%\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Homogenization of a nonlinear equation \hfil EJDE--2001/17} {EJDE--2001/17\hfil A. K. Nandakumaran \& M. Rajesh \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 17, pp. 1--19. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Homogenization of a nonlinear degenerate parabolic differential equation % \thanks{ {\em Mathematics Subject Classifications:} 35B27, 74Q10. \hfil\break\indent {\em Key words:} degenerate parabolic equation, homogenization, two-scale convergence, \hfill\break\indent correctors. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted September 11, 2000. Published March 15, 2001.} } \date{} % \author{ A. K. Nandakumaran \& M. Rajesh } \maketitle \begin{abstract} In this article, we study the homogenization of the nonlinear degenerate parabolic equation $$\partial_t b(\frac{x}{\varepsilon},u_\varepsilon) - \mathop{\rm div} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon}, u_\varepsilon,\nabla u_\varepsilon)=f(x,t),$$ with mixed boundary conditions(Neumann and Dirichlet) and obtain the limit equation as $\varepsilon \to 0$. We also prove corrector results to improve the weak convergence of $\nabla u_\varepsilon$ to strong convergence. \end{abstract} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{rmk}[theorem]{Remark} \newtheorem{coro}[theorem]{Corollary} \newtheorem{propo}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newcommand{\scale}{\;\stackrel{2-s}{\to}\;} \section{Introduction} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with Lipschitz boundary and let $\mathop{\rm T}> 0$ be a constant. Let $\partial \Omega = \Gamma_{1} \cup \Gamma_{2}$, where it is assumed that $\Gamma_{1}$ has positive Hausdorff measure, $H^{n-1}(\Gamma_{1})$. We will denote $\Omega \times [0,T]$ by $\Omega_T$, and $\Gamma_{i} \times [0,T]$ by $\Gamma_{i,T}$, $i=1,2$. We consider the following initial-boundary value problem $$\label{inhomo} \begin{array}{c} \partial_t b(\frac{x}{\varepsilon}, u_\varepsilon) -\mathop{\rm div} \, a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon, \nabla u_\varepsilon) = f(x,t) \quad \mbox{in } \Omega_T,\\ a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon). \nu = 0 \quad \mbox{on } \Gamma_{2,T},\\ u_\varepsilon = g \quad \mbox{on } \Gamma_{1,T},\\ u_\varepsilon(x,0) = u_{0}\quad \mbox{in } \Omega. \end{array}$$ whose diffusion term is a monotone operator. Regarding the existence, uniqueness and regularity results for the above problem, which we will refer to as $(\mathop{\rm P}_{\varepsilon })$, we refer the reader to \cite{AL}. We are interested in the asymptotic behaviour of the problem $(\mathop{\rm P}_{\varepsilon })$ as $\varepsilon \to 0$. The homogenization of such equations with $b(y,s) \equiv s$ or $b(y,s)$ linear in $s$ has been studied quite widely (cf. ~\cite{AvLi,BFM,BLP,DP,CS,M,Ho,OKZ}). However, the case where $b$ is nonlinear has not been studied so much. Recently, H. Jian (cf.~\cite{J}) studied this problem for $b(y,s)$ of the form $b(s)$, assumed to be continuous and non-decreasing in $s$ and satisfying the monotonicity condition. It was shown, under an {\em a priori} assumption on the boundedness of the sequence $u_\varepsilon$ in $L^{\infty}(\Omega_T)$, that the homogenized equation corresponding to this problem is $$\label{homo} \begin{array}{c} \partial_t b(u) - \mathop{\rm div} \, A(u,\nabla u) = f(x,t) \quad\mbox{in }\Omega_T,\\ A(u,\nabla u).\nu = 0\quad \mbox{on } \Gamma_{2,T},\\ u = g \quad \mbox{on } \Gamma_{1,T},\\ u(x,0) = 0 \quad \mbox{in } \Omega \end{array}$$ for a suitable function $A$. That is, the solutions $u_{\varepsilon }$ of the in-homogeneous problem converge in some sense to a solution $u$ of the homogeneous problem. They first obtain a uniform bound, with respect to $\varepsilon$, on $\nabla u_\varepsilon$ in $L^{p}(\Omega_T)$ and hence on $\partial_t b(u_\varepsilon)$ and $a(\frac{x}{\varepsilon}, \frac{t}{\varepsilon},u_\varepsilon, \nabla u_\varepsilon)$ in an appropriate dual space. Thus, the sequences $\partial_t b(u_\varepsilon)$, $a(\frac{x}{\varepsilon}, \frac{t}{\varepsilon}, u_\varepsilon, \nabla u_\varepsilon)$ each have a weak $*$ limit in that space, but to complete the analysis these limits have to be identified as $\partial_t b(u)$ and $A(u,\nabla u)$, respectively. A crucial link in showing this was the fact that $b(u_\varepsilon)$ converges strongly to $b(u)$ in some $L^{q}(\Omega_T)$ and this in turn was used to prove the strong convergence of $u_\varepsilon$ to $u$ in some $L^{r}(\Omega_T)$ ( note that we cannot conclude the strong convergence of $u_\varepsilon$ to $u$ from the uniform bound on the sequence $\nabla u_\varepsilon$ in $L^{p}(\Omega_T)$ because the time derivative is not involved, but this information is hidden in the boundedness of $\partial_tb(u_\varepsilon)$). This is then used to identify the limits of the sequences $\partial_t b(u_\varepsilon)$ and $a(\frac{x}{\varepsilon}, \frac{t}{\varepsilon}, u_\varepsilon, \nabla u_\varepsilon)$. However, for the class of problems that we consider, $b(\frac{x}{\varepsilon},u_\varepsilon)$ can be expected to have only a weak limit in any $L^{q}(\Omega_T)$. This does not help in proving the strong convergence of $u_\varepsilon$ to a $u$ in any $L^{r}(\Omega_T)$, which is crucially needed for identifying the weak limits of the sequences $\partial_t b(\frac{x}{\varepsilon}, u_\varepsilon)$ and $a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)$. However, we are able to prove directly that $u_\varepsilon \to u$ in some $L^{r}(\Omega_T)$ by adapting a technique found in \cite{AL}. From this we can prove that $\partial_t b(\frac{x}{\varepsilon},u_\varepsilon)$ has as its weak limit $\partial_t \overline{b}(u)$. Here, $\overline{b}(s)$ denotes the average of $b(y,s)$ in the variable $y$ in the unit cell $Y=[0,1]^n$. Interestingly, we show that $b(\frac{x}{\varepsilon},u_\varepsilon)-b(\frac{x}{\varepsilon},u)$ strongly converges to $0$ in any $L^{q}(\Omega_T)$, $0 < q < \infty$, which yields the strong convergence of $b(u_\varepsilon)$ to $b(u)$ when $b$ is independent of the variable $y$. The diffusion term in the homogenized problem is the same as in \cite{J}, viz. $\mathop{\rm div} A(u,\nabla u)$ (cf. Theorem (\ref{mt})), but we identify this using the method of two-scale convergence. We also use the two-scale convergence method to prove the corrector results. We prove corrector results under the strong monotonicity assumption on $a$ which in turn, yields a corrector result for the work of H. Jian. That is, we construct suitable strong approximations for $\nabla u_\varepsilon$. The layout of the paper is as follows. In Section 2, we give the weak formulation for the problem $(\mathop{\rm P}_{\varepsilon })$. Then, we state our main results viz. Theorem \ref{mt} and Theorem \ref{corrt}. In Section 3, we prepare the ground for homogenization by obtaining some {\em a priori} estimates and by proving the strong convergence of $u_\varepsilon$ to some $u$ (for a subsequence) in some $L^{r}(\Omega_T)$. In Section 4, we prove our main theorems. \section{Assumptions and Main Results} For $p > 1$, $p^{*}$ will denote the conjugate exponent $p/(p-1)$. Let $V$ be the space, $\{v \in W^{1,p}(\Omega) : v=0 \mbox{ on } \Gamma_{1} \}$ and let $V^{*}$ be the dual of $V$. Let $E = L^{p}(0,T;V)$ and let $W^{1,p}_{per}(Y)$ be the space of elements of $W^{1,p}(Y)$ having the same trace on opposite faces of $Y$. We say, $u_\varepsilon \in g+ E$ is a weak solution of the problem $(\mathop{\rm P}_{\varepsilon })$ if it satisfies: $$\label{d1} b(\frac{x}{\varepsilon},u_\varepsilon) \in L^{\infty}(0,T;L^{1}(\Omega)), \partial_t b(\frac{x}{\varepsilon}, u_\varepsilon) \in L^{p^{*}}(0,T;V^{*})\,,$$ that is $$\label{r1} \int_0^T <\partial_tb(\frac{x}{\varepsilon},u_\varepsilon), \xi(x,t)> \, dt + \int_{\Omega_{T}} (b(\frac{x}{\varepsilon}, u_\varepsilon) - b(\frac{x}{\varepsilon},u_{0})) \partial_t \xi \, dx \, dt = 0$$ for all $\xi \in E \cap W^{1,1}(0,T;L^{\infty}(\Omega))$ with $\xi(T)= 0$ and \begin{eqnarray} \nonumber \int_{0}^{T} <\partial_t b(\frac{x}{\varepsilon},u_\varepsilon), \xi(x,t)> \, dt + \int_{\Omega_{T}} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon). \nabla \xi(x,t) \, dx \, dt \\ \label{weak} =\int_{\Omega_{T}} f(x,t) \xi(x,t) \, dx \, dt \end{eqnarray} for all $\xi \in E$. For the existence of a solution for the weak formulation we make the following assumptions (cf. \cite{AL}). \begin{itemize} \item[(A1)] The function b(y,s) is continuous in $y$ and $s$, $Y$-periodic in $y$ and non-decreasing in $s$ and $b(y,0)=0$. \item[(A2)] There exists a constant $\theta > 0$ such that for every $\delta$ and $R$ with $0< \delta < R$, there exists $C(\delta,R)>0$ such that $$|b(y,s_{1}) - b(y,s_{2})| > C(\delta,M) |s_{1}-s_{2}|^{\theta}$$ for all $y \in Y$ and $s_{1}, s_{2} \in [-R,R]$ with $\delta < |s_{1}|$. \item[(A3)] The mapping $(y,s,\mu,\lambda) \mapsto a(y,s,\mu,\lambda)$ defined from $\mathbb{R}^n \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^n$ to $\mathbb{R}^n$ is measurable in $(y,s)$ and continuous in $(\mu,\lambda)$. Further, it is assumed that there exists three positive constants $\alpha, r, \tau_{0}$ so that for all $(y,s,\mu,\lambda)$ and all $\mu_{1}, \mu_{2} \in \mathbb{R}$ and $\lambda, \lambda_{1}, \lambda_{2} \in \mathbb{R}^n$ one has, \begin{eqnarray} \label{wm0} &a(y,s,\mu,\lambda)(\lambda) \ge \alpha |\lambda|^{p}&\\ \label{wm1} &(a(y,s,\mu,\lambda_{1})-a(y,s,\mu,\lambda_{2})) (\lambda_{1}-\lambda_{2}) > 0, \quad \forall \, \lambda_{1} \neq \lambda_{2}& \\ \label{growth1} &|a(y,s,\mu,\lambda)|\le \alpha^{-1} (1+|\mu|^{p-1}+|\lambda|^{p-1})&\\ \nonumber \lefteqn{|a(y,s,\mu_{1},\lambda)-a(y,s,\mu_{2},\lambda) }\\ & \le \alpha^{-1}|\mu_{1}-\mu_{2}|^{r}(1+|\mu_{1}|^{p-1-r} +|\mu_{2}|^{p-1-r}+|\lambda|^{p-1-r}) &\label{growth2} \end{eqnarray} Also it is assumed that $a(y,s,\mu,\lambda)$ is $Y-\tau_{0}$ periodic in $(y,s)$ for all $(\mu,\lambda)$. \item[(A4)] Assume $g \in L^{p}(0,T;W^{1,p}(\Omega)) \cap L^{\infty}(\Omega_T)$, $\partial_t g \in L^{1}(0,T;L^{\infty}(\Omega))$,\\ $u_{0} \in L^{\infty}(\Omega)$, and $f \in L^{p^{*}}(\Omega_T)$. \item[(A5)] For all $y,s,\mu, \lambda_{1},\lambda_{2}$, $$\label{sm} (a(y,s,\mu,\lambda_{1})-a(y,s,\mu,\lambda_{2}))(\lambda_{1}-\lambda_{2}) \ge \alpha |\lambda_{1} -\lambda_{2}|^{p}.$$ \end{itemize} \begin{rmk}\rm It is to be noted that (A5) implies the conditions (\ref{wm0}) and (\ref{wm1}) in (A3), which alone are sufficient to guarantee the existence of a solution to the weak formulation of $(\mathop{\rm P_{\varepsilon }})$ and for its homogenization. (A5) will be used only in proving the corrector result. \end{rmk} \begin{rmk}\rm The prototype for $b$ is a function of the form $c(y) |s|^{k}\mathop{\rm sgn}(s)$ for some positive real number $k$ and continuous and $Y$-periodic function, $c(.)$, which is non-vanishing on $Y$. \end{rmk} We now state our main theorems. \begin{theorem} Let $u_\varepsilon$ be a family of solutions of $(\mathop{\rm P}_{\varepsilon })$. Assume that there is a constant $C>0$, such that $$\label{a0} \sup_{\varepsilon }\| u_\varepsilon \|_{L^{\infty} (\Omega_T)} \le C$$ Under, the assumptions (A1)-(A4), there exists a subsequence of $\varepsilon$, still denoted by $\varepsilon$, such that for all $q$ with $0 1, \end{eqnarray} and$u$solves, $$\label{homeqn} \begin{array}{c} \partial_t \overline{b}(u) - \mathop{\rm div}A(u,\nabla u) = f \quad\mbox{in } \Omega_T,\\ A(u,\nabla u). \nu = 0 \quad\mbox{on } \Gamma_{2,T},\\ u = g \quad\mbox{on } \Gamma_{1,T},\\ u(x,0) = u_{0} \quad\mbox{in } \Omega, \end{array}$$ where$\overline{b}$and$A$are defined below by (\ref{def1})-(\ref{def2}). \label{mt} \end{theorem} \begin{rmk} \rm Of course, the assumption (\ref{a0}) is true in special cases (see \cite{LU} Ch. 5) and it is reasonable on physical grounds (see \cite{J}). \end{rmk} The functions$\overline{b}$and$A$are defined by $$\label{def1} \overline{b}(s)= \int_{Y} b(y,s) \, dy$$ $$\label{def2} A(\mu,\lambda) = \frac{1}{\tau_{0}} \int^{\tau_{0}}_{0} \int_{Y} a(y,s,\mu,\lambda+\nabla \Phi_{\mu,\lambda}(y,s)) \, dy \, ds$$ where$\Phi_{\mu,\lambda} \in L^{p}(0,\tau_{0};W^{1,p}_{per}(Y))$solves the periodic boundary value problem $$\label{cell1} \int^{\tau_{0}}_{0} \int_{Y} a(y,s,\mu,\lambda+\nabla \Phi_{\mu,\lambda}(y,s)). \nabla \phi(y,s) \, dy \, ds = 0$$ for all$\phi \in L^{p}(0,\tau_{0};W^{1,p}_{per}(Y))$. For the existence of solutions to (\ref{cell1}), we refer the reader to Corollary 1.8, Ch. 3 of \cite{KS}. It can be shown that$A(\mu,\lambda): \mathbb{R}\times \mathbb{R}^n \to \mathbb{R}^n$is continuous and satisfies \begin{eqnarray} \label{p1} &|A(\mu,\lambda)| \le \beta^{-1} (1+ |\mu|^{p-1}+ |\lambda|^{p-1})&\\ \label{p2}& (A(\mu,\lambda_{1})-A(\mu,\lambda_{2})). (\lambda_{1}-\lambda_{2}) > 0, \, \, \forall \lambda_{1} \neq \lambda_{2},&\\ \label{p3} &A(\mu,\lambda). \lambda \ge \beta |\lambda|^{p}& \end{eqnarray} for a positive constant$\beta$which depends only on$\alpha,n,p,\tau_{0}$(cf. Lemmas 2.4-2.6 in \cite{FM}). Note that in (\ref{fconv1.5}) we only have weak convergence of$\nabla u_\varepsilon$in$L^{p}$. We construct some correctors for$\nabla u_\varepsilon$which will improve the weak convergence (\ref{fconv1.5}) to strong convergence. Such results are known as corrector results in the literature of homogenization and are very useful in numerical computations. Let$u(x,t)$be as in Theorem \ref{mt} and let$U_{1} \in L^{p}(\Omega_T \times (0,\tau_0); W^{1,p}_{per}(Y))$be the solution of the variational problem, \begin{eqnarray} \int_{\Omega_{T}} \int_{Y} \int_{0}^{\tau_0} a(y,s,u , \nabla_{x} u + \nabla_{y} U_{1}(x,t,y,s)). \nabla_{y} \Psi(x,t,y,s) \, = 0, \, \, \end{eqnarray} for all$\Psi \in L^{p}(\Omega_T \times (0,\tau_0); W^{1,p}_{per}(Y))$. It will be seen that there is such a function$U_{1}$. The statement of the corrector result is as follows. \begin{theorem} \label{corrt} Let$u_\varepsilon,u$be as in Theorem \ref{mt} and let$U_{1}$be as defined above. We assume all the assumptions in Theorem \ref{mt} and furthermore, the strong monotonicity assumption (A5). Then, if$u, U_{1}$are sufficiently smooth, i.e. belong to$C^{1}(\Omega_T)$and$C(\Omega_T; C_{per}(0,\tau_0) \times C^{1}_{per}(Y ))$, then \begin{eqnarray} \label{sconv1} u_\varepsilon -u - \varepsilon U_{1}(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) & \to & 0 \mbox{ strongly in }L^{p}(\Omega_T) \, \, and,\\ \label{sconv2} \nabla u_\varepsilon - \nabla u - \nabla_{y} U_{1}(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) & \to & 0 \mbox{ strongly in } L^{p}(\Omega_T). \end{eqnarray} \end{theorem} \begin{rmk} \rm Note that we are not claiming$u_\varepsilon -u-\varepsilon U_{1}(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) \to 0$strongly in$L^{p}(0,T;W^{1,p}(\Omega))$as we do not have the full gradient of$U_{1}$with respect to$x$in (\ref{sconv2}). \end{rmk} \section{Preliminaries} We first obtain {\em a priori} bounds under the assumption (\ref{a0}). From now on,$C$will denote a generic positive constant which is independent of$\varepsilon $. \begin{lemma} Let$u_\varepsilon$be a family of solutions of$(\mathop{\rm P}_{\varepsilon })$and assume that (\ref{a0}) holds. Then, \begin{eqnarray} \label{bd1} \sup_{\varepsilon }\|\nabla u_\varepsilon \|_{L^{p} (\Omega_T)} & \le & C\\ \label{bd2} \sup_{\varepsilon }\| a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon) \|_{L^{p^{*}} (\Omega_T)} & \le & C\\ \label{bd3} \sup_{\varepsilon }\| \partial_t b(\frac{x}{\varepsilon},u_\varepsilon) \|_{E^{*}} &\le & C \end{eqnarray} \end{lemma} \paragraph{Proof:} Define the function$B(.,.):\mathbb{R}^n \times \mathbb{R}\to \mathbb{R}$by $$\nonumber B(y,s) =b(y,s)s-\int^{s}_{0}b(y,\tau) \, d\tau$$ The following identity can be deduced as in Lemma 1.5 of Alt and Luckhaus~\cite{AL}. \begin{eqnarray} \nonumber \lefteqn{\int_{\Omega} (B(\frac{x}{\varepsilon}, u_\varepsilon(x,T))-B(\frac{x}{\varepsilon},u_{0}))\, dx }\\ &=& \int_{0}^{T} \langle \partial_t b(\frac{x}{\varepsilon}, u_\varepsilon), (u_\varepsilon -g)\rangle dt - \int_{\Omega_{T}}(b(\frac{x}{\varepsilon},u_\varepsilon)- b(\frac{x}{\varepsilon},u_{0})) \, \partial_t g \, dx \,dt \nonumber\\ &&+\int_{\Omega}(b(\frac{x}{\varepsilon},u_\varepsilon(T)) -b(\frac{x}{\varepsilon},u_{0}))g(T) \,dx \label{E1} \end{eqnarray} Therefore, using (\ref{weak}) we obtain, \begin{eqnarray} \nonumber \lefteqn{\int_{\Omega} B(\frac{x}{\varepsilon},u_\varepsilon(x,T)) \, dx + \int_{\Omega_{T}} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon). \nabla u_\varepsilon \, dx \, dt }\\ &=& \int_{\Omega} B(\frac{x}{\varepsilon},u_{0}) \, dx + \int_{\Omega_{T}} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon). \nabla g \, dx \, dt \nonumber \\ &&+ \int_{\Omega_{T}} f(x,t) (u_\varepsilon -g) \, dx\, dt - \int_{\Omega_{T}}(b(\frac{x}{\varepsilon},u_\varepsilon) -b(\frac{x}{\varepsilon},u_{0})) \, \partial_t g \, dx \, dt \nonumber\\ &&+ \int_{\Omega} (b(\frac{x}{\varepsilon},u_\varepsilon(T)) - b(\frac{x}{\varepsilon},u_{0}))g(T) \, dx. \label{E2} \end{eqnarray} Notice that due to (\ref{a0}), (A1), (A3) and (A4) we obtain \begin{eqnarray} \nonumber \int_{\Omega} B(\frac{x}{\varepsilon},u_\varepsilon(x,T)) \, dx + \int_{\Omega_{T}} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon). \nabla u_\varepsilon \, dx \, dt \\ \le C + C\| \nabla u_\varepsilon \| ^{p-1}_{p,\Omega_T}\|g \|_{p,\Omega_T} \end{eqnarray} Therefore, as$B$is non-negative from its definition, we get using (A3) again that, $$\alpha \| \nabla u_\varepsilon \|^{p}_{p,\Omega_T} \le C + C \| \nabla u_\varepsilon \|^{p-1}_{p,\Omega_T}$$ for all$\varepsilon $. This implies (\ref{bd1}). Then, (\ref{bd2}) follows from (\ref{bd1}) and (A3), while (\ref{bd3}) follows from (\ref{bd1}), (\ref{bd2}) and the weak formulation (\ref{weak}). \hfill$\diamondsuit$\smallskip As a consequence of (\ref{a0}) and the above lemma, we immediately conclude that, for a subsequence of$\varepsilon $( to be denoted by$\varepsilon $again), \begin{eqnarray} \label{conv1}u_\varepsilon & \rightharpoonup & u \mbox{ weakly * in } L^{\infty}(\Omega_T),\\ \label{conv2} \nabla u_\varepsilon & \rightharpoonup & \nabla u \mbox{ weakly in } L^{p}(\Omega_T)\\ \label{conv2.5} b(\frac{x}{\varepsilon},u_\varepsilon) & \rightharpoonup& b^{*}\mbox{ weakly * in } L^{\infty}(\Omega_T)\\ \label{conv3} \partial_t b(\frac{x}{\varepsilon},u_\varepsilon)& \rightharpoonup & w \mbox{ weakly * in } E^{*},\\ \label{conv4} a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon) & \rightharpoonup & A^{*}(x,t) \mbox{ weakly in } L^{p^{*}} (\Omega_T) \end{eqnarray} for some$b^{*} \in L^{\infty}(\Omega_T)$,$w \in E^{*}$and$A^{*} \in L^{p^{*}} (\Omega_T)$. The task is to identify these quantities and obtain the limit equation. We now prove that, for a subsequence,$u_\varepsilon$converges$a. e. $to$u$in$\Omega_T$and this will form a crucial part of the present analysis. This will be found useful in identifying$b^{*}$,$w$and$A^{*}$. \begin{lemma} \label{r2} There exists a continuous, increasing function$\omega$on$\mathbb{R}^{+}$with$\omega(0)=0$, such that, given any$C > 0,\, \delta > \, 0$, if$v_{1},v_{2}$are any two functions in$W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$with$\|v_{i} \|_{\infty,\Omega} \le C$,$i=1,2$, satisfying $$\int_{\Omega}(b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2}))(v_{1}-v_{2})\, dx \le \delta \quad \forall \varepsilon >0$$ then $$\int_{\Omega}|b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2})| \, dx \le \omega(\delta) \quad \forall \varepsilon >0.$$ \end{lemma} \paragraph{Proof:} By the {\em a priori} bounds for$v_{1},v_{2}$in$L^{\infty}$, we can restrict$b$to the domain$Y \times [-C,C]$, where it is uniformly continuous. Now, \begin{eqnarray*} \lefteqn{\int_{\Omega} |b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2})| \,dx }\\ & = & \int_{\Omega \cap \{|v_{1}-v_{2}|<\delta^{\frac{1}{2}}\}} |b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2})| \,dx \\ && + \int_{\Omega \cap \{|v_{1}-v_{2}| \ge \delta^{\frac{1}{2}}\}} |b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2})| \,dx\\ & \le & \omega_{b}(\delta^{\frac{1}{2}})m(\Omega) + \delta^{-\frac{1}{2}}\, \int_{\Omega}(b(\frac{x}{\varepsilon},v_{1})-b(\frac{x}{\varepsilon},v_{2}))(v_{1}-v_{2})\, dx\\ & \le &\omega_{b}(\delta^{\frac{1}{2}})m(\Omega)+\delta^{\frac{1}{2}} \end{eqnarray*} where$\omega_{b}$is the modulus of continuity function for$b$. Thus, we obtain the lemma by taking$\omega(t) \doteq \omega_{b}(t^\frac{1}{2})m(\Omega) + t^\frac{1}{2}$. \hfill$\diamondsuit$\begin{lemma} \label{mlemma} Let$u_\varepsilon$be the solution of (\ref{inhomo}). Then, the sequence$\{ u_\varepsilon \}_{\varepsilon > 0}$is relatively compact in$L^{\theta}(\Omega_T)$, where$\theta$is as in (A2). As a result, there is a subsequence of$u_\varepsilon$such that, $$\label{aeconv} u_\varepsilon \to u \mbox{ a. e. in } \Omega_T$$ \end{lemma} \paragraph{Proof:} {\em Step 1}: Using the arguments from \cite{J}, it can be shown that $$h^{-1} \int^{T-h}_{0} \int_{\Omega} (b(\frac{x}{\varepsilon},u_\varepsilon(t+h))-b(\frac{x}{\varepsilon},u_\varepsilon(t)))(u_\varepsilon(t+h)-u_\varepsilon(t))\, dx \, dt \le C$$ for some constant$C$which is independent of$\varepsilon $and$h$. \noindent{\em Step 2:} We show that $$\int^{T-h}_{0} \int_{\Omega} |b(\frac{x}{\varepsilon},u_\varepsilon(t+h)) - b(\frac{x}{\varepsilon},u_\varepsilon(t))| \, dx \, dt \to 0$$ as$h \to 0$, uniformly with respect to$\varepsilon $. Set, for$R>0$and large, \begin{eqnarray*} E_{\varepsilon ,R} &=& \{ t \in (0,T-h): \|u_\varepsilon(t+h)\|_{W^{1,p}(\Omega)} + \|u_\varepsilon(t)\|_{W^{1,p}(\Omega)} + \| g\|_{W^{1,p}(\Omega)}\\ &&+h^{-1} \int_{\Omega} ( b(\frac{x}{\varepsilon},u_\varepsilon(t+h)) -b(\frac{x}{\varepsilon},u_\varepsilon(t))).(u_\varepsilon(t+h) -u_\varepsilon(t)) \, dx > R \} \end{eqnarray*} From the estimate in Step 1, it follows that$m(E_{\varepsilon ,R}) \le C/R$. Set$E^{'}_{\varepsilon ,R}$to be the complement of$E_{\varepsilon ,R}$in$(0,T-h)$. Hence, for$t \in E^{'}_{\varepsilon ,R}$, by Lemma \ref{r2}, we have $$\label{r3} \int_{\Omega} |b(\frac{x}{\varepsilon},u_\varepsilon(t+h))-b(\frac{x}{\varepsilon},u_\varepsilon(t))|\, dx < \omega(hR).$$ Therefore, \begin{eqnarray*} \lefteqn{\int_{0}^{T-h}\!\! \int_{\Omega}|b(\frac{x}{\varepsilon}, u_\varepsilon(t+h))-b(\frac{x}{\varepsilon},u_\varepsilon(t))| }\\ &=& \int_{E_{\varepsilon ,R}} \!\! \int_{\Omega}|b(\frac{x}{\varepsilon},u_\varepsilon(t+h))-b(\frac{x} {\varepsilon},u_\varepsilon(t))| \\ &&+\int_{E^{'}_{\varepsilon ,R}} \!\! \int_{\Omega}|b(\frac{x}{\varepsilon},u_\varepsilon(t+h)) -b(\frac{x}{\varepsilon},u_\varepsilon(t))| \\ &\le& C/R + T \, \omega(hR) \end{eqnarray*} for all$ \varepsilon , R$and$h$. Now, choose$R $large, fixed so that$C/R$is as small as we please and then let$h \to 0$to complete the proof of Step 2. \noindent{\em Step 3: } By assumption (A2), it follows from Step 2 that $$\label{r4} \int_{0}^{T-h} \int_{\Omega}|u_\varepsilon(t+h)-u_\varepsilon(t)|^{\theta} \, dx \, dt \to 0 \mbox{ as } h \to 0$$ uniformly with respect to$\varepsilon $. \noindent{\em Step 4: } In this crucial step, we demonstrate the relative compactness of the sequence$\{u_\varepsilon\}_{\varepsilon > 0}$in$L^{\theta}(\Omega_T)$. This is an argument to reduce it to the time independent case. Set, $$\left. v_{\varepsilon }(x,t)= \left\{ \begin{array}{ll} u_\varepsilon(x,t) & \mbox{ if } t \in (0,T-h) \setminus E_{\varepsilon ,R}\\ 0 & \mbox{ if } t \in E_{\varepsilon ,R} \cup [T-h,T] \end{array} \right. \right.$$ Choose,$h$so that$T$is an integral multiple of$h$. We have, \begin{eqnarray*} \lefteqn{ \frac{1}{h} \int^{h}_{0} \int^{T}_{0} \int_{\Omega} | u_\varepsilon(t)-\sum_{i=1}^{T/h}\chi_{((i-1)h,ih)} v_\varepsilon((i-1)h+s)|^{\theta} \, dx \, dt \, ds }\\ & = &\frac{1}{h} \sum_{i=1}^{T/h} \int^{ih}_{(i-1)h} \int^{ih}_{(i-1)h} \int_{\Omega} | u_\varepsilon(t)-v_\varepsilon(s)|^{\theta} \, dx \, dt \, ds\\ & \le &\frac{1}{h} \int^{h}_{-h} \int^{min(T,T-s)}_{max(0,-s)} \int_{\Omega} | u_\varepsilon(t)-v_\varepsilon(s+t)|^{\theta} \, dx \, dt \, ds\\ & \le & Sup_{|s|\le h}\int^{min(T,T-s)}_{max(0,-s)} \int_{\Omega} | u_\varepsilon(t)-u_\varepsilon(s+t)|^{\theta} \, dx \, dt \\ &&+ \int_{E_{\varepsilon ,R} \cup (T-h,T)} \int_{\Omega} |u_\varepsilon(t)|^{\theta} \, dx \, dt\\ & \le & T \,w(hR) + C/R \end{eqnarray*} which can be taken small, say less than$\mu$(for all$\varepsilon $), by fixing$h$small and$R=h^{-\frac{1}{2}}$. Therefore, there exists$s_{\varepsilon } \in (0,h)$such that $$\int_{\Omega_{T}} |u_\varepsilon(t) - \sum_{i=1}^{T/h}\chi_{((i-1)h,ih)}v_\varepsilon((i-1)h+s_{\varepsilon }) |^{\theta} \, dx \, dt$$ is small uniformly in$\varepsilon $. Note that the sequences$\{ v_\varepsilon((i-1)h+s_{\varepsilon })\}_{\varepsilon >0}$are independent of time. Therefore, it is enough to show that$\{v_\varepsilon((i-h)+s_{\varepsilon })\}_{\varepsilon >0}$are relatively compact sequences in$L^{\theta}(\Omega_T)$for$i=1,...,T/h$. But, this follows from the compact inclusion of$W^{1,p}(\Omega)$in$L^{p}(\Omega)$as these sequences are bounded in$W^{1,p}(\Omega)$(by the definition of$E_{\varepsilon ,R}$) for each$i$. \hfill$\diamondsuit$\smallskip We end the section by recalling a fact which is quite useful in periodic homogenization. Let$f$be a function in$L^{q}_{loc}(\mathbb{R}^n;C_{per}(Y))$. Then we have the following lemma. \begin{lemma} \label{2scale} The oscillatory function$f(\frac{x}{\varepsilon},x)$converges weakly in$L^{q}_{loc}(\mathbb{R}^n)$to$\int_{Y} f(y,x) \, dy$, for all$q>1$. \end{lemma} \section{Homogenization and Correctors} First, we prove (\ref{fconv1}), (\ref{fconv2}) and (\ref{fconv3}) using Lemma \ref{mlemma} and Lemma \ref{2scale}. Then, we identify$b^{*}$and$A^{*}$given by (\ref{conv4}). Finally, we prove that$u$satisfies the homogenized equation (\ref{homeqn}). By the {\em a priori} bound (\ref{a0}) and (\ref{aeconv}), it follows by the Lebesgue dominated convergence theorem that $$\label{r4.5} u_\varepsilon \to u \mbox{ strongly in } L^{q}(\Omega_T) \, ,$$ for all$q$with$00$. As$b$is continuous, it is uniformly continuous on$Y \times [-M,M]$. Therefore, given$h_{0}>0$, there exists a$\delta > 0$such that, $$|b(y,s)-b(y',s')|0, there exists E \subset \Omega_T such that its Lebesgue measure m(E)0 such that $$\nonumber \|u_\varepsilon -u \|_{\infty,E'}<\delta \, \, \forall \varepsilon < \varepsilon _{1}.$$ Therefore, for \varepsilon < \varepsilon _{1} we have, \begin{eqnarray*} \lefteqn{ \int_{\Omega_{T}} |b(\frac{x}{\varepsilon},u_\varepsilon)-b(\frac{x}{\varepsilon},u)|^{q} \, dx \, dt }\\ & =& \int_{E'}|b(\frac{x}{\varepsilon},u_\varepsilon) -b(\frac{x}{\varepsilon},u)|^{q} \, dx \, dt +\int_{E}|(b(\frac{x}{\varepsilon},u_\varepsilon) -b(\frac{x}{\varepsilon},u))|^{q}\, dx \, dt\\ &\le& h_{0}^{q}m(\Omega_T)+ 2^{q} \sup(|b|^{q})\,m(E)\\ & \le & h_{0}^{q}\,m(\Omega_T)+ 2^{q} \sup (|b|^{q}) \, h_{1}. \end{eqnarray*} This completes the proof as h_{0} and h_{1} can be chosen arbitrarily small. \hfill\diamondsuit \begin{coro} If b(\frac{x}{\varepsilon},u_\varepsilon)=b(u_\varepsilon), then the above proposition shows that b(u_\varepsilon) \to b(u) strongly in L^{q}(\Omega_T), the result of Jian \cite{J}. \end{coro} \begin{coro}\label{corob*} We have the following convergences: \begin{itemize} \item[(i)] b(\frac{x}{\varepsilon},u_\varepsilon) converges to \overline{b}(u) weakly in L^{q}(\Omega_T) for any q \in (1,\infty) and hence b^* = \overline{b}(u). \item[(ii)] \partial_t b(\frac{x}{\varepsilon},u_\varepsilon) \rightharpoonup \partial_t \overline{b}(u) weakly * in E^{*} and thus w=\partial_t \overline{b}(u). \end{itemize} \end{coro} \paragraph{Proof:} (i) We can write, b(\frac{x}{\varepsilon},u_\varepsilon)= (b(\frac{x}{\varepsilon},u_\varepsilon) -b(\frac{x}{\varepsilon},u))+ b(\frac{x}{\varepsilon},u). The result now follows from Proposition \ref{strongb} and Lemma \ref{2scale} and (ii) follows from (i) and (\ref{conv3}). \hfill\diamondsuit\smallskip Finally, we have to show that A^{*}=A(u,\nabla u), which can be proved in a manner similar to that in \cite{J}. We present a different proof of this using the method of {\em two-scale convergence}. Besides, some steps of the proof will be used in proving the corrector result. First, we recall the definition and main results concerning the method of two-scale convergence (cf.~\cite{A,N,Na}). We set the period \tau_0 in the time variable to be 1, for convenience of notation. \begin{definition}\rm Let 1< q <\infty. A sequence of functions v_{\varepsilon } \in L^{q}(\Omega_T) is said to two-scale converge to a function v \in L^{q}(\Omega_T \times Y \times (0,1)) if \begin{eqnarray*} \int_{\Omega_{T}} v_{\varepsilon } \, \psi(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) \, dx \, dt & \stackrel{\varepsilon \to 0}{\to} & \int_{\Omega_{T}} \int_{0}^{1} \int_{Y} v(x,t,y,s) \, \psi(x,t,y,s) \, dy \, ds \, dx \, dt \end{eqnarray*} for all \psi \in L^{q^{*}}(\Omega_T; C_{per}(Y \times (0,1)). We write v_{\varepsilon } \stackrel {2-s}{\to} v. \end{definition} \begin{rmk}\rm From the definition of two-scale convergence, it is easy to see that if v_{\varepsilon } is a sequence of functions in L^{q}(\Omega_T) such that v_{\varepsilon } \stackrel{2-s}{\to} v(x,t,y,s), then v_{\varepsilon } \rightharpoonup \int_{0}^{1} \int_{Y} v(x,t,y,s) \, dy \, ds weakly in L^{q}(\Omega_T). \end{rmk} The following facts about two-scale convergence \cite{A} will be used by us. \begin{theorem} \label{2sth1} If v_{\varepsilon } is a bounded sequence in L^{q}(\Omega_T), then there exists a function v \in L^{q}(\Omega_T \times Y \times (0,1)) such that, up to a subsequence, v_{\varepsilon } \stackrel{2-s}{\to} v(x,t,y,s). \end{theorem} \begin{theorem} \label{2sth2} If v_{\varepsilon }, \nabla v_{\varepsilon } are bounded sequences in L^{q}(\Omega_T), then there exist v \in L^{q}((0,T)\times (0,1);W^{1,q}(\Omega)) and V_{1} \in L^{q}(\Omega_T \times (0,1); W^{1,q}_{per}(Y)) such that, up to a subsequence, \begin{eqnarray*} v_{\varepsilon } &\stackrel{2-s}{\to} & v(x,t,s) \, ,\\ \nabla v_{\varepsilon } & \stackrel{2-s}{\to}& \nabla_{x} v(x,t,s) + \nabla_{y} V_{1}(x,t,y,s). \end{eqnarray*} \end{theorem} The following theorem \cite{A} is useful in obtaining the limit of the product of two two-scale convergent sequences. Let 1 0 and \phi_{0} \in C^{\infty}_{0}(\Omega_T;C^{\infty}_{per}(Y \times (0,1)))^n. Set, \begin{eqnarray} \label{eqn0} \eta_{\varepsilon } & \doteq & \nabla_{x} \phi + (\nabla_{y} \Phi )( x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) + \lambda \phi_{0}(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}),\\ \nonumber a_{\varepsilon } & \doteq & a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)\, , \\ \nonumber d_{\varepsilon } & \doteq & a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi, \eta_{\varepsilon }). \end{eqnarray} We have, \begin{eqnarray} \nonumber J_{\varepsilon } & \doteq & \int_{\Omega_{T}} (a_{\varepsilon }-d_{\varepsilon }). (\nabla u_\varepsilon - \eta_{\varepsilon })\\ \nonumber & = & \int_{\Omega_{T}} a_{\varepsilon }. \nabla u_\varepsilon - \int_{\Omega_{T}} d_{\varepsilon }. \nabla u_\varepsilon - \int_{\Omega_{T}} a_{\varepsilon }. \eta_{\varepsilon } + \int_{\Omega_{T}} d_{\varepsilon }. \eta_{\varepsilon }\\ \label{eqn1}& \doteq & J_{1,\varepsilon } - J_{2,\varepsilon } - J_{3,\varepsilon } + J_{4,\varepsilon } \end{eqnarray} where J_{i,\varepsilon } denotes the respective terms above for i = 1,\cdots,4. Now, \begin{eqnarray} \nonumber J_{1,\varepsilon } & = & \int_{\Omega_{T}} a_{\varepsilon }. \nabla u_\varepsilon \, dx \, dt\\ \nonumber & = & - \int_{0}^{T}\langle \partial_t b(\frac{x}{\varepsilon},u_\varepsilon), u_\varepsilon\rangle \, dt + \int_{\Omega_{T}} f \, u_\varepsilon \, dx \, dt \\ \nonumber &\stackrel{\varepsilon \to 0}{\to} & - \int_{0}^{T} \langle \partial_t \overline{b}(u), u\rangle \, dt + \int_{\Omega_{T}} f \, u \, dx \, dt\\ \label{eqn2} & = & \int_{\Omega_{T}} A^{*}(x,t). \nabla_{x} u \, dx \, dt \end{eqnarray} where the last equality follows from (\ref{strong}). For obtaining the limit of the other terms in the right hand side of (\ref{eqn1}) we will use Theorem \ref{2sth3}. For this we observe that the continuity assumptions on a and the choice of \phi,\Phi,\phi_{0} imply that the sequence \begin{eqnarray*} d_{\varepsilon } &\equiv& a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi, \nabla_{x} \phi + (\nabla_{y} \Phi)(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) + \lambda \phi_{0}(x,t,\frac{x}{\varepsilon},\frac{t}{\varepsilon})) \end{eqnarray*} is of the form \psi(t,x,\frac{x}{\varepsilon},\frac{t}{\varepsilon}) for a \psi \in L^{p^{*}}_{per}(Y \times (0,1);C(\Omega_T)). Thus, d_{\varepsilon } strongly two-scale converges to a(y,s,\phi,\nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0}). Also, it can be seen that \eta_{\varepsilon } is strongly two-scale convergent to \eta(x,t,y,s) \doteq \nabla_{x} \phi(x,t) + \nabla_{y} \Phi(x,t,y,s) + \lambda \phi_{0}(x,t,y,s). Thus, from these observations, Theorem \ref{2sth3} and (\ref{eqn2}), we obtain \begin{eqnarray} \nonumber J_{\varepsilon } & \stackrel{\varepsilon \to 0}{\to} & \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a_{0}(x,t,y,s). \nabla_{x} u \, dy \, ds \, dx \, dt \\ \nonumber & &- \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a(y,s,\phi, \nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0}). (\nabla_{x} u + \nabla_{y} U_{1}) \\ \nonumber & &- \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a_{0}(x,t,y,s). (\nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0}) \\ \nonumber & & + \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a(y,s,\phi,\nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0}). (\nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0}) \end{eqnarray} Note that by setting \phi = 0 in (\ref{2shomo}) we get, \begin{eqnarray}\label{cell} \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a_{0}(x,t,y,s). \nabla_{y} \Phi(x,t,y,s) \, dy \, ds \, dx \, dt & = & 0 \end{eqnarray} for any \Phi \in C^{\infty}_{0}(\Omega_T; C^{\infty}_{per}(Y \times (0,1))). Thus, the above limit can be rewritten as \begin{eqnarray*} \lim_{\varepsilon \to 0} J_{\varepsilon } &=&\int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a_{0}(x,t,y,s).(\nabla_{x} u - \nabla_{x} \phi -\lambda \phi_{0})\, dy \, ds \, dx \, dt\\ &&- \int_{\Omega_{T}} \! \! \int_{Y} \! \! \int_{0}^{1} a(y,s,\phi, \nabla_{x} \phi + \nabla_{y} \Phi + \lambda \phi_{0})\\ && \times (\nabla_{x} u + \nabla_{y} U_{1} - \nabla_{x} \phi- \nabla_{y} \Phi -\lambda \phi_{0}) . \end{eqnarray*} Now, letting \phi \to u strongly in L^{p}(0,T;V) and \Phi \to U_{1} in \\ L^{p}(\Omega_T \times (0,1); W^{1,p}_{per}(Y)) strongly we get, $$\label{eqn3} \lim_{\stackrel{\phi \to u}{\Phi \to U_{1}}} \lim_{\varepsilon \to 0} J_{\varepsilon } = \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} (a(y,s,u,\nabla_{x} u + \nabla_{y} U_{1} + \lambda \phi_{0})-a_{0}(x,t,y,s)). \lambda \phi_{0},$$ where we have used the continuity properties of a. On the other hand, \begin{eqnarray*} J_{\varepsilon } & = & \int_{\Omega_{T}} (a_{\varepsilon } -a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\eta_{\varepsilon })). (\nabla u_\varepsilon - \eta_{\varepsilon }) \, dx \, dt \\ & & + \int_{\Omega_{T}} (a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\eta_{\varepsilon })-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi,\eta_{\varepsilon })).(\nabla u_\varepsilon - \eta_{\varepsilon }) \, dx \, dt\\ &\doteq & L_{1,\varepsilon } + L_{2,\varepsilon }, \end{eqnarray*} where L_{i,\varepsilon }, i=1,2 denotes the respective terms above. By the monotonicity assumption (\ref{wm1}), L_{1,\varepsilon } \ge 0. Therefore, J_{\varepsilon } \ge L_{2,\varepsilon }. Now, by (\ref{growth2}) and generalized H\" older's inequality, \begin{eqnarray*} |L_{2,\varepsilon }| & \le & \int_{\Omega_{T}} |a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\eta_{\varepsilon })-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi,\eta_{\varepsilon })|.|\nabla u_\varepsilon - \eta_{\varepsilon }| \, dx \, dt \\ &\le & \alpha^{-1} \|u_\varepsilon - \phi\|^{r}_{p} (m(\Omega_T)^{\frac{p-1-r}{p}}+\|u_\varepsilon\|^{p-1-r}_{p} +\|\phi\|^{p-1-r}_{p}\\ & & +\|\eta_{\varepsilon }\|^{p-1-r}_{p}) \|\nabla u_\varepsilon - \eta_{\varepsilon } \|_{p}. \end{eqnarray*} Therefore, \begin{eqnarray*} J_{\varepsilon } & \ge & L_{2,\varepsilon }\\ &\ge &-\alpha^{-1}\|u_\varepsilon - \phi\|^{r}_{p} (m(\Omega_T)^{\frac{p-1-r}{p}}+\|u_\varepsilon\|^{p-1-r}_{p} +\|\phi\|^{p-1-r}_{p}\\ & & +\|\eta_{\varepsilon }\|^{p-1-r}_{p}) \|\nabla u_\varepsilon - \eta_{\varepsilon } \|_{p}\\ & \ge &-\alpha^{-1}\|u_\varepsilon - \phi\|^{r}_{p}(C+\|\phi\|^{p-1-r}_{p}+\|\eta_{\varepsilon }\|^{p-1-r}_{p}) (C+ \| \eta_{\varepsilon } \|_{p}) \end{eqnarray*} since u_\varepsilon, \nabla u_\varepsilon are bounded in L^{p}(\Omega_T). We now use the fact that \eta_{\varepsilon } is strongly two-scale convergent to \eta, defined before, to obtain the limit as \varepsilon \to 0 in the above inequality and we get \begin{eqnarray*} \lim_{\varepsilon \to 0} J_{\varepsilon } & \ge & - \alpha^{-1} \|u-\phi \|^{r}_{p} (C+ \| \phi \|^{p-1-r}_{p} + \| \eta \|^{p-1-r}_{p})(C+\| \eta \|_{p,\Omega_T \times Y \times (0,1)}). \end{eqnarray*} Now letting \phi \to u and \Phi \to U_{1} as before, we get $$\label{eqn4} \lim_{\stackrel{\phi \to u}{\Phi \to U_{1}}} \lim_{\varepsilon \to 0} J_{\varepsilon } \ge 0.$$ Therefore, from (\ref{eqn3}) and (\ref{eqn4}), we get $$\int_{\Omega_{T}} \int_{Y} \int_{0}^{1} (a(y,s,u,\nabla_{x} u + \nabla_{y} U_{1} + \lambda \phi_{0})-a_{0}(x,t,y,s)). \lambda \phi_{0} \, dy \, ds \, dx \, dt \ge 0$$ for all \lambda >0 and for all \phi_{0} \in C^{\infty}_{0}(\Omega_T;C^{\infty}_{per}(Y \times (0,1)))^n. Dividing the above inequality and letting \lambda \to 0, we get using the continuity of a, that $$\int_{\Omega_{T}} \int_{Y} \int_{0}^{1} (a(y,s,u,\nabla_{x} u + \nabla_{y} U_{1})-a_{0}(x,t,y,s)).\phi_{0} \, dy \, ds \, dx \, dt \ge 0$$ for all \phi_{0} \in C^{\infty}_{0}(\Omega_T;C^{\infty}_{per}(Y \times (0,1)))^n. By the density of these functions in \linebreak L^{p}(\Omega_T \times Y \times (0,1))^n, we get a_{0}(x,t,y,s) = a(y,s,u,\nabla_{x} u(x,t) + \nabla_{y} U_{1}(x,t,y,s)) \mbox{ a.e. } in \Omega_T \times Y \times (0,1). \hfill\diamondsuit \paragraph{Proof of Theorem \ref{mt}:} The proof follows from Proposition \ref{basic}, Remark \ref{remark1}, Proposition \ref{prop}, (\ref{def2}) and (\ref{cell}). \hfill\diamondsuit \smallskip We now prove corrector results. First, we prove a certain corrector result without any smoothness assumption on (u,U_{1}). Then we deduce Theorem \ref{corrt} from this corrector result. Let \delta > 0 and choose \phi \in C^{1}_{0}(\Omega_T), \Phi \in C_{0}(\Omega_T; C_{per} (0,1) \times C^{1}_{per}(Y)) approximating u,U_{1} respectively, viz. \begin{eqnarray} &\| \phi -u \|_{L^{p}(0,T;W^{1,p}(\Omega))} \le \delta &\\ &\| \Phi - U_{1} \|_{L^{p}(\Omega_T \times (0,1);W^{1,p}_{per}(Y))} \le \delta\,.. & \end{eqnarray} Define, \eta_{\varepsilon } as in (\ref{eqn0}) with \lambda = 0. Then we have the following lemma. \begin{lemma} Let \delta >0 be fixed. Fix \phi, \Phi as above. Under the strong monotonicity assumption (A5), we have $$\limsup_{\varepsilon \to 0} \|\nabla u_\varepsilon - \eta_{\varepsilon }\|_{p,\Omega_T} \le O(\delta^{\frac{r_{0}}{p}})$$ where r_{0}=min(r,1). \end{lemma} \paragraph{Proof:} We will use some of the calculations from Proposition \ref{prop}. For that we observe that the regularity that we have now taken for \phi,\Phi would have been sufficient in the proof of that proposition also. Let J_{\varepsilon } be as in the proof of Proposition \ref{prop}. We have, by the strong monotonicity condition (A5), \begin{eqnarray} \nonumber \alpha \| \nabla u_\varepsilon - \eta_{\varepsilon } \|^{p}_{p,\Omega_T} & \le & \int_{\Omega_{T}}(a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\eta_{\varepsilon })).(\nabla u_\varepsilon -\eta_{\varepsilon }) \, dx \, dt\\ & \doteq & K_{\varepsilon } \end{eqnarray} Now, \begin{eqnarray*} K_{\varepsilon } & = &\int_{\Omega_{T}} ([a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi,\eta_{\varepsilon })] \\ & & + [a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},\phi,\eta_{\varepsilon })-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon })] \\ & & +[a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon })-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\eta_{\varepsilon })]).(\nabla u_\varepsilon -\eta_{\varepsilon }) \, dx\, dt\\ [2mm] & \le &\!\! J_{\varepsilon } + \alpha^{-1}\|u - \phi\|^{r}_{p} [(m(\Omega_T)^{\frac{p-1-r}{p}}+\|u\|^{p-1-r}_{p} +\|\phi\|^{p-1-r}_{p}+\|\eta_{\varepsilon }\|^{p-1-r}_{p})\\ & & \times (\sup_{\varepsilon } \|\nabla u_\varepsilon\|_{p} + \| \eta_{\varepsilon } \|_{p})]\\ & &+ \alpha^{-1}\|u_\varepsilon - u\|^{r}_{p} [(m(\Omega_T)^{\frac{p-1-r}{p}}+\|u_\varepsilon\|^{p-1-r}_{p} + \|u\|^{p-1-r}_{p}+\|\eta_{\varepsilon }\|^{p-1-r}_{p}) \\ &&\times(\sup_{\varepsilon }\|\nabla u_\varepsilon\|_{p} + \| \eta_{\varepsilon } \|_{p})]\\ [2mm] &\le &J_{\varepsilon } + C \delta^{r} (C+\|\eta_{\varepsilon }\|^{p-1-r}_{p})(C+\| \eta_{\varepsilon } \|_{p})\\ & &+ C \|u-u_\varepsilon\|^{r}_{p}(C+\|u_\varepsilon\|^{p-1-r}_{p}+\|\eta_{\varepsilon }\|^{p-1-r}_{p})(C+\| \eta_{\varepsilon } \|_{p}). \end{eqnarray*} Letting \varepsilon \to 0 we get, \begin{eqnarray*} \limsup_{\varepsilon \to 0} K_{\varepsilon } & \le & \lim_{\varepsilon \to 0} J_{\varepsilon } + C \delta^{r}(C+\|\nabla_{x} \phi + \nabla_{y} \Phi \|^{p-1-r}_{p}).(C + \|\nabla_{x} \phi + \nabla_{y} \Phi \|_{p})\\ & \le & \lim_{\varepsilon \to 0} J_{\varepsilon } + C \delta^{r}, \end{eqnarray*} where the last constant C is independent of \delta for 0<\delta \le 1, as the norms of \phi, \Phi are close to the norms of u,U_{1} respectively. Also, we know from the proof of Proposition \ref{prop} that \begin{eqnarray*} \lim_{\varepsilon \to 0} J_{\varepsilon } & = & \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a(y,s,u,\nabla_{x} u + \nabla_{y} U_{1}).(\nabla_{x} u -\nabla_{x} \phi) \, dy \, ds \, dx \, dt\\ & &- \int_{\Omega_{T}} \int_{Y} \int_{0}^{1} a(y,s,\phi,\nabla_{x} \phi + \nabla_{y} \Phi).(\nabla_{x} u -\nabla_{x} \phi + \nabla_{y} U_{1} -\nabla_{y} \Phi) \\ [2mm] &\le & \! \! C \| \nabla_{x} u -\nabla_{x} \phi \|_{p}(1 + \|u \|^{p-1}_{p} + \|\nabla_{x} u\|^{p-1}_{p}+\|\nabla_{y} U_{1}\|^{p-1}_{p})\\ & &\!\! + C(\| \nabla_{x} u -\nabla_{x} \phi \|_{p} + \|\nabla_{y} U_{1} -\nabla_{y} \Phi \|_{p})\\ & & \times (1 + \|\phi \|^{p-1}_{p} + \|\nabla_{x} \phi \|^{p-1}_{p}+\|\nabla_{y} \Phi \|^{p-1}_{p})\\ [2mm] &\le &C \delta \end{eqnarray*} by the choice of \phi, \Phi. Thus,$$ \limsup_{\varepsilon \to 0} \| \nabla u_\varepsilon - \eta_{\varepsilon }\|^{p}_{p} \le \lim_{\varepsilon \to 0} J_{\varepsilon } + C \delta^{r} \le C(\delta + \delta^{r}) \le C \delta^{r_{0}}$$for$0< \delta \le 1$. Hence the lemma. \hfill$\diamondsuit$\smallskip Under the stronger continuity assumption on$a$, viz. $$\label{a5} |a(y,s,\mu,\lambda_{1})-a(y,s,\mu,\lambda_{2})| \le |\lambda_{1}-\lambda_{2}|^{r} (1+|\mu|^{p-1-r}+|\lambda_{1}|^{p-1-r}+|\lambda_{2}|^{p-1-r})$$ for all$(y,s,\mu,\lambda_{1},\lambda_{2})$, we have the following corollary. \begin{coro} Assume (\ref{a5}). Then, we have $$\label{sconv3} \lim_{\varepsilon \to 0} \| a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon })\|_{p^{*}} \le \delta^{\frac{r_{0}^{2}}{p}}$$ \end{coro} \paragraph{Proof:} Note that, \begin{eqnarray*} \lefteqn{\int_{\Omega_{T}} |a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon }|^{p^{*}} \, dx \, dt}\\ & \le & C \int_{\Omega_{T}} |a(\frac{x}{\varepsilon},\frac{t}{\varepsilon}, u_\varepsilon,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon}, \frac{t}{\varepsilon},u,\nabla u_\varepsilon)|^{p^{*}} \, dx \, dt \\ & & +C \int_{\Omega_{T}} |a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\nabla u_\varepsilon)-a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon })|^{p^{*}} \, dx \, dt\\ &\le & C \|u_\varepsilon-u\|^{\frac{r}{p}}_{p,\Omega_T}+ \|\nabla u_\varepsilon-\eta_{\varepsilon } \|^{\frac{r}{p}}_{p,\Omega_T}(C+ \|\eta_{\varepsilon }\|^{p-1-r}_{p,\Omega_T}) \end{eqnarray*} where in the last inequality we have used (\ref{growth2}) and (\ref{a5}) and the fact that the sequences$u_\varepsilon$,$\nabla u_\varepsilon$are bounded. Letting$\varepsilon \to 0$, using Theorem \ref{corrt}, we get \begin{eqnarray*} \lim_{\varepsilon \to 0} \| a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u_\varepsilon,\nabla u_\varepsilon) - a(\frac{x}{\varepsilon},\frac{t}{\varepsilon},u,\eta_{\varepsilon })\|_{p^{*}} & \le & \delta^{\frac{r^{2}_{0}}{p}} (C + \|\nabla_{x} \phi + \nabla_{y} \Phi \|^{p-1-r}_{p})\\ & \le & C \delta^{\frac{r^{2}_{0}}{p}}. \end{eqnarray*} This completes the proof. \hfill$\diamondsuit$\paragraph{Proof of Theorem \ref{corrt}:} If$u,U_{1}$are sufficiently smooth we can take$\phi = u$and$\Phi = U_{1}$in the proof of the previous lemma and (\ref{sconv2}) follows as we can take$\delta \equiv 0$. The convergence in (\ref{sconv1}) is obvious from the strong convergence of$u_\varepsilon$to$u$in$L^{p}(\Omega_T)$. \hfill$\diamondsuit\$ \paragraph{Acknowledgement:} The authors would like to thank the referee for the comments. 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Correctors for flow in a partially fissured medium, {\em Electronic Journal of Differential Equations}, {\bf 1999 No. 27}(1999), 1-15. \end{thebibliography} \noindent{\sc A. K. Nandakumaran } \\ Department of Mathematics \\ Indian Institute of Science \\ Bangalore- 560 012, India \\ e-mail: nands@math.iisc.ernet.in \smallskip \noindent{\sc M. Rajesh } \\ Department of Mathematics \\ Indian Institute of Science \\ Bangalore- 560 012, India \\ e-mail: rajesh@math.iisc.ernet.in \end{document}