\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 210, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/210\hfil Homogenization of a system]
{Homogenization of a system of semilinear diffusion-reaction
equations in an $H^{1,p}$ setting}

\author[H. S. Mahato, M. B\"ohm \hfil EJDE-2013/210\hfilneg]
{Hari Shankar Mahato, Michael B\"ohm}  % in alphabetical order

\address{Hari Shankar Mahato \newline
Center of Industrial Mathematics, University of Bremen,
D-28359, Bremen, Germany}
\email{mahato@math.uni-bremen.de}

\address{Michael B\"ohm \newline
Center of Industrial Mathematics, University of Bremen,
D-28359, Bremen, Germany}
\email{mbohm@math.uni-bremen.de}

\thanks{Submitted July 14, 2013. Published September 19, 2013.}
\subjclass[2000]{35B27, 35K57, 35K58, 46E35, 35D30}
\keywords{Global solution; semilinear parabolic equation;
reversible reactions;\hfill\break\indent 
 Lyapunov functionals; maximal regularity; 
homogenization; two-scale convergence}

\begin{abstract}
 In this article, homogenization of a system of semilinear
 multi-species diffusion-reaction equations is shown.
 The presence of highly nonlinear reaction rate terms on the right-hand
 side of the equations make the model difficult to analyze.
 We obtain some a-priori estimates of the solution which give the strong
 and two-scale convergences of the solution. We homogenize this system
 of diffusion-reaction equations by passing to the limit using
 two-scale convergence.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

The existence of a unique global positive weak solution
$u$ that belongs the space  $[H^{1,p}((0,T);H^{1,q}(\Omega)^{*})\cap L^p((0,T);
H^{1,p}(\Omega))]^I$
is shown in \cite{MB13} (by taking $\vec{q}=0$) for a system of semilinear
diffusion-reaction equations
\begin{gather}
\frac{\partial u}{\partial t}-\nabla (\;D\nabla u-\vec{q}u)
=SR(u)\quad \text{in } (0,T)\times\Omega,\label{en1.1}\\
-D\nabla u\cdot\vec{n}=0\quad \text{on }(0,T)\times\partial \Omega,\label{en1.2}\\
u(0,x)=u_0(x)\quad \text{in } \Omega\label{en1.3}
\end{gather}
under the assumptions:
\begin{itemize}
\item[(i)] $p>n+2$;
\item[(ii)] $u_0\ge 0$, i.e., $u_{{0_i}}\ge 0$ for  $i=1,2,\dots ,I$;
\item[(iii)] $u_{{0_i}}\in (H^{1,q}(\Omega)^{*},H^{1,p}
 (\Omega))_{1-\frac{1}{p}, p}$  for $i=1,2,\dots ,I$,
\item[(iv)] all reactions are linearly independent such that the
stoichiometric matrix $S=(s_{ij})_{1\le j\le J,\, 1\le i \le I}$
has maximal column rank; i.e., $\operatorname{rank}(S)=J$,
\end{itemize}
where $I\in \mathbb{Z}^{+}$, $\frac{1}{p}+\frac{1}{q}=1$,
$\Omega\subset\mathbb{R}^{n}$ a bounded domain with sufficiently
smooth boundary, $D>0$ a constant (see remark \ref{thm1.1}) and $SR(u)$
the reaction rate vector (see \eqref{en1.5}).
Here $u:=(u_1,u_2,\dots ,u_{I})$ is the concentration vector of $I$
chemical species involved in $J$ reactions given by
\begin{equation}
\tau _{1j}X_1+\tau _{2j}X_2+\dots +\tau _{Ij}X_{I}\rightleftharpoons
\nu _{1j}X_1+\nu _{2j}X_2+\dots
+\nu _{Ij}X_{I}, \text{ for }1\le j\le J, \label{en1.4}
\end{equation}
where $X_i$, $1\le i\le I$, denotes the chemical species and the
stoichiometric coefficients $-\tau _{ij}\in \mathbb{Z}^{-}_0$ and
$\nu _{ij}\in\mathbb{Z}^{+}_0$ respectively.
Set $s_{ij}=\nu_{ij}-\tau_{ij}$.
The reaction rate for the \textit{i-th} species is given by
\begin{align}
(SR(u))_i=\sum_{j=1}^{J}s_{ij}\Big(k^{f}_{j}\underset{s_{mj <0}}
{\prod_{m=1}^I}u_{m}^{-s_{mj}}- k^{b}_{j}\underset{s_{mj >0}}
{\prod_{m=1}^I}u_{m}^{s_{mj}}\Big)\quad\text{for }
i=1,2,\dots ,I,\label{en1.5}
\end{align}
where $k^f_j$  $(>0)$ and $k^b_j$ $(>0)$ are the forward and backward reaction 
rate factors respectively.
cf. \cite{Kra08,HSM13,MB13}.

\begin{remark}\label{thm1.1} \rm
The modeling of transport processes in a porous medium very often lead
to the equations of type \eqref{en1.1}-\eqref{en1.3}.
In some situations the advective flux dominates diffusion and even
though diffusion coefficients actually vary from species to species,
 we can consider the same value of the diffusion coefficients for
all the species. However in this paper (also in \cite{MB13,Kra08}),
 due to mathematical technicality we have considered the same diffusion
coefficients for all the species.
\end{remark}

In this article, we consider \eqref{en1.1}-\eqref{en1.3}, assuming
$\vec{q}=0$, in the context of a porous medium and upscale the model
via periodic homogenization in an appropriate function space
setting (see section \ref{sec4}). The global existence of the solution
of \eqref{en1.1}-\eqref{en1.3} for $\vec{q}=0$ considered at the
micro scale follows by the techniques used in \cite{MB13}
(see theorem \ref{thm3.1}).

To fix the ideas, let $Y:=(0,1)^{n}\subset \mathbb{R}^{n}$ be a unit
 representative cell which is composed of a solid part $Y^{s}$
with boundary $\Gamma$ and a pore part $Y^p$ such that
 $Y=Y^{s}\cup Y^p$, $\bar{Y}^{s}\subset Y$ and
$\bar{Y}^{s}\cap\bar{Y}^p=\Gamma$. Suppose that $\Omega$ is a
 porous medium with pore space $\Omega^p$ and solid parts $\Omega ^{s}$
such that $\Omega:=\Omega^p\cup\Omega^{s}$.
The boundary of $\Omega ^{s}$ is denoted by $\Gamma^{*}$ and the outer
boundary of $\Omega$ is denoted by $\partial \Omega$. $\Gamma$,
$\Gamma^{*}$ and $\partial \Omega$ are assumed to be sufficiently smooth.
Assume further that $\Omega$ is periodic (the solid parts in $\Omega$
are periodically distributed) and covered by a finite union of the cell $Y$.
To avoid technical difficulties, we postulate:
\begin{itemize}
\item solid parts do not touch the boundary $\partial \Omega$,
\item solid parts do not touch each other,
\item solid parts do not touch the boundary of $Y$.
\end{itemize}
We use the standard notation (cf. \cite{Mil92,PB08}, e.g.).
Let $\varepsilon>0$ be the scale parameter and $\Omega$ be covered by
a finite union of translated versions of $\varepsilon Y_{k}$ cells
such that $\varepsilon Y_{k}\subset \Omega$ for $k\in \mathbb{Z}^{n}$.
We also define
\begin{gather}
\Omega ^p_{\varepsilon}:= \cup_{k\in \mathbb{Z}^{n}}
\{\varepsilon Y_{k}^p:\varepsilon Y_{k}^p\subset \Omega\},\label{eqn1.6}\\
\Omega ^{s}_{\varepsilon}:= \cup_{k\in \mathbb{Z}^{n}}
\{\varepsilon Y_{k}^{s}:\varepsilon Y^{s}_{k}\subset \Omega\},\label{eqn1.7}\\
\Gamma _{\varepsilon}:= \cup_{k\in \mathbb{Z}^{n}}
\{\varepsilon \Gamma _{k}:\varepsilon \Gamma_{k}\subset \Omega\},\label{eqn1.8}\\
\partial \Omega _{\varepsilon}^p:=\partial \Omega \cup \Gamma _{\varepsilon}.
\label{eqn1.9}
\end{gather}
We denote by $dx$ and $dy$ the volume elements in $\Omega$ and $Y$,
and by $d\sigma_{y}$ and $d\sigma_{x}$ the surface elements on
$\Gamma$ and $\Gamma _\varepsilon$ respectively. The characteristic (indicator)
function of $\Omega _{\varepsilon}^p$ in $\Omega$ denoted by
\begin{equation}
\chi ^{\varepsilon}(x)=\chi(\frac{x}{\varepsilon})\label{eqn1.10}
\end{equation}
is defined as
\begin{equation}
\chi ^{\varepsilon}(x)=\begin{cases}
1 &\text{for }x\in \Omega _{\varepsilon}^p, \\
0 &\text{for }x\in \Omega - \Omega _{\varepsilon}^p.
\end{cases}\label{eqn1.11}
\end{equation}
For $T>0$, $[0,T)$ denotes the time interval. Following the above notation,
the system of diffusion-reaction equations in the pore space at the
micro scale is given by
\begin{gather}
 \frac{\partial u_{\varepsilon}}{\partial t} -\nabla \cdot D
\nabla u_{\varepsilon} = SR(u_{\varepsilon})  \quad
 \text{in }  (0,T)\times \Omega ^p _{\varepsilon}, \label{eqn1.12}\\
 u_{\varepsilon}(0, x) = u_0(x) \quad \text{in }
 \Omega ^p_{\varepsilon},\label{eqn1.13}\\
 -D\nabla u_{{\varepsilon}}\cdot \vec{n} = 0 \quad \text{on }
 (0, T)\times\partial\Omega,\label{eqn1.14}\\
 -D\nabla u_{{\varepsilon}}\cdot \vec{n} = 0 \quad \text{on }
 (0,T)\times\Gamma _{\varepsilon}. \label{eqn1.15}
\end{gather}
We denote the problem \eqref{eqn1.12}-\eqref{eqn1.15} by $(P_{\varepsilon})$.
The derivation of $(P_{\varepsilon})$ at the micro scale is motivated
from the nondimensionalization of \eqref{en1.1}-\eqref{en1.3}, for details
see \cite{DP04,PB08}. Before we begin with the analysis of
\eqref{eqn1.12}-\eqref{eqn1.15}, we make the following assumptions:
\begin{gather}
 p>n+2;\label{eqn1.16}\\
 u_0\ge 0,\text{ i.e., }u_{{0_i}}\ge 0 \text{ for all }i=1,2,\dots ,I;\label{eqn1.17}\\
 u_{{0_i}}\in (H^{1,q}(\Omega _{\varepsilon}^p)^{*},H^{1,p}(\Omega _{\varepsilon}^p))_{1-\frac{1}{p}, p}\text{ for }i=1,2,\dots ,I.\label{eqn1.18}\\
\text{all reactions are linearly independent such that the
stoichiometric matrix} \nonumber \\
S=(s_{ij})_{1\le j\le J,\, 1\le i \le I}
\text{ has maximal column rank, i.e., } \operatorname{rank}(S)=J;\label{eqn1.19}\\
 \sup_{\varepsilon >0} \|u_{{0_i}}\|_{(H^{1,q}
(\Omega _{\varepsilon}^p)^{*},H^{1,p}(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},
 p}}< \infty \text{ for }i=1,2,\dots ,I. \label{eqn1.20}
\end{gather}

\section{Mathematical Preliminaries}\label{sec2}

\subsection{Function Spaces}
\subsubsection{Function Spaces on $\Omega$}\label{sec2.1.1}
Let $1< r,s< \infty$ be such that $\frac{1}{r}+\frac{1}{s}=1$.
Assume that $\Omega \subset \mathbb{R}^{n}$ $(n\ge 2)$ is a bounded
domain with sufficiently smooth boundary $\partial \Omega$.
As usual, $L^{r}(\Omega)$ is the set of all equivalence classes of
real-valued functions $u(.)$ such that $u(x)$ is defined for almost
every $x\in \Omega$, is measurable and $|u(\cdot)|^{r}$ is Lebesgue integrable.
$L^{r}(\Omega)$ is a Banach space with the norm
\begin{equation} \label{eqn2.1}
\|u\|_{L^{r}(\Omega)} = \begin{cases}
\big[\int_{\Omega}|u(x)|^{r}\,dx\big]^{1/r}
 &\text{for }1\le r<\infty, \\
\operatorname{ess\,sup}_{x\in \Omega} |u(x)|
&\text{for } r=\infty.
\end{cases}
\end{equation}
The space $H^{1,r}(\Omega)$ is the usual Sobolev space with the norm
\begin{equation} \label{eqn2.2}
 \|u\|_{H^{1,r}(\Omega)}=\begin{cases}
\big[\|u\|^{r}_{L^{r}(\Omega)}+\|\nabla u\|^{r}_{L^{r}(\Omega)}\big]^{1/r}
&\text{for }1\le r<\infty,  \\
\operatorname{ess\,sup}_{x\in \Omega}[|u(x)|+|\nabla u(x)|]
&\text{for } r=\infty.
\end{cases}
\end{equation}
For a Banach space $X$, $X^{*}$ denotes its dual and the duality pairing
is denoted by $\langle\cdot,\cdot \rangle_{X^{*}\times X}$.
Let $1< p,q< \infty$ be such that $p>n+2$ and $\frac{1}{p}+\frac{1}{q}=1$.
We define the continuous embedding
$L^p(\Omega)\hookrightarrow H^{1,q}(\Omega)^{*}$ as
\begin{equation}
\langle f, v\rangle_{H^{1,q}(\Omega)^{*}\times H^{1,q}(\Omega)}
=\langle f,v\rangle_{L^p(\Omega)\times L^{q}(\Omega)}
\quad\text{for }f\in L^p(\Omega),\; v\in H^{1,q}(\Omega).\label{eqn2.3}
\end{equation}
The symbols $\hookrightarrow$ and $\hookrightarrow\hookrightarrow$ will denote
the  continuous and compact embeddings respectively.
The \textit{Sobolev-Bochner space} is
\begin{equation}
\begin{aligned}
F&:=\big\{u\in L^p((0,T);H^{1,p}(\Omega)):
\frac{du}{dt}\in L^p((0,T);H^{1,q}(\Omega)^{*})\big\} \\
&=H^{1,p}((0,T); H^{1,q}(\Omega)^{*})\cap L^p((0,T);
H^{1,p}(\Omega))\end{aligned} \label{eqn2.4}
\end{equation}
and for any $u\in F$,
\begin{equation}
\|u\|_{F}:=\|u\|_{L^p((0,T); H^{1,p}(\Omega))}
+\|u\|_{L^p((0,T); H^{1,q}(\Omega)^{*})}
+\|\frac{du}{dt}\|_{L^p((0,T); H^{1,q}(\Omega)^{*})},\label{eqn2.5}
\end{equation}
where $\frac{du}{dt}$ is the distributional time derivative of $u$.
For $0<\theta <1$, let
\begin{gather}
\begin{aligned}
(H^{1,q}(\Omega)^{*}, H^{1,p}(\Omega))_{\theta,p}
&\text{ be the real-interpolation space between }\\
&H^{1,q}(\Omega)^{*}\text{ and }H^{1,p}(\Omega),
\end{aligned}\label{eqn2.6}\\
\begin{aligned}
\big[H^{1,q}(\Omega)^{*}, H^{1,p}(\Omega)\big]_{\theta}
&\text{ be the complex-interpolation space between }\\
&H^{1,q}(\Omega)^{*}\text{ and }H^{1,p}(\Omega)
\end{aligned} \label{eqn2.7}
\end{gather}
endowed with one of the usual norms
(cf. \cite{BL76,Tri95,Lun95,Haa06}). Now we introduce the norms on
the vector-valued function spaces. Let $I\in \mathbb{N}$ and
$u:\Omega \to \mathbb{R}^I$. We define
\begin{equation}
[L^p(\Omega)]^I :=
\underbrace{L^p(\Omega)\times L^p(\Omega)\times\dots \times L^p(\Omega)}
_{\rm I-times}\label{eqn2.8}
\end{equation}
and for $u\in [L^p(\Omega)]^I$ the corresponding norm is
\begin{equation}
 \||u\||_{[L^p(\Omega)]^I} :=
\Big[\sum_{i=1}^I\|{u_i}\|_{L^p(\Omega)}^p\Big]^{1/p}.\label{eqn2.9}
\end{equation}
Similarly,
\begin{gather}
\||u\||_{[L^{\infty}(\Omega)]^I}
:= \max_{1\leq i \leq I} \|{u_i}\|_{L^{\infty}(\Omega)},\label{eqn2.10}\\
\||u\||_{[H^{1,p}(\Omega)]^I}
= [\sum_{i=1}^I\|{u_i}\|_{H^{1,p}(\Omega)}^p]^{1/p},\label{eqn2.11} \\
\||u\||_{[H^{1,\infty}(\Omega)]^I}
= \max_{1\leq i \leq I} \|{u_i}\|_{H^{1,\infty}(\Omega)},\label{eqn2.12}\\
\||u\||_{[H^{1,q}(\Omega)^{*}]^{{I}}}
=  [\sum_{i=1}^I\|{u_i}\|^p_{H^{1,q}(\Omega)^{*}}]^{1/p}.\label{eqn2.13}
\end{gather}
We also define
\begin{equation}
F^I:=[H^{1,p}((0,T);H^{1,q}(\Omega)^{*})\cap L^p((0,T); H^{1,p}(\Omega))
]^{{I}}\label{eqn2.14}
\end{equation}
and for $u\in F^I$,
\begin{equation}
\||u\||_{F^I}:= [\sum_{i=1}^{{I}}\|u_i\|^p_{F}]^{1/p}. \label{eqn2.15}
\end{equation}
Similarly,
\begin{equation}
X_{p}^I:=[(H^{1,q}(\Omega)^{*}, H^{1,p}(\Omega ))_{1-\frac{1}{p},p}]^I
\label{eqn2.16}
\end{equation}
and for $u\in X_{p}^I$
\begin{equation}
\||u\||_{X_{p}^I} :=[\sum_{i=1}^{{I}}\|u\|^p_{(H^{1,q}(\Omega)^{*},
H^{1,p}(\Omega))_{1-\frac{1}{p},p}}]^{1/p}.\label{eqn2.17}
\end{equation}

\begin{theorem}\label{thm2.2} %\label{thm2.1.1.1}
Let $p>n+2$, then
$F\hookrightarrow \hookrightarrow L^{\infty}((0,T)\times \Omega)$.
\end{theorem}

For a proof of the above theorem, see \cite[Theorem 2.2]{MB13}.

\begin{theorem}\label{thm2.1.1.2} %\label{thm2.3}
Let $p>n+2$. Then $(H^{1,q}(\Omega)^{*}, H^{1,p}(\Omega))_{1-\frac{1}{p},p}
\hookrightarrow \hookrightarrow L^{\infty}(\Omega)$.
\end{theorem}

For a proof of the above theorem, see \cite[Theorem 2.3]{MB13}.
Let $V$, $H$ and $V^{*}$ be a \textit{Gelfand triple}, where $V$
a Banach space, $H$ a Hilbert space and $V^{*}$ is the dual of $V$.
Let $H$ be identified with its own dual ($H\cong H^{*}$) and
$V\overset{d}{\subset} H$, then $H\overset{d}{\subset} V^{*}$.
Denote $\Xi = \{u\in L^p((0,T);V):\frac{du}{dt}\in L^{q}((0,T);V^{*})\}$.
 We have the following theorem.

\begin{theorem}
Let $V$, $H$ and $V^*$ be as above.
Then $\Xi\subset C([0,T];H)$ and the following rule of integration holds
for any $u$, $v\in \Xi $ and any $0\le t_1\le t_2\le T$:
\begin{equation}
\int_{t_1}^{t_2}\frac{d}{dt}(u(t),v(t))_{H}\,dt
=\int_{t_1}^{t_2} \langle\frac{d u}{dt},v(t)\rangle_{V^{*}\times V}\,dt
+\int_{t_1}^{t_2}\langle u(t),\frac{dv}{dt}\rangle_{V\times V^{*}}\,dt.
\label{eqn2.18}
\end{equation}
\end{theorem}

For a proof of the above theorem, see \cite[lemma 7.3]{Rub05}.

\subsubsection{Function Spaces on $\Omega _{\varepsilon}^p$}\label{sec3.1.2}

The function spaces on the domain $\Omega _{\varepsilon}^p$ are
defined in an analogous way as in section \ref{sec2.1.1}:
we replace $\Omega$ by $\Omega _{\varepsilon}^p$ in the definitions
of the function spaces. The spaces on $\Omega _{\varepsilon}^p$
are endowed with their usual norms as given in \eqref{eqn2.1}-\eqref{eqn2.7}.


From section \ref{sec1}, we notice that the surface area of
$\Gamma _{\varepsilon}$ increases proportionally to
$1/\varepsilon$; i.e., $|\Gamma _\varepsilon |\to \infty$ as
$\varepsilon \to 0$. Keeping this in mind, the $L^p - L^{q}$
duality on $\Gamma _\varepsilon$ can be defined as
\begin{equation}
(u, v)_{L^p(\Gamma _\varepsilon)\times L^{q}(\Gamma _\varepsilon)}
:=\varepsilon \int_{\Gamma _{\varepsilon}}u(x)v(x) \,d\sigma_{x}
\quad \text{for }u\in L^p(\Gamma _\varepsilon) \text{ and }
v\in L^{q}(\Gamma _\varepsilon),\label{eqn2.19}
\end{equation}
and the space $L^p(\Gamma _\varepsilon)$ is furnished with the norm
\begin{equation}
\|\cdot\|^p_{L^p(\Gamma _\varepsilon)}
=\varepsilon \int_{\Gamma _{\varepsilon}}|\cdot|^pd\sigma_{x}\quad \text{and}\quad
 \|\cdot\|_{L^{\infty}(\Gamma _\varepsilon)}
=\underset{x\in \Gamma _{\varepsilon}}{\mathrm{ess\;sup}} |\cdot|.\label{eqn2.20}
\end{equation}
The vector-valued functions and their respective norms on
$\Omega _{\varepsilon}^p$ can be defined in the similar way as
in \eqref{eqn2.8}-\eqref{eqn2.17}. For the sake of simplicity,
we use the following notation:
\begin{gather}
F^I_{\varepsilon}:=[H^{1,p}((0,T); H^{1,q}
(\Omega _{\varepsilon}^p)^{*})\cap L^p((0,T);
 H^{1,p}(\Omega _{\varepsilon}^p))]^I,\label{eqn2.21}
\\
X_{p,{\varepsilon}}^I
:=\big[(H^{1,q}(\Omega _{\varepsilon}^p)^{*}, H^{1,p}
(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}\big]^I,\label{eqn2.22}
\end{gather}
$C$ and $C_i$ are generic nonnegative constants which may be different at
different steps of the inequalities to come.

\subsection{Weak formulation of $(P_{\varepsilon})$}\label{sec2.2}

\begin{definition}\label{thm2.2.1} \rm
A function $u_{\varepsilon}\in F_{\varepsilon}^I$ is said to be a
weak solution of the problem \eqref{eqn1.12}-\eqref{eqn1.15} if it satisfies
\begin{gather}
\begin{aligned}
&\big\langle \frac{\partial u_{\varepsilon}(t)}{\partial t},\phi
\big\rangle_{[H^{1,q}(\Omega _{\varepsilon}^p)^{*}]^I\times [H^{1,q}
(\Omega _{\varepsilon}^p)]^I}
+\int_{\Omega_{\varepsilon}^p}\langle D\nabla u_{\varepsilon}(t,x),
\nabla \phi(x)\rangle_{I}\,dx \\
&=\langle SR(u_{\varepsilon}(t)),\phi
 \rangle_{[H^{1,q}(\Omega _{\varepsilon}^p)^{*}]^I
 \times [H^{1,q}(\Omega _{\varepsilon}^p)]^I}\\
&\quad\text{for every $\phi\in [H^{1,q}(\Omega _{\varepsilon}^p)]^I$
and for a.e. $t$}\,,\label{eqn2.23}
\end{aligned}
\\
 u_{\varepsilon}(0,x)=u_0(x)\quad\text{in }
\Omega_{\varepsilon}^p.\label{eqn2.24}
\end{gather}
\end{definition}

\subsection{Some theorems and lemmas}\label{sec2.3}
\subsubsection{Trace theorems}\label{sec2.3.1}

\begin{lemma}\label{thm2.3.1.1}
Let $\Gamma _\varepsilon$ be as in \eqref{eqn1.8}. Then
\begin{equation}
\varepsilon |\Gamma _{\varepsilon}|
=|\Gamma |\frac{|\Omega|}{|Y|}\label{eqn2.25}.
\end{equation}
\end{lemma}
The proof of the above lemma can be found in \cite[section 2]{ADH}.

\begin{theorem}\label{thm2.3.1.2}
Let $1\le p< \infty$. Let $\Omega ^p_{\varepsilon}$ and
$\Gamma _{\varepsilon}$ be defined as in section \ref{sec1}.
Then there exists a bounded linear operator
$T^{\varepsilon}:H^{1,p}(\Omega _{\varepsilon}^p)
\to L^p(\Gamma _{\varepsilon})$ such that
\begin{gather}
 T^{\varepsilon}u=u|_{\Gamma _\varepsilon}
\quad\text{for }u \in H^{1,p}(\Omega _{\varepsilon}^p)
 \cap C(\bar{\Omega}^p_{\varepsilon})\label{eqn2.26}
\\
 \varepsilon \int_{\Gamma _\varepsilon}|T^{\varepsilon}u(x)|^p\,d\sigma _x
\le C\Big(\int_{\Omega _{\varepsilon}^p}|u(x)|^p\,dx+\varepsilon ^p
\int_{\Omega _{\varepsilon}^p}|\nabla _{x}u(x)|^p\,dx\Big),\label{eqn2.27}
\end{gather}
where the constant $C$ is independent of $\varepsilon$ and $u$.
\end{theorem}

For a proof of the above theorem, see
\cite[Lemma 5.3 (b)]{HoJ91}, and \cite[Lemma 2.7.2]{Rad92}.


\subsubsection{Extension theorems}\label{sec2.3.2}

\begin{theorem}\label{thm2.3.2.1}
Let $1\le p\le \infty$. Suppose that $\Omega ^p_{\varepsilon}$ and
$\Omega $ are defined as in section {\ref{sec1}}.
For $u\in H^{1,p}(\Omega _{\varepsilon}^p)$, there exists a bounded
linear operator $Q^{\varepsilon}: H^{1,p}
(\Omega _{\varepsilon}^p)\to H^{1,p}(\Omega)$ such that
\begin{gather}
 Q^{\varepsilon}u:= u\quad  \text{in }\Omega _{\varepsilon}^p, \label{eqn2.28}\\
\|Q^{\varepsilon}u\|^p_{H^{1,p}(\Omega)} \le
 C\|u\|^p_{H^{1,p}(\Omega _{\varepsilon}^p)}, \label{eqn2.29}
\end{gather}
where the constant $C$ is independent of $\varepsilon$ and $u$ but depends
on $p$.
\end{theorem}

For a proof of the above theorem see \cite[Theorem 5.2]{HoJ91},
also \cite{Tar80}.

Now we prove a theorem similar to theorem \ref{thm2.3.2.1}
for the functions depending on both $t$ and $x$.
Let $1\le p\le \infty$. For $u\in L^p((0,T);H^{1,p}(\Omega _{\varepsilon}^p))$,
we define an operator
$R^{\varepsilon}:L^p((0,T);H^{1,p}(\Omega _{\varepsilon}^p))\to L^p((0,T)
;H^{1,p}(\Omega))$ such that
\begin{equation}
R^{\varepsilon}u(t,x):=[Q^{\varepsilon}u(t,.)](x)\quad
\text{for }u\in L^p((0,T);H^{1,p}(\Omega _{\varepsilon}^p)),\label{eqn2.30}
\end{equation}
where $Q^{\varepsilon}$ is the extension operator from theorem \ref{thm2.3.2.1}.
 Then
\begin{equation*}
\frac{\partial}{\partial t}[R^{\varepsilon}u(t,x)]
=\frac{\partial }{\partial t}[Q^{\varepsilon}u(t,.)](x)
=[Q^{\varepsilon}(\frac{\partial u}{\partial t}(t,.))](x)
=R^{\varepsilon}(\frac{\partial u}{\partial t})(t,x).
\end{equation*}
Based on the above definition we have the following extension theorem for
 the functions depending on $t$ and $x$.

\begin{theorem}\label{thm2.3.2.2}
Let $\Omega$ and $\Omega _{\varepsilon}^p$ be defined as in section \ref{sec1}
and $1\le p,q\le \infty$. Then there exists a bounded linear operator
\begin{align*}
R^{\varepsilon}&:L^{q}((0,T);H^{1,p}(\Omega _{\varepsilon}^p))
\cap H^{1,q}((0,T);L^p(\Omega _{\varepsilon}^p))\\
&\to L^{q}((0,T);H^{1,p}(\Omega))\cap H^{1,q}((0,T);L^p(\Omega))
\end{align*}
such that for all
$u\in L^{q}((0,T);H^{1,p}(\Omega _{\varepsilon}^p))
\cap H^{1,q}((0,T);L^p(\Omega _{\varepsilon}^p))$,
\begin{equation}
\|R^{\varepsilon}u\|_{L^{q}((0,T);H^{1,p}(\Omega))}
\le C\|u\|_{L^{q}((0,T);H^{1,p}(\Omega _{\varepsilon}^p))},\label{eqn2.31}
\end{equation}
where the constant $C$ is independent of $\varepsilon$ and $u$.
\end{theorem}

\begin{proof}
Here we only show the measurability of $R^{\varepsilon}u$.
The inequality \eqref{eqn2.31} follows by scaling.
Since we know that every continuous function is measurable, we
show $R^{\varepsilon}u$ is continuous. But by theorem \ref{thm2.3.2.1}
it can be shown $Q^{\varepsilon}u(t)$ is continuous on $\bar{\Omega}$.
The continuity of $R^{\varepsilon}u$ on $[0,T]\times\bar{\Omega}$
follows from the definition \eqref{eqn2.30}.
\end{proof}

\begin{theorem}\label{thm2.3.2.3}
Let $1< p,q <\infty $ such that $\frac{1}{p}+\frac{1}{q}=1$ and
\[
u\in(H^{1,q}(\Omega _{\varepsilon}^p)^{*},H^{1,p}
(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}.
\]
Then there exists an extension $\bar{u}$ of $u$ such that
$\bar{u}\in (H^{1,q}(\Omega)^{*},H^{1,p}(\Omega))_{1-\frac{1}{p},p}$.
\end{theorem}

\begin{proof}
Let $\theta=1-\frac{1}{p}$. We use the $K$-functional definition for
real interpolation space
$(H^{1,q}(\Omega _{\varepsilon}^p)^{*},H^{1,p}
(\Omega _{\varepsilon}^p))_{\theta,p}$.
To begin with, let $v\in H^{1,q}(\Omega _{\varepsilon}^p)$,
then by theorem \ref{thm2.3.3.2} there exists an extension
$Q^{\varepsilon}v$ of $v$ such that
\begin{gather}
 Q^{\varepsilon}v = v \text{ in }\Omega _{\varepsilon}^p, \label{eqn2.32}\\
\|Q^{\varepsilon}v\|_{H^{1,q}(\Omega)} \leq
 C\|v\|_{H^{1,q}(\Omega _{\varepsilon}^p)},\label{eqn2.33}
\end{gather}
where $C$ is independent of $\varepsilon$ and $v$.
Let $a_0\in H^{1,q}(\Omega _{\varepsilon}^p)^{*}$,
then we define an extension $\bar{a}_0$ of $a_0$ as
\begin{equation}
\langle \bar{a}_0, Q^{\varepsilon}v\rangle_{H^{1,q}(\Omega)^{*}
\times H^{1,q}(\Omega)}:=\langle a_0, v\rangle_{H^{1,q}
(\Omega _{\varepsilon}^p)^{*}\times H^{1,q}
(\Omega _{\varepsilon}^p)}.\label{eqn2.34}
\end{equation}
Therefore,
\begin{align*}
\|\bar{a}_0\|_{H^{1,q}(\Omega)^{*} }
&=\sup_{\|Q^{\varepsilon}v\|_{H^{1,q}(\Omega)}\le 1}
 |\langle \bar{a}_0,Q^{\varepsilon}v\rangle _{H^{1,q}(\Omega)^{*}
 \times H^{1,q}(\Omega)}| \\
&=\sup_{\|v\|_{H^{1,q}(\Omega _{\varepsilon}^p)}\le 1}
 |\langle a_0, v\rangle _{H^{1,q}(\Omega _{\varepsilon}^p)^{*}
 \times H^{1,q}(\Omega _{\varepsilon}^p)}|
\quad\text{by \eqref{eqn2.33} and \eqref{eqn2.34}} \\
&\leq \|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}
\end{align*}
which implies
\begin{equation}
 \|\bar{a}_0\|_{H^{1,q}(\Omega)^*}
\le \|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}.\label{eqn2.35}
\end{equation}
Again assume that $b_0\in H^{1,p}(\Omega _{\varepsilon}^p)$.
Let $\bar{b}_0\in H^{1,p}(\Omega)$ denote the extension of $b_0$ such that
\begin{equation}
\|\bar{b}_0\|_{H^{1,p}(\Omega)} \leq
C\|b_0\|_{H^{1,p}(\Omega _{\varepsilon}^p)}\quad
\text{for } b_0\in H^{1,p}(\Omega _{\varepsilon}^p),\label{eqn2.36}
\end{equation}
where $C$ is independent of $\varepsilon$ and $b_0$.
Let $t>0$. Then
\begin{align*}
\|\bar{a}_0\|_{H^{1,q}(\Omega)^*}+t\|\bar{b}_0\|_{H^{1,p}(\Omega)}
&\le \|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}
 +Ct\|b_0\|_{H^{1,p}(\Omega _{\varepsilon}^p)} \\
&\le \max(1,C)(\|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}
 +t\|b_0\|_{H^{1,p}(\Omega _{\varepsilon}^p)})\,.
\end{align*}
Taking the infimum on both sides, we get successively
\begin{align*}
&\inf_{\underset{\bar{b}_0\in H^{1,p}(\Omega)}
{\bar{a}_0\in H^{1,q}(\Omega)^*}}{\bar{u}=\bar{a}_0+\bar{b}_0}
\Big(\|\bar{a}_0\|_{H^{1,q}(\Omega)^*}
+t\|\bar{b}_0\|_{H^{1,p}(\Omega)}\Big) \\
&\le \max(1,C)\inf_{\underset{\underset{b_0\in H^{1,p}
(\Omega _{\varepsilon}^p)}{a_0\in H^{1,q}(\Omega _{\varepsilon}^p)^{*}}}
{u=a_0+b_0}}
\Big(\|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}+t\|b_0\|_{H^{1,p}
(\Omega _{\varepsilon}^p)}\Big)\,,
\end{align*}
\begin{align*}
&\underbrace{t^{-\theta}
\inf_{\underset{\underset{\bar{b}_0\in H^{1,p}(\Omega)}
{\bar{a}_0\in H^{1,q}(\Omega)^*}}{\bar{u}=\bar{a}_0+\bar{b}_0}}
\Big(\|\bar{a}_0\|_{H^{1,q}(\Omega)^*}+t\|\bar{b}_0\|_{H^{1,p}(\Omega)}\Big)
}_{positive} \\
&\le \max(1,C)\underbrace{t^{-\theta}
\inf{\underset{\underset{b_0\in H^{1,p}
(\Omega _{\varepsilon}^p)}{a_0\in H^{1,q}
(\Omega _{\varepsilon}^p)^*}}{u=a_0+b_0}}
\Big(\|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}+t\|b_0\|_{H^{1,p}
(\Omega _{\varepsilon}^p)}\Big)}_{positive}\,,
\end{align*}
\begin{align*}
&\Big|t^{-\theta}\underset{\underset{\underset{\bar{b}_0
\in H^{1,p}(\Omega)}{\bar{a}_0\in H^{1,q}(\Omega)^*}}{\bar{u}
=\bar{a}_0+\bar{b}_0}}{\inf}(\|\bar{a}_0\|_{H^{1,q}(\Omega)^*}
+t\|\bar{b}_0\|_{H^{1,p}(\Omega)})\Big|^p \\
& \le [\max(1,C)]^p
\Big|t^{-\theta}\inf_{\underset{\underset{b_0\in H^{1,p}
(\Omega _{\varepsilon}^p)}{a_0\in H^{1,q}(\Omega _{\varepsilon}^p)^*}}
{u=a_0+b_0}}
\Big(\|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}
+t\|b_0\|_{H^{1,p}(\Omega _{\varepsilon}^p)}\Big)\Big|^p\,.
\end{align*}
Thus
\begin{align*}
&\int_0^{\infty}\Big|t^{-\theta}
\inf_{\underset{\underset{\bar{b}_0\in H^{1,p}(\Omega)}{\bar{a}_0
\in H^{1,q}(\Omega)^*}}{\bar{u}=\bar{a}_0+\bar{b}_0}}
\Big(\|\bar{a}_0\|_{H^{1,q}(\Omega)^*}+t\|\bar{b}_0\|_{H^{1,p}(\Omega)}\Big)
\Big|^p \frac{dt}{t} \\
&\le [\max(1,C)]^p \int_0^{\infty}|t^{-\theta}
\inf_{\underset{\underset{b_0\in H^{1,p}(\Omega _{\varepsilon}^p)}
{a_0\in H^{1,q}(\Omega _{\varepsilon}^p)^{*}}}{u=a_0+b_0}}
\Big(\|a_0\|_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}}
+t\|b_0\|_{H^{1,p}(\Omega _{\varepsilon}^p)}\Big)\Big|^p \frac{dt}{t}\,,
\end{align*}
\begin{align*}
&\int_0^{\infty}\big|t^{-\theta}K(t,\bar{u}, H^{1,q}(\Omega)^*,
H^{1,p}(\Omega))\big|^p\frac{dt}{t} \\
&\le [\max(1,C)]^p \int_0^{\infty}
\big|t^{-\theta} K(t,u, H^{1,q}(\Omega _{\varepsilon}^p)^{*},
H^{1,p}(\Omega _{\varepsilon}^p))\big|^p\,\frac{dt}{t}\,,
\end{align*}
\[
\|\bar{u}\|_{(H^{1,q}(\Omega)^*, H^{1,p}(\Omega))_{1-\frac{1}{p},p}} \\
\le \max(1,C) \|u\|_{(H^{1,q}(\Omega _{\varepsilon}^p)^{*}, H^{1,p}
(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}}\,,
\]
where the constant $\max(1,C)$ is independent of $\varepsilon$ and $u$.
\end{proof}

\subsubsection{Embedding Theorems}\label{sec2.3.3}

\begin{theorem}\label{thm2.3.3.1}
Let $\Omega$ and $\Omega _{\varepsilon}^p$ be as in section \ref{sec1}.
Assume that $1\le p< n$ and $u\in H^{1,p}(\Omega _{\varepsilon}^p)$.
Then $u\in L^{p^{*}}(\Omega _{\varepsilon}^p)$ and there is a constant $C$
\begin{equation}
\|u\|_{L^{p^{*}}(\Omega _{\varepsilon}^p)}
\le C\|u\|_{H^{1,p}(\Omega _{\varepsilon}^p)},\label{eqn2.37}
\end{equation}
where $p^{*}=np/(n-p)$ and $C$ is independent of $\varepsilon$ and $u$.
In other words, $H^{1,p}(\Omega _{\varepsilon}^p)
\hookrightarrow L^{p^{*}}(\Omega _{\varepsilon}^p)$ with embedding constant
independent of $\varepsilon$.
\end{theorem}

\begin{proof}
Let $u\in H^{1,p}(\Omega _{\varepsilon}^p)$. Then from theorem \ref{thm2.3.2.1},
there exists an extension $Q^{\varepsilon}u$ of $u$ from
$H^{1,p}(\Omega _{\varepsilon}^p)$ to $H^{1,p}(\Omega)$ such that
\begin{equation}
\|Q^{\varepsilon}u\|_{H^{1,p}(\Omega)}\leq
 C\|u\|_{H^{1,p}(\Omega _{\varepsilon}^p)}.\label{eqn2.38}
\end{equation}
Let $v:= Q^{\varepsilon}u$. By assumption $\Omega$ is a bounded domain
with sufficiently smooth boundary, then from
\cite[Theorem 2 of section 5.6.1]{Eva98} we obtain
\begin{equation}
\|v\|_{L^{p^{*}}(\Omega)}
\leq C\|v\|_{H^{1,p}(\Omega)}\text{ for }v\in H^{1,p}(\Omega),\label{eqn2.39}
\end{equation}
where $p^{*}\;=\;\frac{np}{n-p}$ and $C$ depends only on $p$, $n$ and
 $\Omega$ but is independent of $v$. Therefore by \eqref{eqn2.38}
and \eqref{eqn2.39} we obtain
\[
\|u\|_{L^{p^{*}}(\Omega _{\varepsilon}^p)}\le \|v\|_{L^{p^{*}}(\Omega )}
\le C \|v\|_{H^{1,p}(\Omega)}\le C \|u\|_{H^{1,p}(\Omega _{\varepsilon}^p)},
\]
where $C$  is independent of $\varepsilon$ and $u$.
\end{proof}

\begin{theorem}\label{thm2.3.3.2}
Let $1<p,q<\infty$ be such that $p>n+2$ and $\frac{1}{p}+\frac{1}{q}\;=\;1$.
Assume that $u\in (H^{1,q}(\Omega _{\varepsilon}^p)^*,H^{1,p}
(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}$ such that
$\sup_{\varepsilon>0} \|u\|_{(H^{1,q}(\Omega _{\varepsilon}^p)^*,H^{1,p}
(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}}<\infty$.
 Then $u\in L^{\infty}(\Omega _{\varepsilon}^p)$ and
\begin{equation}
\sup_{\varepsilon>0} \|u\|_{L^{\infty}(\Omega _{\varepsilon}^p)}<\infty.
\label{eqn2.40}
\end{equation}
\end{theorem}

\begin{proof}
From theorem \ref{thm2.1.1.2}, we know that for
$u\in (H^{1,q}(\Omega)^*, H^{1,p}(\Omega))_{1-\frac{1}{p},p}$,
$u\in L^{\infty}(\Omega)$ and
\begin{equation}
\|u\|_{L^{\infty}(\Omega)}\leq  C \|u\|_{(H^{1,q}(\Omega)^*,
H^{1,p}(\Omega))_{1-\frac{1}{p},p}},\label{eqn2.41}
\end{equation}
where the constant $C$ is independent of $u$.
Let $u\in (H^{1,q}(\Omega _{\varepsilon}^p)^*,
H^{1,p}(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}$, then
\begin{align*}
\|u\|_{L^{\infty}(\Omega _{\varepsilon}^p)}
&\leq \|u\|_{L^{\infty}(\Omega)} \\
&\leq  C \|u\|_{(H^{1,q}(\Omega)^*, H^{1,p}(\Omega))_{1-\frac{1}{p},p}}
\quad\text{by\eqref{eqn2.41}} \\
&\leq C \|u\|_{(H^{1,q}(\Omega _{\varepsilon}^p)^*, H^{1,p}
 (\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}}\quad
 \text{ by theorem \ref{thm2.3.2.3}} \\
&\leq  C\sup_{\varepsilon>0} \|u\|_{(H^{1,q}(\Omega _{\varepsilon}^p)^*,
H^{1,p}(\Omega _{\varepsilon}^p))_{1-\frac{1}{p},p}} 
< \infty\quad \forall \varepsilon>0,
\end{align*}
where the constant $C$ is independent of $\varepsilon$ and $u$.
Therefore $\sup_{\varepsilon>0} \|u\|_{L^{\infty}(\Omega _{\varepsilon}^p)}
<\infty$.
\end{proof}

From this theorem we notice that for $1\le p<\infty$,
\begin{equation}
\|u\|_{L^p(\Omega_{\varepsilon}^p)}^p=\int_{\Omega_{\varepsilon}^p}|u(x)|^p\,dx
\le |\Omega_{\varepsilon}^p|\|u\|_{L^{\infty}(\Omega_{\varepsilon}^p)}^p
\le |\Omega|\sup_{\varepsilon>0} \|u\|^p_{L^{\infty}(\Omega_{\varepsilon}^p)}
<\infty\quad \forall \varepsilon.\label{eqn2.42}
\end{equation}
\subsection{Two-scale Convergence}\label{sec2.4}

\begin{definition}\label{thm2.4.1} \rm
A sequence of functions $(u_{\varepsilon})_{\varepsilon >0}$ in
$L^p((0,T)\times \Omega)$ is said to two-scale convergent to a limit
$u\in L^p((0,T)\times\Omega \times Y)$ if
\begin{equation}
\lim_{\varepsilon \to 0} \int_0^T\int_{\Omega}u_{\varepsilon}(t,x)
\phi (t,x,\frac{x}{\varepsilon})\,dx\,dt
=\int_0^T\int_{\Omega}\int_{Y}u(t,x,y)\phi (t,x,y)\,dx\,dy\,dt
\label{eqn2.43}
\end{equation}
for all $\phi \in L^{q}((0,T)\times \Omega; C_{per}(Y))$.
\end{definition}

We quote the following theorems whose proofs  can be found in
\cite{All92,Rad92,Cla98}.

\begin{theorem}\label{thm2.4.2}
For every bounded sequence, $(u_{\varepsilon})_{\varepsilon >0}$,
in $L^p((0,T)\times \Omega)$ there exist a subsequence and a
 $u\in L^p((0,T)\times \Omega \times Y)$ such that the subsequence
two-scale converges to $u$.
\end{theorem}

\begin{theorem}\label{thm2.4.3}
Let $(u_{\varepsilon})_{\varepsilon >0}$ be strongly convergent to
$u \in L^p((0,T)\times \Omega)$, then $(u_{\varepsilon})_{\varepsilon >0}$
is two-scale convergent to $u_1(t,x,y)=u(t,x)$.
\end{theorem}

\begin{theorem}\label{thm2.4.4}
Let $(u_\varepsilon)_{\varepsilon >0}$ be a sequence in
$L^p((0,T);H^{1,p}(\Omega))$ such that $u_{\varepsilon}\to u$ weakly
in $L^p((0,T);H^{1,p}(\Omega))$. Then $(u_\varepsilon)_{\varepsilon >0}$
two-scale converges to u and there exist a subsequence $\varepsilon$,
still denoted by same symbol, and a
$u_1\in L^p((0,T)\times \Omega; H^{1,p}_{per}(Y))$ such that
$\nabla _{x}u_{\varepsilon}\overset{2}{\rightharpoonup} \nabla u
+\nabla _{y} u_1$.
\end{theorem}

\section{Global existence and uniqueness of solution to
$(P_{\varepsilon})$}\label{sec3}

The main result of this section is the following existence theorem:

\begin{theorem}\label{thm3.1}
Suppose that the assumptions \eqref{eqn1.16}-\eqref{eqn1.20} are satisfied.
Then there exists a unique positive global weak solution
$u _{\varepsilon}\in F_{\varepsilon}^I$ of the problem $(P_{\varepsilon})$.
\end{theorem}

Theorem \ref{thm3.1} is proved in \cite[heorem 2.4]{MB13} or
\cite[Theorem 4.1.1.1]{HSM13}.
The ingredients of the proof are a Lyapunov functional, Schaefer's fixed
point theorem and \cite[Theorem 2.5]{PS01} which is based on
the maximal regularity of differential operators.
In \cite{MB13} it is also shown that with the help of a Lyapunov
functional we can obtain global in time \textit{a-priori}
estimates of the type
\begin{gather}
\|u_{\varepsilon}(t)\|_{L^{r}(\Omega_{\varepsilon}^p)^I}
\le C_1<\infty\quad \text{for all } r\ge 2
 \text{ and for a.e. }t,\label{eqn3.1}
\\
\|u_{\varepsilon}(t)\|_{L^{\infty}(\Omega_{\varepsilon}^p)^I}
\le C_2<\infty\quad \text{for a.e. }t,\label{eqn3.2}
\end{gather}
where $C_1$ and $C_2$ are independent of $i$, $t$, $\varepsilon$ and
$u_{\varepsilon_i}$. The constant $C_1$ depends only on
$r\in \mathbb{N}$ (cf. \cite{MB13}, see also \cite{HSM13}).

PAGE 10

\section{Homogenization of Problem $(P_{\varepsilon})$}\label{sec4}

\subsection{A-priori estimates} \label{sec4.1}

In this section, we obtain $\varepsilon$-independent \textit{a-priori} 
estimates for the solution $u_\varepsilon$ of $(P_{\varepsilon})$ 
and extend these estimates to all of $(0,T)\times\Omega$. 
The major theorem of this section reads as follows.

\begin{theorem}\label{thm4.1.1}
Let $r\in \mathbb{N}$ ($r\ge 2$). There exists a constant $C>0$ independent 
of $\varepsilon$ such that the extension of $u_{\varepsilon}$ 
(denoted by the same symbol) to all of $(0,T)\times \Omega$ satisfies
\begin{equation}
\begin{aligned}
&\sup_{\varepsilon>0} \Big(\||u_{\varepsilon}\||_{{L^{r}((0,T);L^{r}(\Omega))}^I}
+\||u_{\varepsilon}\||_{{L^{\infty}((0,T);L^{\infty}(\Omega ))}^I}\\
&+\||\nabla u_{\varepsilon}\||_{{L^2((0,T);L^2(\Omega ))}^I}\Big)
\le C<\infty.
\end{aligned}\label{eqn4.1}
\end{equation}
\end{theorem}

We start with the following lemma.

\begin{lemma}\label{thm4.1.2}
Let $r\in \mathbb{N}$ ($r\ge 2$). There exists a constant $C>0$ independent 
of $\varepsilon$ such that the solution $u_{\varepsilon}$ of 
$(P_{\varepsilon})$ satisfies
\begin{equation}
\begin{aligned}
&\sup_{\varepsilon>0} \Big(\||u_{\varepsilon}\||_{{L^{r}((0,T);L^{r}
(\Omega _{\varepsilon}^p))}^I}+\||u_{\varepsilon}\||_{{L^{\infty}((0,T);
L^{\infty}(\Omega _{\varepsilon}^p))}^I}\\
&+\||\nabla u_{\varepsilon}
\||_{{L^2((0,T);L^2(\Omega _{\varepsilon}^p))}^I}\Big) \le C<\infty.
\end{aligned} \label{eqn4.2}
\end{equation}
\end{lemma}

\begin{proof}
By \eqref{eqn3.1} we obtain
\begin{equation}
\begin{aligned}
\||u_{\varepsilon}\||^{r}_{L^{r}((0,T);L^{r}(\Omega _{\varepsilon}^p))^I}
&=\sum_{i=1}^I\int_0^T\|u_{\varepsilon_i}(t)\|^{r}_{L^{r}
(\Omega_{\varepsilon}^p)}\,dt\le C_1\sum_{i=1}^I\int_0^T\,dt\\
&=C_1IT=:C_3<\infty\quad \forall \varepsilon. \label{eqn4.3}
\end{aligned}
\end{equation}
Equation \eqref{eqn3.2} gives
\begin{equation}
\begin{aligned}
\||u_{\varepsilon }\||_{L^{\infty}((0,T);L^{\infty}(\Omega _{\varepsilon}^p))^I}
&=\operatorname{ess\,sup}_{t\in (0,T)} \||u_{\varepsilon}(t)\||_{L^{\infty}
(\Omega _{\varepsilon}^p)^I}\\
&\le \operatorname{ess\,sup}_{t\in (0,T)} C_2
= C_2\quad \forall \varepsilon.\label{eqn4.4}
\end{aligned}
\end{equation}
Testing the \textit{i-th} PDE of \eqref{eqn1.12} with $u_{\varepsilon _i}$,
we obtain\footnote{From \eqref{eqn3.1}, we have
$\|u_{\varepsilon _i}(t)\|_{L^{r}(\Omega _{\varepsilon}^p)}\le C_1$
for all $i$ and for a.e. $t$, where $C_1$ is independent of $\varepsilon$.
This gives $\sup_{\varepsilon>0} \|SR(u_{\varepsilon})_i
\|_{L^p(\Omega _{\varepsilon}^p)}\le C$.
Since $L^p(\Omega _{\varepsilon}^p)\hookrightarrow H^{1,q}
(\Omega _{\varepsilon}^p)^{*}$, from the definition \eqref{eqn2.3}
 we get $\langle SR(u_{\varepsilon})_i,
\phi _i\rangle_{H^{1,q}(\Omega _{\varepsilon}^p)^{*}\times H^{1,q}
(\Omega _{\varepsilon}^p)}=\langle SR(u_{\varepsilon})_i,
 \phi _i\rangle_{L^p(\Omega _{\varepsilon}^p)
\times L^{q}(\Omega _{\varepsilon}^p)}$ for
$\phi_i\in H^{1,q}(\Omega _{\varepsilon}^p)$.}
\begin{align*}
&\int_0^T\langle \frac{\partial u_{\varepsilon_i}(t)}{\partial t},
u_{\varepsilon_i}(t)\rangle _{H^{1,q}(\Omega _{\varepsilon}^p)^{*}
\times H^{1,q}(\Omega _{\varepsilon}^p)}\,dx\,dt \\
&-\int_0^T\langle \nabla \cdot D\nabla u_{\varepsilon_i}(t),
 u_{\varepsilon_i}(t)\rangle _{H^{1,q}(\Omega _{\varepsilon}^p)^{*}
 \times H^{1,q}(\Omega _{\varepsilon}^p)}\,dx\,dt \\
&=\int_0^T\langle SR(u_{\varepsilon}(t))_i,u_{\varepsilon _i}(t)
 \rangle _{H^{1,q}(\Omega _{\varepsilon}^p)^{*}\times H^{1,q}
 (\Omega _{\varepsilon}^p)}\,dt;
\end{align*}
i.e.,
\begin{align*}
&\frac{1}{2}\int_0^T\frac{d}{dt}\|u_{\varepsilon _i}(t)\|^2_{L^2
(\Omega _{\varepsilon}^p)}\,dt +\;\int_0^TD\|\nabla u_{\varepsilon _i}(t)
\|^2_{L^2(\Omega _{\varepsilon}^p)}\,dt \\
&=\int_0^T\langle SR(u_{\varepsilon}(t))_i,u_{\varepsilon _i}(t)
 \rangle_{L^p(\Omega _{\varepsilon}^p)\times L^{q}(\Omega _{\varepsilon}^p) }\,dt
 \\
&\leq \frac{1}{p}\int_0^T\|SR(u_{\varepsilon}(t))_i\|^p_{L^p
 (\Omega _{\varepsilon}^p)}\,dt + \frac{1}{q}\int_0^T
 \|u_{\varepsilon _i}(t)\|_{L^{q}(\Omega _{\varepsilon}^p)}^{q}\,dt;
\end{align*}
i.e.,
\begin{equation}
\begin{aligned}
&\frac{1}{2}\|u_{\varepsilon _i}(T)\|^2_{L^2(\Omega _{\varepsilon}^p)}
+\int_0^TD\|\nabla u_{\varepsilon _i}(t)\|^2_{L^2
(\Omega _{\varepsilon}^p)}\,dt \\
&\le \frac{1}{2}\|u_{0_i}\|^2_{L^2(\Omega _{\varepsilon}^p)}
+\frac{1}{p}\int_0^T\|SR(u_{\varepsilon}(t))_i\|^p_{L^p
(\Omega _{\varepsilon}^p)}\,dt
+ \frac{1}{q}\int_0^T\|u_{\varepsilon _i}(t)\|_{L^{q}
(\Omega _{\varepsilon}^p)}^{q}\,dt.
\end{aligned} \label{eqn4.5}
\end{equation}
Therefore the right-hand side of \eqref{eqn4.5} is bounded by a constant
independent of $\varepsilon$, $i$ and $t$. Let us call this constant
by $\bar{C}$. This gives
\[
\int_0^TD\|\nabla u_{\varepsilon _i}(t)\|^2_{L^2(\Omega _{\varepsilon}^p)}\,dt
\le \bar{C}\quad \text{for all $\varepsilon$, $i$ and for a.e. }t
\]
which implies
\begin{equation}
 \sum_{i=1}^I\int_0^T\|\nabla u_{\varepsilon _i}(t)\|^2_{L^2
(\Omega _{\varepsilon}^p)}\,dt
 \leq \sum_{i=1}^I\frac{\bar{C}}{D}=:C_{4}<\infty\quad
\forall \varepsilon.
\label{eqn4.6}
\end{equation}
Note that $D>0$ is constant in \eqref{eqn4.6}. Adding \eqref{eqn4.3},
\eqref{eqn4.4} and \eqref{eqn4.6} yields
\begin{align*}
&\||u_{\varepsilon}\||_{{L^{r}((0,T);L^{r}
 (\Omega _{\varepsilon}^p))}^I}+\||u_{\varepsilon}\||_{{L^{\infty}((0,T);
 L^{\infty}(\Omega _{\varepsilon}^p))}^I}
 +\||\nabla u_{\varepsilon}\||_{{L^2((0,T);L^2(\Omega _{\varepsilon}^p))}^I}  \\
&\le  C_3^{1/r} + C_2\;+C_{4}^{1/2}
= C<\infty\quad \text{for all }\varepsilon.
\end{align*}
This completes the proof.
\end{proof}

\begin{proof}[Proof of theorem \ref{thm4.1.1}] 
The estimate \eqref{eqn4.2} from  lemma \ref{thm4.1.2} and  
theorem \ref{thm2.3.2.2} accomplish the proof.
\end{proof}

\subsection{Convergence of $u_{\varepsilon}$} \label{sec4.1.2.2}

In this section, we show the \textit{weak}, \textit{strong} and 
\textit{two-scale} convergences of the solution of $(P_{\varepsilon})$.

\begin{theorem}\label{thm4.2.1}
There exists a constant $C$ independent of $\varepsilon$ such that the 
solution $u_{\varepsilon}$ of the problem $(P_{\varepsilon})$ satisfies 
the  estimate
\begin{equation}
\begin{aligned}
&\sup_{\varepsilon>0} \Big(\||u_{\varepsilon}\||_{L^{\infty}((0,T);
L^2(\Omega))^I}+\||u_{\varepsilon}\||_{L^2((0,T);H^{1,2}(\Omega))^I}\\
&+\||\chi^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial t}\||_{L^2((0,T);
H^{1,2}(\Omega)^{*})^I}\Big)  
\le C<\infty,
\end{aligned}\label{eqn4.7}
\end{equation}
where the function $\chi^{\varepsilon}$ is defined in \eqref{eqn1.11}.
\end{theorem}

\begin{proof}
From \eqref{eqn4.1}, we have
\begin{equation}
\begin{aligned}
&\||u_{\varepsilon}\||^2_{L^{\infty}((0,T);L^2(\Omega))^I}\\
&\le |\Omega|\||u_{\varepsilon }\||^2_{L^{\infty}((0,T);
L^{\infty}(\Omega))^I}&\le |\Omega|\sup_{\varepsilon>0}
\||u_{\varepsilon }\||^2_{L^{\infty}((0,T);L^{\infty}(\Omega))^I} \\
&=:C_{5}<\infty\quad \forall \varepsilon.
\end{aligned}\label{eqn4.8}
\end{equation}
Again,
\begin{align*}
&\||u_{\varepsilon}\||_{L^2((0,T);H^{1,2}(\Omega))^I}^2\\
&=\sum_{i=1}^I\|u_{\varepsilon_i}\|_{L^2((0,T);H^{1,2}(\Omega))}^2 \\
&=\sum_{i=1}^I\Big(\|\nabla u_{\varepsilon _i}\|^2_{L^2((0,T);L^2(\Omega))}
 +\|u_{\varepsilon _i}\|^2_{L^2((0,T);L^2(\Omega))}\Big) \\
&\le  \sup_{\varepsilon >0} \sum_{i=1}^I(\|\nabla u_{\varepsilon _i}
 \|^2_{L^2((0,T);L^2(\Omega))}
 +(T|\Omega|)^{1-\frac{2}{r}}\|u_{\varepsilon _i}\|^2_{L^{r}((0,T);
 L^{r}(\Omega))}) 
<\infty,
\end{align*}
by \eqref{eqn4.1}; i.e.,
\begin{equation}
\||u_{\varepsilon}\||_{L^2((0,T);H^{1,2}(\Omega))^I}
\le  C_{6}<\infty\quad \forall \varepsilon. \label{eqn4.9}
\end{equation}
Now, let $\phi \in H^{1,2}_0(0,T)$ and $\psi \in H^{1,2}(\Omega)$.
Then the weak formulation of the \textit{i-th} PDE of the
problem \eqref{eqn1.12}-\eqref{eqn1.15} is
\begin{align*}
&\int_0^T\langle\chi^{\varepsilon}\frac{\partial u_{\varepsilon _i}(t)}
{\partial t}, \phi(t) \psi\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt
+ \int_0^T\int_{\Omega}\phi(t)\chi ^{\varepsilon}(x)\nabla u_{\varepsilon _i}
 (t,x)\nabla \psi (x) \,dx\,dt\\
&= \int_0^T\langle\chi ^{\varepsilon} SR(u_{\varepsilon}(t))_i, \phi(t)
\psi\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt\,dt;
\end{align*}
i.e.,
\begin{align*}
&\Big|\int_0^T\langle\chi^{\varepsilon}
 \frac{\partial u_{\varepsilon _i}(t)}{\partial t},
 \phi(t) \psi\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt\Big|  \\
&\leq \int_0^T\int_{\Omega}|\chi ^{\varepsilon}(x)
||\nabla u_{\varepsilon _i}(t,x)||\nabla \psi(x) ||\phi (t)| \,dx\,dt\\
&\quad +\frac{1}{2}\int_0^T[\|\chi ^{\varepsilon}SR(u_{\varepsilon}(t))_i
\|^2_{L^2(\Omega)}+\|\phi(t) \psi \|^2_{L^2(\Omega)}]\,dt.
\end{align*}
Note that $|\chi ^{\varepsilon}(x)|\le 1$. From \eqref{eqn4.1} the
terms $\sup_{\varepsilon >0}
\|\nabla u_{\varepsilon _i}\|^2_{L^2((0,T);L^2(\Omega))}$ and
$\sup_{\varepsilon >0} \|SR(u_{\varepsilon})_i\|^2_{L^2((0,T);L^2(\Omega))}$
are finite. This gives
\begin{align*}
&\big|\int_0^T\langle\chi^{\varepsilon}\frac{\partial u_{\varepsilon _i}(t)}
{\partial t}, \phi(t) \psi\rangle_{H^{1,2}(\Omega)^{*}
\times H^{1,2}(\Omega)}\,dt\big|\\
&\le  C+\frac{1}{2}\|\phi(t) \|^2_{L^2(0,T)}[\|\nabla \psi \|^2_{L^2(\Omega)}+\|\psi \|^2_{L^2(\Omega)}]\\
&= C +\|\phi \|^2_{L^2(0,T)}\|\psi \|^2_{H^{1,2}(\Omega)}.
\end{align*}
Note that $\phi \in H^{1,2}_0(0,T)$ implies
 $ \|\phi\|_{L^2(0,T)}\le \bar{C}\|\phi\|_{H^{1,2}_0(0,T)}$; i.e.,
$\|\frac{\phi}{\bar{C}}\|_{L^2(0,T)}\le \|\phi\|_{H^{1,2}_0(0,T)}$,
where $\bar{C}>0$ is the embedding constant. Taking the supremum,
over all
$ \|\psi \|_{H^{1,2}(\Omega)}^2\le 1$,
$\|\frac{\phi}{\bar{C}}\|^2_{L^2(0,T)}\le 1$,
$\psi \in H^{1,2}(\Omega)$,
$\frac{\phi}{\bar{C}} \in L^2(0,T)$,
on both  sides of the above inequality yields
\begin{align*}
&\bar{C} \sup 
\Big|\int_0^T\langle\chi^{\varepsilon}
\frac{\partial u_{\varepsilon _i}(t)}{\partial t},
\frac{\phi(t)}{\bar{C}} \psi\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}
\,dt|\\
&\le C+ \bar{C}^2 \sup
\|\psi \|^2_{H^{1,2}(\Omega)}\|\frac{\phi}{\bar{C}} \|^2_{L^2(0,T)}.
\end{align*}
This implies
\[
\|\chi ^{\varepsilon}\frac{\partial u_{\varepsilon _i}}{\partial t}\|_{L^2((0,T);
H^{1,2}(\Omega)^{*})}\le C_{7}
\]
which implies
\[
\sum_{i=1}^I\|\chi ^{\varepsilon}\frac{\partial u_{\varepsilon _i}}
{\partial t}\|_{L^2((0,T);H^{1,2}(\Omega)^{*})}^2\le \; I\;C_{7}^2
\]
which implies
\begin{equation}
\||\chi ^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial t}\||
_{L^2((0,T);H^{1,2}(\Omega)^{*})^I} \le
 (I\;C_{7}^2)^{1/2}\quad \forall \varepsilon.\label{eqn4.10}
\end{equation}
Adding \eqref{eqn4.8}, \eqref{eqn4.9} and \eqref{eqn4.10}, we obtain
\begin{align*}
&\|u_{\varepsilon}\|_{L^{\infty}((0,T);L^2(\Omega))^I}
+\|u_{\varepsilon }\|_{L^2((0,T);H^{1,2}(\Omega))^I}
+\|\chi^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial t}
 \|_{L^2((0,T);H^{1,2}(\Omega)^{*})^I}  \\
&\leq C_{5}+C_{6}+(I\;C_{7}^2)^{1/2}
=: C<\infty\quad \forall \varepsilon.
\end{align*}
Hence the proof is complete.
\end{proof}

The next statement is crucial. It gives the strong convergence of the 
subsequence of the sequence $(u_{\varepsilon _i})_{\varepsilon >0}$. 
This is the main result of Meirmanov \& Zimin in \cite{MZ11}.

\begin{lemma}[{\cite[Theorem 2.1]{MZ11}}]\label{thm4.2.2}
Let $(c_{\varepsilon })_{\varepsilon >0}$ be a bounded sequence in 
the space $L^{\infty}((0,T);L^2(\Omega))\cap L^2((0,T);H^{1,2}(\Omega))$ 
and weakly convergent in $L^2((0,T);L^2(\Omega))\cap L^2((0,T);H^{1,2}(\Omega))$ 
to a function $c$. Suppose further that the sequence 
$(\chi ^{\varepsilon}\frac{\partial}{\partial t}c_{\varepsilon})_{\varepsilon >0}$
 is bounded in $L^2((0,T);H^{1,2}(\Omega)^{*})$. Then the sequence
 $(c_{\varepsilon })_{\varepsilon >0}$ is strongly convergent to the 
function $c$ in $L^2((0,T);L^2(\Omega))$.
\end{lemma}


\begin{theorem}\label{thm4.2.3}
Let $(u_{\varepsilon})_{\varepsilon >0}$ satisfy the estimates \eqref{eqn4.1}
 and \eqref{eqn4.7}. Then there exists a function 
$u \in L^2{((0,T);H^{1,2}(\Omega))^I}$ and a function 
$u^{1}\in L^2((0,T)\times  \Omega;H^{1,2}_{per}(Y)/\mathbb{R})^I$ 
such that up to a subsequence, still denoted by same subscript, 
the following convergence results hold:
\begin{gather}
(u_{\varepsilon})_{\varepsilon >0}
\text{ is weakly convergent to $u$  in }L^2{((0,T);H^{1,2}(\Omega))^I}.
 \label{eqn4.11}\\
(u_{\varepsilon})_{\varepsilon >0} 
\text{ is strongly convergent to $u$ in }L^2((0,T);L^2(\Omega))^I.\label{eqn4.12}
\\
(u_{\varepsilon })_{\varepsilon >0} \textit{ and }
(\nabla _{x}u_{\varepsilon })_{\varepsilon >0} 
\text{ are two-scale convergent to $u$ and 
 $\nabla _{x}u+\nabla _{y}u^{1}$} \nonumber  \\
\text{in the sense of \eqref{eqn2.43} respectively}.\label{eqn4.13}
\end{gather}
\end{theorem}

\begin{proof}
Statement \eqref{eqn4.11} follows from estimate \eqref{eqn4.7}, 
we note that the sequence $(u_{\varepsilon })_{\varepsilon >0}$ 
is bounded in $L^2((0,T);H^{1,2}(\Omega))^I$. 
This implies that, up to a subsequnce, still indexed by the same subscript, 
$(u_{\varepsilon })_{\varepsilon >0}$ is weakly convergent to a 
function $u$ in $L^2((0,T);H^{1,2}(\Omega))^I$. 

For statement \eqref{eqn4.12}, from \eqref{eqn4.7}, it follows that, 
up to a subsequence, still denoted by the same subscript,
 $(u_{\varepsilon })_{\varepsilon >0}$ is weakly convergent to 
$u$ in the space $L^2((0,T);L^2(\Omega))^I\cap L^2((0,T);H^{1,2}(\Omega))^I$ 
and is bounded in 
$L^{\infty}((0,T);L^2(\Omega))^I\cap L^2((0,T);H^{1,2}(\Omega))^I$. 
Also note that from \eqref{eqn4.7} the function
 $(\frac{\partial}{\partial t}
\chi ^{\varepsilon}u_{\varepsilon})_{\varepsilon >0}$ 
is bounded in $L^2((0,T);H^{1,2}(\Omega)^{*})^I$. 
Therefore the subsequence $(u_{\varepsilon })_{\varepsilon >0}$, 
still denoted by the same subscript, is strongly convergent to 
$u$ in $L^2((0,T);L^2(\Omega))^I$.

Statement \eqref{eqn4.12}follows from  estimate \eqref{eqn4.7} and 
theorem \ref{thm2.4.4}.
\end{proof}

\begin{theorem}\label{thm4.2.4}
The limit function $u$ belongs to $L^{\infty}((0,T)\times \Omega \times Y)^I$.
(Note that the function $u$ is independent of the variable $y$.)
\end{theorem}

\begin{proof}
Since $(u_{\varepsilon})_{\varepsilon >0}$ is strongly convergent to $u$ 
in $L^2((0,T);L^2(\Omega))^I$, there exists a subsequence 
$(u_{\varepsilon '})_{\varepsilon '>0}$ which is pointwise 
convergent \cite[Corollary on page 53]{Yos70} to $u$ almost everywhere 
in $(0,T)\times \Omega$; i.e.,
\[
\lim_{\varepsilon ' \to 0} u_{\varepsilon '}(t,x)=u(t,x)\quad
 \text{ a.e. }\quad(t,x)\in (0,T)\times \Omega.
\]
By theorem \ref{thm4.1.1}, we have 
$\|u_{\varepsilon _i}\|_{L^{\infty}((0,T);L^{\infty}( \Omega))}\le C$ 
for all $i$, therefore
\begin{align*}
|u_i(t,x)|^2
&\le  \sum_{i=1}^I|u_i(t,x)|^2
 =\lim_{\varepsilon '\to 0} \sum_{i=1}^I|u_{\varepsilon_i'}(t,x)|^2\\
&\le \sum_{i=1}^I\limsup_{\varepsilon '\to 0} 
 \operatorname{ess\,sup}_{t\in (0,T)} 
\operatorname{ess\,sup}_{x\in \Omega} |u_{\varepsilon_i'}(t,x)|^2 \\
&\le \sum_{i=1}^I\limsup_{\varepsilon '\to 0} C^2 
=C^2 I\quad \text{for a.e. $t$ and $x$}
\end{align*}
which implies
\[
\operatorname{ess\,sup}_{t\in (0,T)} 
\operatorname{ess\,sup}_{x\in \Omega} |u_i(t,x)|^2
\le C^2 I<\infty\quad \text{for all $i$}.
\]
This gives
\begin{align*}
\||u\||^2_{L^{\infty}((0,T)\times \Omega \times Y)^I}
&\le \max_{1\le i\le I} \operatorname{ess\,sup}_{y\in Y}
\operatorname{ess\,sup}_{t\in (0,T)} 
\operatorname{ess\,sup}_{x\in \Omega} |u_i(t,x)|^2\\
&\leq \operatorname{ess\,sup}_{y\in Y}  C^2 I<\infty.
\end{align*}
\end{proof}

\begin{corollary}\label{thm4.2.5}
For all $1\le p\le \infty$, the sequence $(u_{\varepsilon})_{\varepsilon >0}$ 
is strongly convergent to $u$ in $L^p((0,T)\times \Omega)^I$.
\end{corollary}

\begin{proof}
This follows from the straightforward application of Lyapunov's 
interpolation inequality (cf. lemma \ref{thm5.2.1}) and $L^{\infty}$-estimates 
of $u_{\varepsilon}$ and $u$ 
(cf. Lemma \ref{thm4.1.1} and Theorem \ref{thm4.2.4}).
\end{proof}

\begin{theorem}\label{thm4.2.6}
The sequence $(SR(u_\varepsilon))_{\varepsilon >0}$ is strongly convergent 
to $SR(u)$ in space $L^2((0,T)\times \Omega)^I$ as $\varepsilon \to 0$.
\end{theorem}

\begin{proof}
Note that
\begin{equation}
\||SR(u_{\varepsilon})-SR(u)\||^2_{L^2((0,T)\times \Omega)^I}
=\sum_{i=1}^I\|SR(u_{\varepsilon})_i-SR(u)_i\|^2_{L^2((0,T)\times \Omega)}
\label{eqn4.14}
\end{equation}
From \eqref{en1.5}, we have
\begin{gather}
SR(u_\varepsilon)_i=\sum_{j=1}^{J}s_{ij}\bigg( k_{j}^{f}
\underset{s_{mj}<0}{\prod_{m=1}^I}u_{\varepsilon _m}^{-s_{mj}}
-k_{j}^{b}\underset{s_{mj}>0}{\prod_{m=1}^I}u_{\varepsilon _m}^{s_{mj}}
\bigg)\,\label{eqn4.15}
\\
SR(u)_i=\sum_{j=1}^{J}s_{ij}\bigg(k_{j}^{f}\underset{s_{mj}<0}
{\prod_{m=1}^I}u_{m}^{-s_{mj}}-k_{j}^{b}\underset{s_{mj}>0}
{\prod_{m=1}^I}u_{m}^{s_{mj}}\bigg).\label{eqn4.16}
\end{gather}
From \eqref{eqn4.15} and \eqref{eqn4.16},
\begin{equation}
\begin{aligned}
&\|SR(u_\varepsilon)_i-SR(u)_i\|_{L^2((0,T)\times \Omega)} \\
&=\Big\|\sum_{j=1}^{J}s_{ij}\Big(k_{j}^{f}\underset{s_{mj}<0}
{\prod_{m=1}^I}u_{\varepsilon _m}^{-s_{mj}}-k_{j}^{f}\underset{s_{mj}<0}
{\prod_{m=1}^I}u_{m}^{-s_{mj}}\Big) \\
&\quad -\sum_{j=1}^{J}s_{ij}\Big(k_{j}^{b}
\underset{s_{mj}>0}{\prod_{m=1}^I}u_{\varepsilon _m}^{s_{mj}}-k_{j}^{b}
\underset{s_{mj}>0}{\prod_{m=1}^I}u_{m}^{s_{mj}}\Big)
\Big\|_{L^2((0,T)\times \Omega)}
\\
&\le\sum_{j=1}^{J}s_{ij}k^{f}_{j}\Big\|\underset{s_{mj}<0}
{\prod_{m=1}^I}u_{\varepsilon _m}^{-s_{mj}}-\underset{s_{mj}<0}
{\prod_{m=1}^I}u_{m}^{-s_{mj}}\Big\|_{L^2((0,T)\times \Omega)}\\
&\quad + \sum_{j=1}^{J}s_{ij}k_{j}^{b}\Big\|\underset{s_{mj}>0}
{\prod_{m=1}^I}u_{\varepsilon _m}^{s_{mj}}-\underset{s_{mj}>0}
{\prod_{m=1}^I}u_{m}^{s_{mj}}\Big\|_{L^2((0,T)\times\Omega)}.
\end{aligned}\label{eqn4.17}
\end{equation}
By using the strong convergence of $u_{\varepsilon}$ and
 $L^{\infty}$-estimates of $u_{\varepsilon}$ and $u$, it follows
that
\[
\|\underset{s_{mj}<0}{\prod_{m=1}^I}u_{\varepsilon _m}^{-s_{mj}}
-\underset{s_{mj}<0}{\prod_{m=1}^I}u_{m}^{-s_{mj}}
\|_{L^2((0,T)\times \Omega)}\quad\text{and} \quad
\|\underset{s_{mj}>0}{\prod_{m=1}^I}u_{\varepsilon _m}^{s_{mj}}
-\underset{s_{mj}>0}{\prod_{m=1}^I}u_{m}^{s_{mj}}\|_{L^2((0,T)\times \Omega)}
\]
 are  convergent to 0 as $\varepsilon \to 0$.
Therefore $\|SR(u_\varepsilon)_i-SR(u)_i\|_{L^2((0,T)\times \Omega)}\to 0$
as $\varepsilon \to 0$. From \eqref{eqn4.14}, the theorem follows.
\end{proof}

\begin{remark}\label{thm4.2.7}\rm
The strong convergence of $(SR(u_\varepsilon))_{\varepsilon >0}$ 
implies that it is \textit{two-scale} convergent to $SR(u)$ 
in the sense of \eqref{eqn2.43}.
\end{remark}

\subsection{Passage to the limit as $\varepsilon \to 0$} \label{sec4.3}

Let us consider the two functions $\phi _0\in C^{\infty}_0{((0,T)\times\Omega)^I}$ 
and $\phi _1\in C^{\infty}_0(((0,T)\times \Omega);C^{\infty}_{per}(Y))^I$ 
such that $\phi (t,x, \frac{x}{\varepsilon})
:=\phi _0(t,x)+\varepsilon \phi _1(t,x,\frac{x}{\varepsilon})
\in C^{\infty}_0(((0,T)\times \Omega);C^{\infty}_{per}(Y))^I$. 
Using $\phi$ as test function in the weak formulation of \eqref{eqn1.12} 
one obtains
\begin{align*}
&\int_0^T{\langle \frac{\partial u_{\varepsilon}(t)}{\partial t}, 
\phi (t)\rangle _{[H^{1,2}(\Omega _{\varepsilon}^p)^{*}]^I
\times [H^{1,2}(\Omega _{\varepsilon}^p)]^I}}\,dt\\
&-\int_0^T{\langle \nabla \cdot D \nabla u_{\varepsilon}(t), 
\phi(t)\rangle _{[H^{1,2}(\Omega _{\varepsilon}^p)^{*}]^I
\times [H^{1,2}(\Omega _{\varepsilon}^p)]^I}}\,dt   \\
&= \int_0^T{\langle SR(u_{\varepsilon}(t)), \phi(t)\rangle _{[H^{1,2}
(\Omega _{\varepsilon}^p)^{*}]^I\times [H^{1,2}(\Omega _{\varepsilon}^p)]^I}}\,dt;
\end{align*}
i.e., 
\begin{align*}
&\sum_{i=1}^I\int_0^T\langle\frac{\partial u_{\varepsilon _i}(t)}
{\partial t},\phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}
\times H^{1,2}(\Omega _{\varepsilon}^p)}\,dt \\
&-\sum_{i=1}^I\int_0^T\langle\nabla \cdot D\nabla u_{\varepsilon _i}(t),
 \phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}
 \times H^{1,2}(\Omega _{\varepsilon}^p)}\,dt \\
&=\sum_{i=1}^I\int_0^T\langle SR(u_{\varepsilon}(t))_i,
 \phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}\times H^{1,2}
 (\Omega _{\varepsilon}^p)}\,dt;
\end{align*}
i.e., 
\begin{equation}
\begin{aligned}
&\sum_{i=1}^I\int_0^T\langle\frac{\partial u_{\varepsilon _i}(t)}{\partial t},
 \phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}
 \times H^{1,2}(\Omega _{\varepsilon}^p)}\,dt\\
&+\sum_{i=1}^I\int_0^T\int_{\Omega _{\varepsilon}^p} D\nabla u_{\varepsilon _i}
 (t,x)\nabla\phi _i(t,x,\frac{x}{\varepsilon})\,dx\,dt \\
&=\sum_{i=1}^I\int_0^T\langle SR(u_{\varepsilon}(t))_i,
 \phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}\times H^{1,2}
 (\Omega _{\varepsilon}^p)}\,dx\,dt.
\end{aligned}\label{eqn4.18}
\end{equation}
Now we pass to the \textit{two-scale} limit in \eqref{eqn4.18}
term by term.
\begin{align}
&\lim_{\varepsilon \to 0
\sum_{i=1}^I}\int_0^T\langle\frac{\partial u_{\varepsilon}(t)}{\partial t},
\phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}
\times H^{1,2}(\Omega _{\varepsilon}^p)}\,dt \nonumber \\
&=-\lim_{\varepsilon \to 0 } \sum_{i=1}^I \int_0^T
\int_{\Omega _{\varepsilon}^p}{u_{\varepsilon _i}(t,x)}
(\frac{\partial \phi _{0_i}(t,x)}{\partial t}
+\varepsilon \frac{\partial \phi _{1_i}(t,x,\frac{x}{\varepsilon})}{\partial t})
 \,dx\,dt  \nonumber\\
&=-\lim_{\varepsilon \to 0} {\sum_{i=1}^I}
\int_0^T\int_{\Omega}\chi (\frac{x}{\varepsilon})u_{\varepsilon _i}(t,x)
\frac{\partial \phi _{0_i}}{\partial t} \,dx\,dt \nonumber \\
&\quad -\underbrace{ \lim_{\varepsilon \to 0} \varepsilon
{\sum_{i=1}^I}\int_0^T\int_{\Omega}\chi (\frac{x}{\varepsilon})
{u_{\varepsilon _i}(t,x)} \frac{\partial \phi _{1_i}}{\partial t}
\,dx\,dt}_{=0} \nonumber \\
&=-{\sum_{i=1}^I}\int_0^T\int_{\Omega}\int_{Y}\chi (y){u_i}(t,x)
 \frac{\partial \phi _{0_i}(t,x)}{\partial t} \,dx\,dy\,dt \nonumber \\
&=-{\sum_{i=1}^I}\int_0^T\int_{\Omega}\int_{Y^p}{u_i}(t,x)
 \frac{\partial \phi _{0_i}(t,x)}{\partial t} \,dx\,dy\,dt, \quad
\text{ since $\chi(y)=1$ in $Y^p$} \nonumber \\
&= |Y^p|{\sum_{i=1}^I}\int_0^T\langle \frac{\partial u_i(t)}{\partial t},
\phi _{0_i}(t)\rangle _{H^{1,2}(\Omega _{\varepsilon}^p)^{*}
 \times H^{1,2}(\Omega _{\varepsilon}^p)}\,dt  \nonumber \\
&=|Y^p|\int_0^T\langle \frac{\partial u(t)}{\partial t},
\phi _0(t)\rangle _{[H^{1,2}(\Omega _{\varepsilon}^p)^{*}]^I
 \times [H^{1,2}(\Omega _{\varepsilon}^p)]^I}\,dt. \label{eqn4.19}
\end{align} 
Again,
\begin{equation}
\begin{aligned}
&\lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T
 \int_{\Omega _{\varepsilon}^p}D\nabla _{x}u_{\varepsilon _i}
 (t,x)\nabla _{x}\phi _i(t,x,\frac{x}{\varepsilon})\,dx\,dt \\
&=\lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T
 \int_{\Omega _{\varepsilon}^p}D\nabla _{x}u_{\varepsilon _i}(t,x)
 \nabla _{x}(\phi _{0_i}(t,x)+\varepsilon \phi _{1_i}
 (t,x,\frac{x}{\varepsilon}))\,dx\,dt \\
&=\lim_{\varepsilon \to 0} \Big[\sum_{i=1}^I\int_0^T
 \int_{\Omega}\chi (\frac{x}{\varepsilon})D\nabla _{x}u_{\varepsilon _i}
 (t,x)(\nabla _{x}\phi _{0_i}(t,x)+\nabla _{y}\phi _{1_i}
 (t,x,\frac{x}{\varepsilon}))\,dx\,dt \\
&\quad + \varepsilon \sum_{i=1}^I\int_0^T\int_{\Omega}\chi
 (\frac{x}{\varepsilon})D\nabla _{x}u_{\varepsilon _i}
 (t,x)\nabla _{x}\phi _{1_i}(t,x,\frac{x}{\varepsilon})\,dx\,dt\Big]
\\
&= \lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T\int_{\Omega}
 \chi (\frac{x}{\varepsilon})D\nabla _{x}u_{\varepsilon _i}(t,x)
 (\nabla _{x}\phi _{0_i}(t,x)+\nabla _{y}\phi _{1_i}(t,x,\frac{x}{\varepsilon}))
 \,dx\,dt \\
&\quad +\underbrace{\lim_{\varepsilon \to 0} \varepsilon
 \sum_{i=1}^I\int_0^T\int_{\Omega}\chi (\frac{x}{\varepsilon})
 D\nabla _{x}u_{\varepsilon _i}(t,x)\nabla _{x}
 \phi _{1_i}(t,x,\frac{x}{\varepsilon})\,dx\,dt}_{ =0} \\
&=\sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y}\chi (y)
 D(\nabla _{x}u_i(t,x)+\nabla _{y}u_{1_i}(t,x,y))
 (\nabla _{x}\phi _{0_i}(t,x)\\
&\quad +\nabla _{y}\phi _{1_i}(t,x,y))\,dx\,dy\,dt \\
&=\sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y^p}D(\nabla _{x}u_i(t,x)
 +\nabla _{y}u_{1_i}(t,x,y))(\nabla _{x}\phi _{0_i}(t,x)\\
&\quad +\nabla _{y}\phi _{1_i}(t,x,y))\,dx\,dy\,dt
\end{aligned} \label{eqn4.20}
\end{equation}
By \eqref{eqn3.1}, $\sup_{\varepsilon >0}
\|u_{\varepsilon _i}(t)\|_{L^{r}(\Omega _{\varepsilon}^p)}\le C_1$ for all $i$
and for a.e. $t$. Then we have
$\|SR(u_{\varepsilon})_i\|_{L^2(\Omega _{\varepsilon}^p)}\le C$.
Since $L^2(\Omega _{\varepsilon}^p)\hookrightarrow H^{1,2}
(\Omega _{\varepsilon}^p)^{*}$, from \eqref{eqn2.3},
\[
\langle SR(u_{\varepsilon})_i, \phi _i\rangle_{H^{1,2}
(\Omega _{\varepsilon}^p)^{*}\times H^{1,2}(\Omega _{\varepsilon}^p)}
=\langle SR(u_{\varepsilon})_i, \phi _i\rangle_{L^2(\Omega _{\varepsilon}^p)
\times L^2(\Omega _{\varepsilon}^p)},
\quad \phi_i\in H^{1,2}(\Omega _{\varepsilon}^p).
\]
 Thus
\begin{align}
&\lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T\langle SR(u_\varepsilon(t))_i,
 \phi _i(t)\rangle_{H^{1,2}(\Omega _{\varepsilon}^p)^{*}\times H^{1,2}
 (\Omega _{\varepsilon}^p)}\,dt \nonumber \\
&=\lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T\langle SR(u_\varepsilon(t))_i,
 \phi _i(t)\rangle_{L^2(\Omega _{\varepsilon}^p)
 \times L^2(\Omega _{\varepsilon}^p)}\,dt \nonumber\\
&=\lim_{\varepsilon \to 0} \sum_{i=1}^I
 \int_0^T\int_{\Omega _{\varepsilon}^p}SR(u_\varepsilon(t,x))_i
 \phi _i(t,x)\,dx\,dt \nonumber\\
&=\lim_{\varepsilon \to 0} \sum_{i=1}^I\int_0^T\int_{\Omega}
 \chi (\frac{x}{\varepsilon})SR(u_{\varepsilon})_i\phi_{0_i}(t,x)\,dx\,dt 
\nonumber\\
&\quad +\underbrace{\lim_{\varepsilon \to 0} \varepsilon
 \sum_{i=1}^I\int_0^T\int_{\Omega}\chi (\frac{x}{\varepsilon})
 SR(u_{\varepsilon})_i\phi_{1_i}(t,x,\frac{x}{\varepsilon})\,dx\,dt}_{=0} 
\nonumber \\
&=\sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y}\chi(y)SR(u(t,x))_i
 \phi _{0_i}(t,x)\,dx\,dy\,dt \nonumber \\
&=\sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y^p}SR(u(t,x))_i
 \phi _{0_i}(t,x)\,dx\,dy\,dt \nonumber\\
&=|Y^p|\int_0^T\langle SR(u(t)),\phi _0(t)
\rangle _{[H^{1,2}(\Omega)^{*}]^I\times [H^{1,2}(\Omega)]^I}\,dt. \label{eqn4.21}
\end{align}
Combining \eqref{eqn4.19}-\eqref{eqn4.21}, we obtain
\begin{equation}
\begin{aligned}
&|Y^p|\int_0^T\langle \frac{\partial u(t)}{\partial t},
 \phi _0(t)\rangle _{[H^{1,2}(\Omega)^{*}]^I\times [H^{1,2}(\Omega)]^I}\,dt 
+ \sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y^p}D(\nabla _{x}u_i(t,x)\\
&+\nabla _{y}u_{1_i}(t,x,y)) (\nabla _{x}\phi _{0_i}(t,x)
 +\nabla _{y}\phi _{1_i}(t,x,y))\,dx\,dy\,dt \\
&=|Y^p|\int_0^T\langle SR(u(t)),\phi _0(t)\rangle _{[H^{1,2}(\Omega)^{*}]^I
\times [H^{1,2}(\Omega)]^I}\,dt.
\end{aligned}\label{eqn4.22}
\end{equation}
Now choosing $\phi _0(t,x)\equiv 0$, i.e., $\phi _{0_i}(t,x)\equiv 0$
for all $i=1,2,\dots , I$, then $\phi (t,x)=\phi _1(t,x,\frac{x}{\varepsilon})$
 and the equation \eqref{eqn4.22} reduces to
\begin{equation}
\sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y^p}D(\nabla _{x}u_i(t,x)
+\nabla _{y}u_{1_i}(t,x,y)) \nabla _{y}\phi _{1_i}(t,x,y)\,dx\,dy\,dt=0.\label{eqn4.23}
\end{equation}
Let us choose $u_{1_i}(t,x,y)=\sum_{j=1}^{n}
\frac{\partial u_i(t,x)}{\partial x_{j}}a_{j}(t,x,y)+c_i(x)$, for all
$i=1,2,\dots ,I$, where $c(x)$ is any arbitrary function of $x$.
The equation \eqref{eqn4.23} is satisfied by each $u_{1_i}$ if $a_{j}$,
for $j=1,2,\dots ,n$, is the solution of the \textit{Cell-Problem}
\begin{gather}
-\nabla _{y}\cdot(D(\nabla _{y}a_{j}(t,x,y)+e_{j}))=0\quad
\text{ for }(t,x,y)\in (0,T)\times \Omega \times Y^{P},\label{eqn4.24}\\
-D(\nabla _{y}a_{j}(t,x,y)+e_{j})\cdot \vec{n}=0\quad
\text{ for }(t,x,y)\in (0,T)\times \Omega \times \Gamma,\label{eqn4.25}\\
y\mapsto a_{j}(y) \text{ is }Y-\text{periodic}.\label{eqn4.26}
\end{gather}
On the other hand, if $a_{j}$ is the solution of the cell-problem
\eqref{eqn4.24}-\eqref{eqn4.26}, the equation \eqref{eqn4.23}
is satisfied if
$u_{1_i}(t,x,y)=\sum_{j=1}^{n}\frac{\partial u_i(t,x)}{\partial x_{j}}a_{j}(t,x,y)
+c_i(x)$.
Setting $\phi _1(t,x,\frac{x}{\varepsilon})\equiv 0$; i.e.,
$\phi _{1_i}(t,x,\frac{x}{\varepsilon})\equiv 0$ for all $i$.
Then the equation \eqref{eqn4.22} reduces to
\begin{align*}
&|Y^p|\int_0^T\langle \frac{\partial u(t)}{\partial t},
\phi _0(t)\rangle _{[H^{1,2}(\Omega)^{*}]^I\times [H^{1,2}(\Omega)]^I}\,dt 
+ \sum_{i=1}^I\int_0^T\int_{\Omega}\int_{Y^p}D(\nabla _{x}u_i(t,x)\\
& +\nabla _{y}u_{1_i}(t,x,y)) (\nabla _{x}\phi _{0_i}(t,x)
 +\nabla _{y}\phi _{1_i}(t,x,y))\,dx\,dy\,dt \\
&=|Y^p|\int_0^T\langle SR(u(t)),\phi _0(t)\rangle _{[H^{1,2}(\Omega)^{*}]^I
 \times [H^{1,2}(\Omega)]^I}\,dt;
\end{align*}
i.e.,
\begin{equation}
\begin{aligned}
&{\sum_{i=1}^I}\int_0^T\langle \frac{\partial u_i(t)}{\partial t},
\phi _{0_i}(t)\rangle _{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt \\
&+ {\sum_{i=1}^I}\int_0^T\int_{\Omega}\int_{Y^p}\frac{D}{|Y^p|}
 (\nabla _{x}u_i(t,x)+\nabla _{y}u_{1_i}(t,x,y))
 \nabla _{x}\phi _{0_i}(t,x)\,dx\,dy\,dt  \\
&= {\sum_{i=1}^I}\int_0^T\langle SR(u(t))_i,\phi _{0_i}(t)
\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt.
\end{aligned}\label{eqn4.27}
\end{equation}
Substituting $u_{1_i}(t,x,y)=\vec{a}(t,x,y)\cdot \nabla _{x}u_i(t,x)+c(x)$;
 i.e., $\nabla _{y}u_{1_i}=\sum_{j=1}^{n}\nabla _{y}a_{j}
\frac{\partial u_i}{\partial x_{j}}$ in \eqref{eqn4.27}, then we obtain
\begin{align*}
&{\sum_{i=1}^I}\int_0^T\langle \frac{\partial u_i(t)}{\partial t},
  \phi _{0_i}(t)\rangle _{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt \\
&+ {\sum_{i=1}^I}\int_0^T\int_{\Omega}\int_{Y^p}\frac{D}{|Y^p|}
 \Big(\nabla _{x}u_i(t,x)+{\sum_{j=1}^{n}}\nabla _{y}a_{j}
 \frac{\partial u_i(t,x,y)}{\partial x_{j}}\Big) \nabla _{x}
 \phi _{0_i}(t,x)\,dx\,dy\,dt \\
&= {\sum_{i=1}^I}\int_0^T\langle SR(u(t))_i,\phi _{0_i}(t)
 \rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt,
\intertext{i.e., }\;&{\sum_{i=1}^I}\int_0^T
 \langle \frac{\partial u_i(t)}{\partial t}, \phi _{0_i}(t)
 \rangle _{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt \\
&+ \sum_{i=1}^I\int_0^T\int_{\Omega}{\sum_{j,k=1}^{n}}
\big\{\frac{D}{|Y^p|}\int_{Y^p}(\delta _{jk}
+\frac{\partial a_{j}}{\partial y_{k}})\,dy\big\}
{\frac{\partial u_i(t,x)}{\partial x_{j}}}
{\frac{\partial \phi _{0_i}(t,x)}{\partial {x_k}}}\,dx\,dt \\
&= {\sum_{i=1}^I}\int_0^T\langle SR(u(t))_i,\phi _{0_i}(t)
\rangle_{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt;
\end{align*}
i.e.,
\begin{equation}
\begin{aligned}
&{\sum_{i=1}^I}\int_0^T\langle \frac{\partial u_i(t)}{\partial t},
\phi _{0_i}(t)\rangle _{H^{1,2}(\Omega)^{*}\times H^{1,2}(\Omega)}\,dt
+\sum_{i=1}^I\int_0^T\int_{\Omega}\Upsilon\;\nabla _{x}u_i(t,x)
\nabla \phi _{0_i}(t,x)\,dx\,dt  \\
&={\sum_{i=1}^I}\int_0^T\langle SR(u(t))_i,
\phi _{0_i}(t)\rangle_{H^{1,2}(\Omega)^{*}
\times H^{1,2}(\Omega)}\,dt,
\end{aligned}\label{eqn4.28}
\end{equation}
where $\Upsilon$ is a second order tensor whose components are given as
\begin{equation}
\rho_{jk}=\int_{Y^p}\frac{D}{|Y^p|}(\delta _{jk}
+\frac{\partial a_{j}}{\partial y_{k}})\,dy \quad\text{for all }
j,k=1,2,\dots , n,\label{eqn4.29}
\end{equation}
where $a_{j}$ is the solution of the cell-problem
\eqref{eqn4.24}-\eqref{eqn4.26}. Similarly the boundary condition
simplifies to
\begin{equation}
\Upsilon \nabla u\cdot \vec{n}=0 \quad \text{on }
 (0,T)\times \partial \Omega. \label{eqn4.30}
\end{equation}
Therefore the strong form of the complete homogenized problem is
\begin{gather}
 \frac{\partial u}{\partial t} -\nabla  \Upsilon \nabla u= SR(u)
\quad \text{in } (0,T)\times \Omega,\label{eqn4.31}\\
 -\Upsilon \nabla u\cdot \vec{n} = 0 \quad \text{on }
 (0, T)\times\partial\Omega, \label{eqn4.32}\\
  u(0, x) = u_0(x)\quad \text{in }  \Omega. \label{eqn4.33}
\end{gather}
Let us denote this problem by $(P)$.

\begin{proposition}[\cite{HoJ91}]\label{thm4.3.1}
The tensor $\Upsilon=(\rho_{jk})_{\underset{1\le k\le n}{1\le j\le n}}$ 
is a second-order positive definite symmetric tensor.
\end{proposition}


\begin{theorem}\label{thm4.3.2}
There exists a unique solution 
$u\in F^I\cap L^{\infty}((0,T);L^{\infty}(\Omega))^I$ 
of the homogenized problem \eqref{eqn4.31}--\eqref{eqn4.33}.
\end{theorem}

\begin{proof}
From \eqref{eqn4.1} and \eqref{eqn4.7}, it follows that the 
\textit{two-scale} limit $u$ belongs to the space
$[H^{1,2}((0,T);H^{1,2}(\Omega)^{*})\cap L^2((0,T);H^{1,2}(\Omega))
\cap L^{\infty}((0,T)\times \Omega)]^I$. It remains to prove
\begin{itemize}
\item Uniqueness of the solution of \eqref{eqn4.31}--\eqref{eqn4.33} and
\item $u\in F^I$.
\end{itemize}

Given the positive definiteness of $\Upsilon$ and the $L^{\infty}$-estimate 
of $u$, the uniqueness follows by a straightforward application of 
Gronwall's inequality. Now, the reformulation of  
problem \eqref{eqn4.31}-\eqref{eqn4.33} is given by
\begin{gather}
\frac{d u(t)}{d t}+Au(t)=f(t),\label{eqn4.34}\\
u(0,x)=u_0(x),\label{eqn4.35}
\end{gather}
where $f(t)=SR(u(t))+\kappa u(t)$, $\kappa >0$ and the operator 
$A:H^{1,p}(\Omega )^I\to [H^{1,q}(\Omega)^{*}]^I$ is defined as 
$Au_{\varepsilon}:=(A_1u_{\varepsilon _1},A_2u_{\varepsilon _2},
\dots ,A_{I}u_{\varepsilon _I})$ such that for $1\le i\le I$,
\[
\langle A_iu_i,w_i\rangle :=\int_{\Omega }\nabla u_i(x)\Upsilon
 \nabla w_i(x)\,dx 
 +\kappa \int_{\Omega }u_i(x)w_i(x)\,dx 
\]
for $u_i\in H^{1,p}(\Omega )$  and $w_i\in H^{1,q}(\Omega )$.
Operator $A$ has maximal parabolic regularity on
 $[H^{1,q}(\Omega)^{*}]^I$ by section \ref{sec5.1}. 
Since $u\in L^{\infty}((0,T)\times \Omega)^I$, 
$SR(u)+\kappa u\in L^p((0,T);L^p(\Omega))^I$. 
The embedding $L^p(\Omega)\hookrightarrow H^{1,q}(\Omega)^{*}$ 
implies $SR(u)+\kappa u\in L^p((0,T);H^{1,q}(\Omega)^{*})^I$. 
Furthermore, theorem \ref{thm2.3.2.3} shows that $u_{{0}}$ is in 
$ X^I_{p}$. Therefore, by \cite[Theorem 2.5]{PS01}, 
there exists a unique solution $u$ in $ F^I$ of the problem 
\eqref{eqn4.34}-\eqref{eqn4.35} such that
\begin{equation}
\||u\||_{F^I}\le \tilde{C}\Big(\||u_0\||_{X_{p}^I}
+\||f\||_{L^p((0,T);H^{1,q}(\Omega)^{*})^I}\Big),\label{eqn4.36}
\end{equation}
where $\tilde{C}>0$ depends only on $p$ but independent of $u$, $u_0$
and $f$. In other words, the problem $(P)$ has a unique
positive global weak solution $u$ in $ F^I$.
\end{proof}

\section{Appendix}\label{sec5}
\subsection{Maximal regularity}\label{sec5.1}

\begin{definition} \rm
Let $1<p<\infty$, $X$ be a Banach space and $A: X\to X$ be a closed, 
not necessarily bounded, operator, where the domain $D(A)$ of $A$ 
is dense in $X$. $A$ is said to have the maximal $L^p$-regularity 
if for every $f\in L^p((0,T);X)$ there exists a unique solution 
$u\in L^p((0,T);D(A))\cap H^{1,p}((0,T);X)$ of the problem
\begin{gather}
\frac{du(t)}{dt}+Au(t)=f(t)\quad  \text{for }t > 0, \label{eqn5.1}\\
u(0)=0,\label{eqn5.2}
\end{gather}
which satisfies
\begin{equation}
\|u\|_{L^p((0,T);X)}+\|u_{t}\|_{L^p((0,T);X)}+\|u\|_{L^p((0,T);D(A))}
\le C\|f\|_{L^p((0,T);X)},\label{eqn5.3}
\end{equation}
where $C>0$ is a constant independent of $f$.
\end{definition}
For a detailed overview on maximal regularity, we refer to 
 \cite{ACFP07,Mon09,Pru02,DR09,KuW04} and references therein.
 Now we set $D(A):=H^{1,p}(\Omega)$ and $X:=H^{1,q}(\Omega)^{*}$. 
Clearly, $D(A)\overset{d}{\subseteq}X$.
($A\overset{d}{\subseteq}B$ means that $A$ is dense in $B$.)
 Let $\mu=(\mu_{ij})_{1\le j\le n,\, 1\le i\le n}$ be a positive 
definite symmetric matrix-field, where $\mu _{ij}\in C(\bar{\Omega})$ 
and there is a constant $C>0$
\begin{equation}
\sum_{i,j=1}^{n}\mu _{ij}(x)\zeta _i\zeta _{j}\ge C|\zeta|^2 \quad
\text{for all }\zeta \in \mathbb{R}^{n} \text{ and }x\in \Omega.\label{eqn5.4}
\end{equation}
We define a sesquilinear form 
$a(u,v):H^{1,p}(\Omega)\times H^{1,q}(\Omega)\to \mathbb{R}$ by
\begin{equation}
a(u,v):=\int_{\Omega}\mu \nabla u \cdot \nabla v\,dx
+\kappa \int_{\Omega}uv\,dx \quad \text{for }u\in H^{1,p}(\Omega)
\text{ and } v\in H^{1,q}(\Omega),\label{eqn5.5}
\end{equation}
where $\kappa >0$. We further define an operator 
$A:H^{1,p}(\Omega)\to H^{1,q}(\Omega)^{*}$ associated with the form $a(u,v)$ by
\begin{equation}
\langle Au,v\rangle:=a(u,v)\quad \text{for }u\in H^{1,p}(\Omega)\text{ and } 
v\in H^{1,q}(\Omega).\label{eqn5.6}
\end{equation}
In \cite{CL94} and \cite{DR09}, it is shown that: 
(i) $\|A^{is}\|_{L(X)}\le Ke^{\theta |s|}$ for some $0<\theta< \frac{\pi}{2}$,
 $s\in \mathbb{R}$, where $K>0$,
 and (ii) $(-\infty, 0]\subset \rho(A)$ 
(resolvent of $A$) and $\|(\lambda +A)^{-1}\|_{L(X)}\le \frac{C}{1+|\lambda|}$ 
for every $\lambda \in [0,\infty)$, where $C>0$.
 By a theorem of Dore and Venni (cf. \cite{DV87}), $A$ has maximal 
$L^p$-regularity on $H^{1,q}(\Omega)^{*}$.

\subsection{Some Inequalities}\label{sec5.2}

\begin{lemma}[Lyapunov's interpolation inequality]\label{thm5.2.1}
Let $1\le p\le q\le r \le \infty$ and $0<\theta <1 $ be such that 
$\frac{1}{q}=\frac{\theta}{p}+\frac{1-\theta}{r}$.
Assume also that $u\in L^p(\Omega)\cap L^{r}(\Omega)$. 
Then $u\in L^{q}(\Omega)$ and satisfies
\begin{equation}
\|u\|_{L^{q}(\Omega)}\le \|u\|_{L^p(\Omega)}^{\theta}
\|u\|_{L^{r}(\Omega)}^{1-\theta}.
\end{equation}
\end{lemma}

The prof of the above lemma can be found in \cite[Inequality B.2.h]{Eva98}.


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\end{document}