\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 $$\label{11} \begin{gathered} \frac{\partial u}{\partial t}-\triangle u = \lambda \frac{f(u)}{( \int_{\Omega} f(u)\, dx )^2} , \quad \mbox{in } \Omega \times ]0;T[, \\ u = 0 \quad \mbox{on } \partial \Omega \times ]0;T[, \\ u(0)= u_0 \quad \mbox{in } \Omega, \end{gathered}$$ 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 $$\label{12} \begin{gathered} u_{t}= \nabla .(k(u)\nabla u) + \sigma (u)|\nabla \varphi |^2, \\ \nabla ( \sigma (u) \nabla \varphi )= 0, \end{gathered}$$ 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 $$\label{13} 0< \sigma _{1}\leq \sigma (s) \leq \sigma _{2}, \forall s\in \mathbb{R}.$$ 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: $$\label{13a} \begin{gathered} U^{n}-\tau \triangle U^{n}= U^{n-1}+ \lambda \tau \frac{f(U^{n})}{ \big ( \int_{\Omega} f(U^{n})\, dx \big )^2} , \mbox{ in } \Omega , \\ U^{n} = 0 \quad \mbox{on } \partial \Omega, \\ U^{0}= u_0 \quad \mbox{in } \Omega. \end{gathered}$$ 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 $$\label{21} k_{0} \geq \max \big ( 0,\frac{\alpha N}{2}-2 \big ).$$ Then, there exists $d_{0}>0$ such that if $\| u_{0}\|_{k_{0}+2}0$ \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 $$\label{22a} \begin{gathered} \langle u_{m}',w_{j}\rangle +(u_{m},w_{j})= \frac{\lambda } { \big ( \int_{\Omega} f(u_{m})\, dx \big )^{2} } \, \langle f(u_{m}),w_{j} \rangle , \, 1\leq j \leq m , \\ u_{m}(0)=u_{om}. \end{gathered}$$ 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 $$\label{25} \frac{1}{k+2}\frac{d}{dt}\| u_{m}\|_{k+2}^{k+2} +\frac{4}{(k+2)^{2}} \| \nabla |u_{m}|^{\frac{k}{2}} u_{m}\|_{2}^{2} \leq c_{5} \| u_{m}\|_{k+\alpha +2}^{k+\alpha +2}+ c_{6}.$$ By using well-known Sobolev's and Gagliardo-Nirenberg's inequalities, we have $$\label{26} \| u_{m}\|_{k_{0}+\alpha +2}^{k_{0}+\alpha +2}\leq c_{7} \| u_{m}\|_{k_{0}+2}^{\alpha } \| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2},$$ Thus, from \eqref{25} and \eqref{26}, we obtain $$\label{27} \frac{1}{k_{0}+2}\frac{d}{dt}\| u_{m}\|_{k_{0}+2}^{k_{0}+2} \leq ( c_{8} \| u_{m}\|_{k_{0}+2}^{\alpha }- \frac{4}{(k_{0}+2)^{2}})\| \nabla |u_{m}|^{\gamma }u_{m}\|_{2}^{2} + c_{6}.$$ We shall make the following compatibility condition on $u_{0}$ $$\label{28} \| u_{0}\|_{k_{0}+2} < \Big( \frac{4}{c_{8}(k_{0}+2)^{2}} \Big )^{1/\alpha}=d_{0}.$$ Then, there exists a small $\tau >0$ such that $$\label{29} \| u_{m}(t)\|_{k_{0}+2} < d_{0} \mbox{ for } t\in ]0,\tau [.$$ Hence \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 00$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 $$\label{21a} \begin{gathered} U-\tau \triangle U= h(x) + \lambda \frac{f(U)}{( \int_{\Omega} f(U)\, dx)^2} , \quad \mbox{in } \Omega , \\ U = 0 \quad \mbox{on } \partial \Omega, \end{gathered}$$ \textbf{Existence.} Define the map$S(\mu , .)$by$U=S(\mu ,v), \mu \in [0, 1]$if $$\label{22} \begin{gathered} U - \tau \triangle U= \mu g(x, v) \quad\mbox{in } \Omega , \\ U = 0 \quad \mbox{on } \partial \Omega , \\ U^{0}=\mu u_{0}, \end{gathered}$$ 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 2and define $$\delta =\begin{cases} \frac{2d}{d-2} & \mbox{if } 20 depending on the data but not on N such that for any n\in \{1, \dots, N\} \begin{gather*} |U^{n}|_{L^{\infty}(\Omega)} \leq c(T,u_{0}), \label{ea}\\ |U^{n}|_{2}^2+\tau \sum _{k=1}^{n} |\nabla U^{k}|_{2}^2 \leq c(T,u_{0}), \label{eb}\\ \sum _{k=1}^{n}|U^{k}-U^{k-1}|_{2}^2 \leq c(T,u_{0}). \label{ec} \end{gather*} \end{theorem} \begin{proof} (i) Multiplying \eqref{13} by |U^{k}|^{m}U^{k} for some integer m\geq 1, using lemma \ref{22} and H\^older's inequality, we obtain after simplification $$\label{31} |U^{k}|_{m+2} \leq |U^{k-1}|_{m+2}+c_{31}\tau .$$ By induction and taking the limit in the resulting inequality as m \to +\infty, we get$$ |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}: $$\label{41} (U^{n}-U^{n-1}, \varphi ) +\tau (\nabla U^{n}, \nabla \varphi ) = \frac{\lambda \tau} { \big ( \int_{\Omega} f(U^{n})\, dx \big )^2} (f(U^{n}), \varphi ),$$ for all \varphi \in H^{1}_{0}(\Omega). Integrating the continuous problem \eqref{11} over I_{n}, we get $$\label{42} (u^{n}-u^{n-1}, \varphi ) +\tau (\nabla \overline{u}^{n}, \nabla \varphi ) = \lambda \tau \int_{I_{n}}\frac{(f(u^{n}), \varphi )} {\big ( \int_{\Omega} f(u^{n})\, dx \big )^2}, \quad \forall \, \varphi \in H^{1}_{0}(\Omega)$$ Subtracting \eqref{42} from \eqref{41} and adding from n=1 to m with m\leq N, we obtain \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} Let (-\triangle)^{-1} the green operator satisfying (\nabla (-\triangle)^{-1}\psi , \nabla \varphi)= (\psi, \varphi)_{H^{-1}(\Omega), H^{1}_{0}(\Omega)} $$for all \psi \in H^{-1}(\Omega), \varphi \in H^{1}_{0}(\Omega). Choosing \varphi = (-\triangle)^{-1}(e^{n}) as test function, we then obtain $$\label{44} I_{1}+I_{2} \leq I_{3} + I_{4} ,$$ where \begin{gather*} I_{1}= \sum_{n=1}^{m}(e^{n}-e^{n-1}, (-\triangle)^{-1}(e^{n})), \quad I_{2}= \tau \sum_{n=1}^{m}( e_{u}^{n}, e^{n}), \\ I_{3} \leq c_{37} \tau |\sum_{n=1}^{m}(\overline{f(u)}^{n}-f(U^{n}), (-\triangle)^{-1}(e^{n}) )| ,\\ I_{4}= c_{38} \tau |\sum_{n=1}^{m}(f(U^{n}),(-\triangle)^{-1}(e^{n}) )| . \end{gather*} With the aid of the elementary identity 2a(a-b)=a^2-b^2+(a-b)^2 and the property of (-\triangle)^{-1}, I_{1} reduces after straightforward calculations to$$ 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 \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} 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} \begin{thebibliography}{00} \bibitem{al} W. Allegretto, Y. Lin and A. Zhou: \emph{A box scheme for coupled systems resulting from Microsensor Thermistor Problems}, Dynamics of Continuous Discrete and Impulsive systems {\bf{5}}, (1999) 209-223. \bibitem{bl} J. W. Bebernes, A. A. Lacey: \emph{Global existence and finite-time blow-up for a class of nonlocal parabolic problems}, Advances in Differencial Equations, Vol. {\bf{2}}, No. {\bf{6}}, pp. 927-953 November 1997. \bibitem{car} H. 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