geq n_{\tau}.
$$
Therefore, the existence of  absorbing sets in $H_{0}^{1}(\Omega)$ is
proved. Applying Temam \cite[Theorem 1.1]{tem}, we therefore
get the result.
\end{proof}

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\end{document}