\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 06, pp. 1--27.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2012/06\hfil Homogenization and correctors] {Homogenization and correctors for composite media with coated and highly anisotropic fibers} \author[A. Boughammoura \hfil EJDE-2012/06\hfilneg] {Ahmed Boughammoura} \address{Ahmed Boughammoura \newline Institut Sup\'erieur d'Informatique et de Math\'ematiques de Monastir, Tunisia} \email{Ahmed.Boughammoura@laposte.net} \thanks{Submitted August 17, 2011. Published January 10, 2012.} \subjclass[2000]{35B27, 35B40, 35K65, 76M50} \keywords{Homogenization; correctors; monotone problem; composite media; \hfill\break\indent coatings; highly anisotropic fibers} \begin{abstract} This article presents the homogenization of a quasilinear elliptic-parabolic problem in an $\varepsilon$-periodic medium consisting of a set of highly aniso\-tropic fibers surrounded by coating layers, the whole being embedded in a third material having an order $1$ conductivity. The conductivity along the fibers is of order $1$ but the conductivities in the transverse directions and in the coatings are scaled by $\mu=o(\varepsilon)$ and $\varepsilon^p$, as $\varepsilon\to 0$, respectively. The heat flux are quasilinear, monotone functions of the temperature gradient. The heat capacities of the medium components are bounded but may vanish on certain subdomains, so the problem may become degenerate. By using the two-scale convergence method, we can derive the two-scale homogenized systems and prove some corrector-type results depending on the critical value $\gamma=\lim_{\varepsilon\searrow 0}\varepsilon^p/\mu$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of the problem} Homogenization of problems, in composite media with fibers, has been considered in \cite{Bouchite-Bellieud2,Cailleria-Dinari,Briane,Sili} and further references therein. Most of the previous works dealt with the case of the fiber-reinforced composite materials without coatings. Motivated by the study of the effects of the combination of the insulating coatings and the high anisotropy of fibers in the overall behavior of composite media, we propose here, a special class of fibrous structure exhibiting non-standard effective models. Especially, in the present work, we consider the homogenization of a quasilinear elliptic-parabolic problem in a three-phase conducting composite. One of the constituent materials corresponds to a set of fibers surrounded by a second material which works as an insulating or coated layers, and the whole is being embedded in a third material termed matrix. The fibers are considered to be highly anisotropic, with a longitudinal order $1$ conductivity and a very low conductivity in the transverse directions. The conductivity of the matrix is of order $1$ but becomes very small in the coatings. We shall refer to such material as a composite medium with coated and highly anisotropic fibers. In \cite{bough2008}, the author has dealt with the linear case. Here, we continue this investigation by studying the case where the heat flux are non-linear functions of the temperature gradient. One common peculiarity of \cite{bough2008} and the present work is that the heat capacities $c_j, j=1,2,3$ are assumed to degenerate at some subdomains and even to vanish in the whole domain. Thus, our problem covers the quasilinear elliptic equation as well as the quasilinear parabolic one in a composite medium with coated and highly anisotropic fibers. The geometry of the medium is the same as in \cite{bough2008}. We shall recall it and keep globally the same notations. We denote by $\widetilde{Y}$ and ${Y}$ the cubes $]-\frac{1}{2},\frac{1}{2}[^{2}$ and $]-\frac{1}{2},\frac{1}{2}[^{3}$ respectively, thus ${Y}=\widetilde{Y}\times {\mathrm I}$, ${\mathrm I}=]-\frac{1}{2},\frac{1}{2}[$. We assume that $\widetilde{Y}$is partitioned as $\widetilde{Y}=\widetilde{Y}_{1}\cup \widetilde{Y}_{13} \cup \widetilde{Y}_{3}\cup \widetilde{Y}_{23}\cup \widetilde{Y}_{2}$ where $\widetilde{Y}_{1}, \widetilde{Y}_{2}, \widetilde{Y}_{3}$ are three connected open subsets such that $ \widetilde{Y}_{1}\cap \widetilde{Y}_{2}=\emptyset$, $\partial\widetilde{Y}\cap\widetilde{Y}_{3}=\emptyset$ and where $\widetilde{Y}_{\alpha 3},\ \alpha=1,2$ is the interface between $\widetilde{Y}_{\alpha}$ and $\widetilde{Y}_{3}$; thus $\widetilde{Y}_{3}$ separates $\widetilde{Y}_{1}$ and $\widetilde{Y}_{2}$ (see Figure 1). For $i=1,2,3$ we denote $\chi_i $ the characteristic function of $Y_i:=\widetilde{Y_i}\times {\mathrm I}$ and $\theta_1,\theta_2,\theta_3$ their respective Lebesgue measures which are supposed to be of the same magnitude order. Let $ \widetilde E_i$ the $\mathbb {Z}^{2}$-translates of $\widetilde Y_i$ (i.e., $\widetilde E_i:= \widetilde Y_i+\mathbb {Z}^{2}$) and $\widetilde\Gamma_{\alpha 3}$, $\alpha=1,2$ the surface separating $\widetilde E_\alpha$ and $\widetilde E_3$. \emph{We shall assume that only $\widetilde E_2$ is connected}. We introduce the contracted sets $ \widetilde Y_i^{\varepsilon}:={\varepsilon}\widetilde Y_i$, $\widetilde E_i^{\varepsilon}:={\varepsilon} \widetilde E_i$, $i=1,2,3$ and $\widetilde \Gamma_{\alpha 3}^{\varepsilon} :={\varepsilon} \widetilde\Gamma_{\alpha 3}$, $\alpha=1,2,$ where ${\varepsilon}$ is a small positive parameter. Now, let $\widetilde\Omega$ be a regular bounded domain in $\mathbb {R}^{2}$. We denote by $\widetilde\Omega_i^{\varepsilon}:=\widetilde\Omega \cap \widetilde E_i^{\varepsilon}$, and $\widetilde S_{\alpha 3}^{\varepsilon}:= \widetilde\Omega\cap\widetilde\Gamma_{\alpha 3}$. Finally, let $\Omega:= \widetilde\Omega\times {\mathrm I}$ be the cylinder having a base $\widetilde\Omega$ and a height $1$ and $\Omega_i^{\varepsilon}:=\widetilde\Omega_i^{\varepsilon}\times {\mathrm I}$, $i=1,2,3$. Henceforth, $x=(\widetilde x,x_3)$ and $y=(\widetilde y,y_3)$ denote points of $\mathbb {R}^3$ and $Y$ respectively and by $\widetilde{y}$ and $\widetilde{x}$ we denote the transverse vectors $(y_1,y_{2})$ and $(x_1, x_{2})$ respectively. We use the notation $\partial_{x_i}$ for the partial derivative with respect to $x_i$. 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\linethickness{0.3mm} \multiput(81.24,39.38)(0.13,0.12){5}{\line(1,0){0.13}} \linethickness{0.3mm} \multiput(70,23.12)(0.12,0.12){109}{\line(0,1){0.12}} \linethickness{0.3mm} \multiput(70.62,22.5)(0.12,0.12){104}{\line(1,0){0.12}} \linethickness{0.3mm} \multiput(72.5,23.12)(0.12,0.12){89}{\line(0,1){0.12}} \linethickness{0.3mm} \multiput(73.74,23.12)(0.12,0.12){78}{\line(1,0){0.12}} \linethickness{0.3mm} \multiput(75.62,23.76)(0.12,0.12){57}{\line(1,0){0.12}} \linethickness{0.3mm} \multiput(77.5,24.38)(0.12,0.12){31}{\line(0,1){0.12}} \end{picture} \caption{A typical basic cell $\widetilde{Y}$} \label{fig1} \end{figure} Let $p>1$ be a real number and let $p'$ its conjugate: $p=p'(p-1)$. For $k=1,2,3$, let $c_k \in L^\infty (\mathbb{R}^{3})$ be the heat capacity of the $k$-th component. These functions are $Y$-periodic with respect to $y$ with a period $Y$ and satisfy the following assumption: \begin{itemize} \item[(A1)] $0 \leq c_k(y)$ a.e. $y \in Y$, $k=1,2,3$. %\label{a1} \end{itemize} The corresponding $\varepsilon$-periodic coefficients are defined by \begin{equation} c_k^\varepsilon(x)=c_k(\frac{x}{\varepsilon}),\quad x \in \Omega_k^\varepsilon,\; k=1,2,3. \end{equation} Concerning the heat flux, we shall suppose that they are given by three non-linear $Y$-periodic vectorial functions \begin{equation} \mathbb{A}_k(y,\xi) : \mathbb{R}^{3}\times\mathbb{R}^{3}\to \mathbb{R}^{3},\quad k=1,2,3, \end{equation} satisfying the following assumptions \begin{itemize} \item[(A2)] for all $\xi\in\mathbb{R}^{3}$, the function $ y\mapsto \mathbb{A}_k (y,\xi)$ is measurable and $Y$-periodic, \item[(A3)] for a.e. $y\in Y$, the function $ \xi\mapsto \mathbb{A}_k (y,\xi)$ is continuous, \label{a2} \item[(A4)] there exist a constant $c_0>0$ and $p>2$ such that, for all $\xi\in\mathbb{R}^{3}$, $$ 0\leq c_0|\xi|^p\leq \mathbb{A}_k(y,\xi).\xi $$ \item[(A5)] there exist a constant $c>0$ and $p>2$ such that for all $\xi\in\mathbb{R}^{3}$, $$ |\mathbb{A}_k(y,\xi)|\leq c(1+|\xi|^{p-1}), $$ \item[(A6)] the operators $\mathbb{A}_k$ are strictly monotone; i.e., for a.e. $y\in Y$, $$ (\mathbb{A}_k(y,\xi)-\mathbb{A}_k(y,\eta)). (\xi-\eta)>0,\quad \forall \xi\ne \eta\quad \text{in } \mathbb{R}^{3}. $$ \end{itemize} To prove the corrector results, we need to assume stronger hypotheses of monotonicity: \begin{itemize} \item[(A5')] there exist a constant $K_1>0$ such that, for $\xi,\eta\in \mathbb{R}^{3}$ and a.e. $y\in Y$, $$ |\mathbb{A}_k(y,\xi)- \mathbb{A}_k(y,\eta)|\leq K_1 (|\xi|+|\eta|)^{p-2}|\xi-\eta|, $$ \item[(A6')] there exist a constant $K_2>0$ such that, for $\xi,\eta\in \mathbb{R}^{3}$ and a.e. $y\in Y$, $$ (\mathbb{A}_k(y,\xi)-\mathbb{A}_k(y,\eta)).(\xi-\eta) \geq K_2(|\xi|+|\eta|)^{p-2}|\xi-\eta|^2. $$ \item[(A7)] The function $\mathbb{A}_1$ is independent of the vertical coordinate and has the following form $$ \mathbb{A}_1(y,\xi):=\mathbb{A}_1(\widetilde y,\xi) =\begin{pmatrix} \widetilde{\mathbb{A}}_1(\widetilde y,\widetilde\xi)\\ \mathbb{A}_{13}(\widetilde y,\xi_3) \end{pmatrix}. $$ \end{itemize} Obviously, the functions \begin{gather*} \widetilde{\mathbb{A}}_1(\widetilde y,\widetilde\xi) : \mathbb{R}^2\times\mathbb{R}^2\to \mathbb{R}^2,\\ \mathbb{A}_{13}(\widetilde y,\xi_3) :\ \mathbb{R}^2\times\mathbb{R}\to \mathbb{R} \end{gather*} satisfy assumptions (A4)--(A6) by choosing $\xi=(\widetilde\xi,0)$ and $(0,\xi_3)$ respectively. An example of $\mathbb{A}_k$ satisfy the assumptions (A2)--(A7) is $$ \mathbb{A}_1(y,\xi) =\begin{pmatrix} |\widetilde\xi|^{p-2}\widetilde\xi\\ |\xi_3|^{p-2}\xi_3 \end{pmatrix},\quad \mathbb{A}_\alpha(y,\xi) =|\xi|^{p-2}\xi,\quad \alpha=2,3; $$ i.e., the corresponding $p$-Laplacian operators. Then, the diffusion through the material filling the sets $E_1^\varepsilon$, $E_2^\varepsilon$ and $E_3^\varepsilon$ is, respectively, $$ \mathbb{A}_1^{\varepsilon} (x,\xi):=\begin{pmatrix} \mu(\varepsilon){\widetilde{\mathbb{A}}_1}^\varepsilon(x,\widetilde\xi)\\ \mathbb{A}_{13}^\varepsilon(x,\xi_3) \end{pmatrix},\quad \mathbb{A}_2^\varepsilon(x,\xi):=\mathbb{A}_2(\frac{x}{\varepsilon},\xi),\quad \mathbb{A}_3^\varepsilon(x,\xi):=\mathbb{A}_3(\frac{x}{\varepsilon},\xi), $$ where $$ {\widetilde{\mathbb{A}}_1}^\varepsilon(x,\widetilde\xi) ={\widetilde{\mathbb{A}}_1}(\frac{\widetilde x}{\varepsilon},\widetilde\xi),\quad \mathbb{A}_{13}^\varepsilon(x,\xi_3) =\mathbb{A}_{13}(\frac{\widetilde x}{\varepsilon},\xi_3). $$ The global diffusion and the heat capacity of the medium is respectively \begin{gather*} \mathbb{A}^{\varepsilon} (x,\xi)=\sum_{k=1}^2\chi_k^\varepsilon(x)\mathbb{A}_k^{\varepsilon} (x,\xi)+\varepsilon^p\chi_3^\varepsilon(x)\mathbb{A}_3^{\varepsilon} (x,\xi),\\ c^\varepsilon(x)=\sum_{k=1}^3\chi_k^\varepsilon(x)c_k^\varepsilon(x). \end{gather*} Let us assume that the lateral and bottom boundaries of $\Omega$ are maintained at a fixed temperature (homogeneous Dirichlet condition), while the top boundary is insulated (homogeneous Neumann condition), and that the initial distribution of the temperature on $\Omega$ is given for every $\varepsilon$ as $$ u_0^\varepsilon(x)=\sum_{k=1}^3 \chi_k^\varepsilon(x)u_{0k}^\varepsilon(x). $$ Then, the evolution of the temperature $u^\varepsilon(t,x)$ is governed by the following initial boundary value problem, being in fact, a sequence of problems $({\mathcal P}_\varepsilon)$ indexed by $\varepsilon$: \begin{equation}\label{Pep} %({\mathcal P}_\varepsilon) \begin{gathered} \frac{\partial}{\partial t}(c^\varepsilon (x )u^\varepsilon (t,x )) = \operatorname{div}(\mathbb{A}^\varepsilon (x,\nabla u^\varepsilon (t,x ))) + f^\varepsilon (t,x ), \quad x \in \Omega,\; t>0 , \\ u^\varepsilon(t, x ) = 0, \ x \in \partial\Omega \cap\{-\frac{1}{2}\leq x_3 < \frac{1}{2}\}=: \Gamma_{LB} , \quad t>0, \\ {\mathbb{A}}^\varepsilon(x,\nabla u^\varepsilon (t, x )).n=0, \quad x \in \partial\Omega\setminus \Gamma_{LB} ,\quad t>0, \\ u^\varepsilon(0,x ) = u_0^\varepsilon(x ), \quad x \in \Omega, \end{gathered} \end{equation} where $n$ denotes the outward normal to the boundary of $\Omega$, the subscript $L$ (resp. $B$) stands for lateral (resp. bottom) boundary and $f^\varepsilon\in L^{p'}(0,T;L^{p'}(\Omega))$ represents a given time-dependent heat source. The precise meaning of the initial condition will be done in the following section. In the linear context, models of particular interest are developed by Mabrouk-Samadi \cite{msam}, Mabrouk-Boughammoura \cite{mbou2} and Showalter-Visarraga \cite{showvisa} for the so-called highly heteregeneous medium which consits of two connected ``hard" components having comparable conductivities, separated by a third ``soft" material having a much lower conductivity. The common point of these works is that the three phases have only highly contrasting isotropic conductivities. These models do not display a directional dependence of the effective fields in the resulting limit problems. However, in the present model and in \cite{bough2008}, one of the phases (the fibers) have also highly anisotropic conductivity. This ``partially" highly anisotropy in the fibers leads to some kind of directional dependence on the macro and micro variables. Mathematically, the combination of the ``partially" highly anisotropy and the insulating coatings poses an interesting challenge in the homogenization process. In particular, we will see, in the case $ \gamma\in \mathbb{R}^*_+$, that the resulting two-scale homogenized systems is ``strongly" influenced by this combination : the effective temperature field is obtained by solving a homogenized problem in the domain $\Omega$ and an auxiliary problem in the coated fiber $\overline{Y_1}\cup Y_3$ with a non-standard boundary conditions across the interface between the fiber and the coating (see \eqref{hom1-gammaY13} and Remark \ref{rkp}). Hence, the main feature of the present work is to provide ``rigorous" models for quasilinear heat transfer problem in fibrous composite materials taking in account the influence of the physical properties, at the micro-scale, of the coating and the fiber. In particular, we derive some new effective interface conditions which describe the interaction between the heat transfer processes of conduction in the fibers and the coatings (see \eqref{transmission2} and \eqref{transmission1}). Furthermore, we improve these models by some corrector-type results. Finally, the closest work, as far as we know, to ours was done by Mabrouk \cite{Mabrouk}, in which the author studied the homogenization of a nonlinear degenerate heat transfer problem in a highly heterogeneous medium. Although the mathematical framework used in \cite{Mabrouk} is closely similar, the two situations are clearly distinct in the geometry of the microstructure. Moreover, the homogenized results of \cite{Mabrouk} are recovered, here, when $\gamma:=\lim_{\varepsilon\to 0}\varepsilon^p/\mu=0$ by replacing, formally, the operator $\partial_{x_3}$ by $\nabla$. However, our results in the case $0<\gamma<\infty$ can not be obtained by the physical setting considered in \cite{Mabrouk}. The corrector results are not addressed in \cite{Mabrouk}, that is only weakly convergent results are proved. Yet, here we shall prove strong convergence of the gradients of temperature as well as the heat flux by adding some correctors (see Section 5). Thus, the present study is actually quite different and can be considered as an improvement of \cite{Mabrouk} and a generalization of \cite{bough2008} to quasilinear (monotone operators in the gradient) heat transfer problem in composite materials with coated and highly anisotropic fibers. \section{Mathematical framework} Hereafter, various spaces of functions on $\Omega$ will be used. For each $p>1$, $L^p(\Omega)$ and $W^{1,p}(\Omega)$ are the usual Lebesgue space and Sobolev space respectively. If $\mathbb{R}$ is a Banach space, we denote $\mathbb{R}'$ its dual ; the value of $x'\in \mathbb{R}'$ at $x\in \mathbb{R}$ is denoted $x'(x)$ or sometimes $\langle x',x\rangle_{\mathbb{R}',\mathbb{R}}$. If ${\mathbb H}$ is a Hilbert space, we denote its scalar product $(.,.)_{\mathbb H}$, the dot denotes the usual scalar product in $\mathbb{R}^{3}$. If $\mathbb{R}$ is a Banach space and $X$ is a topological one, $\mathcal{C}(X;\mathbb{R})$ is the space of continuous $\mathbb{R}$-valued functions on $X$ with the sup-norm. For any measure space $\Omega$, $L^p(\Omega;\mathbb{R})$ is the space of $p$-th power norm-summable functions on $\Omega$ with values in $\mathbb{R}$. If $\Omega=(0,T)$ is the time space, we shall often write $L^p(0,T;\mathbb{R})$. In particular, spaces of $Y$-periodic functions will be denoted by a subscript $\sharp$. For example, $\mathcal{C}_\sharp(Y)$ is the Banach space of functions which are defined on $\mathbb{R}^{3}$, continuous and $Y$-periodic. Similarly, ${L^p_\sharp}(Y)$ is the Banach space of functions in ${L^p}_{\mathrm{loc}}(\mathbb{R}^{3})$ which are $Y$-periodic. We endow this space with the norm of ${L^p}(Y)$ and remark that it can be identified with the space of $Y$-periodic extensions to $\mathbb{R}^{3}$ of the functions in ${L^p}(Y)$. Similarly, we define the Banach space $W^{1,p}_\sharp(Y)$ with the usual norm of $W^{1,p}(Y)$. As in \cite{bough2008}, to have a weak formulation of the above problem we shall use the convenient mathematical model built in \cite{Mabrouk}, using the functional framework, developed by Showalter for degenerate parabolic equations (see \cite{showalter}, Section III.6). Let us recall the precise meaning of the weak formulation of the problem we investigate. For more details see \cite{mbou, Mabrouk, bough2008}. Let $p\geq 2$ and $p'$ its conjugate. We define the following Banach spaces \begin{gather*} V=W^{1,p}_{\Gamma_{LB}}(\Omega):=\{ u \in W^{1,p}(\Omega): u=0 \text{ on } \Gamma_{LB}\},\quad \mathcal{V}= L^p(0,T;V),\\ V' , \mathcal{V}'=L^{p'}(0,T;V') \end{gather*} be their dual spaces. For $\varepsilon>0$, let $ C^\varepsilon,A^\varepsilon:\ V\to V'$ be continuous operators, which are defined by the continuous bilinear forms on $V\times V$: \begin{gather*} \langle C^\varepsilon u,v\rangle_{V',V} = c^\varepsilon (u,v):= \int_{\Omega} c^\varepsilon(x)u(x)v(x)dx, \\ \langle A^\varepsilon u,v\rangle_{V',V} = a^\varepsilon (u,v):= \int_{\Omega} \mathbb{A}^\varepsilon(x,\nabla u(x))\nabla v(x) dx. \end{gather*} Let $V_c^\varepsilon $ be the completion of $V$ with the semi-scalar product, defined by the form $c^\varepsilon$ and let $V_c^{'\varepsilon}$ be its dual. Then, we have $V_c^\varepsilon=\{ u : (c^\varepsilon)^{1/2} u \in L^2({\Omega})\}$ and ${V_c^\varepsilon}'= \{ (c^\varepsilon)^{1/2}u,\ u\in L^2({\Omega}) \}$. The operator $C^\varepsilon$ admits a continuous extension from $V_c^\varepsilon$ into ${V_c^\varepsilon}'$ denoted also by $C^\varepsilon$. Given $ f^\varepsilon \in L^{p'}(0,T;L^{p'}({\Omega}))$ or more generally $f^\varepsilon$ in $\mathcal{V}'$ and $w_0^\varepsilon $ in ${V_c^\varepsilon}'$, we are now able to give a weak formulation of the above initial-boundary value problem as the following abstract Cauchy problem \begin{equation}\label{cauchy} \text{Find } u\in \mathcal{V}:\; \frac{d}{dt}{\mathcal{C}}^\varepsilon u + {\mathcal{A}}^\varepsilon u = f^\varepsilon \in \mathcal{V}',\quad {\mathcal{C}}^\varepsilon u(0)= w_0^\varepsilon \in {V_c^\varepsilon}'. \end{equation} Here, ${\mathcal{A}}^\varepsilon$ and ${\mathcal{C}}^\varepsilon$ are the realization of $A^\varepsilon$ and $C^\varepsilon$ as operators from $\mathcal{V}$ to $\mathcal{V}'$, that is precisely $({\mathcal{A}}^\varepsilon u(t),{\mathcal{C}}^\varepsilon u(t)) = (A^\varepsilon (u(t)), C^\varepsilon (u(t)))$ for a.e. $t\in (0,T)$. Let us underline that, in the abstract formulation above, we implicitly require that $\frac{d}{dt}{\mathcal{C}}^\varepsilon u $ belongs to $\mathcal{V}'$. This allows us to give a precise meaning to the initial condition ${\mathcal{C}}^\varepsilon u(0)$. Thus, given $u_0^\varepsilon$ in $V_c^\varepsilon$ and $w_0^\varepsilon$ in ${V_c^\varepsilon}'$ related by $w_0^\varepsilon=c^\varepsilon u_0^\varepsilon$, we can express the initial condition by one of the two equivalent equalities \begin{equation} ({\mathcal{C}}^\varepsilon u^\varepsilon)(0)= C^\varepsilon u^\varepsilon(0) =w_0^\varepsilon \in {V_c^\varepsilon}' \Longleftrightarrow (c^\varepsilon)^{1/2}u^\varepsilon(0)= (c^\varepsilon)^{1/2}u^\varepsilon_0 \in L^2( {\Omega}). \end{equation} We define the Banach space $W_p^\varepsilon (0,T):=\{ u\in \mathcal{V}: \frac{d}{dt}\mathcal{C}^\varepsilon u\in \mathcal{V}' \}$, then, the abstract Cauchy problem can, thereby, be written more explicitly as: Find $u$ in $W_p^\varepsilon(0,T)$ such that \begin{equation} \label{PCep} \begin{gathered} \frac{d}{dt} {\mathcal{C}}^\varepsilon u(t)+{\mathcal{A}}^\varepsilon u(t) = f^\varepsilon(t) \in V' \quad \text{for a.e. } t\in (0,T),\\ {\mathcal{C}}^\varepsilon u(0) = w_0^\varepsilon \quad \text{in } {V_c^\varepsilon}'. \end{gathered} \end{equation} The initial condition is meaningful since $u$ is in $W_p^\varepsilon(0,T)$ then ${\mathcal{C}}^\varepsilon u\in \mathcal{C}(0,T;{V_c^\varepsilon}')$ by \cite[Proposition 6.3]{showalter}. For the present study, we need to recall some equivalent variational formulations of the problem \eqref{PCep} from \cite[Proposition 1.2]{Mabrouk}. \begin{proposition}\label{var-equiv} The following statements are equivalent: \begin{enumerate} \item $u$ is the solution of \eqref{PCep}. \item $u\in W_p^\varepsilon(0,T)$ and for all $v\in W_p^\varepsilon(0,T)$ with $v(T)=0$, we have \begin{equation}\label{equiv2} -\int_0^T \langle u(t), v'(t) \rangle_{V^\varepsilon_c}dt +\int_0^T a^\varepsilon(u(t),v(t))dt=\int_0^Tf^\varepsilon(t)(v(t))dt+w_0^\varepsilon (v(0)), \end{equation} this, by density, holds for all $v\in L^p(0,T; V)$ such that $v'\in L^{p'}(0,T; V^\varepsilon_c)$. \item $u\in { L^p(0,T;V)}$ and for all $v\in W^{1,p}(0,T;V),$ we have \begin{equation} \label{formulation-faible} \begin{split} &-\int_Q c^\varepsilon u v'dx\,dt + \mu(\varepsilon) \int_{Q_1^\varepsilon} {\widetilde{\mathbb{A}}_1}^\varepsilon({\widetilde x}, \nabla_{\widetilde x} u).\nabla_{\widetilde x}v dx\,dt\\ &+ \int_{Q_1^\varepsilon} \mathbb{A}_{13}^\varepsilon (\widetilde x,\partial_{{x_3}} { u}).\partial_{{x_3}} {v} dx\,dt \\ &+ \int_{Q_2^\varepsilon} \mathbb{A}_2^\varepsilon({x},\nabla_{ x} u). \nabla_{ x}v\,dx\,dt + \varepsilon^p \int_{Q_3^\varepsilon} \mathbb{A}_3^\varepsilon({x},\nabla_{x} u).\nabla_{x}v\, dx\,dt \\ & = \int_Q fv\,dx\,dt-\int_\Omega c^\varepsilon u(T,x) v(T,x)dx +\int_\Omega c^\varepsilon u_0^\varepsilon v(0,x)dx. \end{split} \end{equation} \end{enumerate} \end{proposition} \begin{remark}{\rm For each $\varepsilon>0$, the operator $\mathcal{A}^\varepsilon: \mathcal{V}\to \mathcal{V}'$ is continuous, monotone, coercive and bounded, and the operator $\mathcal{C}^\varepsilon: \mathcal{V}\to \mathcal{V}'$ is continuous, linear, symmetric and monotone. Hence, the Cauchy problem \eqref{PCep} admits, for each $\varepsilon>0$, a unique solution $u\in W_p^\varepsilon(0,T)$ by \cite[Corollary 6.3]{showalter}. }\end{remark} Throughout this work, we shall assume that $$ u_0^\varepsilon(x)=u_0(x)\in L^p(\Omega). $$ Hereafter, let $ f\in L^{p'}(0,T;L^{p'}({\Omega}))$ be fixed. Our objective is to study the behavior of the sequence $\{ u^\varepsilon \}$ as $ \varepsilon \to 0$ moreover, we prove a corrector results for the gradients and flux under the strong monotonicity conditions (A5') and (A6'). This will be achieved below, in particular, we will show that the limit depends on the critical value $\gamma=\lim_{\varepsilon\downarrow 0} \frac{\varepsilon^p}{\mu }$. Our further analysis will be, as in \cite{bough2008}, based on the method of the two-scale convergence \cite{allaire,nguetseng1}. For the sake of clarity, we recall its definition. \begin{definition}\rm A function $\phi(t,x,y)\in L^p(Q\times Y,\mathcal{C}_\sharp(Y))$ satisfying \begin{equation}\label{cv12sp} \lim_{\varepsilon\to 0}\int_Q \phi(t,x,\frac{x}{\varepsilon})^p\,dt\,dx =\int_Q\int_Y \phi(t,x,y)^p\,dt\,dx\,dy. \end{equation} is called admissible test function. \end{definition} \begin{definition} \rm A sequence $u^\varepsilon$ in $L^p(Q)$ two-scale converges to a function $u^0\in L^p(Q\times Y)$, and we denote this $u^\varepsilon\stackrel{\mathit{2s , p}}{\to }u^0$ ($u^\varepsilon\stackrel{2s}{\to }u^0$ if p=2 ), if, for any $\phi(t,x,y)\in {\mathcal D}(Q,\mathcal{C}_\sharp(Y))$, \begin{equation}\label{cv2sp} \lim_{\varepsilon\to 0}\int_Qu^\varepsilon(t,x)\phi(t,x,\frac{x}{\varepsilon})\,dt\,dx =\int_Q\int_Yu^0(t,x,y)\phi(t,x,y)\,dt\,dx\,dy. \end{equation} \end{definition} Throughout the paper, we denote by C a constant not depending on $\varepsilon$ and whose value may vary from one line to the next. From a bounded sequence in a Lebesgue space, we can take a subsequence that converges weakly, but virtually all the subsequences converge to the same limit as the limiting equations have a unique solution, so we normally ignore to mention the term ``subsequence". \section{A priori estimates} First, we recall the fundamental lemma which generalizes to the case $p>2$ lemma 2.1. of \cite{bough2008}, proved for $p=2$. We shall not give the proof since it involves only minor modifications of the case $p=2$. \begin{lemma}\label{Mabrouk-samadi} There exists a constant $C$ such that, for every $v\in V$, we have \begin{equation} \|v\|^p_{L^p(\Omega)}\leq C(\|\partial_{x_3} v\|^p_{L^p(\Omega_1^\varepsilon)}+ \|\nabla v\|^p_{L^p( \Omega_2^\varepsilon)} + \varepsilon^{p}\|\nabla v\|^p_{L^p(\Omega_3^\varepsilon)}). \end{equation} \end{lemma} The above lemma is used for proving the following a priori estimates. \begin{lemma}\label{lemma-estimates} Let $f^\varepsilon=f$, then, here exists a constant $C$ such that \begin{gather}\label{estimate1} \|u^\varepsilon\|_{L^p(Q)} \leq C, \\ \label{estimate2} (\mu^{1/p}\|\nabla_{\widetilde x} u^\varepsilon\|_{L^p(Q_1^\varepsilon)}, \| \partial_{x_3} u^\varepsilon \|_{L^p(Q_1^\varepsilon)}) \leq C, \\ \label{estimate3} \Big(\| \nabla u^\varepsilon\|_{L^p(Q_2^\varepsilon)}, \varepsilon\| \nabla u^\varepsilon\|_{L^p(Q_3^\varepsilon)}\Big) \leq C, \\ \label{estimate22} \Big(\mu^{1/p'}\|\widetilde{\mathbb{A}}_1^\varepsilon (x,\nabla_{\widetilde x} u^\varepsilon)\|_{L^{p'} (Q_1^\varepsilon)},\|\mathbb{A}_{13}^\varepsilon(x, \partial_{x_3} u^\varepsilon) \|_{L^{p'}(Q_1^\varepsilon)}\Big) \leq C, \\ \label{estimate33} (\|\mathbb{A}_2^\varepsilon(x, \nabla u^\varepsilon)\|_{L^{p'}(Q_2^\varepsilon)}, \varepsilon^{\frac{p}{p'}}\|\mathbb{A}_3^\varepsilon(x, \nabla u^\varepsilon)\|_{L^{p'}(Q_3^\varepsilon)}) \leq C. \end{gather} Moreover, if $00$, \[ \int_0^T\int_\Omega |f(s,x)\|(u^\varepsilon(s,x))|\,dx\,ds \leq \frac{\eta}{p}\|f\|^{p'}_{L^{p'}(Q)} +\frac{1}{p\eta^{p-1}}\|u^\varepsilon\|^{p}_{L^{p}(Q)}. \] Thus, using assumption (A5), \begin{align*} &\mu \int_{Q_1^\varepsilon} |\nabla_{\widetilde x}u^\varepsilon|^p\,dx\,dt +\int_{Q_1^\varepsilon} |\partial_{{x_3}} {u^\varepsilon}|^p \,dx\,dt\\ &+\int_{Q_2^\varepsilon} |\nabla_{ x}u^\varepsilon|^p\,dx\,dt +\varepsilon^p \int_{Q_3^\varepsilon} |\nabla_{x}u^\varepsilon)|^p \,dx\,dt \\ &\leq \frac{\eta}{p}\|f\|^{p'}_{L^{p'}(Q)} +\frac{1}{p\eta^{p-1}}\|u^\varepsilon\|^{p}_{L^{p}(Q)} +C. \end{align*} Since $u^\varepsilon(t,.)\in V$, using lemma \ref{Mabrouk-samadi} in the right hand side, we obtain \begin{align*} &\mu \int_{Q_1^\varepsilon} |\nabla_{\widetilde x}u^\varepsilon|^p\,dx\,dt + \int_{Q_1^\varepsilon} |\partial_{{x_3}} {u^\varepsilon}|^p \,dx\,dt + \int_{Q_2^\varepsilon} |\nabla_{ x}u^\varepsilon|^p\,dx\,dt + \varepsilon^p \int_{Q_3^\varepsilon} |\nabla_{x}u^\varepsilon)|^p \,dx\,dt \\ &\leq C\frac{\eta}{p}+\frac{C}{p\eta^{p-1}}(\|\partial_{x_3} u^\varepsilon\|^p_{L^p(\Omega_1^\varepsilon)} + \|\nabla u^\varepsilon\|^p_{L^p( \Omega_2^\varepsilon)} + \varepsilon^{p}\|\nabla u^\varepsilon\|^p_{L^p(\Omega_3^\varepsilon)}) +C, \end{align*} we can absorb the right-hand side by choosing ${\eta^{p-1}} >1$. Thus \begin{gather*} \|u^\varepsilon\|_{L^p(Q)}\leq C, \\ (\mu\|\nabla_{\widetilde x} u^\varepsilon\|^p_{L^p(Q_1^\varepsilon)}, \| \partial_{x_3} u^\varepsilon \|_{L^p(Q_1^\varepsilon)}, \| \nabla u^\varepsilon\|_{L^p(Q_2^\varepsilon)}, \varepsilon\| \nabla u^\varepsilon\|_{L^p(Q_3^\varepsilon)}) \leq C. \end{gather*} The bounds of $\widetilde{\mathbb{A}}_1^\varepsilon$, $\mathbb{A}_{13}^\varepsilon$ and $\mathbb{A}_{\alpha}^\varepsilon,\ \alpha=2,3$ are obtained using H\"older's inequality and assumption (A6). \end{proof} As a consequence of the a priori estimates mentioned above, we have the following result. \begin{lemma}\label{lemmacv} Let $\gamma:=\lim_{\varepsilon\to 0}\frac{\varepsilon^p}{\mu}$. Assume that $\gamma<+\infty$ and $f^\varepsilon =f$. There exists \begin{gather*} u_2 \in L^p(0,T; V),\quad v_1 \in L^p\big(Q; W^{1,p}_\sharp(\widetilde Y_1)/\mathbb{R}\big),\quad z\in L^p(Q\times Y), \\ (v_2,v_3)\in \prod _{i=2}^3 L^p(Q; W^{1,p}_\sharp(Y_i)/\mathbb{R}),\quad g_k\in L^{p'}(Q\times Y),\ u^*_k\in L^{2}(Q\times Y),\ k=1,2,3, \end{gather*} such that we have the following two-scale convergence holds: \begin{gather*} u^\varepsilon(t,x) \stackrel{\mathit{2s , p}}{\to} \chi_1(y) v_1(t,x,{\widetilde y})+\chi_2(y) u_2(t,x) +\chi_3(y) v_3(t,x,y), \\ \chi_1^\varepsilon(x) (u^\varepsilon(t,x), \varepsilon \nabla_{\widetilde x} u^\varepsilon(x)) \stackrel{\mathit{2s , p}}{\to} \chi_1(y) (v_1(t,x,{\widetilde y}), \nabla_{\widetilde y} v_1(t,x,{\widetilde y})), \\ \chi_1^\varepsilon(x) \partial_{x_3} u^\varepsilon(x) \stackrel{\mathit{2s , p}}{\to} \chi_1(y) z(t,x, y), \quad\text{such that } \partial_{x_3} v_1(t,x,{\widetilde y}) = \int_{\mathrm I} z(t,x,y)dy_N, \\ \chi_2^\varepsilon(x)(u^\varepsilon(t,x), \nabla_x u^\varepsilon(t,x)) \stackrel{\mathit{2s , p}}{\to } \chi_2(y)(u_2(t,x),[\nabla_x u_2(t,x)+ \nabla_y v_2(t,x,y)]), \\ \chi_3^\varepsilon(x)(u^\varepsilon(t,x),\varepsilon \nabla_x u^\varepsilon(t,x)) \stackrel{\mathit{2s , p}}{\to } \chi_3(y) (v_3(t,x,y), \nabla_y v_3 (t,x,y)), \\ \mu^{1/p'}\chi_1^\varepsilon(x)\widetilde{\mathbb{A}}_1^\varepsilon(x,\nabla_{\widetilde x}u^\varepsilon(t,x)) \stackrel{\mathit{2s , p'}}{\to } \chi_1(y) \widetilde g_1(t,x,{ y}), \\ \chi_1^\varepsilon(x)\mathbb{A}_{13}^\varepsilon(x,\partial_{x_3}u^\varepsilon(t,x) ) \stackrel{\mathit{2s , p'}}{\to } \chi_1(y) g_{13 }(t,x,{ y}), \\ \chi_2^\varepsilon(x) \mathbb{A}_2^\varepsilon(x,\nabla_{x}u^\varepsilon(t,x)) \stackrel{\mathit{2s , p'}}{\to } \chi_2(y) g_2(t,x,{ y}), \\ \varepsilon^{\frac{p}{p'}} \chi_3^\varepsilon(x) \mathbb{A}_3^\varepsilon(x,\nabla_{x}u^\varepsilon(t,x)) \stackrel{\mathit{2s , p'}}{\to } \chi_3(y) g_3(t,x,{ y}), \\ \chi_k^\varepsilon(x)(c^\varepsilon_k)^{1/2}u^\varepsilon(T,x) \stackrel{\mathit{2s}}{\to } \chi_k(y)u^*_k(x,y),\quad k=1,2,3,\\ (c^\varepsilon)^{1/2}u^\varepsilon(T,x) \stackrel{\mathit{2s}}{\to } u^*(x,y):= \sum_{k=1}^3\chi_k(y)u^*_k(x,y). \end{gather*} Moreover, there exists a unique function $ w_3\in L^p(Q;W^{1,p}_\sharp(Y_3))$ such that \begin{equation} \begin{gathered} v_3(t,x,y)= u_2(t,x)+ w_3(t,x,y)\ \text{in }Y_3 \\ w_3(t,x,y)= v_1(t,x,{\widetilde y})- u_2(t,x) \ \text{on }{ Y_{13}:={\widetilde Y_{13}}\times {\mathrm I}}\\ w_3(t,x,y)=0\ \text{on }{ Y_{23}:={\widetilde Y_{23}}\times {\mathrm I}} \end{gathered} \end{equation} and $ u^\varepsilon $ converges weakly in $L^p(Q)$ to the function $$ U(t,x)=(1-\theta_1) u_2(t,x)+\int_{{\widetilde Y}_1} v_1(t,x,{\widetilde y})d{\widetilde y}+\int_{Y_3} w_3(t,x,y)dy. $$ \end{lemma} The proof of the above lemma is the same as that of \cite[Lemma 2.3]{bough2008}, we omit it. \begin{remark}{\rm If $\gamma=\infty$, the sequence $$ \varepsilon\chi_1^{\varepsilon}\nabla_{\widetilde x}{u^\varepsilon} =\frac{\varepsilon}{{\mu^{1/p}}}{\mu^{1/p}} \chi_1^{\varepsilon}\nabla_{\widetilde x}{u^\varepsilon} $$ is not bounded in $L^p(Q,\mathbb{R}^2)$ in general. The scaled sequence $ \frac{{\mu^{1/p}}}{\varepsilon}\chi_1^{\varepsilon}{u^\varepsilon}$ converges strongly to zero in $L^p(Q)$ as $\varepsilon\to 0$ since $\|\chi_1^{\varepsilon}{u^\varepsilon}\|_{L^p(Q)}\leq C$. Thus, hereafter, we shall consider only the most interesting cases $\gamma=0$ and $0<\gamma<\infty$. }\end{remark} \section{Homogenization in the case $\gamma=0$} Since $\gamma=0$ and $\sup_\varepsilon\Bigl(\mu\|\nabla {u^\varepsilon}\|_{Q_1^\varepsilon}^p\Bigr) \leq C$, the function $$ \varepsilon\chi_1(\frac{x}{\varepsilon})\nabla_{\widetilde x}{u^\varepsilon}(t,x) =\frac{\varepsilon}{{\mu^{1/p}}}{\mu^{1/p}}\chi_1(\frac{x}{\varepsilon}) \nabla_{\widetilde x}{u^\varepsilon}(t,x) $$ converges strongly to zero in $ L^p(Q;\mathbb{R}^2)$. Thus $\chi_1(y)\nabla_{\widetilde y} v_1(t,x,{\widetilde y})=0$, then $$ \chi_1(y) v_1(t,x,{\widetilde y}):=u_1(t,x) $$ and especially \begin{equation} \label{Ugamma=0} U(t,x)=\theta_1u_1(t,x) +(1-\theta_1) u_2(t,x)+\int_{Y_3} w_3(t,x,y)dy. \end{equation} Moreover, the sequence $$ \mu \chi_1^\varepsilon\widetilde{\mathbb{A}}_{1}(x,\nabla_{\widetilde x} u^\varepsilon) =\mu^{1/p}\mu^{1/p'}\chi_1^\varepsilon\widetilde{\mathbb{A}}_{1} (x,\nabla_{\widetilde x}u^\varepsilon)\to 0 $$ strongly in $L^{p'}(Q)$. Now, for every datum $Z\in\mathbb{R}^{3}$, let \begin{equation}\label{a1hom} \mathbb{A}_{13}^{\mathrm{hom}}(Z):=\int_{\widetilde{Y_1}} \mathbb{A}_{13}(\widetilde y, Z)d{\widetilde y}, \end{equation} and let $w_{2,Z}$ be the unique solution ($v_2$) of the following cellular problem \begin{equation}\label{cell12} \begin{gathered} -\operatorname{div}_{y}\Bigl(\mathbb{A}_2 (y,Z+ \nabla_y{w_{2,Z}}) \Bigr)=0 \quad \text{in }Y_2 \\ \mathbb{A}_2 (y,Z+ \nabla_y{w_{2,Z}}).n(y)=0 \quad \text{on }{Y_{2 3}}\\ y\mapsto{w_{2,Z}}(y),\quad \mathbb{A}_2 (y,Z+ \nabla_y{w_{2,Z}}).n(y) \big|_{\partial Y_2\cap \partial Y} \quad Y-\text{periodic}, \end{gathered} \end{equation} we define the function \begin{equation} \label{a2hom} \mathbb{A}_{2}^{\mathrm{hom}}(Z):=\int_{Y_2} \mathbb{A}_{2}(y,Z +\nabla_y{w_{2,Z}}(t,x,y))dy. \end{equation} \begin{theorem}\label{thm1} The functions $(u_\alpha,w_3)\in L^p(0,T; V)\times L^p(Q; W^{1,p}_\sharp(Y_3))$, $\alpha=1,2$ are the unique solutions of the homogenized coupled problems \begin{equation}\label{hom11} \begin{gathered} \begin{aligned} &\widetilde {c_1}\frac{\partial { u_1}}{\partial t}(t,x) -\partial_{x_3}\big(\mathbb{A}_{13}^{\mathrm{hom}} (\partial_{x_3}{u_1}(t,x))\big)\\ &+ \int_{Y_{13}} \mathbb{A}_3(y,\nabla_yw_3(t,x,y)) n_3(y) dS(y) = \theta_1 f \quad \text{in }Q \end{aligned}\\ \begin{aligned} &\widetilde {c_2}\frac{\partial { u_2}}{\partial t}(t,x) -\operatorname{div}_x\big(\mathbb{A}_{2}^{\mathrm{hom}}(\nabla_x{u_2}(t,x))\big)\\ &+ \int_{Y_{23}} \mathbb{A}_3(y,\nabla_yw_3(t,x,y)) n_3(y) dS(y) = \theta_2 f \quad \text{in }Q \end{aligned}\\ \widetilde {c_\alpha}{u_\alpha}(0,x)=\widetilde {c_\alpha}u_0(x) \quad \text{in }\Omega ,\quad \widetilde{c_\alpha}=\int_{Y_\alpha} c_\alpha(y)dy, \\ {u_\alpha}(t,x)=0 \quad \text{on }\Gamma_{LB} \end{gathered} \end{equation} and \begin{equation}\label{homY3var} \begin{gathered} { c_3} (y)(\frac{\partial { u_2}}{\partial t}(t,x) +\frac{\partial { w_3}}{\partial t}(t,x,y)) - \operatorname{div}_y\Bigl( \mathbb{A}_3(y,\nabla_yw_3(t,x,y)) \Bigr) = f \quad \text{in }Y_3 \\ {w_3}(t,x,y)=u_1(t,x)-u_2(t,x)\quad \text{on }Y_{13}\\ {w_3}(t,x,y)=0 \quad \text{on }Y_{23} \\ {c_3}(y){w_3}(0,x,y)={c_3}(y)(u_0(x)-u_2(0,x)),\quad y\in Y_3\\ y\mapsto \mathbb{A}_3(y,\nabla_yw_3(t,x,y)).n(y) \big|_{\partial Y\cap \partial Y_3} \quad Y-\text{periodic} \end{gathered} \end{equation} \end{theorem} \begin{remark} {\rm Let us comment on these results. These problems involve, roughly speaking, three coupled fields : two macroscopic functions ($u_1,u_2$) and a microscopic one $w_3$. Notice that, only the longitudinal heat flux in the fiber is shown to be the unique factor contributing on the effective behavior of the composite (see second term of the first equation in \eqref{hom11}). Moreover, the auxiliary problem \eqref{homY3var} is defined on the surrounding coating ($Y_3$) of the coated fiber ($\overline{Y_1}\cup Y_3$). Besides, there is no heat flux exchange across the fiber-coating interface. }\end{remark} \subsection*{Proof of Theorem \ref{thm1}} Let $C^1_{LB}(\overline{\Omega}) =\{v \in C^1(\overline{\Omega}):v=0\text{ on }\Gamma_{LB}\}$. We shall consider test functions $\psi_\alpha,\psi,\phi_2$ defined as follows: \begin{gather*} \psi_\alpha(t,x)\in W^{1,p}(0,T; C^1_{LB}(\overline{\Omega})) \\ \psi(t,x,y)\in W^{1,p}(0,T; C^1_{LB}(\overline{\Omega}; \mathcal{C}^\infty_\sharp(Y))),\quad \psi(t,x,y)= \psi_\alpha(t,x)\quad\text{in $Y_\alpha$ a.e.} \\ \phi_2(t,x,y)\in {\mathcal D}(Q;\mathcal{C}^\infty_\sharp(Y)) \end{gather*} We define the function $v^\varepsilon(t,x)=\psi(t,x,\frac{x}{\varepsilon}) +\varepsilon\phi_2(t,x,\frac{x}{\varepsilon})$. Then $v^\varepsilon\in W^{1,p}(0,T; V)$, hence $v^\varepsilon$ is an allowable test function. By putting it in the formulation \eqref{formulation-faible}, using the fact that $\mu=\mu^{1/p'}\mu^{1/p}$ and letting $\varepsilon\to 0$, we obtain \begin{align} &-\sum_{\alpha=1}^2 \int_{Q} \int_{Y_\alpha} c_\alpha (y)u_\alpha(t,x) \psi_\alpha'(t,x) \,dt\,dx\,dy \nonumber\\ &- \sum_{\alpha=1}^2 \int_{\Omega} \int_{Y\alpha} c_\alpha (y)u_0(x) \psi_\alpha(0,x) \,dx\,dy \nonumber \\ &+ \sum_{\alpha=1}^2 \int_{\Omega} \int_{Y\alpha} c_\alpha^{1/2} (y)u_\alpha^*(x,y) \psi_\alpha(T,x) \,dx\,dy \nonumber \\ & - \int_{Q} \int_{Y_3} c_3 (y)\Bigl[u_2(t,x)+w_3(t,x,y) \Bigr]\psi'(t,x,y)\,dt\,dx\,dy \nonumber \\ &- \int_{\Omega} \int_{Y_3} c_3 (y)u_0(x)\psi(0,x,y) \,dx\,dy \label{cv}\\ &+ \int_{\Omega} \int_{Y_3} c_3^{1/2} (y)u_3^*(x,y)\psi(T,x,y) \,dx\,dy + \int_{Q} \int_{ Y_1} g_{13}(t,x, y).\partial_{x_3}\psi_1(t,x) \,dt\,dx\,dy \nonumber \\ &+ \int_{Q} \int_{Y_2} g_2(t,x,y).\Bigl[ \nabla_x \psi_2(t,x) +\nabla_y\phi_2(t,x,y)\Bigr] \,dt\,dx\,dy \nonumber \\ &+\int_{Q} \int_{Y_3} g_3(t,x,y). \nabla_y\psi(t,x,y) \,dt\,dx\,dy \nonumber \\ &=\sum_\alpha\int_{Q} \int_{Y_\alpha} f (t,x) \psi_\alpha(t,x) \,dt\,dx\,dy +\int_{Q} \int_{Y_3} f (t,x)\psi(t,x,y)\,dt\,dx\,dy. \nonumber \end{align} (i) Take $\psi_1=0= \psi_2$ and $\phi_2=0$. Then \begin{align*} &- \int_{Q} \int_{Y_3} c_3(y)\Bigl[u_2(t,x)+w_3(t,x,y)\Bigr] \psi'(t,x,y)\,dt\,dx\,dy \\ &- \int_{\Omega} \int_{Y_3} c_3(y)u_0(x)\psi(0,x,y) \,dx\,dy + \int_{\Omega} \int_{Y_3} c_3^{1/2} (y)u_3^*(x,y)\psi(T,x,y) \,dx\,dy \\ & + \int_{Q} \int_{Y_3} g_3(t,x,y). \nabla_y\psi(t,x,y)\,dt\,dx\,dy\\ &=\int_{\Omega_T} \int_{Y_3} f (t,x)\psi(t,x,y)\,dt\,dx\,dy \end{align*} for all $\psi\in W^{1,p}(0,T; C^1_{LB}(\overline{\Omega}; \mathcal{C}^\infty_\sharp(Y)))$ with $\psi(t,x,.)=0$ in $Y_1\cup Y_2$. This remains true, by density, for all $\psi \in W^{1,p}_{LB}(\Omega;W^{1,p}_\sharp(Y)) $, $\psi=0$ on $Y_1\cup Y_2$. For a.e. $x\in\Omega$, we have, thus, a cellular problem on $Y_3$: Find $w_3=w_3(.,x,.)\in L^{p}((0,T); W^{1,p}_\sharp(Y))$ such that \begin{equation}\label{hom11Y3proof} \begin{split} &- \int_0^{T} \int_{Y_3} c_3(y)\Bigl[u_2(t,x)+w_3(t,x,y)\Bigr]\psi'(t,y)\,dt\,dy\\ &- \int_{Y_3} c_3(y)u_0(x)\psi(0,y)dy\\ &+ \int_{Y_3}c_3^{1/2} (y)u_3^*(x,y)\psi(T,y) dy + \int_0^{T} \int_{Y_3} g_3(t,x,y). \nabla_y \psi(t,y) \,dt\,dy \\ &= \int_0^{T} \int_{Y_3} f (t,x)\psi(t,y)\,dt\,dy, \end{split} \end{equation} for all $\psi(t,y) \in W^{1,p}((0,T); W^{1,p}_\sharp(Y))$, with $\psi=0$ on $Y_1\cup Y_2$. Integrating by parts in $t$ and in $y$ successively, we obtain \begin{align*} &\int_0^{T} \int_{Y_3} c_3(y)\frac{\partial}{\partial t} \Bigl[u_2(t,x)+w_3(t,x,y)\Bigr]\psi(t,y)\,dt\,dy \\ &+ \int_{Y_3} c_3(y)(u_0(x)-(u_2(0,x)+w_3(0,x,y)))\psi(0,y)dy\\ &-\int_{Y_3} c_3(y)(u_2(T,x)+w_3(T,x,y))\psi(T,y)dy + \int_{Y_3} c_3^{1/2} (y)u_3^*(x,y)\psi(T,y) dy\\ &- \int_0^{T} \int_{Y_3} \operatorname{div}_y\Bigl( g_3(t,x,y) \Bigr) \psi(t,y) \,dt\,dy \\ &+\int_0^{T} \int_{\partial Y\cap \partial Y_3} \Bigl( g_3(t,x,y) \Bigr).n(y) \psi(t,y) dtdS(y)\\ &= \int_0^{T} \int_{Y_3} f (t,x)\psi(t,y)\,dt\,dy. \end{align*} This is the variational form of an evolution problem on $Y_3$ which we write in a more explicit form ($x$ is a parameter): Find $w_3\in L^p(Q; W^{1,p}_\sharp(Y_3))$ such that $c_3(y)w_3'\in L^p(Q; (W^{1,p}_\sharp(Y_3)))$ and \begin{gather} c_3 (y)\Big(\frac{\partial { u_2}}{\partial t}(t,x) +\frac{\partial { w_3}}{\partial t}(t,x,y)\Big) - \operatorname{div}_y\Bigl( g_3(t,x,y) \Bigr) = f \quad \text{in }Y_3 \nonumber \\ {w_3}(t,x,y)=u_1(t,x)-u_2(t,x)\quad \text{on }Y_{13} \nonumber \\ {w_3}(t,x,y)=0 \quad \text{on }Y_{23} \label{hom11Y3proofvar} \\ {c_3}(y){w_3}(0,x,y)={c_3}(y)(u_0(x)-u_2(0,x))\quad y\in Y_3 \nonumber \\ y\mapsto g_3(t,x,y).n(y)\big|_{\partial Y\cap \partial Y_3} \quad Y-\text{periodic} \nonumber \end{gather} and the final condition $$ c_3^{1/2} (y)u_3^*(x,y)=c_3(y)(u_2(T,x)+w_3(T,x,y)) $$ which is however not a part of the problem. It will be only used below to identify the functions $ g_{13},g_2,g_3$. (ii) Taking now $\phi_2=0$ and $ \psi=0$ in $Y_3\cup Y_1$, and using an integration by parts and the initial and final conditions satisfied by $w_3$, we have \begin{equation}\label{hom11Y2proof} \begin{split} &- \int_{Q} \int_{Y_2} c_2(y)u_2(t,x) \psi_2'(t,x) \,dt\,dx\,dy - \int_{\Omega} \int_{Y_2} c_2(y)u_0(x) \psi_2(0,x)\,dx\,dy \\ &+ \int_{\Omega} \int_{Y_2} c_2^{1/2}(y)u_2^*(x,y) \psi_2(T,x)\,dx\,dy\\ &+ \int_{Q} \int_{Y_2} g_2(t,x,y)\bigl[ \nabla_x{{\psi_2(t,x)}} +\nabla_y{{\phi_2(t,x,y)}}\bigr] \,dt\,dx\,dy \\ &+ \int_{Q} \int_{Y_{23}} g_3(t,x,y).n(y)\psi_2\,dt\,dx\,dS(y)\\ &= \int_{Q} \int_{Y_2} f \psi_2 \,dt\,dx\,dy. \end{split} \end{equation} Taking $\phi_2=0$ and $\psi_2$ arbitrary in $ W^{1,p}(0,T;V)$, we obtain the variational form of the following initial-boundary value problem in $Q$: \begin{gather} \widetilde {c_2}\frac{\partial { u_2}}{\partial t}(t,x) -\operatorname{div}_x\Bigl(\int_{ Y_{2}} g_2(t,x,y) dy\Bigr) \int_{Y_{23}} g_3(t,x,y).n(y) dS(y) = \theta_2 f \quad \text{in }Q \nonumber \\ \widetilde {c_2}{u_2}(0,x)=\widetilde {c_2}u_0(x)\quad \text{in }\Omega,\quad \tilde{c_2}=\int_{Y_2}{c_2}(y)dy \label{hom11proof}\\ {u_2}(t,x)=0 \quad \text{on }\partial \Omega \nonumber \end{gather} and the final condition $$ \widetilde {c_2}{u_2}(T,x)=\int_{Y_2}c^{1/2}(y)u_2^*(x,y)dy. $$ Now, taking $\psi_2=0$ and $\phi_2$ arbitrary in $ {\mathcal D}(Q;W^{1,p}_{\sharp}(Y_2))$, we have $$ \int_{Q} \int_{Y_2} g_2(t,x,y) \nabla_y\phi_2(t,x,y) \,dt\,dx\,dy=0, $$ by integration by parts in $y$, we obtain $$- \int_{ Y_{2}} \operatorname{div}_y\Bigl(g_2(t,x,y) \Bigr)dy + \int_{Q} \int_{\partial Y_2} g_2(t,x,y).n(y) \phi_2(t,x,y) \,dt\,dx\,dS(y)=0 $$ for a.e. $(t,x)\in Q$. This remains true, by density, for all $\phi_2\in L^p(Q;W^{1,p}_{\sharp}(Y_2))$ and gives for each $(t,x)\in Q$ the variational formulation of a cellular problem on $Y_2$, \begin{equation}\label{hom11celly2} \begin{gathered} -\operatorname{div}_y\Bigl(g_2(t,x,y) \Bigr)=0\quad\text{in }Y_2\\ \Bigl( g_2(t,x,y) \Bigr).n(y)\big|_{{\partial Y_2\cap \partial Y_3}} =0\\ y\mapsto g_2(t,x,y).n(y)\big|_{{\partial Y\cap \partial Y_3}} \quad Y-\text{periodic} \end{gathered} \end{equation} Similarly, for all $\psi_1\in W^{1,p}(0,T;W_{LB}^{1,p}(\Omega))$, we obtain \begin{equation}\label{hom11Y1proof} \begin{split} &- \int_{Q} \int_{Y_1} c_1(y)u_1(t,x) \psi_1'(t,x) \,dt\,dx\,dy - \int_{\Omega} \int_{Y_1} c_1(y)u_0(x) \psi_1(0,x)\,dx\,dy \\ &+ \int_{\Omega} \int_{Y_1} c_1^{1/2}(y)u_1^*(x,y) \psi_1(T,x) \,dx\,dy\\ &+ \int_{Q} \int_{\widetilde Y_1} g_{13}(t,x, y) \partial_{x_3}{\psi_1}(t,x) \,dt\,dxd{\widetilde y}\\ &+ \int_{Q} \int_{Y_{13}} g_3(t,x,y).n(y) dS(y)\\ &= \int_{Q} \int_{Y_1} f \psi_1 \,dt\,dx\,dy, \end{split} \end{equation} which is the variational formulation of the following initial-boundary value problem in $Q$. \begin{equation}\label{hom1Y1proof} \begin{gathered} \widetilde {c_1}\frac{\partial { u_1}}{\partial t}(t,x) - \partial_{x_3}({ g_{13}}(t,x,y)) + \int_{Y_{13}} g_3(t,x,y).n(y) dS(y) = \theta_1 f \quad \text{in }\Omega \\ \widetilde {c_1}{u_1}(0,x)=\widetilde {c_1}u_0(x)\quad \text{in }\Omega, \quad \tilde{c_1}=\int_{Y_1}{c_1}(y)dy \\ {u_1}(t,x)=0 \quad \text{on }\partial \Omega \end{gathered} \end{equation} and the final condition $$ \widetilde {c_1}{u_1}(T,x)=\int_{Y_1}c^{1/2}(y)u_1^*(x,y)dy. $$ From this, we obtain the homogenized problem (\ref{hom11}). It remains to identify $g_k$ in terms of $v_k,u_2$. Before proceeding, we prove the following useful identity. \begin{lemma}\label{conservationofenery} \begin{align*} &\sum_{\alpha=1}^2\frac{1}{2} \widetilde c_\alpha \int_\Omega|u_\alpha(T,x)|^2dx+ \frac{1}{2} \int_\Omega\int_{Y_3}c_3(y)|v_3(T,x, y)|^2 dx\,dy \\ &-\frac{1}{2}\sum_{k=1}^3 \widetilde c_k\int_\Omega|u_0(x)|^2dx + \int_Q\int_{Y_1} g_{13}(t,x,y){\partial}_{x_3} u_1(t,x )\,dt\,dx\,dy\\ &+ \int_Q\int_{Y_2} g_2(t,x,y)\Bigl(\nabla_{ x}u_2(t,x) +\nabla_{ y} v_2(t,x, y) \Bigr)\,dt\,dx\,dy\\ &+\int_{Q} \int_{Y_3} g_3(t,x,y). \nabla_yv_3(t,x,y) \,dt\,dx\,dy\\ & =\int_Q f(t,x)U(t,x)dt\,dx. \end{align*} \end{lemma} \begin{proof} We start from the two-scale homogenized problem \eqref{cv} and we consider the sequences $$ \psi_{k, n},\ k=1,2,3,\quad \phi_{2,n},\ \phi_{3,n} $$ such that \begin{enumerate} \item $\psi_{\alpha, n}\to u_\alpha$ in $L^p(0,T;V)$, $\frac{\partial}{\partial t}\psi_{1, n}\to \frac{\partial}{\partial t} u_\alpha1$ in $L^{p'}(0,T;V')$, $\alpha=1,2,$ \item $\psi_{3,n}\to v_3$ in $L^p(0,T;W^{1,p}_\sharp(Y_3))$, $\frac{\partial}{\partial t}\psi_{ 3,n}\to \frac{\partial}{\partial t} v_3$ in $L^{p'}(0,T;W^{1,p}_\sharp(Y_3)')$, \item $\nabla_y\phi_{2, n}\to \nabla_y v_2$ in $L^p(Q\times Y_2)$, $\nabla_y\phi_{3, n}\to \nabla_y v_3$ in $L^p(Q\times Y_3)$. \end{enumerate} Note that the smoothness of the above sequences $\psi_{k, n}$, $k=1,2,3$, $\phi_{2,n}$, $\phi_{3,n}$ implies their two-scale convergence in strong sense to the corresponding limits \cite[Theorem 1.8]{allaire}. Therefore, passing to the limit with respect to $n$ and taking in account of the final conditions, we obtain \begin{align*} &-\sum_{\alpha=1}^2 \int_{Q} \int_{Y_\alpha} c_\alpha (y)u_\alpha(t,x) u_\alpha'(t,x) \,dt\,dx\,dy - \sum_{k=1}^3 \int_{\Omega} \int_{Y\alpha} c_k (y)u_0(x)^2 \,dx\,dy \\ &+ \sum_{\alpha=1}^2 \int_{\Omega} \int_{Y\alpha} c_\alpha (y) u_\alpha(T,x)^2 \,dx\,dy \\ &- \int_{Q} \int_{Y_3} c_3 (y)v_3(t,x,y)v_3'(t,x,y)\,dt\,dx\,dy + \int_{\Omega} \int_{Y_3} c_3 (y)v_3(T,x,y)^2 \,dx\,dy\\ &+ \int_{Q} \int_{ Y_1} g_{13}(t,x, y).\partial_{x_3}u_1(t,x) \,dt\,dx\,dy \\ &+ \int_{Q} \int_{Y_2} g_2(t,x,y).\Bigl[ \nabla_x u_2(t,x) +\nabla_yv_2(t,x,y)\Bigr] \,dt\,dx\,dy \\ &+ \int_{Q} \int_{Y_3} g_3(t,x,y). \nabla_yv_3(t,x,y) \,dt\,dx\,dy\\ &= \int_{Q} f (t,x) U(t,x) \,dt\,dx. \end{align*} Integrating the above equality with respect to the $t$ variable, we obtained the states result. \end{proof} We are now equipped to identify $ g_{13},g_2$ and $g_3$. \subsection*{Identification of $ g_{13},g_2$ and $g_3$} Let $\phi$ and $\Phi$ be in $\mathcal{C}^\infty_0(Q; \mathcal{C}^\infty_\sharp(Y))^N$ and \\ $\mathcal{C}^\infty_0(Q; \mathcal{C}^\infty_\sharp(Y))$ respectively. For $\varepsilon>0$ and $h>0$ we define the test function \begin{equation} \begin{split} \eta^\varepsilon(t,x)&=\chi_1^\varepsilon(x) \begin{pmatrix} 0\\ \partial_{x_3} \end{pmatrix} u_1(t,x)+\chi_2^\varepsilon(x)\nabla_{ x}u_2(t,x)\\ &\quad +\varepsilon\nabla_{ x}\phi(t,x,\frac{x}{\varepsilon}) + h\Phi(t,x,\frac{x}{\varepsilon}). \end{split} \end{equation} Note that $ \eta^\varepsilon$ and (by the continuity assumption) $\mathbb{A}_k^\varepsilon(x, \eta^\varepsilon):=\mathbb{A}_k^\varepsilon(\frac{x}{\varepsilon}, \eta^\varepsilon(t,x))$, $ k=1,2,3$ are admissible test functions (in $L^p(Q)$) for the two-scale convergence and \begin{align*} \eta^\varepsilon(t,x)\stackrel{\mathit{2s , p'}}{\to }\eta(t,x,y) &=:\chi_1(y)\begin{pmatrix} 0\\ \partial_{x_3} \end{pmatrix} u_1(t,x)+\chi_2(y)\nabla_x u_2(t,x)\\ &\quad +\nabla_y\phi(t,x,y)+h\Phi(t,x,y). \end{align*} The monotonicity condition (A6) yields \begin{equation}\label{monotony} \int_Q\Bigl( \mathbb{A}^\varepsilon(x,\nabla_x u^\varepsilon(t,x)) -\mathbb{A}^\varepsilon(x,\eta^\varepsilon(t,x))\Bigr)\Bigl(\nabla_x u^\varepsilon(t,x) -\eta^\varepsilon(t,x) \Bigr)\,dt\,dx\geq 0. \end{equation} Expanding this expression and employing \eqref{tt} yields \begin{align*} &\int_0^T\int_\Omega f(t,x)u^\varepsilon(t,x)\,dx\,dt - \frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega c^\varepsilon(x) (u^\varepsilon_0)^2dx\\ & -\int_Q\Bigl(\mathbb{A}^\varepsilon(x,\nabla_x u^\varepsilon(t,x))\eta^\varepsilon(t,x) + \mathbb{A}^\varepsilon(x,\eta^\varepsilon(t,x))(\nabla_x u^\varepsilon(t,x) -\eta^\varepsilon(t,x))\Bigr)\,dt\,dx\geq 0. \end{align*} Letting $\varepsilon\to 0$, the two-scale convergence of $u^\varepsilon$ and $\chi_k^\varepsilon\mathbb{A}_k^\varepsilon$ and the continuity of $\mathbb{A}_k$ give in the limit \begin{align} &\int_Q f(t,x)U(t,x)\,dx\,dt-\liminf_{\varepsilon\to 0} \frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega\int_Y c(y)dy (u_0)^2dx \nonumber\\ &- \int_Q\int_{Y_1} g_{13}(t,x,y)\Bigl(\partial_{x_3}u_1(t,x) +h\Phi_{N}(t,x,y) \Bigr)\,dt\,dx\,dy \nonumber \\ &- \int_Q\int_{Y_2}g_{2}(t,x,y)\Bigl( \nabla_xu_2(t,x) +\nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \nonumber \\ &- \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl( \nabla_{ y}\phi(t,x,{ y}) +h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \label{eta}\\ &+ \int_Q\int_{Y_1}\mathbb{A}_{13}(x,\eta_{N}(t,x,y)) \Bigl( \partial_{x_3}u_1(t,x)+h\Phi_{N}(t,x,y) \Bigr)\,dt\,dx\,dy \nonumber\\ &+ \int_Q\int_{Y_2}\mathbb{A}_{2}(x,\eta(t,x,y))\Bigl( -\nabla_xu_2(t,x) +\nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \nonumber\\ &+ \int_Q\int_{Y_3}\mathbb{A}_{3}(x,\eta(t,x,y)) \Bigl( \nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \geq 0. \nonumber \end{align} Since $ \mathbb{A}_k $ is continuous we may replace $\phi_\beta,\beta=2,3$ by a sequence converging strongly in $L^p(Q;W^{1,p}_\sharp(Y_\beta)/\mathbb{R})$ to $v_\beta$; thus replacing $\eta(t,x,y)$ in \eqref{eta} with $\nabla_x u_2+\nabla_{y}v_2 +h\Phi $ and $ \nabla_{y}v_3 +h\Phi $ successively and using Lemma \ref{conservationofenery}, the above sum simplified to \begin{align*} &\int_Q\int_{Y_1}\Bigl[\mathbb{A}_{13}\bigl(\widetilde y,\partial_{x_3} u_1(t,x)+h\Phi(t,x,y)\bigr)- g_{13}(t,x,y)\Bigr] h\Phi_{N}(t,x,y) \,dt\,dx\,dy\\ &+ \int_Q\int_{Y_2}\Bigl[\mathbb{A}_{2}\bigl(x,\nabla_x u_2 +\nabla_{y}v_2+h\Phi(t,x,y)\bigr)-g_{2}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy\\ &+ \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl[\mathbb{A}_{3} \bigl(x,\nabla_{y}v_3+h\Phi(t,x,y)\bigr)-g_{3}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy\\ &\geq \frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega\int_Y c(y)dy (u_0)^2dx\\ &\quad +\liminf_{\varepsilon\to 0}\Bigl[\frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx+\frac{1}{2} \int_\Omega\int_Y c(y)dy (u_0)^2dx\Bigr]. \end{align*} Thus, dividing by $h$ and letting $h\to 0$ we see that for every $\Phi$, \begin{equation} \label{etah} \begin{split} &\int_Q\int_{Y_1}\Bigl[\mathbb{A}_{13}\bigl(\widetilde y, \partial_{x_3}u_1(t,x)- g_{13}(t,x,y)\Bigr] h\Phi_{N}(t,x,y) \,dt\,dx\,dy\\ &+ \int_Q\int_{Y_2}\Bigl[\mathbb{A}_{2}\bigl(y,\nabla_x u_2 +\nabla_{y}v_2\bigr)-g_{2}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy\\ &+ \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl[\mathbb{A}_{3} \bigl(y,\nabla_{y}v_3\bigr)-g_{3}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy \geq 0. \end{split} \end{equation} We therefore have proved the desired results, namely, that \begin{gather*} g_{13}(t,x,y)=\mathbb{A}_{13}(\widetilde y,\partial_{ x_3}u_1(t,x))\\ g_{2}(t,x,y)=\mathbb{A}_{2}(y,\nabla_x u_2(t,x)+\nabla_{ y}v_2(t,x,{ y}))\\ g_{3}(t,x,y)=\mathbb{A}_{3}(y,\nabla_{ y}v_3(t,x,{ y})) \end{gather*} Hence equations \eqref{hom11}-\eqref{homY3var} are satisfied. To complete the proof it suffices to show that $\{u_2,v_1,v_2,v_3\}$ is the unique solution of \eqref{hom11}-\eqref{homY3var}. In fact, the uniqueness is a consequence of the strict monotonicity of $\mathbb{A}_k,\ k=1,2,3$. Indeed, if $\{u_2^1,v_1^1,v_2^1,v_3^1\}$ and $\{u_2^2,v_1^2,v_2^2,v_3^2\}$ are two solutions of \eqref{hom11}-\eqref{homY3var}, using \eqref{cv}, by difference we obtain \begin{align*} &-\sum_{\alpha=1}^2 \int_{Q} \int_{Y_\alpha} c_\alpha (y)(u_\alpha^1-u_\alpha^2) \psi_\alpha' \,dt\,dx\,dy\\ &+ \sum_{\alpha=1}^2 \int_{\Omega} \int_{Y\alpha} c_\alpha^{1/2} (y)({u_\alpha^*}^1-{u_\alpha^*}^2) \psi_\alpha(T,x) \,dx\,dy \\ &- \int_{Q} \int_{Y_3} c_3 (y)(v_3^1-v_3^2)\psi' \,dt\,dx\,dy + \int_{\Omega} \int_{Y_3} c_3^{1/2} (y) ({v_3^*}^1-{v_3^*}^2)\psi \,dx\,dy \\ &+ \int_{Q} \int_{ Y_1} (\mathbb{A}_{13}(\widetilde y, \partial_{ x_3}u_1^1 )-\mathbb{A}_{13}(\widetilde y, \partial_{ x_3}u_1^2 )).\partial_{x_3}\psi_1 \,dt\,dx\,dy \\ &+ \int_{Q} \int_{Y_2} (\mathbb{A}_{2}(y,\nabla_x u_2^1 +\nabla_{ y}v_2^1 )-\mathbb{A}_{2}(y,\nabla_x u_2^2 +\nabla_{ y}v_2^2 )).(\nabla_x \psi_2 +\nabla_y\phi_2) \,dt\,dx\,dy\\ &+ \int_{Q} \int_{Y_3} (\mathbb{A}_{3}(y,\nabla_{ y}v_3^1 ) -\mathbb{A}_{3}(y,\nabla_{ y}v_3^2 )). \nabla_y\psi \,dt\,dx\,dy=0. \end{align*} In particular, for $\psi_\alpha=u_\alpha^1-u_\alpha^2$, $\phi_2=v_2^1-v_2^2$ and $\psi=v_3^1-v_3^2$, we obtain, in view of the initial and final conditions satisfied by $u_\alpha,v_3$ and by the strict monotonicity of $\mathbb{A}_k$, $k=1,2,3$: $$ \partial_{ x_3}(u_1^1-u_1^2)=0,\quad \nabla_{ y}(v_3^1-v_3^2)=0\quad \text{in } Q\times {Y_3}, $$ since $(u_1^1-u_1^2)=0$ on $(0,T)\times \Gamma_{LB}$ and $(v_3^1-v_3^2)=0$ on $(0,T)\times {Y_{23}}$, thus $u_1^1=u_1^2$ and $v_3^1=v_3^2$. As a consequence, $$ \int_{Q} \int_{Y_2} (\mathbb{A}_{2}(y,\nabla_x u_2^1 +\nabla_{ y}v_2^1 )-\mathbb{A}_{2}(y,\nabla_x u_2^2 +\nabla_{ y}v_2^2 )).(\nabla_x \psi_2 +\nabla_y\phi_2) \,dt\,dx\,dy=0, $$ so this problem has a unique solution in the space $ W^{1,p}_{LB}(\Omega)\times L^p(\Omega; W^{1,p}_\sharp(Y_2))$ by an application of Lax-Milligram lemma, then $u_2^1=u_2^2$ and $v_2^1=v_2^2$. \section{Homogenization in the case $0<\gamma<\infty$} In this case we shall proof the following result. \begin{theorem}\label{thm2} Functions $(u_2,v_1,w_3)\in L^p(0,T; W^{1,p}_{LB}(\Omega)) \times L^p(Q; W^{1,p}_\sharp(\widetilde Y_1)/\mathbb{R}) \times L^p(Q; W^{1,p}_\sharp(Y_3))$ are the unique solutions of the two-scale homogenized problems \begin{equation} \label{hom1-gammaY2} \begin{gathered} \begin{aligned} &\widetilde {c_2}\frac{\partial { u_2}}{\partial t} -\operatorname{div}_x(\mathbb{A}_{2}^{\mathrm{hom}}(\nabla_x{u_2}(t,x)))\\ & + \int_{Y_{23}} \mathbb{A}_3(y,\nabla_yw_3(t,x,y)).n_3(y) dS(y) = \theta_2 f \quad \text{in }Q \end{aligned} \\ \widetilde {c_2}{u_2}(0,x)=\widetilde {c_2}u_0(x)\quad \text{in }\Omega ,\quad \widetilde{c_2}=\int_{Y_2} c_2(y)dy \\ {u_2}(t,x)=0 \quad \text{on } (0,T)\times \partial \widetilde\Omega\times[0,1[ \end{gathered} \end{equation} where $\mathbb{A}_{2}^{\mathrm{hom}}$ is define by \eqref{a2hom}. \begin{equation}\label{hom1-gammaY13} \begin{gathered} \begin{aligned} & \langle c_1 \rangle_{\mathrm I}({\widetilde y}) \frac{\partial { v_1}}{\partial t}(t,x,{\widetilde y}) - \frac{1}{\gamma^{1/p}} \operatorname{div}_{\widetilde y} \big(\widetilde{\mathbb{A}}_{1}(\widetilde y, \nabla_{\widetilde y}{v_1}(t,x,{\widetilde y}))\big)\\ & - \partial_{x_3}( \mathbb{A}_{13}(\widetilde y, \partial_{x_3}v_1(t,x,{\widetilde y}))) = f\quad \text{in } {\widetilde Y_1} \end{aligned} \\ { c_3} (y)\Big(\frac{\partial { u_2}}{\partial t}(t,x) +\frac{\partial { w_3}}{\partial t}(t,x,y)\Big) - \operatorname{div}_y (\mathbb{A}_3(y,\nabla_yw_3(t,x,y))) = f\quad \text{in } Y_3 \\ \widetilde{\mathbb{A}}_{1}(\widetilde y, \nabla_{\widetilde y}{v_1}).n(\widetilde y)={\gamma}^{1/p} \langle \mathbb{A}_3(y,\nabla_y w_3).n(y) \rangle_{\mathrm I}\quad \text{on }\widetilde Y_{13} \\ {w_3}(t,x,y)=v_1(t,x,\widetilde y)-u_2(t,x)\quad \text{on }Y_{13}, \quad {w_3}(t,x,y)=0 \quad \text{on }Y_{23} \\ \langle c_1\rangle _{\mathrm I} (\widetilde y)v_1 =\langle c_1\rangle_{\mathrm I} (\widetilde y) u_0,\quad \widetilde y\in \widetilde Y_1 \\ {c_3}(y){w_3}(0,x,y)={c_3}(y)(u_0(x)-u_2(0,x)),\quad y\in Y_3\\ y\mapsto \mathbb{A}_3(y,\nabla_yw_3(t,x,y)).n(y) \big|_{{\partial Y\cap \partial Y_3}} \quad Y-\text{periodic} \end{gathered} \end{equation} where $\langle .\rangle_{\mathrm I}$ denotes the integration with respect to $ y_3$ over ${\mathrm I}$. \end{theorem} \begin{remark} \label {rkp}{\rm Contrary to the previous case, here the problems \eqref{hom1-gammaY2}-\eqref{hom1-gammaY13} involve a unique macroscopic function $u_2$ and two microscopic functions $v_1,w_3$. The functions $v_1, w_3$ are ``strongly" coupled via the following non-standard boundary conditions \begin{gather}\label{transmission2} {w_3}(t,x,y)-v_1(t,x,\widetilde y)=u_2(t,x)\quad \text{on }(0,T)\times Y_{13}, \\ \label{transmission1} \widetilde{\mathbb{A}}_{1}(\widetilde y, \nabla_{\widetilde y}{v_1}).n(\widetilde y)={\gamma}^{1/p} \langle \mathbb{A}_3(y,\nabla_y w_3).n(y)\rangle_{\mathrm I} \quad \text{on }(0,T)\times \widetilde Y_{13}. \end{gather} As a consequence, the above interface conditions exhibit a remarkable temperature jump and a transverse heat flux continuity. This, might be interpreted as the combined effects of fiber coatings together with the high anisotropy of the fibers in the overall behavior of the composite. In addition, it should be noted that the auxiliary problem \eqref{hom1-gammaY13} is defined in the coated fiber $\overline{Y_1}\cup Y_3$ and involves both the longitudinal and the transverse thermal conductivities of the fiber. }\end{remark} \subsection*{Proof of Theorem \ref{thm2}} Let $\psi_2,\psi,\phi_2$ be test functions as defined in the proof of Theorem \ref{thm1} and let $$ \psi_1(t,x,\widetilde y)\in W^{1,p}(0,T; {C_{LB}^1} (\overline{\Omega};\mathcal{C}^\infty_\sharp(\widetilde Y))) $$ be a test function such that $ \psi(t,x,y)= \psi_1(t,x,\widetilde y)$ in $Y_1$ a.e. As in the previous case, we take $v^\varepsilon(t,x)= \psi(t,x,\frac{x}{\varepsilon}) +\varepsilon\phi_2(t,x,\frac{x}{\varepsilon})$ in \eqref{formulation-faible} and letting $\varepsilon\to 0$, we deduce the following two-scale limit \begin{align*} &-\int_{Q} \int_{Y_1} c_1 (y)v_1(t,x,\widetilde y) \psi_1'(t,x,\widetilde y) \,dt\,dx\,dy - \int_{\Omega} \int_{Y_1} c_1 (y)u_0(x) \psi_1(0,x,\widetilde y) \,dx\,dy\\ &+ \int_{\Omega} \int_{Y_1} c_1^{1/2} (y)u_1^*(x, y) \psi_1(T,x,\widetilde y) \,dx\,dy\\ &- \int_{Q} \int_{Y_2} c_2 (y)u_2(t,x) \psi_2'(t,x) \,dt\,dx\,dy - \int_{\Omega} \int_{Y2} c_2 (y)u_0(x) \psi_2(0,x) \,dx\,dy\\ &+ \int_{\Omega} \int_{Y_2} c_2^{1/2} (y)u_2^*(x,y)\psi_2(T,x) \,dx\,dy\\ &- \int_{Q} \int_{Y_3} c_3 (y)\Bigl[u_2(t,x)+w_3(t,x,y)\Bigr] \psi'(t,x,y)\,dt\,dx\,dy\\ & - \int_{\Omega} \int_{Y_3} c_3 (y)u_0(x)\psi(0,x,y) \,dx\,dy\\ &+ \int_{\Omega} \int_{Y_3} c_3^{1/2} (y)u_3^*(x,y)\psi(T,x,y) \,dx\,dy\\ &+\frac{1}{\gamma^{1/p}}\int_{Q} \int_{Y_1} \widetilde g_1(t,x,y)\nabla_{\widetilde y}\psi_1(t,x,\widetilde y) \,dt\,dxd{ y}\\ & +\int_{Q} \int_{ Y_1} g_{13}(t,x, y)\partial_{x_3} \psi_1(t,x,\widetilde y) \,dt\,dxd{ y} \\ &+ \int_{Q} \int_{Y_2} g_2(t,x,y)\Bigl[ \nabla_x \psi_2(t,x) +\nabla_y\phi_2(t,x,y)\Bigr] \,dt\,dx\,dy\\ & +\int_{Q} \int_{Y_3} g_3(t,x,y) \nabla_y\psi(t,x,y) \,dt\,dx\,dy \\ &=\int_{Q} \int_{ Y_1} f (t,x) \psi_1(t,x) \,dt\,dxd {y}+ \int_{Q} \int_{Y_2} f (t,x) \psi_2(t,x) \,dt\,dx\,dy\\ &\quad +\int_{Q} \int_{Y_3} f (t,x)\psi(t,x,y)\,dt\,dx\,dy. \end{align*} (i) We choose $\psi_1=0= \psi_2$ and $\phi_2=0$. Then, we have the cellular problem in $Y_3$. (ii) Taking $\phi_2=0$ and $\psi_2=0$ and an integration by parts with respect to $x_3$, we obtain \begin{align*} &\int_{Q} \int_{ Y_1} g_{13}(t,x,y)\partial_{x_3}\psi_1 (t,x,\widetilde y) \,dt\,dx\,dy\\ &= \int_{Q} \int_{\widetilde Y_1} (\int_{\mathrm I} g_{13}(t,x,y)d{y_3})\partial_{x_3}\psi_1(t,x,\widetilde y) \,dt\,dx\,d{\widetilde y}\\ &= -\int_{Q} \int_{\widetilde Y_1} \partial_{x_3}(\int_{\mathrm I} g_{13}(t,x,y)d{y_3}) \psi_1(t,x,\widetilde y) \,dt\,dx\,d{\widetilde y}. \end{align*} After integration by parts with respect to $t$ and ${\widetilde y}$ successively we have \begin{align} &\int_{Q} \int_{\widetilde Y_1} \langle c_1\rangle _{\mathrm I} (\widetilde y)v_1'(t,x,\widetilde y) \psi_1 \,dt\,dxd{\widetilde y} - \int_{\Omega} \int_{\widetilde Y_1} \langle c_1\rangle _{\mathrm I} (\widetilde y)\Bigl(u_0(x)-v_1(0,x,\widetilde y)\Bigr) \psi_1 dx\,d{\widetilde y} \nonumber\\ &+ \int_{\Omega} \int_{Y_1} \Bigl( c_1^{1/2} (y)u_1^*(x, y) - \langle c_1\rangle _{\mathrm I} (\widetilde y)v_1(T,x,\widetilde y) \Bigr)\psi_1(T,x,\widetilde y) \,dx\,dy \nonumber\\ &-\frac{1}{\gamma^{1/p}} \int_{Q} \int_{\widetilde Y_1} \operatorname{div}_{\widetilde y} \bigl(\widetilde g_1(t,x, y)\bigr)\psi_1 d{\widetilde y}\,dt\,dx \nonumber \\ & -\frac{1}{\gamma^{1/p}}\int_{Q} \int_{\widetilde Y_{13}} \widetilde g_1(t,x,y).n(y)\psi_1 dS({\widetilde y}) \,dt\,dx \label{hom1-gammaY2proof} \\ & -\int_{Q} \int_{\widetilde Y_1} \partial_{x_3} \langle g_{13} (t,x,y)\rangle_{\mathrm I} (\widetilde y) \psi_1 \,dt\,dx\,d{\widetilde y} \nonumber \\ &+ \int_{Q} \int_{\widetilde Y_{13}} \langle g_3(t,x,y) .n(y) \rangle_{\mathrm I} (\widetilde y)\psi_1\,dt\,dx\,dS(\widetilde y) \nonumber\\ &= \int_{Q} \int_{\widetilde Y_1} f \psi_1 \,dt\,dx\,d{\widetilde y} \nonumber \end{align} for all $\psi_1\in W^{1,p}\big(0,T;C^1_{LB}(\overline{\Omega}; \mathcal{C}^\infty_\sharp(\widetilde Y_1))\big)$. Thus, we get the cell problem in ${\widetilde Y_1}$ \eqref{hom1-gammaY13}. (iii) Taking $(\psi_1=0$, $\phi_2=0)$, then $(\psi_1=0, \psi_2=0)$ we obtain the initial-boundary value problem \eqref{hom1-gammaY2}. It remains to identify $\widetilde g_1$ and $ g_{13}$. This is done as in the preceding case. More precisely, let $\phi(t,x,y)$, $\Phi(t,x,y)$ and $\psi_1(t,x,\widetilde y)$ be in $\mathcal{C}^\infty_0(Q; \mathcal{C}^\infty_\sharp(Y))^N$, $\mathcal{C}^\infty_0(Q; \mathcal{C}^\infty_\sharp(Y))$ and $\mathcal{C}^\infty_0(Q; \mathcal{C}^\infty_\sharp(\widetilde Y_1))$ respectively. For $\varepsilon>0$ and $h>0$ we define the following test function \begin{align*} \eta^\varepsilon(t,x) &=\chi_1^\varepsilon(x)\nabla^\varepsilon_x\psi_1(t,x,\frac{\widetilde x}{\varepsilon}) +\chi_2^\varepsilon(x)\nabla_{ x}u_2(t,x)\\ &\quad +\varepsilon(1-\chi_1^\varepsilon(x))\nabla_{ x}\phi(t,x,\frac{x}{\varepsilon}) + h\Phi(t,x,\frac{x}{\varepsilon}), \end{align*} where \begin{equation}\label{delta-epsilon} \nabla^\varepsilon_x=\begin{pmatrix} \varepsilon \nabla_{\widetilde x} \\ \partial_{x_3} \end{pmatrix}. \end{equation} Note that $\mathbb{A}_k^\varepsilon(x, \eta^\varepsilon):=\mathbb{A}_k^\varepsilon(\frac{x}{\varepsilon}, \eta^\varepsilon(t,x))$, $k=1,2,3$ are admissible test functions (in $L^p(Q)$) for the two-scale convergence and $$ \eta^\varepsilon(t,x)\stackrel{\mathit{2s , p'}}{\to }\eta(t,x,y), $$ where \begin{align*} \eta(t,x,y) &= \chi_1(y)\begin{pmatrix} \nabla_{\widetilde y} \\ \partial_{x_3} \end{pmatrix} \psi_1(t,x,\widetilde y)+\chi_2(y)\nabla_x u_2(t,x)\\ &\quad +(1-\chi_1(y))\nabla_y\phi(t,x,y)+h\Phi(t,x,y). \end{align*} Using monotonicity condition (A6) and letting $\varepsilon\to 0$, exactly as in the previous case, we obtain \begin{align*} %\label{eta2} &\int_Q f(t,x)U(t,x)\,dx\,dt-\liminf_{\varepsilon\to 0} \frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega\int_Y c(y)dy (u_0)^2dx \\ &-\frac{1}{\gamma^{1/p}} \int_Q\int_{Y_1}\widetilde g_{1}(t,x,y) \Bigl(\nabla_{\widetilde y} \psi_1(t,x,{\widetilde y} ) +h\widetilde\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \\ & - \int_Q\int_{Y_1} g_{13}(t,x,y)\Bigl(\partial_{x_3} \psi_1(t,x,\widetilde y) +h\phi_3(t,x,y) \Bigr)\,dt\,dx\,dy \\ &- \int_Q\int_{Y_2}g_{2}(t,x,y)\Bigl( \nabla_xu_2(t,x) +\nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \\ & - \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl( \nabla_{ y}\phi(t,x,{ y}) +h\Phi(t,x,y) \Bigr)\,dt\,dx\,dy \\ &+ \frac{1}{\gamma^{1/p}}\int_Q\int_{Y_1} \widetilde{\mathbb{A}}_{1}(x,\eta(t,x,y))\Bigl( \nabla_{\widetilde y} \psi_1(t,x,{\widetilde y} ) +h\widetilde\Phi(t,x,y) \Bigr)\,dt\,dx\,dy\\ &+ \int_Q\int_{Y_1}\mathbb{A}_{13}(x,\eta(t,x,y)) \Bigl( -\partial_{x_3}\psi_1(t,x,\widetilde y) +h\phi_3(t,x,y) \Bigr)\,dt\,dx\,dy \\ & + \int_Q\int_{Y_2}\mathbb{A}_{2}(x,\eta(t,x,y)) \Bigl( -\nabla_xu_2(t,x)+\nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y) \Bigr) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_3}\mathbb{A}_{3}(x,\eta(t,x,y)) \Bigl( \nabla_{ y}\phi(t,x,{ y})+h\Phi(t,x,y)\Bigr)\,dt\,dx\,dy \geq 0. \end{align*} Now, we may replace $\psi_1$ and $\phi_\beta,\beta=2,3$ by a sequence converging strongly in $L^p(Q;W^{1,p}_\sharp(\widetilde Y_1)/\mathbb{R})$ and $L^p(Q;W^{1,p}_\sharp(Y_\beta)/\mathbb{R})$ to $v_1$ and $v_\beta$ respectively, thus replacing $\eta(t,x,y)$ in \eqref{eta} with $\begin{pmatrix} \nabla_{\widetilde y}v_1 \\ \partial_{x_3}v_1 \end{pmatrix}$, $\nabla_x u_2+\nabla_{y}v_2 +h\Phi $ and $ \nabla_{y}v_3 +h\Phi $ successively. Using the conservation of energy given by the lemma \ref{conservationofenery} adapted to the present case, the above sum simplified to \begin{align*} &\frac{1}{\gamma^{1/p}}\int_Q\int_{Y_1} \Bigl[\widetilde{\mathbb{A}}_{1}\bigl(\widetilde y, \nabla_{\widetilde y}v_1(t,x,\widetilde y) +h\widetilde\Phi(t,x,y)\bigr)-\widetilde g_{1}(t,x,y)\Bigr] h\widetilde\Phi(t,x,y) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_1}\Bigl[\mathbb{A}_{13}\bigl(\widetilde y, \partial_{x_3}v_1(t,x,\widetilde y)+h\phi_3(t,x,y)\bigr) - g_{13}(t,x,y)\Bigr] h\phi_3(t,x,y) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_2}\Bigl[\mathbb{A}_{2}\bigl(x,\nabla_x u_2 +\nabla_{y}v_2+h\Phi(t,x,y)\bigr)-g_{2}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy\\ &+ \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl[\mathbb{A}_{3} \bigl(x,\nabla_{y}v_3+h\Phi(t,x,y)\bigr)-g_{3}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy\\ &\geq \frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega\int_Y c(y)dy (u_0)^2dx \\ &\quad +\liminf_{\varepsilon\to 0}\Biggl[\frac{1}{2} \int_\Omega c^\varepsilon(x)u^\varepsilon(T,x)^2dx +\frac{1}{2}\int_\Omega\int_Y c(y)dy (u_0)^2dx\Biggr]. \end{align*} Thus dividing by $h$ and letting $h\to 0$ we see that for every $\Phi$, \begin{align*} &\frac{1}{\gamma^{1/p}} \int_Q\int_{Y_1} \Bigl[\widetilde{\mathbb{A}}_{1}\bigl(\widetilde y, \nabla_{\widetilde y}v_1(t,x,\widetilde y)\bigr) -\widetilde g_{1}(t,x,y)\Bigr] h\widetilde\Phi(t,x,y) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_1}\Bigl[\mathbb{A}_{13}\bigl(\widetilde y, \partial_{x_3}v_1(t,x,\widetilde y)\bigr)-g_{13}(t,x,y)\Bigr] h\phi_3(t,x,y) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_2}\Bigl[\mathbb{A}_{2}\bigl(y,\nabla_x u_2 +\nabla_{y}v_2\bigr)-g_{2}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy \\ &+ \int_Q\int_{Y_3}g_{3}(t,x,y)\Bigl[\mathbb{A}_{3} \bigl(y,\nabla_{y}v_3\bigr)-g_{3}(t,x,y)\Bigr] h\Phi(t,x,y) \,dt\,dx\,dy \geq 0. \end{align*} Thus, \begin{gather*} \langle \widetilde g_{1}(t,x,y)\rangle_{\mathrm I} =\widetilde{\mathbb{A}}_{1}(\widetilde y,\nabla_{ \widetilde y} v_1(t,x,\widetilde y)),\\ \langle g_{13}(t,x,y)\rangle_{\mathrm I}=\mathbb{A}_{13} (\widetilde y,\partial_{ x_3}v_1(t,x,\widetilde y)). \end{gather*} Therefore we have proved the desired results. Now, to complete the proof of Theorem \ref{thm2} we shall show the uniqueness of the solution following exactly the same lines as in the proof of Theorem \ref{thm1}. \section{Corrector results} Now, we prove corrector results for the gradients of temperature and the corresponding flux under the stronger hypotheses (A5')--(A6') of monotonicity. Let $u^\varepsilon$ be the solution of the problem \eqref{Pep}. Let $v_1,u_2,v_2,v_3$ be as in Theorem \ref{thm1} (when $\gamma=0$ we recall that $v_1(t,x,\widetilde y)=u_1(t,x)$). We define the sequences of functions \begin{equation} \label{corrector1} \begin{gathered} \xi_1(t,x, y):=\chi_1(y)\nabla_{\widetilde y,x_3} v_1(t,x,{\widetilde y}),\\ \xi_2(t,x,y):=\chi_2(y)(\nabla_x u_2(t,x)+\nabla_y v_2(t,x,y)), \\ \xi_3(t,x,y):=\chi_3(y) \nabla_y v_3(t,x,y),\\ \xi_k^\varepsilon(t,x):=\chi_k^\varepsilon(x) \xi_k^\varepsilon(t,x,\frac{x}{\varepsilon}),\ k=1,2,3,\\ \mathbb{B}_1^\varepsilon(x,\xi):=\chi_1^\varepsilon(x) \begin{pmatrix} \frac{\mu}{\varepsilon}{\widetilde{\mathbb{A}}}_1^\varepsilon(\widetilde x,\widetilde\xi) \\ \quad \mathbb{A}_{13}^\varepsilon(\widetilde x,\xi_3) \end{pmatrix}, \\ \mathbb{B}_2^\varepsilon(x,\xi) := \chi_2^\varepsilon(x){\mathbb{A}}_2^\varepsilon (\widetilde x,\widetilde\xi),\\ \mathbb{B}_3^\varepsilon(x,\xi) := \chi_3^\varepsilon(x) {\varepsilon}^{p-1} {\mathbb{A}}_3^\varepsilon(\widetilde x,\widetilde\xi), \end{gathered} \end{equation} where $\nabla_{\widetilde y,x_3}=\begin{pmatrix} \nabla_{\widetilde y} \\ \partial_{x_3} \end{pmatrix}$. Note that $\mathbb{B}_k^\varepsilon$ satisfies the strong monotonicity condition (A5'), since $\frac{\mu}{\varepsilon}=\frac{\mu^{1/p}}{\varepsilon}\mu^{1/p'}\to 0$; thus, for example, we have \begin{equation} \label{bm} \begin{split} |\mathbb{B}_3^\varepsilon(x,\xi)-\mathbb{B}_3^\varepsilon(x,\eta)| &= |{\varepsilon}^{p-1}\mathbb{A}_3^\varepsilon(x,\xi) - {\varepsilon}^{p-1}\mathbb{A}_3^\varepsilon(x,\eta)|\\ &\leq K_1 {\varepsilon}^{p-1}(|\xi|+|\eta|)^{p-2}|\xi-\eta|\\ &\leq K_1 (|\xi|+|\eta|)^{p-2}|\xi-\eta|. \end{split} \end{equation} In a similar manner, we get the same inequality for $\mathbb{B}_1^\varepsilon$ and $\mathbb{B}_2^\varepsilon$. \begin{theorem}\label{thm1-corrector} If the functions, $\nabla _{\widetilde y} v_1$, $\nabla _y v_2$ and $\nabla _y v_3$ are admissible (cf. Definition \ref{cv12sp}), then \begin{gather}\label{e11} \limsup_{\varepsilon \searrow 0}\|\chi_1^\varepsilon (\nabla^\varepsilon_xu^\varepsilon - \xi_1^\varepsilon) \|_{L^{p}(Q)}=0, \\ \label{e12} \limsup_{\varepsilon \searrow 0}\|\chi_2^\varepsilon (\nabla_xu - \xi_2^\varepsilon) \|_{L^{p}(Q)}=0, \\ \label{e13} \limsup_{\varepsilon \searrow 0}\|\chi_3^\varepsilon (\varepsilon\nabla_xu^\varepsilon - \xi_3^\varepsilon) \|_{L^{p}(Q)}=0. \\ \label{e23} \limsup_{\varepsilon \searrow 0}\|\chi_k^\varepsilon (\mathbb{B}_k^\varepsilon(x, \nabla_xu^\varepsilon) - \mathbb{B}_k^\varepsilon(x,\xi_k^\varepsilon)) \|_{L^{p'}(Q)}=0,\quad k=1,2,3. \end{gather} Where $\nabla^\varepsilon$ is defined by \eqref{delta-epsilon}. \end{theorem} Let us mention that the convergence \eqref{e11}-\eqref{e23} means that, under hypotheses (A5')-(A6') and $\nabla _{\widetilde y} v_1$, $\nabla _y v_2$ and $\nabla _y v_3$ are admissible, the oscillations of the sequences, in the above, are all contained in the corresponding two-scale limits. Moreover, the proof of this theorem is motivated by the approach based on the two-scale convergence. Now, let us introduce some more notation, functions and quantities which we will use hereafter. We will use $M$ to denote a generic constant which does not depend on $\varepsilon$, but probably on $p$, $K_1$, $K_2$, $c_0$, $c$ and the $L^{p'}$ (resp. $L^p$) norm of the data $f$ (resp. $u_0$). Let $\kappa\in ]0,1[$ be a constant and $\psi_k(t,x,y),\ k=1,2,3,$ be admissible test functions such that \begin{equation}\label{admissible1} \|\nabla_{\widetilde y,x_3}v_1-\psi_1\|^p_{L^p(Q\times{Y_1})} +\sum_{\alpha=2}^3\|\nabla_{y}v_\alpha -\psi_\alpha\|^p_{L^p(Q\times{Y_\alpha})}\leq \kappa. \end{equation} Define the following functions: \begin{equation} \label{admissible2} \begin{gathered} \eta_1^\varepsilon(t,x) := \chi_1^\varepsilon(x)(\nabla^\varepsilon_x u^\varepsilon(t,x) +\psi_1(t,x,\frac{x}{\varepsilon})), \\ \eta_2^\varepsilon(t,x) := \chi_2^\varepsilon(x)(\nabla_x u^\varepsilon(t,x) +\psi_2(t,x,\frac{x}{\varepsilon})), \\ \eta_3^\varepsilon(t,x):= \chi_3^\varepsilon(x) \psi_3(t,x,\frac{x}{\varepsilon}). \end{gathered} \end{equation} Note that the functions $ \eta_k^\varepsilon$ and $\mathbb{B}_k^\varepsilon(x, \eta_k^\varepsilon)$, $k=1,2,3$ arise from admissible test functions and we have the following two-scale convergence (cf. Lemma \ref{lemmacv}): \begin{gather*} \eta_1^\varepsilon(t,x) \stackrel{\mathit{2s , p}}{\to } \eta_1(t,x,y):=\chi_1(y)(\nabla_{\widetilde y,x_3}v_1(t,x,y)+\psi_1(t,x,y)), \\ \eta_2^\varepsilon(t,x) \stackrel{\mathit{2s , p}}{\to } \eta_2(t,x,y):=\chi_2(y) (\nabla_xu_2(t,x)+\psi_2(t,x,y)), \\ \eta_3^\varepsilon(t,x) \stackrel{\mathit{2s , p}}{\to } \eta_3(t,x,y):=\chi_3(y) \psi_3(t,x,y), \\ \mathbb{B}_k^\varepsilon(x,\eta_k^\varepsilon) \stackrel{\mathit{2s , p'}}{\to } \chi_k(y) \mathbb{B}_k(y,\eta_k(t,x,y)),\ k=1,2,3, \end{gather*} where $\mathbb{B}_1(y,\eta)= \begin{pmatrix} \nu{\widetilde{\mathbb{A}}_1}({\widetilde y},{\widetilde \eta}) \\ \mathbb{A}_{13}({\widetilde y},\eta_{1N}) \end{pmatrix}$, ($\nu=0$ if $\gamma=0$ and $\nu=\frac{1}{\gamma^{1/p}}$ else), $\mathbb{B}_2=\mathbb{A}_2$ and $\mathbb{B}_3=\mathbb{A}_3$. \begin{lemma}\label{c1} \[ \sum_{k=1}^3\|\xi_k\|^p_{L^p(Q\times {Y_k})} \leq\frac{c}{c_0}(\|f\|^{p'}_{L^{p'}(Q)}+\|u_0\|^{p}_{L^{p}(\Omega)}). \] \end{lemma} The proof of the above lemma follows from the identity in Lemma \ref{conservationofenery} and assumption (A5). \begin{lemma}\label{c2} Let $\xi_k$, $\eta_k$, $\mathbb{B}_k^\varepsilon$, $\xi^\varepsilon_k$, $\eta^\varepsilon_k$, $\mathbb{B}_k$, $k=1,2,3$ be functions as defined above. Then \begin{gather*} \limsup_{\varepsilon \searrow 0}\int_{Q_1^\varepsilon}(\mathbb{B}_1^\varepsilon (x,\nabla_xu^\varepsilon)- \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)). (\nabla^\varepsilon u^\varepsilon -\eta_1)\,dx\,dt\leq {\mathbb E},\\ \limsup_{\varepsilon \searrow 0}\int_{Q_2^\varepsilon}(\mathbb{B}_2^\varepsilon (x,\nabla_xu^\varepsilon)- \mathbb{B}_2^\varepsilon(x,\eta_2^\varepsilon)). (\nabla u^\varepsilon -\eta_2)\,dx\,dt\leq {\mathbb E},\\ \limsup_{\varepsilon \searrow 0}\int_{Q_3^\varepsilon}(\mathbb{B}_3^\varepsilon (x,\nabla_xu^\varepsilon)- \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon)). (\varepsilon\nabla u^\varepsilon -\eta_3)\,dx\,dt\leq {\mathbb E}, \end{gather*} where \[ {\mathbb E}:=\sum_{k=1}^3\int_{Q\times Y_{k}} (\mathbb{B}_k(y,\xi_k)- \mathbb{B}_k(y,\eta_k)).(\xi_k -\eta_k)dy\,dx\,dt. \] \end{lemma} \begin{proof} Firstly, we denote the integrals appearing in the left-side of the above inequalities by ${\mathbb E}^\varepsilon_1$, ${\mathbb E}^\varepsilon_2$ and ${\mathbb E}^\varepsilon_3$ respectively. Secondly, we put \begin{gather*} \mathbb{D}^\varepsilon(T) =\int_\Omega c^\varepsilon(x) u^\varepsilon(T,x)^2dx,\quad \mathbb{D}^\varepsilon(0)=\int_\Omega c^\varepsilon(x) u_0(x)^2dx,\\ \mathbb{D}^0(T)=\sum_{\alpha=1}^2\widetilde c_\alpha\int_\Omega|u_\alpha(T,x)|^2dx + \int_\Omega\int_{Y_3}c_3(y)|v_3(T,x, y)|^2\,dx\,dy \\ \mathbb{D}^0(0)=\sum_{k=1}^3 \widetilde c_k\int_\Omega|u_0(x)|^2dx. \end{gather*} Then, for $k=1,2,3$, using \eqref{tt}, we obtain \begin{align*} {\mathbb E}^\varepsilon_k & \leq \sum_{j=1}^3{\mathbb E}^\varepsilon_j = \int_Qf(t,x)u^\varepsilon(t,x)\,dt\,dx+\frac{1}{2}\mathbb{D}^\varepsilon(0) -\frac{1}{2}\mathbb{D}^\varepsilon(T)\\ &\quad -\sum_{j=1}^3 \int_{Q_3^\varepsilon} \mathbb{B}_j^\varepsilon(x,\nabla_xu^\varepsilon). \eta_j^\varepsilon \,dt\,dx -\int_{Q_1^\varepsilon} \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon) . (\nabla^\varepsilon_xu^\varepsilon-\eta_1^\varepsilon) \,dt\,dx\\ &\quad -\int_{Q_2^\varepsilon} \mathbb{B}_2^\varepsilon(x,\eta_2^\varepsilon) . (\nabla_xu^\varepsilon-\eta_2^\varepsilon) \,dt\,dx - \int_{Q_3^\varepsilon} \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon) . (\varepsilon\nabla_xu^\varepsilon-\eta_3^\varepsilon) \,dt\,dx. \end{align*} Now, using the two-scale convergence, we deduce \begin{align*} \limsup_{\varepsilon \searrow 0} \sum_{j=1}^3{\mathbb E}^\varepsilon_j &\leq \int_{Q\times Y}f(t,x)U(t,x)\,dt\,dx\,dy +\frac{1}{2}\mathbb{D}^0(0)-\liminf_{\varepsilon \searrow 0}\frac{1}{2} \mathbb{D}^\varepsilon(T)\\ &\quad - \sum_{j=1}^3\int_{Q\times{Y_j}} \mathbb{B}_j(y,\xi_j) . \eta_j \,dt\,dx\,dy\\ &\quad - \sum_{j=1}^3\int_{Q\times{Y_j}} \mathbb{B}_j(y,\eta_j) . (\xi_j-\eta_j) \,dt\,dx\,dy. \end{align*} The right-hand side can be written as \begin{align*} &\int_{Q\times Y}f(t,x)U(t,x)\,dt\,dx\,dy +\frac{1}{2}\mathbb{D}^0(0)-\liminf_{\varepsilon \searrow 0} \frac{1}{2}\mathbb{D}^\varepsilon(T)\\ &-\sum_{j=1}^3\int_{Q\times{Y_j}} \mathbb{B}_j(y,\xi_j) . \xi_j \,dt\,dx\,dy \\ &+ \sum_{j=1}^3\int_{Q\times{Y_j}} (\mathbb{B}_j(y,\xi_j)-\mathbb{B}_j(y,\eta_j)).(\xi_j-\eta_j) \,dt\,dx\,dy. \end{align*} From the energy identity (cf. Lemma \ref{conservationofenery}) we obtain \begin{align*} &\sum_{j=1}^3\int_{Q\times{Y_j}} \mathbb{B}_j(y,\xi_j) . \xi_j \,dt\,dx\,dy\\ &= \int_{Q\times Y}f(t,x)U(t,x)\,dt\,dx\,dy +\frac{1}{2}\mathbb{D}^0(0)+ \frac{1}{2}\mathbb{D}^0(T)\\ &\geq \int_{Q\times Y}f(t,x)U(t,x)\,dt\,dx\,dy +\frac{1}{2}\mathbb{D}^0(0) -\liminf_{\varepsilon \searrow 0}\frac{1}{2}\mathbb{D}^\varepsilon(T). \end{align*} This completes the proof. \end{proof} \begin{lemma}\label{c3} Let $\xi_k$, $\eta_k$, $\kappa$ be as defined above. Then $$ {\mathbb E}:=\sum_{k=1}^3\int_{Q\times Y_{k}}(\mathbb{B}_k(y,\xi_k) - \mathbb{B}_k(y,\eta_k).(\xi_k -\eta_k)dy\,dx\,dt \leq M\kappa^{2/p}. $$ \end{lemma} \begin{proof} Using (A5') and H\"older's inequality, \begin{align*} {\mathbb E}&\leq \sum_{k=1}^3\int_{Q\times Y_{k}} (\mathbb{B}_k(y,\xi_k)- \mathbb{B}_k(y,\eta_k) (\xi_k -\eta_k)dy\,dx\,dt \\ &\leq K_1\sum_{k=1}^3\int_{Q\times Y_{k}}(|\xi_k| +|\eta_k|)^{p-2} |\xi_k -\eta_k|^2dy\,dx\,dt \\ &\leq K_1\sum_{k=1}^3\Big(\int_{Q\times Y_{k}}(|\xi_k| + |\eta_k|)^pdy\,dx\,dt\Big)^{\frac{p-2}{p}} \|\xi_k -\eta_k\|_{L^p }^2 \\ &\leq K_1\sum_{k=1}^3(\|\xi_k\|_{L^p }+ \|\eta_k\|_{L^p })^{p-2} \|\xi_k -\eta_k\|_{L^p }^2\\ &\leq K_1\Big(\sum_{k=1}^3 (\|\xi_k\|_{L^p }+ \|\eta_k\|_{L^p } )^{p}\Big)^{\frac{p-2}{p}}\Big(\sum_{k=1}^3 (\|\xi_k -\eta_k\|_{L^p }^p) \Big)^{2/p}\\ &\leq K_1\Big(\sum_{k=1}^3 (2\|\xi_k\|_{L^p }+ \|\xi_k -\eta_k\|_{L^p })^{p}\Big)^{\frac{p-2}{p}}\Big(\sum_{k=1}^3 (\|\xi_k -\eta_k\|_{L^p }^p)\Big)^{2/p}\\ &\leq K_1\Big(\sum_{k=1}^3 2^p(2^p\|\xi_k\|_{L^p }^p+ \|\xi_k -\eta_k\|_{L^p }^p)\Big)^{\frac{p-2}{p}}\Big(\sum_{k=1}^3 (\|\xi_k -\eta_k\|_{L^p }^p)\Big)^{2/p}. \end{align*} By the estimate proved in Lemma \ref{c1}, we deduce the result. \end{proof} Now, we prove some preliminary corrector results. \begin{theorem}\label{thm1bis-corrector} Under the same assumption as in Theorem \ref{thm1-corrector}, we have: \begin{gather} \label{limsup1} \limsup_{\varepsilon \searrow 0}\|\chi_1^\varepsilon\nabla^\varepsilon_xu^\varepsilon - \eta_1^\varepsilon \|_{L^{p}(Q)}\leq M\kappa^{2/p}, \\ \label{limsup2} \limsup_{\varepsilon \searrow 0}\|\chi_2^\varepsilon\nabla_xu^\varepsilon - \eta_2^\varepsilon \|_{L^{p}(Q)}\leq M\kappa^{2/p}, \\ \label{limsup3} \limsup_{\varepsilon \searrow 0}\|\chi_3^\varepsilon\varepsilon\nabla_xu^\varepsilon - \eta_3^\varepsilon \|_{L^{p}(Q)}\leq M\kappa^{2/p}. \\ \label{limsup11} \limsup_{\varepsilon \searrow 0}\|\mathbb{B}_k^\varepsilon(x,\nabla_xu^\varepsilon) - \mathbb{B}_k^\varepsilon(x,\eta_k^\varepsilon) \|_{L^{p'}(Q)} \leq M\kappa^{2/(p-1)},\quad k=1,2,3. \end{gather} \end{theorem} \begin{proof} From \eqref{bm}, we obtain \begin{gather*} |\chi_1^\varepsilon\nabla_x^\varepsilon u^\varepsilon(t,x)- \eta_1^\varepsilon(t,x)|^p \leq \frac{1}{K_2}(\mathbb{B}_1^\varepsilon(x,\nabla^\varepsilon_xu^\varepsilon) - \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)).( \nabla^\varepsilon_xu^\varepsilon- \eta_1^\varepsilon),\\ |\chi_2^\varepsilon\nabla_xu^\varepsilon(t,x)- \eta_2^\varepsilon(t,x)|^p \leq \frac{1}{K_2}(\mathbb{B}_2^\varepsilon(x,\nabla_xu^\varepsilon) - \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon)).(\nabla_xu^\varepsilon- \eta_3^\varepsilon), \\ |\chi_3^\varepsilon\varepsilon\nabla_xu^\varepsilon(t,x)- \eta_3^\varepsilon(t,x)|^p \leq \frac{1}{K_2}(\mathbb{B}_3^\varepsilon(x,\nabla_xu^\varepsilon) - \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon)).(\varepsilon\nabla_xu^\varepsilon- \eta_3^\varepsilon). \end{gather*} Therefore, \begin{gather*} \|\chi_1^\varepsilon\nabla_xu^\varepsilon- \eta_1^\varepsilon\|^p_{L^p(Q_1^\varepsilon)} \leq \frac{1}{K_2}\int_{Q_1^\varepsilon}(\mathbb{B}_1^\varepsilon(x,\nabla^\varepsilon_xu^\varepsilon) - \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)).(\nabla^\varepsilon_xu^\varepsilon- \eta_1^\varepsilon)\,dt\,dx,\\ \|\chi_2^\varepsilon\nabla_xu^\varepsilon - \eta_2^\varepsilon \|^p_{L^p(Q_2^\varepsilon)} \leq \frac{1}{K_2}\int_{Q_2^\varepsilon}(\mathbb{B}_2^\varepsilon(x,\nabla_xu^\varepsilon) - \mathbb{B}_2^\varepsilon(x,\eta_2^\varepsilon)). (\nabla_xu^\varepsilon- \eta_2^\varepsilon)\,dt\,dx,\\ \|\chi_3^\varepsilon\varepsilon\nabla_xu^\varepsilon - \eta_3^\varepsilon \|^p_{L^p(Q_3^\varepsilon)} \leq \frac{1}{K_2}\int_{Q_3^\varepsilon}(\mathbb{B}_3^\varepsilon(x,\varepsilon\nabla_xu^\varepsilon) - \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon)).(\varepsilon\nabla_xu^\varepsilon- \eta_3^\varepsilon)\,dt\,dx. \end{gather*} Now, let $$ \mathbb{G}^\varepsilon:=\|\chi_1^\varepsilon\nabla_x^\varepsilon u^\varepsilon - \eta_1^\varepsilon\|^p_{L^p(Q_1^\varepsilon)}+\|\chi_2^\varepsilon\nabla_xu^\varepsilon - \eta_2^\varepsilon \|^p_{L^p(Q_2^\varepsilon)}+\|\chi_3^\varepsilon\varepsilon\nabla_xu^\varepsilon - \eta_3^\varepsilon \|^p_{L^p(Q_3^\varepsilon)}, $$ and \begin{align*} \mathbb{F}^\varepsilon &:=\int_{Q_1^\varepsilon}(\mathbb{B}_1^\varepsilon(x,\nabla^\varepsilon_xu^\varepsilon) - \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)).(\nabla^\varepsilon_xu^\varepsilon- \eta_1^\varepsilon)\,dt\,dx\\ &\quad +\int_{Q_2^\varepsilon}(\mathbb{B}_2^\varepsilon(x,\nabla_xu^\varepsilon) - \mathbb{B}_2^\varepsilon(x,\eta_2^\varepsilon)).(\nabla_xu^\varepsilon- \eta_2^\varepsilon)\,dt\,dx\\ &\quad +\int_{Q_3^\varepsilon}(\mathbb{B}_3^\varepsilon(x,\varepsilon\nabla_xu^\varepsilon) - \mathbb{B}_3^\varepsilon(x,\eta_3^\varepsilon)).(\nabla_xu^\varepsilon- \eta_3^\varepsilon)\,dt\,dx. \end{align*} From th above estimates, we have $\mathbb{G}^\varepsilon\leq \mathbb{F}^\varepsilon/c_0$. Therefore, passing to the limit-sup as $\varepsilon\to 0$ and using Lemmas \ref{c2} and \ref{c3}, we obtain $$ \limsup_{\varepsilon \searrow 0}\mathbb{G}^\varepsilon\leq M\kappa^{2/p}. $$ This concludes the proof of \eqref{limsup1}-\eqref{limsup3}. To achieve the proof, we will only prove the estimate \eqref{limsup11} for $k=1$, the others are proved in a similar manner. Let $q=p'=\frac{p}{p-1}$, then by \eqref{bm} we have \begin{align*} &\int_{Q_1^\varepsilon}|\mathbb{B}_1^\varepsilon(x,\nabla^\varepsilon_xu^\varepsilon) - \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)|^q\,dt\,dx\\ &\leq K_1 \int_{Q_1^\varepsilon}(|\nabla^\varepsilon_xu^\varepsilon| +|\eta_1^\varepsilon|)^{(p-2)q}|\nabla^\varepsilon_xu^\varepsilon- \eta_1^\varepsilon|^q\,dt\,dx. \end{align*} Since $$ \frac{1}{p-1}+\frac{p-2}{p-1}=1, $$ by H\"older's inequality, \begin{align*} &\int_{Q_1^\varepsilon}|\mathbb{B}_1^\varepsilon(x,\nabla^\varepsilon_xu^\varepsilon) - \mathbb{B}_1^\varepsilon(x,\eta_1^\varepsilon)|^q\,dt\,dx\\ & \leq K_1 \Big(\int_{Q_1^\varepsilon}(|\nabla^\varepsilon_xu^\varepsilon| +|\eta_1^\varepsilon|)^p\,dt\,dx\Big)^{\frac{p-2}{p-1}} \Big(\int_{Q_1^\varepsilon}|\nabla^\varepsilon_xu^\varepsilon-\eta_1^\varepsilon|^p\,dt\,dx \Big)^{\frac{1}{p-1}}\\ & \leq M \|\chi_1^\varepsilon(\nabla^\varepsilon_xu^\varepsilon- \eta_1^\varepsilon) \|_{L^{p}(Q)}^q. \end{align*} Now, using \eqref{limsup1}, we obtain the desired estimate. \end{proof} \begin{proof}[\textbf{Proof of Theorem \ref{thm1-corrector}}] Since the functions $\nabla _y v_2$ and $\nabla _y v_3$ are assumed to be admissible test functions, we can choose $\psi_1=\nabla _y v_2$ and $\psi_2=\nabla _y v_3$. 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