# Homogenization of Smoluchowski-type equations with transmission boundary
conditions

Bruno Franchi1 and Silvia Lorenzani2

(1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.
Donato 5, 40126 Bologna, Italy
2 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci
32, 20133 Milano, Italy)

###### Abstract

In this work, we prove a two-scale homogenization result for a set of
diffusion-coagulation Smoluchowski-type equations with transmission boundary
conditions. This system is meant to describe the aggregation and diffusion of
pathological tau proteins in the cerebral tissue, a process associated with
the onset and evolution of a large variety of tauopathies (such as Alzheimer’s
disease). We prove the existence, positivity and boundedness of solutions to
the model equations derived at the microscale (that is the scale of single
neurons). Then, we study the convergence of the homogenization process to the
solution of a macro-model asymptotically consistent with the microscopic one.

Keywords: Smoluchowski equation, transmission boundary conditions, two-scale
homogenization, tauopathies

MSC 2020: 35K55, 80M40, 92B05

##  1 Introduction

Let us consider a bounded open set $\Omega$ in $\mathbb{R}^{3}$ with a smooth
boundary $\partial\Omega$. This domain is decomposed into long cylindrical
cavities, periodically distributed with period $\epsilon$, having generators
parallel to the $z$-axis. More precisely, given a bounded domain $D$ in
$\mathbb{R}^{2}$, let us set $\Omega:=D\times]0,L[$ with $x\in D$ and
$z\in]0,L[$. We denote by $\Gamma_{L}:=\partial D\times]0,L[$ the lateral
boundary and by $\Gamma_{B}:=\overline{D}\times\\{0,L\\}$ the upper and lower
sides of $\partial\Omega$. Let $Y=[0,1]\times[0,1]$ be the unit periodicity
cell in $\mathbb{R}^{2}$ having the paving property, i.e. the disjoint union
of translated copies of $Y$ can indeed cover the whole space. Let $X$ be an
open subset of $Y$ with a smooth boundary $R=\partial X$, such that
$\overline{X}\subset\mathrm{Int}\,Y$, and $Z=Y\setminus X$. Then, we define
$G_{\epsilon}$ to be the set of all translated images of
$\epsilon\,\overline{X}$:

$G_{\epsilon}:=\cup\bigg{\\{}\epsilon(k+\overline{X})\mid\epsilon(k+\overline{X})\subset
D,k\in\mathbb{Z}^{2}\bigg{\\}}$

and $D_{\epsilon}=D\setminus G_{\epsilon}$, as well as

$R_{\epsilon}:=\cup\bigg{\\{}\partial(\epsilon(k+\overline{X}))\mid\epsilon(k+\overline{X})\subset
D,k\in\mathbb{Z}^{2}\bigg{\\}}.$

Thus, the geometric structure of the domain in $\mathbb{R}^{3}$ is given by:
$\Pi_{\epsilon}:=G_{\epsilon}\times]0,L[,$
$\Omega_{\epsilon}:=D_{\epsilon}\times]0,L[,$ and
$\Gamma_{\epsilon}:=R_{\epsilon}\times]0,L[.$

Throughout this paper, $\epsilon$ will denote the general term of a sequence
of positive reals which converges to zero. Let us introduce two nonnegative
vector-valued functions,
$u^{\epsilon}:[0,T]\times\Pi_{\epsilon}\rightarrow\mathbb{R}^{M}$,
$u^{\epsilon}(t,x,z)=(u_{1}^{\epsilon},\ldots,u_{M}^{\epsilon})$ and
$v^{\epsilon}:[0,T]\times\Omega_{\epsilon}\rightarrow\mathbb{R}^{M}$,
$v^{\epsilon}(t,x,z)=(v_{1}^{\epsilon},\ldots,v_{M}^{\epsilon})$, which solve
the following system of discrete Smoluchowski-type equations [40] with
transmission boundary conditions.

For $1\leq m\leq M$ we have:

$\displaystyle\begin{cases}\frac{\displaystyle\partial{u_{m}^{\epsilon}}}{\displaystyle\partial
t}-{\epsilon}^{2}\,D_{m}\,\Delta_{x}u_{m}^{\epsilon}-\tilde{D}_{m}\,\partial_{z}^{2}u_{m}^{\epsilon}\\\
=L_{m}(u^{\epsilon})+f_{m}^{\epsilon}(t,x,z)&\text{in
}[0,T]\times\Pi_{\epsilon}\\\ \\\
\frac{\displaystyle\partial{v_{m}^{\epsilon}}}{\displaystyle\partial
t}-d_{m}\,\Delta v_{m}^{\epsilon}=N_{m}(v^{\epsilon})&\text{in
}[0,T]\times\Omega_{\epsilon}\\\ \\\ \\\
\epsilon\,D_{m}\,\nabla_{x}u_{m}^{\epsilon}\cdot\nu_{\epsilon}=-c_{m}(x,z)(u_{m}^{\epsilon}-v_{m}^{\epsilon})_{+}&\text{on
}[0,T]\times\Gamma_{\epsilon}\\\ \\\
d_{m}\,\nabla_{x}v_{m}^{\epsilon}\cdot\nu_{\epsilon}={\epsilon}^{2}\,D_{m}\,\nabla_{x}u_{m}^{\epsilon}\cdot\nu_{\epsilon}&\text{on
}[0,T]\times\Gamma_{\epsilon}\\\ \\\
\nabla_{x}v_{m}^{\epsilon}\cdot\nu_{\epsilon}=0&\text{on
}[0,T]\times\Gamma_{L}\\\ \\\
\partial_{z}u_{m}^{\epsilon}=\partial_{z}v_{m}^{\epsilon}=0&\text{on
}[0,T]\times\Gamma_{B}\\\ \\\
u_{m}^{\epsilon}(0,x,z)=U_{m}^{\epsilon}(x,z)&\text{in }\Pi_{\epsilon}\\\
v_{m}^{\epsilon}(0,x,z)=0&\text{in }\Omega_{\epsilon}\end{cases}$ (1.1)

where

$L_{1}(u^{\epsilon})=-\sum_{j=1}^{M}a_{1,j}u_{1}^{\epsilon}u_{j}^{\epsilon}$
(1.2)

$L_{m}(u^{\epsilon})=\frac{1}{2}\,\sum_{j=1}^{m-1}a_{j,m-j}\,u_{j}^{\epsilon}\,u_{m-j}^{\epsilon}-\sum_{j=1}^{M}a_{m,j}u_{m}^{\epsilon}u_{j}^{\epsilon}\;\;\;(1<m<M)$
(1.3)

$L_{M}(u^{\epsilon})=\frac{1}{2}\,\sum_{\begin{subarray}{c}j+k\geq M\\\ k<M\\\
j<M\end{subarray}}a_{j,k}\,u_{j}^{\epsilon}\,u_{k}^{\epsilon}$ (1.4)

$N_{1}(v^{\epsilon})=-\sum_{j=1}^{M}b_{1,j}v_{1}^{\epsilon}v_{j}^{\epsilon}$
(1.5)

$N_{m}(v^{\epsilon})=\frac{1}{2}\,\sum_{j=1}^{m-1}b_{j,m-j}\,v_{j}^{\epsilon}\,v_{m-j}^{\epsilon}-\sum_{j=1}^{M}b_{m,j}v_{m}^{\epsilon}v_{j}^{\epsilon}\;\;\;(1<m<M)$
(1.6)

$N_{M}(v^{\epsilon})=\frac{1}{2}\,\sum_{\begin{subarray}{c}j+k\geq M\\\ k<M\\\
j<M\end{subarray}}b_{j,k}\,v_{j}^{\epsilon}\,v_{k}^{\epsilon}$ (1.7)

and

$f_{m}^{\epsilon}(t,x,z)=\begin{cases}f^{\epsilon}(t,x,z)&\text{if }m=1\\\
0&\text{if }1<m\leq M\end{cases}$
$U_{m}^{\epsilon}(x,z)=\begin{cases}U_{1}^{\epsilon}(x,z)>0&\text{if }m=1\\\
0&\text{if }1<m\leq M\end{cases}$

In the system of equations above, $\nu_{\epsilon}$ is the outer normal on
$\Gamma_{\epsilon}$ with respect to $\Omega_{\epsilon}$;
$f^{\epsilon}(t,x,z)\in C^{1}([0,T]\times\Pi_{\epsilon})$, $c_{i}(x,z)\in
L^{\infty}(\Gamma_{\epsilon})(i=1,\ldots,M)$ and $U_{1}^{\epsilon}(x,z)\in
L^{\infty}(\Pi_{\epsilon})$ are given positive functions. In (1.1), the
unknowns $u_{m}^{\epsilon}$ and $v_{m}^{\epsilon}$ ($1\leq m<M$) represent the
concentration of $m$-clusters, that is, clusters consisting of $m$ identical
elementary particles (monomers), while $u_{M}^{\epsilon}$ and
$v_{M}^{\epsilon}$ take into account aggregations of more than $M-1$ monomers.
We assume that the only reaction allowing clusters to coalesce to form larger
clusters is a binary coagulation mechanism, while the approach of two clusters
leading to aggregation results only from a diffusion process. In particular,
we indicate with $D_{i}$, $\tilde{D}_{i}$, $d_{i}$ ($1\leq i\leq M$) the
positive diffusion coefficients of $i$-clusters in different regions of the
domain and along different directions. The kinetic coefficients $a_{i,j}$ and
$b_{i,j}$ represent a reaction in which an ($i+j$)-cluster is formed from an
$i$-cluster and a $j$-cluster. Therefore, they can be interpreted as
’coagulation rates’ and are symmetric $a_{i,j}=a_{j,i}>0$, $b_{i,j}=b_{j,i}>0$
($i,j=1,\ldots,M$).

Our main statement shows that it is possible to homogenize the system of
equations (1.1) as $\epsilon\rightarrow 0$.

######  Theorem 1.1.

Let $v_{m}^{\epsilon}(t,x,z)$ and $u_{m}^{\epsilon}(t,x,z)$ ($1\leq m\leq M$)
be a family of weak solutions to the system (1.1) (see Definition 2.1). The
sequences $v_{m}^{\epsilon}$, $\nabla_{x}v_{m}^{\epsilon}$,
$u_{m}^{\epsilon}$, $\epsilon\nabla_{x}u_{m}^{\epsilon}$ ($1\leq m\leq M$),
two-scale converge to: $v_{m}(t,x,z)$,
$\nabla_{x}v_{m}(t,x,z)+\nabla_{y}\tilde{v}_{m}(t,x,y,z)$, $u_{m}(t,x,y,z)$,
$\nabla_{y}u_{m}(t,x,y,z)$, respectively. The limiting functions $v_{m}\in
L^{2}(0,T;H^{1}(\Omega))\bigcap H^{1}(0,T;L^{2}(\Omega))$, $\tilde{v}_{m}\in
L^{2}([0,T]\times\Omega;H^{1}_{\sharp}(Y)/\mathbb{R})$, $u_{m}\in
L^{2}([0,T]\times\Omega;H^{1}_{\sharp}(Y))\bigcap H^{1}(0,T;L^{2}(\Omega\times
Y))$ are the unique solutions of the following two-scale homogenized systems.

For $1\leq m\leq M$ we have:

$\begin{split}&|Z|\,\frac{\displaystyle\partial v_{m}}{\displaystyle\partial
t}(t,x,z)-\text{div}_{x}\bigg{[}d_{m}\,A\,\nabla_{x}v_{m}(t,x,z)\bigg{]}-d_{m}\,|Z|\,\partial_{z}^{2}v_{m}(t,x,z)\\\
&=\displaystyle\int_{\Gamma}c_{m}(x,z)\,(u_{m}(t,x,y,z)-v_{m}(t,x,z))_{+}\,d\sigma(y)+|Z|\,N_{m}(v)\,\,\,\;\;\;\;\;\;\;\;\text{
in}\,\,[0,T]\times\Omega\end{split}$ (1.8)

$[A\,\nabla_{x}v_{m}(t,x,z)]\cdot\nu=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
on}\,\,[0,T]\times\Gamma_{L}$ (1.9)

$\partial_{z}v_{m}(t,x,z)=0\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
on}\,\,[0,T]\times\Gamma_{B}$ (1.10)

$v_{m}(t=0,x,z)=0\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in}\,\,\Omega$ (1.11)

where

$N_{1}(v)=-\sum_{j=1}^{M}b_{1,j}v_{1}(t,x,z)\,v_{j}(t,x,z)$
$N_{m}(v)=\frac{1}{2}\,\sum_{j=1}^{m-1}b_{j,m-j}\,v_{j}(t,x,z)\,v_{m-j}(t,x,z)-\sum_{j=1}^{M}b_{m,j}v_{m}(t,x,z)\,v_{j}(t,x,z)\;\;\;(1<m<M)$
$N_{M}(v)=\frac{1}{2}\,\sum_{\begin{subarray}{c}j+k\geq M\\\ k<M\\\
j<M\end{subarray}}b_{j,k}\,v_{j}(t,x,z)\,v_{k}(t,x,z)$

In (1.8) and (1.9), $A$ is a matrix with constant coefficients defined by

$A_{ij}=\displaystyle\int_{Z}(\nabla_{y}w_{i}+\hat{e}_{i})\cdot(\nabla_{y}w_{j}+\hat{e}_{j})\,dy$
(1.12)

with $\hat{e}_{i}$ being the $i$th unit vector in $\mathbb{R}^{3}$, and
$(w_{i})_{1\leq i\leq 3}$ the family of solutions of the cell problem

$\displaystyle\begin{cases}-\text{div}_{y}[\nabla_{y}w_{i}+\hat{e}_{i}]=0\,\,\,\qquad\;\;\;\;\;\text{
in}\,\,Z\\\
(\nabla_{y}w_{i}+\hat{e}_{i})\cdot\nu=0\,\,\,\qquad\;\;\;\;\;\;\;\;\text{
on}\,\,\Gamma\\\ y\rightarrow w_{i}(y)\;\;\;\;\;Y-\text{periodic}\end{cases}$
(1.13)

appearing in the limiting function

${\tilde{v}}_{m}(t,x,y,z)=\sum_{i=1}^{3}\,w_{i}(y)\,\frac{\displaystyle\partial
v_{m}}{\displaystyle\partial x_{i}}(t,x,z)\;\;\;\;(1\leq m\leq M)$ (1.14)

Furthermore, for $1\leq m\leq M$ we have:

$\begin{split}&\frac{\displaystyle\partial u_{m}}{\displaystyle\partial
t}(t,x,y,z)-D_{m}\,\triangle_{y}u_{m}(t,x,y,z)-\tilde{D}_{m}\,\partial_{z}^{2}u_{m}(t,x,y,z)\\\
&=L_{m}(u)+f_{m}(t,x,y,z)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;y\in
X\end{split}$ (1.15)

$\begin{split}&D_{m}\,\nabla_{y}u_{m}(t,x,y,z)\cdot\nu\\\
&=-c_{m}(x,z)\,(u_{m}(t,x,y,z)-v_{m}(t,x,z))_{+}\;\;\;\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;y\in\Gamma\end{split}$
(1.16)

$\partial_{z}u_{m}(t,x,y,z)=0\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\qquad
t>0,\;\;(x,z)\in\overline{D}\times\\{0,L\\},\;\;y\in X$ (1.17)

$u_{m}(t=0,x,y,z)=U_{m}(x,y,z)\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x,z)\in\Omega,\;\;y\in
X$ (1.18)

where

$L_{1}(u)=-\sum_{j=1}^{M}a_{1,j}u_{1}(t,x,y,z)\,u_{j}(t,x,y,z)$
$L_{m}(u)=\frac{1}{2}\,\sum_{j=1}^{m-1}a_{j,m-j}\,u_{j}(t,x,y,z)\,u_{m-j}(t,x,y,z)-\sum_{j=1}^{M}a_{m,j}u_{m}(t,x,y,z)\,u_{j}(t,x,y,z)\;\;\;(1<m<M)$
$L_{M}(u)=\frac{1}{2}\,\sum_{\begin{subarray}{c}j+k\geq M\\\ k<M\\\
j<M\end{subarray}}a_{j,k}\,u_{j}(t,x,y,z)\,u_{k}(t,x,y,z)$

and

$f_{m}(t,x,y,z)=\begin{cases}f(t,x,y,z)&\text{if }m=1\\\ 0&\text{if }1<m\leq
M\end{cases}$ $U_{m}(x,y,z)=\begin{cases}U_{1}(x,y,z)>0&\text{if }m=1\\\
0&\text{if }1<m\leq M\end{cases}$

###  1.1 Motivation

The system of equations in (1.1), known under the generic name of discrete
Smoluchowski equations with diffusion, is meant to model the aggregation and
diffusion of the pathological tau protein in the brain, a process associated
with the development of a large variety of cerebral diseases called
tauopathies. Indeed, pathological accumulations of hyperphosphorylated tau
protein aggregates, known as neurofibrillary tangles (NFTs), are detected in
several neurodegenerative tauopathies, including Alzheimer’s disease (AD) [21,
22, 25, 36]. Tau is a highly soluble, natively unfolded protein which is
predominantly located in the axons of neurons of the central nervous system.
Here, its physiological function is to support assembly and stabilization of
axonal microtubules. Under pathological conditions, tau can assume abnormal
conformations, due to two transformations: hyperphosphorylation and
misfolding. In particular, hyperphosphorylation has a negative impact on the
biological function of tau proteins since it inhibits the binding to
microtubules, compromising their stabilization and axonal transport and
promotes self-aggregation. Thus, misfolded tau monomers constitute the
building unit for the formation of oligomers which in turn lead to highly
structured and insoluble fibrils. For many years, cell autonomous mechanisms
were believed to be responsible for the evolution of neurodegenerative
diseases, implying that the same aggregation events occur independently in
different brain cells. However, accumulating evidence now demonstrates that
the progression of tau pathology reflects cell-to-cell propagation of the
disease, achieved through the release of tau into the extracellular space and
the uptake by surrounding healthy neurons [15, 42]. Extracellular tau then
seeds physiological tau in the recipient cells to propagate the pathological
process through neural pathways made by bundles of axons called tracts. This
mechanism, often referred to as ’prion-like’ propagation of tau pathology, has
led to the idea that extracellular tau could be a novel therapeutic target to
halt the spread of the disease [37]. For this reason, in the present work, we
focus on a model which describes in a simplified way the tau diffusion along
tracts starting from the microscopic release and uptake mechanism.

In this context, the set $\Pi_{\epsilon}$ represents a bundle of axons in the
white matter (a tract), while the domain $\Omega_{\epsilon}$ indicates the
extracellular region filled by cerebrospinal fluid. The variables
$u_{m}^{\epsilon}\geq 0$ and $v_{m}^{\epsilon}\geq 0$ refer to the
concentration of $m$-clusters of hyperphosphorylated tau spreading within the
neuronal axons (represented by the long cylindrical cavities) and in
extracellular space, respectively. Concerning the diffusion process, the
choice made in our model to define two coefficients reflects the biological
properties of the tau protein that diffuses differently in extracellular space
and in the axon [21]. Moreover, within the neuronal axon, we further
distinguish between two different diffusion coefficients, since tau propagates
preferentially along the $z$-axis of the cylinder that represents the axon.
The initial condition $U_{1}^{\epsilon}(x,z)$, given at $t=0$ by the
concentration of monomers diffusing within the axons, represents the amount of
endogenous misfolded tau protein, while the source term $f^{\epsilon}(t,x,z)$,
in the system (1.1) for $m=1$, indicates the production of hyperphosphorylated
tau monomers.

###  1.2 Background of this work

There is a large literature related to the use of the Smoluchowski equation in
various physical contexts (e.g., [41, 30, 31, 14, 39, 2]), but only a few
works concerning its application in the biomedical field [9, 5]. Recently, the
important role of the Smoluchowski equations in modelling at different scales
the evolution of neurodegenerative diseases, such as AD, has been investigated
in [32, 1, 4, 5, 38, 18, 16, 17] . In fact, the present work is part of a
broader research effort carried on by various groups of researchers with
diverse collaborations on mathematical models of the progression of AD, and
represents an initial bridge between microscopic models of tau diffusion
developed in biology and macroscopic mathematical models based on graph
theory. To this end, the homogenization technique, introduced by the
mathematicians in the seventies to carry out a sort of averaging process on
the solutions of partial differential equations with rapidly varying
coefficients or describing media with microstructures, has been applied [12,
11, 13, 20, 27, 19, 10].

It is nowadays generally accepted that tau protein, in synergetic combination
with another protein, the so-called $\beta$-amyloid peptide, plays a key role
in the development of AD (see [8]). We refer for instance to [5] for a
discussion on macroscopic mathematical modeling of this interaction. Here, our
interest is focused on the tau protein that diffuses through the neural
pathway, whereas we ignore deliberately the action of the $\beta$-amyloid.

Unlike the approach proposed for instance in [38], [5], where the modeling of
tau coagulation-diffusion processes has been carried out on a large scale,
that is the scale of the neural network characterized by the connectivity of
different regions through bundles of axons (tracts), in this paper only the
mesoscopic dynamics within a portion of tract has been investigated. Starting
from the derivation of model equations valid at the microscale, by using the
so-called two-scale homogenization technique we have proved that the solution
two-scale converges to the solution of a macromodel asymptotically consistent
with the original one. The notion of two-scale homogenization has been first
introduced by Nguetseng [35] and Allaire [3] in the deterministic periodic
setting, and later generalized to the stochastic framework by Zhikov and
Piatnitsky [43]. Unlike other homogenization techniques (see [13], [12] for a
review), the two-scale convergence method is self-contained in that, in a
single step, one can derive the homogenized equations and prove the
convergence of the sequence of solutions to the problem at hand.

To stress the novelty of the present paper, it is worth noting that two-scale
homogenization techniques have been already used by the authors to pass from
microscopic to macroscopic model of the diffusion of toxic proteins in the
cerebral parenchyma affected by AD, but in utterly different biological
perspectives and geometries. Indeed, in [16] and [17], the authors aimed to
describe production, aggregation and diffusion of $\beta$-amyloid peptide in
the cerebral tissue (macroscopic scale), a process associated with the
development of AD, starting from the derivation of a model at the single
neuron level (microscopic scale). Thus, the present paper differs from our
previous works both for the biological meaning and, consequently, for the
geometry of the problem and the boundary conditions introduced in order to
take into account the peculiarities of tau protein propagation.

There is a large literature devoted to the study of transmission boundary
conditions somehow akin to those of (1.1), especially in the framework of
porous media [26]. The main results in this respect, which are relevant to our
work, can be found in [24, 34, 28]. In particular, in [24], it is assumed that
the porous medium is composed of periodically arranged cubic cells of size
$\epsilon$, split up into a solid part (a ball surrounded by semi-permeable
membranes) and a fluid part. In this setting, the diffusion and reactions of
chemical species in the fluid and in the solid part are studied, while
transmission boundary conditions are imposed on the interface. The same
geometry is considered also in [28], where deposition effects under the
influence of thermal gradients are analyzed. The model takes into account the
motion of populations of colloidal particles dissolved in the water
interacting together via Smoluchowski coagulation terms. The colloidal matter
cannot penetrate the solid grain boundary, but it deposits there. This process
is again described by transmission boundary conditions. A domain decomposed
into long cylindrical cavities periodically distributed has been defined in
[34] to model a porous medium consisting of a fluid part and solid bars.
Chemical substances, dissolved in the fluid, are transported by diffusion and
adsorbed on the surface of the bars (through transmission boundary conditions)
where chemical reactions take place.

###  1.3 Outline of the paper

The paper is organized as follows. In Section 2 we prove the existence of weak
solutions to the system of Smoluchowski-type equations (1.1), while Section 3
is devoted to the proof of their positivity and boundedness. A priori
estimates on the derivatives of the solutions are also obtained and reported
at the end of Section 3. In Section 4 we study the convergence of the
homogenization process and prove our main result concerning the two-scale
limit of the solutions to the set of equations (1.1). Finally, in the
Appendices we recall some basic Theorems related to functional analysis and on
the two-scale convergence method.

##  2 Existence of solutions

Let us consider the following truncation of the nonlinear terms in the
Smoluchowski-type equations (1.1) for $1\leq i\leq M$ [28]:

$L_{i}^{\tilde{M}}(u^{\epsilon}):=L_{i}(\sigma_{\tilde{M}}(u_{1}^{\epsilon}),\sigma_{\tilde{M}}(u_{2}^{\epsilon}),\dots,\sigma_{\tilde{M}}(u_{M}^{\epsilon}))$
(2.1)

$N_{i}^{\tilde{M}}(v^{\epsilon}):=N_{i}(\sigma_{\tilde{M}}(v_{1}^{\epsilon}),\sigma_{\tilde{M}}(v_{2}^{\epsilon}),\dots,\sigma_{\tilde{M}}(v_{M}^{\epsilon}))$
(2.2)

where

$\sigma_{\tilde{M}}(s):=\begin{cases}0,\;\;\;\;s<0\\\
s,\;\;\;\;s\in[0,{\tilde{M}}]\\\ {\tilde{M}},\;\;\;\;s>{\tilde{M}}\end{cases}$
(2.3)

with ${\tilde{M}}>0$ being a fixed threshold. If ${\tilde{M}}$ is large
enough, the estimates derived later in this paper will give bounds that will
remain below ${\tilde{M}}$. This means that the results obtained in the
following hold also for the uncutted coagulation terms.

######  Definition 2.1.

The functions $u_{i}^{\epsilon}\in H^{1}([0,T];L^{2}(\Pi_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Pi_{\epsilon}))$ and
$v_{i}^{\epsilon}\in H^{1}([0,T];L^{2}(\Omega_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Omega_{\epsilon}))$ ($1\leq i\leq M$) are solutions to
the problem (1.1) if the following relations hold, a.e. in $[0,T]$ and for a
fixed value of $\epsilon>0$.

If $1\leq i\leq M$:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}u_{i}^{\epsilon}\,\psi_{i}\,dx\,dz+\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{i}\nabla_{x}u_{i}^{\epsilon}\cdot\nabla_{x}\psi_{i}\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{i}\,\partial_{z}u_{i}^{\epsilon}\cdot\partial_{z}\psi_{i}\,dx\,dz$
$\displaystyle+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,\psi_{i}\,d\sigma_{\epsilon}=\displaystyle\int_{\Pi_{\epsilon}}L_{i}^{\tilde{M}}(u^{\epsilon})\,\psi_{i}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}f_{i}^{\epsilon}(t,x,z)\,\psi_{i}\,dx\,dz$
(2.4)

for all $\psi_{i}\in H^{1}(\Pi_{\epsilon})$, and

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}v_{i}^{\epsilon}\,\phi_{i}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{i}\nabla
v_{i}^{\epsilon}\cdot\nabla\phi_{i}\,dx\,dz-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,\phi_{i}\,d\sigma_{\epsilon}$
$\displaystyle=\displaystyle\int_{\Omega_{\epsilon}}N_{i}^{\tilde{M}}(v^{\epsilon})\,\phi_{i}\,dx\,dz$
(2.5)

for all $\phi_{i}\in H^{1}(\Omega_{\epsilon})$, along with the initial
conditions $u_{i}^{\epsilon}(0,x,z)$ and $v_{i}^{\epsilon}(0,x,z)$.

######  Remark 2.1.

In Eqs. (2.1)-(2.1) the integrals over the boundary $\Gamma_{\epsilon}$ are
well defined thanks to the regularity of the functions $u_{i}^{\epsilon}$ and
$v_{i}^{\epsilon}$, as stated in Definition 2.1, and the existence of the
interpolation-trace inequality given by Eq.(A.7) in Appendix A. With an abuse
of notation, in Eqs. (2.1)-(2.1) we have indicated with $u_{i}^{\epsilon}$ and
$v_{i}^{\epsilon}$ also the trace of these functions.

######  Lemma 2.1.

For a given small $\epsilon>0$, the system (1.1) has a solution
$u_{i}^{\epsilon}\in H^{1}([0,T];L^{2}(\Pi_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Pi_{\epsilon}))$ and $v_{i}^{\epsilon}\in
H^{1}([0,T];L^{2}(\Omega_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Omega_{\epsilon}))$ ($1\leq i\leq M$) in the sense of
the Definition 2.1.

###### Proof.

Let {$\xi_{j}$} be an orthonormal basis of $H^{1}(\Pi_{\epsilon})$ and
{$\eta_{j}$} of $H^{1}(\Omega_{\epsilon})$. We denote by $u_{i,n}^{\epsilon}$
and $v_{i,n}^{\epsilon}$ ($1\leq i\leq M$) the Galerkin approximations of
$u_{i}^{\epsilon}$ and $v_{i}^{\epsilon}$, respectively, that is

$u_{i,n}^{\epsilon}(t,x,z):=\sum_{j=1}^{n}\alpha_{i,j}^{n}(t)\,\xi_{j}(x,z)$
(2.6)

for all $t\in[0,T]$, $(x,z)\in\Pi_{\epsilon}$ and

$v_{i,n}^{\epsilon}(t,x,z):=\sum_{j=1}^{n}\beta_{i,j}^{n}(t)\,\eta_{j}(x,z)$
(2.7)

for all $t\in[0,T]$, $(x,z)\in\Omega_{\epsilon}$. Let now $i=1$.

Since {$\xi_{j}$} is an orthonormal basis of $H^{1}(\Pi_{\epsilon})$, for each
$n\in\mathbb{N}$, there exists

$u_{1,n}^{\epsilon,0}(x,z)=\sum_{j=1}^{n}\alpha_{1,j}^{0,n}\,\xi_{j}(x,z)$
(2.8)

so that $u_{1,n}^{\epsilon,0}\rightarrow U_{1}^{\epsilon}(x,z)$ in
$H^{1}(\Pi_{\epsilon})$ as $n\rightarrow\infty$.

Likewise, for each $n\in\mathbb{N}$, there exists

$v_{1,n}^{\epsilon,0}(x,z)=\sum_{j=1}^{n}\beta_{1,j}^{0,n}\,\eta_{j}(x,z)$
(2.9)

so that $v_{1,n}^{\epsilon,0}\rightarrow 0$ in $H^{1}(\Omega_{\epsilon})$ as
$n\rightarrow\infty$.

To derive the coefficients of the Galerkin approximations, we impose that the
functions $u_{1,n}^{\epsilon}$ and $v_{1,n}^{\epsilon}$ satisfy Eqs. (2.1) and
(2.1). Therefore, for all $\psi_{1}\in{\mathrm{s}pan}\\{\xi_{j}\\}_{j=1}^{n}$,
Eq. (2.1) can be written as:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}u_{1,n}^{\epsilon}\,\psi_{1}\,dx\,dz+\epsilon^{2}\,\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,\nabla_{x}u_{1,n}^{\epsilon}\cdot\nabla_{x}\psi_{1}\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}u_{1,n}^{\epsilon}\cdot\partial_{z}\psi_{1}\,dx\,dz$
$\displaystyle+\epsilon\,\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,\psi_{1}\,d\sigma_{\epsilon}=\displaystyle\int_{\Pi_{\epsilon}}L_{1}^{\tilde{M}}(u_{i,n}^{\epsilon})\,\psi_{1}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,\psi_{1}\,dx\,dz$
(2.10)

By testing Eq. (2) with $\psi_{1}=\xi_{k}$, we get

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}\bigg{[}\sum_{j=1}^{n}\alpha_{1,j}^{n}(t)\,\xi_{j}(x,z)\bigg{]}\,\xi_{k}\,dx\,dz+\epsilon^{2}\,\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,\bigg{[}\sum_{j=1}^{n}\alpha_{1,j}^{n}(t)\,\nabla_{x}\xi_{j}(x,z)\bigg{]}\cdot\nabla_{x}\xi_{k}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\bigg{[}\sum_{j=1}^{n}\alpha_{1,j}^{n}(t)\,\partial_{z}\xi_{j}(x,z)\bigg{]}\cdot\partial_{z}\xi_{k}\,dx\,dz$
$\displaystyle+\epsilon\,\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,\bigg{[}\sum_{j=1}^{n}\alpha_{1,j}^{n}(t)\,\xi_{j}(x,z)-\sum_{j=1}^{n}\beta_{1,j}^{n}(t)\,\eta_{j}(x,z)\bigg{]}_{+}\,\xi_{k}\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Pi_{\epsilon}}\xi_{k}\,\sum_{a=1}^{M}a_{1,a}\,\sigma_{\tilde{M}}\bigg{[}\sum_{b=1}^{n}\alpha_{1,b}^{n}(t)\,\xi_{b}(x,z)\bigg{]}\,\sigma_{\tilde{M}}\bigg{[}\sum_{c=1}^{n}\alpha_{a,c}^{n}(t)\,\xi_{c}(x,z)\bigg{]}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,\xi_{k}\,dx\,dz.$
(2.11)

Hence, for $k\in\\{1,\ldots,n\\}$:

$\displaystyle\partial_{t}\alpha_{1,k}^{n}(t)+\sum_{j=1}^{n}A_{1jk}\,\alpha_{1,j}^{n}(t)=-F_{1,k}(\alpha_{1}^{n}(t),\beta_{1}^{n}(t))$
$\displaystyle-\displaystyle\int_{\Pi_{\epsilon}}\xi_{k}\,\sum_{a=1}^{M}a_{1,a}\,\sigma_{\tilde{M}}\bigg{[}\sum_{b=1}^{n}\alpha_{1,b}^{n}(t)\,\xi_{b}(x,z)\bigg{]}\,\sigma_{\tilde{M}}\bigg{[}\sum_{c=1}^{n}\alpha_{a,c}^{n}(t)\,\xi_{c}(x,z)\bigg{]}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,\xi_{k}\,dx\,dz$
(2.12)

where the coefficients $A_{1jk}$ are defined by

$A_{1jk}:=\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,\nabla_{x}\xi_{j}\cdot\nabla_{x}\xi_{k}\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}\xi_{j}\cdot\partial_{z}\xi_{k}\,dx\,dz$
(2.13)

and

$F_{1,k}(\alpha_{1}^{n}(t),\beta_{1}^{n}(t)):=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,\xi_{k}\,\,d\sigma_{\epsilon}$
(2.14)

We choose now $v_{1,n}^{\epsilon}$ satisfying Eq. (2.1) for all
$\phi_{1}\in{\mathrm{s}pan}\\{\eta_{j}\\}_{j=1}^{n}$, i.e.:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}v_{1,n}^{\epsilon}\,\phi_{1}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}\,\nabla
v_{1,n}^{\epsilon}\cdot\nabla\phi_{1}\,dx\,dz$
$\displaystyle-\epsilon\,\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,\phi_{1}\,d\sigma_{\epsilon}=\displaystyle\int_{\Omega_{\epsilon}}N_{1}^{\tilde{M}}(v_{i,n}^{\epsilon})\,\phi_{1}\,dx\,dz$
(2.15)

By testing Eq. (2) with $\phi_{1}=\eta_{k}$, we get

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}\bigg{[}\sum_{j=1}^{n}\beta_{1,j}^{n}(t)\,\eta_{j}(x,z)\bigg{]}\,\eta_{k}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}\,\bigg{[}\sum_{j=1}^{n}\beta_{1,j}^{n}(t)\,\nabla\eta_{j}(x,z)\bigg{]}\cdot\nabla\eta_{k}\,dx\,dz$
$\displaystyle-\epsilon\,\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,\bigg{[}\sum_{j=1}^{n}\alpha_{1,j}^{n}(t)\,\xi_{j}(x,z)-\sum_{j=1}^{n}\beta_{1,j}^{n}(t)\,\eta_{j}(x,z)\bigg{]}_{+}\,\eta_{k}\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Omega_{\epsilon}}\eta_{k}\,\sum_{a=1}^{M}b_{1,a}\,\sigma_{\tilde{M}}\bigg{[}\sum_{b=1}^{n}\beta_{1,b}^{n}(t)\,\eta_{b}(x,z)\bigg{]}\,\sigma_{\tilde{M}}\bigg{[}\sum_{c=1}^{n}\beta_{a,c}^{n}(t)\,\eta_{c}(x,z)\bigg{]}\,dx\,dz$
(2.16)

Hence, for $k\in\\{1,\ldots,n\\}$:

$\displaystyle\partial_{t}\beta_{1,k}^{n}(t)+\sum_{j=1}^{n}B_{1jk}\,\beta_{1,j}^{n}(t)=G_{1,k}(\alpha_{1}^{n}(t),\beta_{1}^{n}(t))$
$\displaystyle-\displaystyle\int_{\Omega_{\epsilon}}\eta_{k}\,\sum_{a=1}^{M}b_{1,a}\,\sigma_{\tilde{M}}\bigg{[}\sum_{b=1}^{n}\beta_{1,b}^{n}(t)\,\eta_{b}(x,z)\bigg{]}\,\sigma_{\tilde{M}}\bigg{[}\sum_{c=1}^{n}\beta_{a,c}^{n}(t)\,\eta_{c}(x,z)\bigg{]}\,dx\,dz$
(2.17)

where the coefficients $B_{1jk}$ are defined by

$B_{1jk}:=\displaystyle\int_{\Omega_{\epsilon}}d_{1}\,\nabla\eta_{j}\cdot\nabla\eta_{k}\,dx\,dz$
(2.18)

and

$G_{1,k}(\alpha_{1}^{n}(t),\beta_{1}^{n}(t)):=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,\eta_{k}\,\,d\sigma_{\epsilon}$
(2.19)

Eqs.(2) and (2) represent a system of $2n$ ordinary differential equations for
the coefficients $\alpha_{1}^{n}=(\alpha_{1,k}^{n})_{k=1,\ldots,n}$,
$\beta_{1}^{n}=(\beta_{1,k}^{n})_{k=1,\ldots,n}$. By taking into account
formulas (2.8) and (2.9), we assume as initial conditions:

$\alpha_{1,k}^{n}(0)=\alpha_{1,k}^{0,n};\;\;\;\;\beta_{1,k}^{n}(0)=\beta_{1,k}^{0,n}.$
(2.20)

Since the left-hand side of this system of ordinary differential equations is
linear and the right-hand side has a sublinear growth (thanks to the Lipschitz
property of the functions), one can conclude that the Cauchy problem (2), (2),
(2.20) has a unique solution extended to the whole interval $[0,T]$ (see the
generalization of the Picard-Lindelöf theorem in [23], Theorem 5.1, p. 156).
Moreover, $\alpha_{1}^{n}=(\alpha_{1,k}^{n})_{k=1,\ldots,n}(t)$,
$\beta_{1}^{n}=(\beta_{1,k}^{n})_{k=1,\ldots,n}(t)\in H^{1}([0,T])$ for
$t\in[0,T]$. We prove in the following the global Lipschitz property of
$F_{1,k}$. The proof of the Lipschitz continuity of $G_{1,k}$ is similar.

Let $(u_{1,n}^{\epsilon},v_{1,n}^{\epsilon})$ and
$(\tilde{u}_{1,n}^{\epsilon},\tilde{v}_{1,n}^{\epsilon})$ be of the form (2.6)
and (2.7), with coefficients
($\alpha_{1}^{n}(t)=(\alpha_{1,k}^{n})_{k=1,\ldots,n}$,
$\beta_{1}^{n}(t)=(\beta_{1,k}^{n})_{k=1,\ldots,n}$) and
($\tilde{\alpha}_{1}^{n}(t)=(\tilde{\alpha}_{1,k}^{n})_{k=1,\ldots,n}$,
$\tilde{\beta}_{1}^{n}(t)=(\tilde{\beta}_{1,k}^{n})_{k=1,\ldots,n}$),
respectively.

One has:

$\displaystyle
F_{1,k}(\alpha_{1}^{n}(t),\beta_{1}^{n}(t))-F_{1,k}(\tilde{\alpha}_{1}^{n}(t),\tilde{\beta}_{1}^{n}(t))$
$\displaystyle=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,\bigg{[}(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}-(\tilde{u}_{1,n}^{\epsilon}-\tilde{v}_{1,n}^{\epsilon})_{+}\bigg{]}\xi_{k}(x,z)\,\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\,C_{1}\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,|(u_{1,n}^{\epsilon}-\tilde{u}_{1,n}^{\epsilon})-(v_{1,n}^{\epsilon}-\tilde{v}_{1,n}^{\epsilon})|\,|\xi_{k}(x,z)|\,d\sigma_{\epsilon}$
$\displaystyle=\epsilon\,C_{1}\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\bigg{|}\sum_{j=1}^{n}(\alpha_{1,j}^{n}(t)-\tilde{\alpha}_{1,j}^{n}(t))\xi_{j}(x,z)$
$\displaystyle-\sum_{j=1}^{n}(\beta_{1,j}^{n}(t)-\tilde{\beta}_{1,j}^{n}(t))\eta_{j}(x,z)\bigg{|}|\xi_{k}(x,z)|\,\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\,C_{1}\sum_{j=1}^{n}|\alpha_{1,j}^{n}(t)-\tilde{\alpha}_{1,j}^{n}(t)|\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,|\xi_{j}(x,z)|\,|\xi_{k}(x,z)|\,\,d\sigma_{\epsilon}$
$\displaystyle+\epsilon\,C_{1}\sum_{j=1}^{n}|\beta_{1,j}^{n}(t)-\tilde{\beta}_{1,j}^{n}(t)|\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,|\eta_{j}(x,z)|\,|\xi_{k}(x,z)|\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\,C_{1}\max\\{c_{jk}\\}\sum_{j=1}^{n}|\alpha_{1,j}^{n}(t)-\tilde{\alpha}_{1,j}^{n}(t)|$
$\displaystyle+\epsilon\,C_{1}\max\\{d_{jk}\\}\sum_{j=1}^{n}|\beta_{1,j}^{n}(t)-\tilde{\beta}_{1,j}^{n}(t)|$
(2.21)

where the coefficients ($c_{jk}$) and ($d_{jk}$) are given by

$c_{jk}:=\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)|\xi_{j}(x,z)|\,|\xi_{k}(x,z)|\,\,d\sigma_{\epsilon}$
(2.22)

$d_{jk}:=\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)|\eta_{j}(x,z)|\,|\xi_{k}(x,z)|\,\,d\sigma_{\epsilon}$
(2.23)

for $j,k=1,\ldots,n$. Hence, we get

$|F_{1,k}(\alpha_{1}^{n},\beta_{1}^{n})-F_{1,k}(\tilde{\alpha}_{1}^{n},\tilde{\beta}_{1}^{n})|\leq\epsilon\,C_{2}(n)\bigg{[}|\alpha_{1}^{n}-\tilde{\alpha}_{1}^{n}|+|\beta_{1}^{n}-\tilde{\beta}_{1}^{n}|\bigg{]}.$
(2.24)

The same conclusions can be drawn also when $1<i\leq M$ by applying exactly
the arguments considered above.

Uniform estimates

Let us now prove uniform estimates in $n$ for $u_{i,n}^{\epsilon}$ and
$v_{i,n}^{\epsilon}$ ($1\leq i\leq M$). In the case $i=1$, we take in Eq. (2)
$\psi_{1}=u_{1,n}^{\epsilon}(t,\cdot)$ as test function:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}(u_{1,n}^{\epsilon})^{2}\,dx\,dz+\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{1}|\nabla_{x}u_{1,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}|\partial_{z}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,u_{1,n}^{\epsilon}\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,\sigma_{\tilde{M}}(u_{1,n}^{\epsilon})\sigma_{\tilde{M}}(u_{j,n}^{\epsilon})\bigg{]}\,u_{1,n}^{\epsilon}\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,u_{1,n}^{\epsilon}\,dx\,dz$
(2.25)

Since the first term on the right-hand side is always negative due to the
truncation of the coagulation terms (see Eq. (2.3)), one obtains:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\epsilon^{2}\,D_{1}\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\tilde{D}_{1}\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle\leq-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,u_{1,n}^{\epsilon}\,d\sigma_{\epsilon}+\frac{\displaystyle
1}{\displaystyle 2}\|f^{\epsilon}(t,x,z)\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+\frac{\displaystyle 1}{\displaystyle
2}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$ (2.26)

where we have applied the Hölder inequality to the last term on the right-hand
side of Eq. (2). Since the function $f^{\epsilon}(t,x,z)$ is bounded in
$L^{2}([0,T]\times\Pi_{\epsilon})$, Eq. (2) reads:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\epsilon^{2}\,D_{1}\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\tilde{D}_{1}\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle\leq-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,u_{1,n}^{\epsilon}\,d\sigma_{\epsilon}+C_{f}+\frac{\displaystyle
1}{\displaystyle 2}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$ (2.27)

where $C_{f}$ is a positive constant. By testing now Eq. (2) with
$\phi_{1}=v_{1,n}^{\epsilon}(t,\cdot)$, we get

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}(v_{1,n}^{\epsilon})^{2}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}|\nabla
v_{1,n}^{\epsilon}|^{2}\,dx\,dz-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,v_{1,n}^{\epsilon}\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,\sigma_{\tilde{M}}(v_{1,n}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j,n}^{\epsilon})\bigg{]}\,v_{1,n}^{\epsilon}\,dx\,dz$
(2.28)

Since again the term on the right-hand side is negative, we conclude:

$\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+d_{1}\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,v_{1,n}^{\epsilon}\,d\sigma_{\epsilon}$
(2.29)

Adding the inequalities (2) and (2.29), it follows that

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\partial_{t}\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\epsilon^{2}\,D_{1}\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\tilde{D}_{1}\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+d_{1}\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C_{f}+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(v_{1,n}^{\epsilon}-u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
$\displaystyle+\frac{\displaystyle 1}{\displaystyle
2}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$ (2.30)

Let us estimate the second term on the right-hand side of (2):

$\displaystyle
I=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(v_{1,n}^{\epsilon}-u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)|u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon}|^{2}\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\|c_{1}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\displaystyle\int_{\Gamma_{\epsilon}}(|u_{1,n}^{\epsilon}|^{2}+|v_{1,n}^{\epsilon}|^{2})\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\|c_{1}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\bigg{(}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Gamma_{\epsilon})}+\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Gamma_{\epsilon})}\bigg{)}$
(2.31)

Applying the generalized interpolation-trace inequality (A.7) in Appendix A to
each term inside the round brackets, one has:

$I\leq C_{1}\,\epsilon\,\eta\bigg{(}\|\nabla
u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{)}+C_{2}\,\epsilon\,{\eta}^{-1}\bigg{(}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{)}$
(2.32)

where $\eta$ is a small positive constant. If we take into account the
estimate (2.32), the inequality (2) reads:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\partial_{t}\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+(\epsilon^{2}\,D_{1}-C_{1}\,\epsilon\,\eta)\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+(\tilde{D}_{1}-C_{1}\,\epsilon\,\eta)\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+(d_{1}-C_{1}\,\epsilon\,\eta)\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq C_{f}$
$\displaystyle+\bigg{(}C_{2}\,\epsilon\,{\eta}^{-1}+\frac{\displaystyle
1}{\displaystyle
2}\bigg{)}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+C_{2}\,\epsilon\,{\eta}^{-1}\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
(2.33)

If one chooses $\eta<\min\\{\frac{\epsilon
D_{1}}{C_{1}},\frac{\tilde{D}_{1}}{\epsilon\,C_{1}},\frac{d_{1}}{\epsilon\,C_{1}}\\}$,
the last three terms on the left-hand side are positive and Eq. (2) reduces
to:

$\partial_{t}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\partial_{t}\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
2\,C_{f}+2\,C_{2}\,\epsilon\,{\eta}^{-1}\bigg{(}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{)}$
(2.34)

Integrating Eq. (2.34) over $[0,t]$ with $t\in[0,T]$, we get

$\displaystyle\|u_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq\|u_{1,n}^{\epsilon,0}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon,0}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle+2\,C_{f}\,T+2\,C_{2}\,\epsilon\,{\eta}^{-1}\displaystyle\int_{0}^{t}ds\bigg{(}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{)}$
(2.35)

Since the sequences $(u_{1,n}^{\epsilon,0})_{n\in\mathbb{N}}$ and
$(v_{1,n}^{\epsilon,0})_{n\in\mathbb{N}}$ converge in $H^{1}(\Pi_{\epsilon})$
and $H^{1}(\Omega_{\epsilon})$, respectively, they are bounded in $L^{2}$.
Therefore, Eq. (2) reads

$\displaystyle\|u_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C$
$\displaystyle+2\,C_{2}\,\epsilon\,{\eta}^{-1}\displaystyle\int_{0}^{t}ds\bigg{(}\|u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{)}$
(2.36)

Applying Gronwalls’s inequality, we obtain

$\|u_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}(t)\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C+2\,C\,C_{2}\,\epsilon\,{\eta}^{-1}\displaystyle\int_{0}^{t}e^{2\,C_{2}\,\epsilon\,{\eta}^{-1}(t-s)}\,ds$
(2.37)

Therefore, given $\epsilon\in[0,1]$, $\eta$ is fixed and for $t\in[0,T]$ we
get:

$\|u_{1,n}^{\epsilon}(t,\cdot)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1,n}^{\epsilon}(t,\cdot)\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C_{3}$ (2.38)

where $C_{3}$ is a positive constant independent of $n$ and $\epsilon$.

By testing Eqs. (2.1) and (2.1), in the case $1<i\leq M$, with
$\psi_{i}=u_{i,n}^{\epsilon}(t,\cdot)$ and
$\phi_{i}=v_{i,n}^{\epsilon}(t,\cdot)$, respectively, one gets:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\epsilon^{2}\,D_{i}\|\nabla_{x}u_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\tilde{D}_{i}\|\partial_{z}u_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle\leq-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i,n}^{\epsilon}-v_{i,n}^{\epsilon})_{+}\,u_{i,n}^{\epsilon}\,d\sigma_{\epsilon}+C+\frac{\displaystyle
1}{\displaystyle 4}\|u_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$ (2.39)

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|v_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+d_{i}\|\nabla
v_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i,n}^{\epsilon}-v_{i,n}^{\epsilon})_{+}\,v_{i,n}^{\epsilon}\,d\sigma_{\epsilon}$
$\displaystyle+\tilde{C}+\frac{\displaystyle 1}{\displaystyle
4}\|v_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}$ (2.40)

where $C$ and $\tilde{C}$ are two positive constants. Adding the inequalities
(2) and (2), and exploiting the estimate (2) (which holds also when $1<i\leq
M$) along with the interpolation-trace inequality (A.7), we obtain that also
the functions $u_{i,n}^{\epsilon}$ and $v_{i,n}^{\epsilon}$ ($1<i\leq M$)
satisfy Eq. (2.34). The rest of the proof carries over verbatim, leading to
Eq. (2.38) also for the case $1<i\leq M$.

Thus, we can conclude that:

$\\{u_{i,n}^{\epsilon}\\}$ is bounded in
$L^{\infty}([0,T];L^{2}(\Pi_{\epsilon}))$, and $\\{v_{i,n}^{\epsilon}\\}$ is
bounded in $L^{\infty}([0,T];L^{2}(\Omega_{\epsilon}))$.

Let us now derive uniform estimates in $n$ for
$\partial_{t}u_{i,n}^{\epsilon}$, $\nabla u_{i,n}^{\epsilon}$ and
$\partial_{t}v_{i,n}^{\epsilon}$, $\nabla v_{i,n}^{\epsilon}$ ($1\leq i\leq
M$). In the case $i=1$, we take in Eq. (2)
$\psi_{1}=\partial_{t}u_{1,n}^{\epsilon}(t,\cdot)$ as test function:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}|\partial_{t}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,\partial_{t}(|\nabla_{x}u_{1,n}^{\epsilon}|^{2})\,dx\,dz$
$\displaystyle+\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{t}(|\partial_{z}u_{1,n}^{\epsilon}|^{2})\,dx\,dz+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Pi_{\epsilon}}w_{\tilde{M}}^{\epsilon}\,(\partial_{t}u_{1,n}^{\epsilon})\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,(\partial_{t}u_{1,n}^{\epsilon})\,dx\,dz$
(2.41)

where
$w_{\tilde{M}}^{\epsilon}=:\sum_{j=1}^{M}a_{1,j}\,\sigma_{\tilde{M}}(u_{1,n}^{\epsilon})\,\sigma_{\tilde{M}}(u_{j,n}^{\epsilon})$.

By taking into account the Hölder and Young inequalities, Eq. (2) becomes

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}|\partial_{t}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle\epsilon^{2}\,D_{1}}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\,\partial_{t}(|\nabla_{x}u_{1,n}^{\epsilon}|^{2})\,dx\,dz+\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\,\partial_{t}(|\partial_{z}u_{1,n}^{\epsilon}|^{2})\,dx\,dz$
$\displaystyle\leq\frac{\displaystyle 1}{\displaystyle
2}\|w_{\tilde{M}}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
$\displaystyle+\eta^{-1}\|f^{\epsilon}(t,x,z)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\eta\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
(2.42)

Exploiting the truncation of the coagulation terms and choosing
$\eta=\frac{\displaystyle 1}{\displaystyle 4}$, we get:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{1}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle\leq
C^{1}_{\tilde{M}}-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}+4\,\|f^{\epsilon}(t,x,z)\|^{2}_{L^{2}(\Pi_{\epsilon})}$
(2.43)

where $C^{1}_{\tilde{M}}$ is a positive constant which depends on $\tilde{M}$.

Let us now test Eq. (2) with the function
$\phi_{1}=\partial_{t}v_{1,n}^{\epsilon}(t,\cdot)$:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}|\partial_{t}v_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle
d_{1}}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}(|\nabla
v_{1,n}^{\epsilon}|^{2})\,dx\,dz$
$\displaystyle-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}v_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
$\displaystyle=-\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,\sigma_{\tilde{M}}(v_{1,n}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j,n}^{\epsilon})\bigg{]}\,(\partial_{t}v_{1,n}^{\epsilon})\,dx\,dz$
(2.44)

By applying once again the Hölder and Young inequalities to the right-hand
side as above, and exploiting Eq. (2.3), we end up with the following
expression:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\|\partial_{t}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\frac{\displaystyle
d_{1}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}|^{2}\,dx\,dz$ $\displaystyle\leq
C^{2}_{\tilde{M}}+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}v_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
(2.45)

where $C^{2}_{\tilde{M}}$ is a positive constant which depends on $\tilde{M}$.
Adding (2) and (2), and taking into account the $L^{2}$-boundedness of
$f^{\epsilon}(t,x,z)$, one obtains:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\|\partial_{t}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{1}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle+\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle
d_{1}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}|^{2}\,dx\,dz$ $\displaystyle\leq
C+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,(\partial_{t}v_{1,n}^{\epsilon}-\partial_{t}u_{1,n}^{\epsilon})\,d\sigma_{\epsilon}$
(2.46)

where $C$ is a positive constant. If we decompose now the function
$\partial_{t}(v_{1,n}^{\epsilon}-u_{1,n}^{\epsilon})$ on the right-hand side
in its positive and negative parts, Eq. (2) can be rewritten as:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\|\partial_{t}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{1}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle+\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle
d_{1}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}|^{2}\,dx\,dz$ (2.47) $\displaystyle\leq
C-\frac{\displaystyle\epsilon}{\displaystyle
2}\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\partial_{t}[(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})^{2}_{+}]\,d\sigma_{\epsilon}$
(2.48)

Integrating over $[0,t]$ with $t\in[0,T]$, we deduce:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\displaystyle\int_{0}^{t}ds\,\|\partial_{s}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\displaystyle\int_{0}^{t}ds\,\|\partial_{s}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{1}\,\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}|^{2}\,dx\,dz-\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{1}\,\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz$
$\displaystyle+\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\,\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}|^{2}\,dx\,dz-\frac{\displaystyle\tilde{D}_{1}}{\displaystyle
2}\,\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz$
$\displaystyle+\frac{\displaystyle d_{1}}{\displaystyle
2}\,\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}|^{2}\,dx\,dz-\frac{\displaystyle d_{1}}{\displaystyle
2}\,\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz\leq C\,T$
$\displaystyle+\frac{\displaystyle\epsilon}{\displaystyle
2}\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})^{2}_{+}(0)\,d\sigma_{\epsilon}-\frac{\displaystyle\epsilon}{\displaystyle
2}\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})^{2}_{+}\,d\sigma_{\epsilon}$
(2.49)

Hence, taking into account that the last term on the right-hand side of Eq.(2)
is negative, one has

$\displaystyle\displaystyle\int_{0}^{t}ds\,\|\partial_{s}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+2\,\displaystyle\int_{0}^{t}ds\,\|\partial_{s}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+2\,{\displaystyle\epsilon^{2}}\,D_{1}\,\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+2\,\tilde{D}_{1}\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+2\,d_{1}\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq 4\,C\,T$
$\displaystyle+2\,D_{1}\,\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz+2\,\tilde{D}_{1}\,\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz$
$\displaystyle+2\,d_{1}\,\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{1,n}^{\epsilon}(0)|^{2}\,dx\,dz+2\,\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})^{2}_{+}(0)\,d\sigma_{\epsilon}$
(2.50)

Since the sequences $u_{1,n}^{\epsilon}(0)$ and $v_{1,n}^{\epsilon}(0)$ are
bounded, it follows:

$\displaystyle\displaystyle\int_{0}^{T}ds\,\|\partial_{s}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+2\,\displaystyle\int_{0}^{T}ds\,\|\partial_{s}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+2\,\epsilon^{2}\,D_{1}\,\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+2\,\tilde{D}_{1}\,\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+2\,d_{1}\,\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq\tilde{C}\;\;\;\;\;\;\mbox{for}\;\;t\in[0,T]$
(2.51)

where $\tilde{C}$ is a positive constant independent of $n$ and $\epsilon$.

In the case $1<i\leq M$, by testing Eqs. (2.1) and (2.1) with
$\psi_{i}=\partial_{t}u_{i,n}^{\epsilon}(t,\cdot)$ and
$\phi_{i}=\partial_{t}v_{i,n}^{\epsilon}(t,\cdot)$, respectively, one obtains:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\|\partial_{t}u_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle\epsilon^{2}}{\displaystyle
2}\,D_{i}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\nabla_{x}u_{i,n}^{\epsilon}|^{2}\,dx\,dz+\frac{\displaystyle\tilde{D}_{i}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Pi_{\epsilon}}|\partial_{z}u_{i,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle\leq
C-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i,n}^{\epsilon}-v_{i,n}^{\epsilon})_{+}\,(\partial_{t}u_{i,n}^{\epsilon})\,d\sigma_{\epsilon}$
(2.52)

$\displaystyle\frac{\displaystyle 1}{\displaystyle
4}\|\partial_{t}v_{i,n}^{\epsilon}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\frac{\displaystyle
d_{i}}{\displaystyle
2}\,\partial_{t}\displaystyle\int_{\Omega_{\epsilon}}|\nabla
v_{i,n}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle\leq\tilde{C}+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i,n}^{\epsilon}-v_{i,n}^{\epsilon})_{+}\,(\partial_{t}v_{i,n}^{\epsilon})\,d\sigma_{\epsilon}$
(2.53)

due to the boundedness of the coagulation terms given by (2.3). In Eqs. (2)
and (2), $C$ and $\tilde{C}$ are two positive constants which depend on
$\tilde{M}$. Adding the two inequalities (2) and (2), and applying exactly the
same arguments considered for $i=1$, we obtain that also the functions
$u_{i,n}^{\epsilon}$ and $v_{i,n}^{\epsilon}$ ($1<i\leq M$) satisfy Eq. (2).

Thus, combining the estimates (2.38) and (2), one concludes that:

$\\{u_{i,n}^{\epsilon}\\}$ is bounded in
$H^{1}([0,T];L^{2}(\Pi_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Pi_{\epsilon}))$, and $\\{v_{i,n}^{\epsilon}\\}$ is
bounded in $H^{1}([0,T];L^{2}(\Omega_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Omega_{\epsilon}))$. Hence, we can state the following
proposition.

######  Proposition 2.1.

Since $(u_{i,n}^{\epsilon})_{n\in\mathbb{N}}$ is bounded in
$L^{\infty}([0,T];H^{1}(\Pi_{\epsilon}))$, by the Banach-Alaoglu theorem we
may assume that, up to a subsequence, $u_{i,n}^{\epsilon}\rightharpoonup
u_{i}^{\epsilon}$ weakly* in $L^{\infty}([0,T];H^{1}(\Pi_{\epsilon}))$, i.e.
for all $\psi\in L^{1}([0,T];H^{1}(\Pi_{\epsilon}))$

$\begin{split}\int_{0}^{T}\int_{\Pi_{\epsilon}}&\big{(}\psi(t,x,z)u_{i,n}^{\epsilon}(t,x,z)+\nabla\psi(t,x,z)\cdot\nabla
u_{i,n}^{\epsilon}(t,x,z)\big{)}\,dt\,dx\,dz\\\
&\longrightarrow\int_{0}^{T}\int_{\Pi_{\epsilon}}\big{(}\psi(t,x,z)u_{i}^{\epsilon}(t,x,z)+\nabla\psi(t,x,z)\cdot\nabla
u_{i}^{\epsilon}(t,x,z)\big{)}\,dt\,dx\,dz\end{split}$ (2.54)

as $n\to\infty$. Since $L^{2}([0,T];H^{1}(\Pi_{\epsilon}))\subset
L^{1}([0,T];H^{1}(\Pi_{\epsilon}))$, formula (2.54) holds also for $\psi\in
L^{2}([0,T];H^{1}(\Pi_{\epsilon}))$.

In addition, $(u_{i,n}^{\epsilon})_{n\in\mathbb{N}}$ is bounded in
$H^{1}([0,T];L^{2}(\Pi_{\epsilon}))$, thus we can also assume that
$u_{i,n}^{\epsilon}\rightharpoonup u_{i}^{\epsilon}$ weakly in
$H^{1}([0,T];L^{2}(\Pi_{\epsilon}))$ as $n\to\infty$.

Finally, since in particular

$\mbox{$(u_{i,n}^{\epsilon})_{n\in\mathbb{N}}$ is bounded in}\quad
H^{1}([0,T];L^{2}(\Pi_{\epsilon}))\cap
L^{\infty}([0,T];H^{1}(\Pi_{\epsilon})),$

by the Aubin-Lions-Simon theorem ([6], Theorem II.5.16, p. 102) (see Appendix
B) we can infer that

$u_{i,n}^{\epsilon}\rightarrow u_{i}^{\epsilon}$ strongly in
$C^{0}([0,T];L^{2}(\Pi_{\epsilon}))$ as $n\to\infty$.

An analogous proposition can be proved also for the sequence
$(v_{i,n}^{\epsilon})_{n\in\mathbb{N}}$.

Now, integrating Eq. (2) with respect to time and using as a test function
$\psi_{1}=\phi(t)\zeta_{1}(x,z)$, with $\phi\in{\mathcal{D}}([0,T])$ and
$\zeta_{1}\in H^{1}(\Pi_{\epsilon})$, we get

$\displaystyle\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}\partial_{t}u_{1,n}^{\epsilon}\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz+\epsilon^{2}\,\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}D_{1}\,\nabla_{x}u_{1,n}^{\epsilon}\cdot\nabla_{x}\zeta_{1}\,\phi(t)\,dt\,dx\,dz$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}u_{1,n}^{\epsilon}\cdot\partial_{z}\zeta_{1}\,\phi(t)\,dt\,dx\,dz$
$\displaystyle+\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1,n}^{\epsilon}-v_{1,n}^{\epsilon})_{+}\,\phi(t)\,\zeta_{1}(x,z)\,dt\,d\sigma_{\epsilon}$
$\displaystyle=\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}L_{1}^{\tilde{M}}(u_{i,n}^{\epsilon})\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz$
(2.55)

where we denote by ${\mathcal{D}}([0,T])$ the set of indefinitely
differentiable functions whose support is a compact set.

Exploiting the convergence results stated in Proposition 2.1 we can pass to
the limit as $n\rightarrow\infty$ to obtain:

$\displaystyle\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}\partial_{t}u_{1}^{\epsilon}\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz+\epsilon^{2}\,\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}D_{1}\,\nabla_{x}u_{1}^{\epsilon}\cdot\nabla_{x}\zeta_{1}\,\phi(t)\,dt\,dx\,dz$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}u_{1}^{\epsilon}\cdot\partial_{z}\zeta_{1}\,\phi(t)\,dt\,dx\,dz$
$\displaystyle+\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,\phi(t)\,\zeta_{1}(x,z)\,dt\,d\sigma_{\epsilon}$
$\displaystyle=\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}L_{1}^{\tilde{M}}(u_{i}^{\epsilon})\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz$
$\displaystyle+\displaystyle\int_{0}^{T}\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,\phi(t)\,\zeta_{1}(x,z)\,dt\,dx\,dz$
(2.56)

where we have taken into account that the term
$L_{1}^{\tilde{M}}(u_{i}^{\epsilon})$ is Lipschitz continuous. Eq. (2), which
holds for arbitrary $\phi(t)\in{\mathcal{D}}([0,T])$ and $\zeta_{1}(x,z)\in
H^{1}(\Pi_{\epsilon})$, is exactly the variational equation (2.1).

By using the same arguments handled above for $u_{1}^{\epsilon}$, we can prove
that the functions $v_{1}^{\epsilon}$ and $u_{i}^{\epsilon}$,
$v_{i}^{\epsilon}$ ($1<i\leq M$) satisfy Eqs. (2.1), (2.1).

It remains to show that the initial conditions hold. By the Aubin-Lions-Simon
theorem it follows that $u_{1,n}^{\epsilon}\rightarrow u_{1}^{\epsilon}$
strongly in $C^{0}([0,T];L^{2}(\Pi_{\epsilon}))$ as $n\to\infty$. Since
$u_{1,n}^{\epsilon}(t=0)=u_{1,n}^{\epsilon,0}$ and
$u_{1,n}^{\epsilon,0}\rightarrow U^{\epsilon}_{1}(x,z)$ strongly in
$H^{1}(\Pi_{\epsilon})$ as $n\to\infty$, we conclude that
$u_{1}^{\epsilon}(0,x,z)=U^{\epsilon}_{1}(x,z)$. Following the same line of
arguments, we also obtain $v_{1}^{\epsilon}(0,x,z)=0$ along with
$u_{i}^{\epsilon}(0,x,z)=v_{i}^{\epsilon}(0,x,z)=0$ ($1<i\leq M$).

∎

##  3 Positivity and boundedness of solutions

######  Lemma 3.1.

For a given small $\epsilon>0$, let $u_{i}^{\epsilon}(t,x,z)$ and
$v_{i}^{\epsilon}(t,x,z)$ ($1\leq i\leq M$) be solutions of the system (1.1)
in the sense of the Definition 2.1. Then $0\leq u_{i}^{\epsilon}(t,x,z)<M_{i}$
a.e. in $(0,T)\times\Pi_{\epsilon}$ and $0\leq
v_{i}^{\epsilon}(t,x,z)<\overline{M}_{i}$ a.e. in
$(0,T)\times\Omega_{\epsilon}$, where $M_{i}>0$ and $\overline{M}_{i}>0$ are
constants independent of $\tilde{M}$ and $\epsilon$.

###### Proof.

In the case $i=1$, let us test Eq. (2.1) with the function
$\psi_{1}=-u_{1}^{\epsilon,-}(t,\cdot)$:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}u_{1}^{\epsilon}\,(-u_{1}^{\epsilon,-})\,dx\,dz+\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{1}\nabla_{x}u_{1}^{\epsilon}\cdot\nabla_{x}(-u_{1}^{\epsilon,-})\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}u_{1}^{\epsilon}\cdot\partial_{z}(-u_{1}^{\epsilon,-})\,dx\,dz+\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(-u_{1}^{\epsilon,-})\,d\sigma_{\epsilon}$
$\displaystyle=\displaystyle\int_{\Pi_{\epsilon}}L_{1}^{\tilde{M}}(u^{\epsilon})\,(-u_{1}^{\epsilon,-})\,dx\,dz+\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,(-u_{1}^{\epsilon,-})\,dx\,dz$
(3.1)

Decomposing the function $u_{1}^{\epsilon}$ in its positive and negative
parts, we get:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\frac{\partial
u_{1}^{\epsilon,-}}{\partial
t}\,u_{1}^{\epsilon,-}\,dx\,dz+\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,|\nabla_{x}u_{1}^{\epsilon,-}|^{2}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,|\partial_{z}u_{1}^{\epsilon,-}|^{2}\,dx\,dz=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,u_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,\sigma_{\tilde{M}}(u_{1}^{\epsilon})\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\bigg{]}\,u_{1}^{\epsilon,-}\,dx\,dz-\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,u_{1}^{\epsilon,-}\,dx\,dz$
(3.2)

Since the last but one term on the right-hand side is always zero and the last
one is negative, one obtains:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\epsilon^{2}D_{1}\,\|\nabla_{x}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\tilde{D}_{1}\,\|\partial_{z}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,u_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
(3.3)

Let us now take $\phi_{1}=-v_{1}^{\epsilon,-}(t,\cdot)$ as test function in
Eq. (2.1) and decompose the function $v_{1}^{\epsilon}$ in its positive and
negative parts:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\frac{\partial
v_{1}^{\epsilon,-}}{\partial
t}\,v_{1}^{\epsilon,-}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}\,|\nabla
v_{1}^{\epsilon,-}|^{2}\,dx\,dz=-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,v_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
$\displaystyle+\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,\sigma_{\tilde{M}}(v_{1}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\bigg{]}\,v_{1}^{\epsilon,-}\,dx\,dz$
(3.4)

Since the last term on the right-hand side is always zero, Eq. (3) reduces to:

$\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}+d_{1}\,\|\nabla
v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}=-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,v_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
(3.5)

Adding Eqs. (3) and (3.5) we obtain:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\partial_{t}\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}+\epsilon^{2}D_{1}\,\|\nabla_{x}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+\tilde{D}_{1}\,\|\partial_{z}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+d_{1}\,\|\nabla
v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(u_{1}^{\epsilon,-}-v_{1}^{\epsilon,-})\,d\sigma_{\epsilon}$
(3.6)

Let us now estimate the term on the right-hand side of Eq. (3):

$\displaystyle\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(u_{1}^{\epsilon,-}-v_{1}^{\epsilon,-})\,d\sigma_{\epsilon}\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,{\mathcal{H}}(u_{1}^{\epsilon}-v_{1}^{\epsilon})(u_{1}^{\epsilon}-v_{1}^{\epsilon})\,u_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,{\mathcal{H}}(u_{1}^{\epsilon}-v_{1}^{\epsilon})(u_{1}^{\epsilon,-}\,v_{1}^{\epsilon,-})\,d\sigma_{\epsilon}\leq\epsilon\|c_{1}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\displaystyle\int_{\Gamma_{\epsilon}}u_{1}^{\epsilon,-}\,v_{1}^{\epsilon,-}\,d\sigma_{\epsilon}$
(3.7)

where we denote by ${\mathcal{H}}(\cdot)$ the Heaviside function:

${\mathcal{H}}(s):=\begin{cases}0,\;\;\;\;s<0\\\ 1,\;\;\;\;s\geq
0.\end{cases}$

Exploiting the Hölder and Young inequalities along with the generalized
interpolation-trace inequality (A.7), Eq. (3) becomes

$\displaystyle\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(u_{1}^{\epsilon,-}-v_{1}^{\epsilon,-})\,d\sigma_{\epsilon}$
$\displaystyle\leq\frac{\displaystyle\epsilon}{\displaystyle
2}\|c_{1}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\bigg{[}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Gamma_{\epsilon})}+\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Gamma_{\epsilon})}\bigg{]}$
$\displaystyle\leq C_{1}\,\epsilon\,\eta\bigg{[}\|\nabla
u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|\nabla
v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{]}$
$\displaystyle+C_{2}\,\epsilon\,\eta^{-1}\bigg{[}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{]}$
(3.8)

where $\eta$ is a small constant. Finally from Eqs. (3) and (3) it follows
that

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\partial_{t}\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}+(\epsilon^{2}D_{1}-C_{1}\,\epsilon\,\eta)\,\|\nabla_{x}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}$
$\displaystyle+(\tilde{D}_{1}-C_{1}\,\epsilon\,\eta)\,\|\partial_{z}u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+(d_{1}-C_{1}\,\epsilon\,\eta)\,\|\nabla
v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}$ $\displaystyle\leq
C_{2}\,\epsilon\,\eta^{-1}\bigg{[}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{]}$
(3.9)

If one chooses
$\eta<min\\{\frac{\epsilon\,D_{1}}{C_{1}},\frac{\tilde{D}_{1}}{\epsilon\,C_{1}},\frac{d_{1}}{\epsilon\,C_{1}}\\}$,
Eq. (3) reduces to:

$\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\frac{\displaystyle
1}{\displaystyle
2}\partial_{t}\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C_{2}\,\epsilon\,\eta^{-1}\bigg{[}\|u_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1}^{\epsilon,-}\|^{2}_{L^{2}(\Omega_{\epsilon})}\bigg{]}$
(3.10)

Setting the initial conditions: $u_{1}^{\epsilon,-}(0)\equiv 0$ and
$v_{1}^{\epsilon,-}(0)\equiv 0$, Gronwall’s lemma gives:

$\|u_{1}^{\epsilon,-}(t,\cdot)\|^{2}_{L^{2}(\Pi_{\epsilon})}+\|v_{1}^{\epsilon,-}(t,\cdot)\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
0$ (3.11)

that is, $u_{1}^{\epsilon}\geq 0$ a.e. in $\Pi_{\epsilon}$ and
$v_{1}^{\epsilon}\geq 0$ a.e. in $\Omega_{\epsilon}$, for all $t\in[0,T]$.

In the case $1<i\leq M$, by testing Eqs. (2.1) and (2.1) with
$\psi_{i}=-u_{i}^{\epsilon,-}(t,\cdot)$ and
$\phi_{i}=-v_{i}^{\epsilon,-}(t,\cdot)$, respectively, and decomposing the
functions $u_{i}^{\epsilon}$ and $v_{i}^{\epsilon}$ in the positive and
negative parts, one gets:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\frac{\partial
u_{i}^{\epsilon,-}}{\partial
t}\,u_{i}^{\epsilon,-}\,dx\,dz+\epsilon^{2}\displaystyle\int_{\Pi_{\epsilon}}D_{i}\,|\nabla_{x}u_{i}^{\epsilon,-}|^{2}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{i}\,|\partial_{z}u_{i}^{\epsilon,-}|^{2}\,dx\,dz=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,u_{i}^{\epsilon,-}\,d\sigma_{\epsilon}$
$\displaystyle-\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\,\sigma_{\tilde{M}}(u_{i-j}^{\epsilon})\bigg{]}\,u_{i}^{\epsilon,-}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{M}a_{i,j}\,\sigma_{\tilde{M}}(u_{i}^{\epsilon})\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\bigg{]}\,u_{i}^{\epsilon,-}\,dx\,dz$
(3.12)

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\frac{\partial
v_{i}^{\epsilon,-}}{\partial
t}\,v_{i}^{\epsilon,-}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{i}\,|\nabla
v_{i}^{\epsilon,-}|^{2}\,dx\,dz$
$\displaystyle=-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,v_{i}^{\epsilon,-}\,d\sigma_{\epsilon}$
$\displaystyle-\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\,\sigma_{\tilde{M}}(v_{i-j}^{\epsilon})\bigg{]}\,v_{i}^{\epsilon,-}\,dx\,dz$
$\displaystyle+\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{i,j}\,\sigma_{\tilde{M}}(v_{i}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\bigg{]}\,v_{i}^{\epsilon,-}\,dx\,dz$
(3.13)

Since in both Eqs. (3) and (3) the last but one term on the right-hand side is
negative and the last one always zero, we obtain that also the functions
$u_{i}^{\epsilon,-}$ and $v_{i}^{\epsilon,-}$ ($1<i\leq M$) satisfy Eqs. (3)
and (3.5), respectively. Therefore, applying exactly the same arguments
considered for $i=1$, we conclude that $u_{i}^{\epsilon}\geq 0$ a.e. in
$\Pi_{\epsilon}$ and $v_{i}^{\epsilon}\geq 0$ a.e. in $\Omega_{\epsilon}$, for
all $t\in[0,T]$.

Let us now prove the boundedness of solutions. In the case $i=1$, we test Eq.
(2.1) with $\psi_{1}=p(u_{1}^{\epsilon})^{p-1}(p\geq 2)$:

$\displaystyle
p\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}u_{1}^{\epsilon}\,(u_{1}^{\epsilon})^{p-1}\,dx\,dz+\epsilon^{2}\,p\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,\nabla_{x}u_{1}^{\epsilon}\cdot\nabla_{x}(u_{1}^{\epsilon})^{p-1}\,dx\,dz$
$\displaystyle+p\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,\partial_{z}u_{1}^{\epsilon}\cdot\partial_{z}(u_{1}^{\epsilon})^{p-1}\,dx\,dz+\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(u_{1}^{\epsilon})^{p-1}d\sigma_{\epsilon}$
$\displaystyle=-p\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,\sigma_{\tilde{M}}(u_{1}^{\epsilon})\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\bigg{]}\,(u_{1}^{\epsilon})^{p-1}\,dx\,dz$
$\displaystyle+p\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,(u_{1}^{\epsilon})^{p-1}\,dx\,dz$
(3.14)

Since the last term on the left-hand side is positive and the first term on
the right-hand side is negative, Eq. (3) reduces to:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\frac{\partial}{\partial
t}(u_{1}^{\epsilon})^{p}\,dx\,dz+\epsilon^{2}\,p\,(p-1)\displaystyle\int_{\Pi_{\epsilon}}D_{1}\,(u_{1}^{\epsilon})^{p-2}\,|\nabla_{x}u_{1}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle+p\,(p-1)\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{1}\,(u_{1}^{\epsilon})^{p-2}\,|\partial_{z}u_{1}^{\epsilon}|^{2}\,dx\,dz\leq
p\displaystyle\int_{\Pi_{\epsilon}}f^{\epsilon}(t,x,z)\,(u_{1}^{\epsilon})^{p-1}\,dx\,dz$
(3.15)

Taking into account that the last two terms on the left-hand side are positive
and integrating over $[0,t]$ with $t\in[0,T]$, we obtain:

$\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}\frac{\partial}{\partial
s}(u_{1}^{\epsilon})^{p}\,ds\,dx\,dz\leq
p\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}f^{\epsilon}(s,x,z)\,(u_{1}^{\epsilon})^{p-1}\,ds\,dx\,dz$
(3.16)

Hence,

$\displaystyle\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon})^{p}\,dx\,dz\leq\displaystyle\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon}(0))^{p}\,dx\,dz+p\,\|f^{\epsilon}\|_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon})^{p-1}\,ds\,dx\,dz$
(3.17)

In order to estimate the last term on the right-hand side of Eq. (3.17), it is
now convenient to use Young’s inequality in the following form [7]:

$a\,b\leq\eta\,a^{p^{\prime}}+\eta^{1-p}\,b^{p},\;\;\;\forall a\geq 0,\;b\geq
0,\;\;p^{\prime}=\frac{p}{p-1}$ (3.18)

with: $a=(u_{1}^{\epsilon})^{p-1}$,
$b=p\,\|f^{\epsilon}\|_{L^{\infty}([0,T]\times\Pi_{\epsilon})}$ and for an
arbitrary $\eta>0$. Therefore, we get:

$\displaystyle
p\,\|f^{\epsilon}\|_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon})^{p-1}\,ds\,dx\,dz$
$\displaystyle\leq\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}p^{p}\,\|f^{\epsilon}\|^{p}_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\,\eta^{1-p}\,dxdz+\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}\eta\,(u_{1}^{\epsilon})^{p}\,dx\,dz$
$\displaystyle\leq
p^{p-1}\,\|f^{\epsilon}\|^{p}_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\,\eta^{1-p}\,|\Pi_{\epsilon}|\,T+\eta\,\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon})^{p}\,dx\,dz$
(3.19)

Choosing $\eta=p$ and using the inequality (3) in (3.17), we conclude that:

$\|u_{1}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}\leq\|U_{1}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}+\|f^{\epsilon}\|^{p}_{L^{\infty}([0,T]\times\Pi_{\epsilon})}|\Pi_{\epsilon}|\,T+p\displaystyle\int_{0}^{t}ds\,\|u_{1}^{\epsilon}(s)\|^{p}_{L^{p}(\Pi_{\epsilon})}$
(3.20)

Applying Gronwall’s inequality it follows that:

$\|u_{1}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}\leq\bigg{[}\|U_{1}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}+\|f^{\epsilon}\|^{p}_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\,|\Pi_{\epsilon}|\,T\bigg{]}\,e^{pt}$
(3.21)

And finally

$\sup_{t\in[0,T]}\lim_{p\rightarrow\infty}\bigg{[}\displaystyle\int_{\Pi_{\epsilon}}(u_{1}^{\epsilon})^{p}\,dx\,dz\bigg{]}^{1/p}\leq\bigg{[}\|U_{1}^{\epsilon}\|_{L^{\infty}(\Pi_{\epsilon})}+\|f^{\epsilon}\|_{L^{\infty}([0,T]\times\Pi_{\epsilon})}\bigg{]}\,e^{T}\leq
M_{1}$ (3.22)

where $M_{1}$ is a positive constant due to the boundedness of the initial
condition $U_{1}^{\epsilon}(x,z)$ and of the source term
$f^{\epsilon}(t,x,z)$.

We test now Eq. (2.1) with
$\phi_{1}=(v_{1}^{\epsilon}(t,\cdot)-\overline{M}_{1})^{+}$, where
$\overline{M}_{1}$ is a positive constant:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}(v_{1}^{\epsilon}-\overline{M}_{1})\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}\nabla(v_{1}^{\epsilon}-\overline{M}_{1})\cdot\nabla(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\,dx\,dz$
$\displaystyle-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle=\displaystyle\int_{\Omega_{\epsilon}}N_{1}^{\tilde{M}}(v^{\epsilon})\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\,dx\,dz$
(3.23)

Decomposing the function $(v_{1}^{\epsilon}-\overline{M}_{1})$ in its positive
and negative parts, one obtains:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}|^{2}\,dx\,dz+\displaystyle\int_{\Omega_{\epsilon}}d_{1}|\nabla(v_{1}^{\epsilon}-\overline{M}_{1})^{+}|^{2}\,dx\,dz$
$\displaystyle=\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle-\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,\sigma_{\tilde{M}}(v_{1}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\bigg{]}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\,dx\,dz$
(3.24)

Since the last term on the right-hand side is negative, we conclude:

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}+d_{1}\|\nabla(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
(3.25)

Let us now estimate the term on the right-hand side of (3):

$\displaystyle\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z){\mathcal{H}}(u_{1}^{\epsilon}-v_{1}^{\epsilon})\,(u_{1}^{\epsilon}-M_{1})\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle-\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z){\mathcal{H}}(u_{1}^{\epsilon}-v_{1}^{\epsilon})\,(v_{1}^{\epsilon}-M_{1})\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z){\mathcal{H}}(u_{1}^{\epsilon}-v_{1}^{\epsilon})\,(u_{1}^{\epsilon}-M_{1})^{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
(3.26)

if one chooses $M_{1}\leq\overline{M}_{1}$. By applying the Hölder and Young
inequalities, along with (A.7), one gets:

$\displaystyle\epsilon\displaystyle\int_{\Gamma_{\epsilon}}c_{1}(x,z)(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,(v_{1}^{\epsilon}-\overline{M}_{1})^{+}d\sigma_{\epsilon}$
$\displaystyle\leq\frac{\displaystyle\epsilon}{\displaystyle
2}\|c_{1}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\bigg{[}\|(u_{1}^{\epsilon}-M_{1})^{+}\|^{2}_{L^{2}(\Gamma_{\epsilon})}+\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Gamma_{\epsilon})}\bigg{]}$
$\displaystyle\leq
C_{1}\,\epsilon\,\eta\,\|\nabla(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega{\epsilon})}+C_{2}\,\epsilon\,\eta^{-1}\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega{\epsilon})}$
(3.27)

where $\eta$ is a small constant and $u_{1}^{\epsilon}$ satisfies the
inequality (3.22). Therefore, from Eq. (3):

$\displaystyle\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}+(d_{1}-C_{1}\epsilon\eta)\|\nabla(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle\leq
C_{2}\,\epsilon\,\eta^{-1}\,\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
(3.28)

If we choose $\eta<\frac{d_{1}}{\epsilon\,C_{1}}$, Eq. (3) reduces to:

$\frac{\displaystyle 1}{\displaystyle
2}\partial_{t}\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq
C_{2}\,\epsilon\,\eta^{-1}\,\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
(3.29)

and the Gronwall lemma gives:

$\|(v_{1}^{\epsilon}(t,\cdot)-\overline{M}_{1})^{+}\|^{2}_{L^{2}(\Omega_{\epsilon})}\leq\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}(0)\|^{2}_{L^{2}(\Omega_{\epsilon})}\,\exp(2\,C_{2}\,\epsilon\,\eta^{-1}\,T)$
(3.30)

for all $t\in[0,T]$. Since $v_{1}^{\epsilon}(0)=0$, then:

$\|(v_{1}^{\epsilon}-\overline{M}_{1})^{+}(0)\|^{2}_{L^{2}(\Omega_{\epsilon})}=0$

and from Eq. (3.30) it follows:

$v_{1}^{\epsilon}\leq\overline{M}_{1}.$ (3.31)

In the case $1<i\leq M$ we proceed by induction and test Eq. (2.1) with
$\psi_{i}=p(u_{i}^{\epsilon})^{p-1}(p\geq 2)$:

$\displaystyle
p\displaystyle\int_{\Pi_{\epsilon}}\partial_{t}u_{i}^{\epsilon}\,(u_{i}^{\epsilon})^{p-1}\,dx\,dz+\epsilon^{2}\,p\displaystyle\int_{\Pi_{\epsilon}}D_{i}\,\nabla_{x}u_{i}^{\epsilon}\cdot\nabla_{x}(u_{i}^{\epsilon})^{p-1}\,dx\,dz$
$\displaystyle+p\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{i}\,\partial_{z}u_{i}^{\epsilon}\cdot\partial_{z}(u_{i}^{\epsilon})^{p-1}\,dx\,dz+\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,(u_{i}^{\epsilon})^{p-1}d\sigma_{\epsilon}$
$\displaystyle=-p\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{M}a_{i,j}\,\sigma_{\tilde{M}}(u_{i}^{\epsilon})\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,dx\,dz$
$\displaystyle+\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\,\sigma_{\tilde{M}}(u_{i-j}^{\epsilon})\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,dx\,dz$
(3.32)

Since the last term on the left-hand side is positive and the first term on
the right-hand side is negative, Eq. (3) reduces to:

$\displaystyle\displaystyle\int_{\Pi_{\epsilon}}\frac{\partial}{\partial
t}(u_{i}^{\epsilon})^{p}\,dx\,dz+\epsilon^{2}\,p\,(p-1)\displaystyle\int_{\Pi_{\epsilon}}D_{i}\,(u_{i}^{\epsilon})^{p-2}\,|\nabla_{x}u_{i}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle+p\,(p-1)\displaystyle\int_{\Pi_{\epsilon}}\tilde{D}_{i}\,(u_{i}^{\epsilon})^{p-2}\,|\partial_{z}u_{i}^{\epsilon}|^{2}\,dx\,dz$
$\displaystyle\leq\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\,\sigma_{\tilde{M}}(u_{i-j}^{\epsilon})\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,dx\,dz$
(3.33)

Taking into account that the last two terms on the left-hand side are positive
and integrating over $[0,t]$ with $t\in[0,T]$, we obtain:

$\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}\frac{\partial}{\partial
s}(u_{i}^{\epsilon})^{p}\,ds\,dx\,dz\leq\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,\sigma_{\tilde{M}}(u_{j}^{\epsilon})\,\sigma_{\tilde{M}}(u_{i-j}^{\epsilon})\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,ds\,dx\,dz$
(3.34)

Exploiting the boundedness of $u_{j}^{\epsilon}(1\leq j\leq i-1)$ in
$L^{\infty}(0,T;L^{\infty}(\Pi_{\epsilon}))$ and setting the initial
conditions, one gets:

$\displaystyle\int_{\Pi_{\epsilon}}(u_{i}^{\epsilon})^{p}\,dx\,dz\leq\frac{\displaystyle
p}{\displaystyle
2}\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,ds\,dx\,dz$
(3.35)

where $K_{j}(1\leq j\leq i-1)$ are positive constants. In order to estimate
the term on the right-hand side of Eq. (3.35), we use the Young inequality
(3.18) with: $a=(u_{i}^{\epsilon})^{p-1}$ and
$b=p\,[\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}]$. We find:

$\displaystyle
p\,\displaystyle\int_{0}^{t}\int_{\Pi_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,(u_{i}^{\epsilon})^{p-1}\,ds\,dx\,dz$
$\displaystyle\leq\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}p^{p}\,\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,\eta^{1-p}\,dxdz+\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}\eta\,(u_{i}^{\epsilon})^{p}\,dx\,dz$
$\displaystyle\leq
p^{p-1}\,\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,\eta^{1-p}\,|\Pi_{\epsilon}|\,T+\eta\,\displaystyle\int_{0}^{t}ds\,\int_{\Pi_{\epsilon}}(u_{i}^{\epsilon})^{p}\,dx\,dz$
(3.36)

Choosing $\eta=p$ and using the inequality (3) in (3.35), we obtain:

$\|u_{i}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}\leq\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,|\Pi_{\epsilon}|\,T+p\displaystyle\int_{0}^{t}ds\,\|u_{i}^{\epsilon}(s)\|^{p}_{L^{p}(\Pi_{\epsilon})}$
(3.37)

The Gronwall lemma applied to (3.37) leads to the estimate:

$\|u_{i}^{\epsilon}\|^{p}_{L^{p}(\Pi_{\epsilon})}\leq\bigg{[}\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,|\Pi_{\epsilon}|\,T\,e^{pt}$
(3.38)

Hence,

$\sup_{t\in[0,T]}\lim_{p\rightarrow\infty}\bigg{[}\displaystyle\int_{\Pi_{\epsilon}}(u_{i}^{\epsilon})^{p}\,dx\,dz\bigg{]}^{1/p}\leq\sum_{j=1}^{i-1}a_{j,i-j}\,K_{j}\,K_{i-j}\,e^{T}\leq
M_{i}$ (3.39)

where $M_{i}$ is a positive constant.

We test now Eq. (2.1) with
$\phi_{i}=p[(v_{i}^{\epsilon}(t,\cdot)-\overline{m}_{i})^{+}]^{p-1}$ ($p\geq
2$), where $\overline{m}_{i}$ is a positive constant:

$\displaystyle
p\,\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}(v_{i}^{\epsilon}-\overline{m}_{i})\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
$\displaystyle+p\,\displaystyle\int_{\Omega_{\epsilon}}d_{i}\nabla(v_{i}^{\epsilon}-\overline{m}_{i})\cdot\nabla[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
$\displaystyle-\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle=-p\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{i,j}\,\sigma_{\tilde{M}}(v_{i}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
$\displaystyle+\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\,\sigma_{\tilde{M}}(v_{i-j}^{\epsilon})\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
(3.40)

Decomposing the function $(v_{i}^{\epsilon}-\overline{m}_{i})$ in its positive
and negative parts and taking into account the boundedness of
$v_{j}^{\epsilon}\,(1\leq j\leq i-1)$ in
$L^{\infty}(0,T;L^{\infty}(\Omega_{\epsilon}))$ due to the inductive
hypothesis, one obtains:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,dx\,dz+p\,(p-1)\displaystyle\int_{\Omega_{\epsilon}}d_{i}\,|\nabla(v_{i}^{\epsilon}-\overline{m}_{i})^{+}|^{2}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-2}\,dx\,dz$
$\displaystyle\leq\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle+\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
$\displaystyle-p\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{M}b_{i,j}\,\sigma_{\tilde{M}}(v_{i}^{\epsilon})\,\sigma_{\tilde{M}}(v_{j}^{\epsilon})\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
(3.41)

where $K_{j}(1\leq j\leq i-1)$ are positive constants. Since the last term on
the left-hand side is positive and the one on the right-hand side is negative,
Eq. (3) reduces to:

$\displaystyle\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,dx\,dz\leq\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-v_{i}^{\epsilon})_{+}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle+\frac{\displaystyle p}{\displaystyle
2}\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
(3.42)

Let us now estimate the first term, $I_{1}$, on the right-hand side of Eq.
(3):

$\displaystyle
I_{1}=\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z){\mathcal{H}}(u_{i}^{\epsilon}-v_{i}^{\epsilon})\,(u_{i}^{\epsilon}-v_{i}^{\epsilon})[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z){\mathcal{H}}(u_{i}^{\epsilon}-v_{i}^{\epsilon})\,(u_{i}^{\epsilon}-\overline{m}_{i})\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle-\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z){\mathcal{H}}(u_{i}^{\epsilon}-v_{i}^{\epsilon})\,(v_{i}^{\epsilon}-\overline{m}_{i})\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
$\displaystyle\leq\epsilon\,p\displaystyle\int_{\Gamma_{\epsilon}}c_{i}(x,z)(u_{i}^{\epsilon}-\overline{m}_{i})^{+}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}d\sigma_{\epsilon}$
(3.43)

where the last step in inequality (3 ) has been obtained decomposing the
functions $(u_{i}^{\epsilon}-\overline{m}_{i})$ and
$(v_{i}^{\epsilon}-\overline{m}_{i})$ in positive and negative parts. By
applying the Hölder inequality and exploiting the generalized interpolation-
trace inequality (A.7) along with the the boundedness of $u_{i}^{\epsilon}$ in
$L^{\infty}(0,T;L^{\infty}(\Pi_{\epsilon}))$, one concludes that:

$0<I_{1}\leq\epsilon\,p\,\|c_{i}(x,z)\|_{L^{\infty}(\Gamma_{\epsilon})}\,\|(u_{i}^{\epsilon}-\overline{m}_{i})^{+}\|_{L^{2}(\Gamma_{\epsilon})}\,\|[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\|_{L^{2}(\Gamma_{\epsilon})}=0$
(3.44)

Hence, $I_{1}=0$.

Therefore, Eq. (3) reduces to:

$\displaystyle\int_{\Omega_{\epsilon}}\partial_{t}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,dx\,dz\leq
p\,\displaystyle\int_{\Omega_{\epsilon}}\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}\,dx\,dz$
(3.45)

Integrating over $[0,t]$ with $t\in[0,T]$, and estimating the term on the
right-hand side by using Young’s inequality in the form (3.18), with:
$a=[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p-1}$ and
$b=p\,\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}$, we get

$\displaystyle\displaystyle\int_{0}^{t}\int_{\Omega_{\epsilon}}\frac{\partial}{\partial
s}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,ds\,dx\,dz$
$\displaystyle\leq
p^{p-1}\,\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,\eta^{1-p}\,|\Omega_{\epsilon}|\,T+\eta\,\displaystyle\int_{0}^{t}\int_{\Omega_{\epsilon}}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,ds\,dx\,dz$

Choosing $\eta=p$ and setting the initial conditions, it follows that:

$\displaystyle\|(v_{i}^{\epsilon}-\overline{m}_{i})^{+}\|^{p}_{L^{p}(\Omega_{\epsilon})}\leq\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,|\Omega_{\epsilon}|\,T$
$\displaystyle+p\,\displaystyle\int_{0}^{t}ds\,\|(v_{i}^{\epsilon}-\overline{m}_{i})^{+}\|^{p}_{L^{p}(\Omega_{\epsilon})}$
(3.46)

Finally, the Gronwall Lemma leads to the estimate

$\|(v_{i}^{\epsilon}-\overline{m}_{i})^{+}\|^{p}_{L^{p}(\Omega_{\epsilon})}\leq\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}^{p}\,|\Omega_{\epsilon}|\,T\,e^{p\,t}$
(3.47)

Hence,

$\sup_{t\in[0,T]}\lim_{p\rightarrow\infty}\bigg{[}\displaystyle\int_{\Omega_{\epsilon}}[(v_{i}^{\epsilon}-\overline{m}_{i})^{+}]^{p}\,dx\,dz\bigg{]}^{1/p}\leq\bigg{[}\sum_{j=1}^{i-1}b_{j,i-j}\,K_{j}\,K_{i-j}\bigg{]}\,e^{T}$
(3.48)

Therefore, since the positive part of the function
$(v_{i}^{\epsilon}-\overline{m}_{i})$ is bounded, it follows that:
$v_{i}^{\epsilon}(t,x,z)\leq\overline{M}_{i}$, where $\overline{M}_{i}$ is a
positive constant.

∎

######  Lemma 3.2.

For a given small $\epsilon>0$, let $u_{i}^{\epsilon}(t,x,z)$ and
$v_{i}^{\epsilon}(t,x,z)$ ($1\leq i\leq M$) be solutions of the system (1.1)
in the sense of the Definition 2.1. Then, the following estimates hold:

$\epsilon\|\nabla_{x}u_{i}^{\epsilon}\|_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}\leq
C_{i}^{x}$ (3.49)

$\|\partial_{z}u_{i}^{\epsilon}\|_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}\leq
C_{i}^{z}$ (3.50)

$\|\partial_{t}u_{i}^{\epsilon}\|_{L^{2}(0,T;L^{2}(\Pi_{\epsilon}))}\leq
C_{i}^{t}$ (3.51)

$\|\nabla v_{i}^{\epsilon}\|_{L^{\infty}(0,T;L^{2}(\Omega_{\epsilon}))}\leq
D_{i}$ (3.52)

$\|\partial_{t}v_{i}^{\epsilon}\|_{L^{2}(0,T;L^{2}(\Omega_{\epsilon}))}\leq
D_{i}^{t}$ (3.53)

where $1\leq i\leq M$ and $C_{i}^{x},C_{i}^{z},C_{i}^{t},D_{i},D_{i}^{t}$ are
positive constants independent of $\tilde{M}$ and $\epsilon$.

###### Proof.

In the case $i=1$, let $u_{1,n}^{\epsilon}$ and $v_{1,n}^{\epsilon}$ be the
approximate solutions defined in the proof of Lemma 2.1. Then the inequality
(2) holds with a constant $\tilde{C}\geq 0$ independent of $n$ and $\epsilon$.
Using the convergence results reported in Proposition 2.1 and the lower-
semicontinuity of the norm from Theorem B.2 [12] (see Appendix B), we get from
Eq. (2):

$\displaystyle\|\partial_{t}u_{1}^{\epsilon}\|^{2}_{L^{2}(0,T;L^{2}(\Pi_{\epsilon}))}+2\,\|\partial_{t}v_{1}^{\epsilon}\|^{2}_{L^{2}(0,T;L^{2}(\Omega_{\epsilon}))}+2\,\epsilon^{2}\,D_{1}\,\|\nabla_{x}u_{1}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}$
$\displaystyle+2\,\tilde{D}_{1}\,\|\partial_{z}u_{1}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}+2\,d_{1}\,\|\nabla
v_{1}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega_{\epsilon}))}$
$\displaystyle\leq\lim\inf_{n\rightarrow\infty}\|\partial_{t}u_{1,n}^{\epsilon}\|^{2}_{L^{2}(0,T;L^{2}(\Pi_{\epsilon}))}+\lim\inf_{n\rightarrow\infty}2\,\|\partial_{t}v_{1,n}^{\epsilon}\|^{2}_{L^{2}(0,T;L^{2}(\Omega_{\epsilon}))}$
$\displaystyle+\lim\inf_{n\rightarrow\infty}2\,\epsilon^{2}\,D_{1}\,\|\nabla_{x}u_{1,n}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}+\lim\inf_{n\rightarrow\infty}2\,\tilde{D}_{1}\,\|\partial_{z}u_{1,n}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Pi_{\epsilon}))}$
$\displaystyle+\lim\inf_{n\rightarrow\infty}2\,d_{1}\,\|\nabla
v_{1,n}^{\epsilon}\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega_{\epsilon}))}\leq\tilde{C}$
(3.54)

For the case $1<i\leq M$, the proof carries over verbatim, since also the
functions $u_{i,n}^{\epsilon}$ and $v_{i,n}^{\epsilon}$ ($1<i\leq M$) satisfy
the inequality (2).

∎

##  4 Homogenization

The behavior of the solutions $u_{i}^{\epsilon},v_{i}^{\epsilon}$ $(1\leq
i\leq M)$ of the set of Eqs. (1.1) as $\epsilon\rightarrow 0$ will now be
studied. In order to pass to the limit, it is necessary to obtain equations
and estimates in $\Omega$.

######  Lemma 4.1.

Let us consider the sets defined in Section 1.

1\. There exists a linear continuous extension operator

${\tilde{P}}:H^{1}(X)\rightarrow H^{1}(Y)$ (4.1)

such that

${\tilde{P}}u=u\quad\text{in}\;X$ (4.2)

and

$\|{\tilde{P}}\,u\|_{L^{2}(Y)}\leq C\|u\|_{L^{2}(X)}$ (4.3)

$\|\nabla({\tilde{P}}\,u)\|_{L^{2}(Y)}\leq C\|\nabla u\|_{L^{2}(X)}$ (4.4)

where $C$ is a positive constant.

2\. There exists a family of linear continuous extension operators

${\tilde{P}}_{\epsilon}:H^{1}(G_{\epsilon})\rightarrow H^{1}(D)$ (4.5)

such that

${\tilde{P}}_{\epsilon}u^{\epsilon}=u^{\epsilon}\quad\text{in}\;G_{\epsilon}$
(4.6)

and

$\|{\tilde{P}}_{\epsilon}u^{\epsilon}\|_{L^{2}(D)}\leq
C\|u^{\epsilon}\|_{L^{2}(G_{\epsilon})}$ (4.7)

$\|\nabla({\tilde{P}}_{\epsilon}u^{\epsilon})\|_{L^{2}(D)}\leq C\|\nabla
u^{\epsilon}\|_{L^{2}(G_{\epsilon})}$ (4.8)

where the constant $C>0$ does not depend on $\epsilon$.

###### Proof.

1\. First we extend $u$ into a neighbourhood $X_{0}$ of $R=\partial X$ with
smooth boundary, such that $\text{clos}(X)\subset X_{0}$ and
$\text{clos}(X_{0})\subset\text{int}(Y)$. Since we have assumed that $R$ is
sufficiently smooth, we can construct a diffeomorphism as follows:

$\displaystyle\Phi:R\times]-\delta,\delta[\rightarrow X_{0}$ (4.9)
$\displaystyle\Phi(y,\lambda)=x$ (4.10)

Exploiting this coordinate transformation, the function $u$ can be extended by
reflection:

$u^{*}(x)=u^{*}(\Phi(y,\lambda))=\begin{cases}u(\Phi(y,\lambda))&\lambda\geq
0\\\ u(\Phi(y,-\lambda))&\lambda<0\end{cases}$ (4.11)

Let us now consider the following smooth function

$\Psi:Y\rightarrow[0,1]$ (4.12)

such that $\text{supp}\Psi\subseteq Z$ and $\Psi=1$ in $Z\setminus X_{0}$.
Then, we define

$\tilde{u}(x):=(1-\Psi)(u^{*}(x)-m)+m$ (4.13)

where

$m:=\frac{\displaystyle 1}{\displaystyle|X|}\displaystyle\int_{X}u(y)\,dy$
(4.14)

Let us prove that $\tilde{u}(x)$ is an extension of $u(x)$, which satisfies
(4.3) and (4.4). One gets:

$\displaystyle\|\tilde{u}\|^{2}_{L^{2}(Z)}=\displaystyle\int_{Z}|(1-\Psi)(u^{*}-m)+m|^{2}\,dx$
$\displaystyle=\displaystyle\int_{Z\cap X_{0}}|(1-\Psi)u^{*}+\Psi
m|^{2}\,dx+\displaystyle\int_{Z\setminus X_{0}}m^{2}\,dx$ (4.15)

Taking into account the following estimates

$\displaystyle\int_{Z\cap X_{0}}\bigg{[}\frac{\displaystyle
1}{\displaystyle|X|}\displaystyle\int_{X}u(y)\,dy\bigg{]}^{2}\,dx\leq\displaystyle\int_{Z\cap
X_{0}}\frac{\displaystyle
1}{\displaystyle|X|}\bigg{(}\displaystyle\int_{X}u^{2}\,dy\bigg{)}\,dx\leq\frac{\displaystyle|Z|}{\displaystyle|X|}\|u\|^{2}_{L^{2}(X)}$
(4.16)

$\displaystyle\int_{Z\setminus X_{0}}\bigg{[}\frac{\displaystyle
1}{\displaystyle|X|}\displaystyle\int_{X}u(y)\,dy\bigg{]}^{2}\,dx\leq\displaystyle\int_{Z\setminus
X_{0}}\frac{\displaystyle
1}{\displaystyle|X|}\bigg{(}\displaystyle\int_{X}u^{2}\,dy\bigg{)}\,dx\leq\frac{\displaystyle|Z|}{\displaystyle|X|}\|u\|^{2}_{L^{2}(X)}$
(4.17)

and the Minkowski inequality, Eq. (4) can be rewritten as

$\displaystyle\|\tilde{u}\|^{2}_{L^{2}(Z)}\leq 2\displaystyle\int_{Z\cap
X_{0}}(1-\Psi)^{2}\,|u^{*}|^{2}\,dx+2\displaystyle\int_{Z\cap
X_{0}}\Psi^{2}\,m^{2}\,dx+\frac{\displaystyle|Z|}{\displaystyle|X|}\|u\|^{2}_{L^{2}(X)}$
$\displaystyle\leq C(Z,X,\Psi)\|u\|^{2}_{L^{2}(X)}$ (4.18)

For the derivative we obtain:

$\displaystyle\|\nabla\tilde{u}\|^{2}_{L^{2}(Z)}=\displaystyle\int_{Z}|\nabla[(1-\Psi)(u^{*}-m)]|^{2}\,dx$
$\displaystyle=\displaystyle\int_{Z}|(u^{*}-m)\nabla(1-\Psi)+(1-\Psi)\nabla
u^{*}|^{2}\,dx$ (4.19)

By using the Minkowski and Poincaré inequalities, Eq. (4) becomes:

$\displaystyle\|\nabla\tilde{u}\|^{2}_{L^{2}(Z)}\leq 2\displaystyle\int_{Z\cap
X_{0}}|(u^{*}-m)\nabla(1-\Psi)|^{2}\,dx+2\displaystyle\int_{Z\cap
X_{0}}|(1-\Psi)\nabla u^{*}|^{2}\,dx$ $\displaystyle\leq
c\displaystyle\int_{Z\cap
X_{0}}|(u^{*}-m)|^{2}\,dx+c^{*}\displaystyle\int_{Z\cap X_{0}}|\nabla
u^{*}|^{2}\,dx\leq C_{1}(\Psi)\|\nabla u\|^{2}_{L^{2}(X)}$ (4.20)

2\. The construction of $\tilde{P}_{\epsilon}$ is obvious by summation over
the individual cells.

∎

######  Lemma 4.2.

Let us consider the sets defined in Section 1.

1\. There exists a linear continuous extension operator

${P}:H^{1}(Z)\rightarrow H^{1}(Y)$ (4.21)

such that

${P}v=v\quad\text{in}\;Z$ (4.22)

and

$\|{P}\,v\|_{L^{2}(Y)}\leq C\|v\|_{L^{2}(Z)}$ (4.23)

$\|\nabla({P}\,v)\|_{L^{2}(Y)}\leq C\|\nabla v\|_{L^{2}(Z)}$ (4.24)

where $C$ is a positive constant.

2\. There exists a family of linear continuous extension operators

${P}_{\epsilon}:H^{1}(D_{\epsilon})\rightarrow H^{1}(D)$ (4.25)

such that

${P}_{\epsilon}v^{\epsilon}=v^{\epsilon}\quad\text{in}\;D_{\epsilon}$ (4.26)

and

$\|{P}_{\epsilon}v^{\epsilon}\|_{L^{2}(D)}\leq
C\|v^{\epsilon}\|_{L^{2}(D_{\epsilon})}$ (4.27)

$\|\nabla({P}_{\epsilon}v^{\epsilon})\|_{L^{2}(D)}\leq C\|\nabla
v^{\epsilon}\|_{L^{2}(D_{\epsilon})}$ (4.28)

where the constant $C>0$ does not depend on $\epsilon$.

###### Proof.

This Lemma can be proved by applying the same arguments considered in the
proof of Lemma 4.1 and in Ref. [33] (p. $25$).

∎

######  Remark 4.1.

Analogous extension theorems hold also for $u_{i}^{\epsilon}(t,x,z)$, defined
in $[0,T]\times\Pi_{\epsilon}$, and $v_{i}^{\epsilon}(t,x,z)$, defined in
$[0,T]\times\Omega_{\epsilon}$ ($1\leq i\leq M$), since $z$ and $t$ can be
considered as parameters, because of the geometry of the domain.

From now on, we identify $v_{i}^{\epsilon}$ with its extension
$P_{\epsilon}v_{i}^{\epsilon}$ according to Lemma 4.2 and $u_{i}^{\epsilon}$
with its extension ${\tilde{P}}_{\epsilon}u_{i}^{\epsilon}$ according to Lemma
4.1. In order to study the limiting behavior of the set of Eqs. (1.1), we use
the notion of two-scale convergence. Some definitions and results on two-scale
convergence, introduced in Refs. [35], [3], are reported in Appendix C.

######  Proposition 4.1.

Let $v_{i}^{\epsilon}$ ($1\leq i\leq M$) be the extension of the solutions to
the system (1.1). Then, up to a subsequence:

$v_{i}^{\epsilon}(t,x,z)\rightharpoonup v_{i}(t,x,z)\;\;\;\text{ weakly \,
in}\;\;\;L^{2}(0,T;H^{1}(\Omega))$ (4.29)

$\partial_{t}v_{i}^{\epsilon}(t,x,z)\rightharpoonup\partial_{t}v_{i}(t,x,z)\;\;\;\text{
weakly \, in}\;\;\;L^{2}([0,T]\times\Omega)$ (4.30)

$v_{i}^{\epsilon}(t,x,z)\rightarrow v_{i}(t,x,z)\;\;\;\text{ strongly \,
in}\;\;\;C^{0}([0,T];L^{2}(\Omega))$ (4.31)

###### Proof.

The convergence results (4.29)-(4.31) follow immediately from the a priori
estimates given in Lemma 3.2.

∎

######  Remark 4.2.

Since $v_{i}^{\epsilon}(t,x,z)$ converges weakly to $v_{i}(t,x,z)$ ($1\leq
i\leq M$) in $L^{2}(0,T;H^{1}(\Omega))\cap H^{1}(0,T;L^{2}(\Omega))$, Theorem
C.5 in Appendix C implies the two-scale convergence to the same
$v_{i}(t,x,z)$, and there exists a function ${\tilde{v}}_{i}\in
L^{2}([0,T]\times\Omega;H^{1}_{\sharp}(Y)/\mathbb{R})$ such that up to a
subsequence $\nabla_{x}v_{i}^{\epsilon}(t,x,z)$ two-scale converges to
$\nabla_{x}v_{i}(t,x,z)+\nabla_{y}\tilde{v}_{i}(t,x,y,z)$ (where $y\in Y$ is
the microscopic variable).

Moreover, the interpolation-trace inequality (A.7) in Appendix A and Theorem
C.7 in Appendix C allow one to infer the two-scale convergence of
$v_{i}^{\epsilon}$ on the boundary $\Gamma_{\epsilon}$.

Next we prove the convergence of $u_{i}^{\epsilon}$ ($1\leq i\leq M$).

######  Proposition 4.2.

Let $u_{i}^{\epsilon}(t,x,z)$ ($1\leq i\leq M$) be the extension of the
solutions to the system (1.1). Then, up to a subsequence:

$\displaystyle u_{i}^{\epsilon}(t,x,z)\rightarrow{u}_{i}(t,x,y,z)$
$\displaystyle\partial_{t}u_{i}^{\epsilon}(t,x,z)\rightarrow\partial_{t}{u}_{i}(t,x,y,z)$
$\displaystyle\partial_{z}u_{i}^{\epsilon}(t,x,z)\rightarrow\partial_{z}{u}_{i}(t,x,y,z)$
$\displaystyle\epsilon\nabla_{x}u_{i}^{\epsilon}(t,x,z)\rightarrow\nabla_{y}{u}_{i}(t,x,y,z)$
(4.32)

in the two-scale sense with ${u}_{i}(t,x,y,z)\in
L^{2}([0,T]\times\Omega;H^{1}_{\sharp}(Y))\cap H^{1}(0,T;L^{2}(\Omega\times
Y))$ (where $y\in Y$ is the microscopic variable).

###### Proof.

The convergence results (4.2) follow immediately from the a priori estimates
given in Lemma 3.2 and Theorem C.6 in Appendix C.

∎

Proof of the main Theorem 1.1

In the case $m=1$, let us rewrite Eq. (2.1) in the form:

$\begin{split}&\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}v_{1}^{\epsilon}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\phi_{1}\,dt\,dx\,dz+d_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\nabla_{x}v_{1}^{\epsilon}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\nabla_{x}\phi_{1}\,dt\,dx\,dz\\\
&+d_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{z}v_{1}^{\epsilon}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\partial_{z}\phi_{1}\,dt\,dx\,dz-\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,\phi_{1}\,dt\,d\sigma_{\epsilon}\\\
&=-\displaystyle\int_{0}^{T}\int_{\Omega}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,v_{1}^{\epsilon}\,v_{j}^{\epsilon}\bigg{]}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\phi_{1}\,dt\,dx\,dz\end{split}$ (4.33)

where $\chi(\frac{\displaystyle x}{\displaystyle\epsilon})$ is the
characteristic function of $\Omega_{\epsilon}$. Inserting in Eq. (4.33) the
following test function:

$\phi_{1}:=\phi_{1}^{0}(t,x,z)+\epsilon{\tilde{\phi}}_{1}\bigg{(}t,x,\frac{x}{\epsilon},z\bigg{)}$

where $\phi_{1}^{0}\in C^{1}([0,T]\times\overline{\Omega})$ and
$\tilde{\phi}_{1}\in C^{1}([0,T]\times\overline{\Omega};C_{\\#}^{\infty}(Y))$,
we obtain:

$\begin{split}&\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}v_{1}^{\epsilon}(t,x,z)\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,[\phi_{1}^{0}(t,x,z)+\epsilon\,\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)]\,dt\,dx\,dz\\\
&+d_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\nabla_{x}v_{1}^{\epsilon}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\bigg{[}\nabla_{x}\phi_{1}^{0}+\epsilon\nabla_{x}\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)+\nabla_{y}\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)\bigg{]}\,dt\,dx\,dz\\\
&+d_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{z}v_{1}^{\epsilon}(t,x,z)\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,[\partial_{z}\phi_{1}^{0}(t,x,z)+\epsilon\partial_{z}\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)]\,dt\,dx\,dz\\\
&-\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}(t,x,z)-v_{1}^{\epsilon}(t,x,z))_{+}\,[\phi_{1}^{0}+\epsilon\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)]\,dt\,d\sigma_{\epsilon}\\\
&=-\displaystyle\int_{0}^{T}\int_{\Omega}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,v_{1}^{\epsilon}(t,x,z)\,v_{j}^{\epsilon}(t,x,z)\bigg{]}\,\chi(\frac{\displaystyle
x}{\displaystyle\epsilon})\,[\phi_{1}^{0}+\epsilon\tilde{\phi}_{1}(t,x,\frac{x}{\epsilon},z)]\,dt\,dx\,dz\end{split}$
(4.34)

Passing to the two-scale limit we get

$\begin{split}&\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{Z}\partial_{t}v_{1}(t,x,z)\,\phi_{1}^{0}(t,x,z)\,d{y}\\\
&+d_{1}\,\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{Z}[\nabla_{x}v_{1}+\nabla_{y}\tilde{v}_{1}(t,x,{y},z)]\,[\nabla_{x}\phi_{1}^{0}+\nabla_{y}\tilde{\phi}_{1}(t,x,{y},z)]\,d{y}\\\
&+d_{1}\,\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{Z}\partial_{z}v_{1}(t,x,z)\,\partial_{z}\phi_{1}^{0}(t,x,z)\,d{y}\\\
&-\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{\Gamma}c_{1}(x,z)\,(u_{1}(t,x,{y},z)-v_{1}(t,x,z))_{+}\,\phi_{1}^{0}(t,x,z)\,d\sigma({y})\\\
&=-\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{Z}\bigg{[}\sum_{j=1}^{M}b_{1,j}\,v_{1}(t,x,z)\,v_{j}(t,x,z)\bigg{]}\,\phi_{1}^{0}(t,x,z)\,d{y}\end{split}$
(4.35)

An integration by parts shows that Eq. (4.35) is a variational formulation
associated with the following homogenized system:

$-div_{y}[d_{1}(\nabla_{x}v_{1}(t,x,z)+\nabla_{y}{\tilde{v}}_{1}(t,x,{y},z))]=0,\;\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;{y}\in
Z$ (4.36)
$[\nabla_{x}v_{1}(t,x,z)+\nabla_{y}{\tilde{v}}_{1}(t,x,{y},z)]\cdot\nu=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;{y}\in\Gamma$
(4.37) $\begin{split}&|Z|\frac{\displaystyle\partial
v_{1}}{\displaystyle\partial
t}(t,x,z)-div_{x}\bigg{[}d_{1}\displaystyle\int_{Z}d{y}\,(\nabla_{x}v_{1}(t,x,z)+\nabla_{y}\tilde{v}_{1}(t,x,{y},z))\bigg{]}\\\
&-d_{1}\,|Z|\,\partial_{z}^{2}v_{1}(t,x,z)+|Z|\sum_{j=1}^{M}b_{1,j}\,v_{1}(t,x,z)\,v_{j}(t,x,z)\\\
&=\displaystyle\int_{\Gamma}c_{1}(x,z)\,(u_{1}(t,x,{y},z)-v_{1}(t,x,z))_{+}\,d\sigma(y)\;\;\;\text{
in}\,\,[0,T]\times\Omega\end{split}$ (4.38)
$\bigg{[}\displaystyle\int_{Z}(\nabla_{x}v_{1}(t,x,z)+\nabla_{y}{\tilde{v}}_{1}(t,x,{y},z))\,d{y}\bigg{]}\cdot\nu=0\;\;\;\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
on}\,\,[0,T]\times\Gamma_{L}$ (4.39)
$\partial_{z}v_{1}(t,x,z)=0\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
on}\,\,[0,T]\times\Gamma_{B}$ (4.40)

By continuity we have that

$v_{1}(t=0,x,z)=0\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
in}\,\,\Omega.$ (4.41)

Since we have assumed that the diffusion coefficient is constant and we have
proved that the limiting function $v_{1}(t,x,z)$ does not depend on the
microscopic variable $y$, Eqs. (4.36) and (4.37) reduce to:

$\triangle_{y}{\tilde{v}}_{1}(t,x,{y},z)=0,\,\,\,\qquad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;{y}\in
Z$ (4.42)
$\nabla_{y}{\tilde{v}}_{1}(t,x,{y},z)\cdot\nu=-\nabla_{x}v_{1}(t,x,z)\cdot\nu,\,\,\,\qquad\;\;\;\;\;t>0,\;\;(x,z)\in\Omega,\;\;{y}\in\Gamma$
(4.43)

Then, ${\tilde{v}}_{1}(t,x,{y},z)$ satisfying Eqs. (4.42) and (4.43) can be
written as

${\tilde{v}}_{1}(t,x,{y},z)=\sum_{i=1}^{3}\,w_{i}({y})\,\frac{\displaystyle\partial
v_{1}}{\displaystyle\partial x_{i}}(t,x,z)$ (4.44)

where $(w_{i})_{1\leq i\leq 3}$ is the family of solutions of the cell problem

$\displaystyle\begin{cases}-\text{div}_{y}[\nabla_{y}w_{i}+\hat{e}_{i}]=0\,\,\,\qquad\;\;\;\;\;\text{
in}\,\,Z\\\
(\nabla_{y}w_{i}+\hat{e}_{i})\cdot\nu=0\,\,\,\qquad\;\;\;\;\;\;\;\;\text{
on}\,\,\Gamma\\\ y\rightarrow
w_{i}({y})\;\;\;\;\;Y-\text{periodic}\end{cases}$ (4.45)

with $\hat{e}_{i}$ being the $i$th unit vector in ${\mathbb{R}}^{3}$.

Inserting the relation (4.44) in Eqs. (4.38) and (4.39), we get

$\begin{split}&|Z|\,\frac{\displaystyle\partial v_{1}}{\displaystyle\partial
t}(t,x,z)-\text{div}_{x}\bigg{[}d_{1}\,A\,\nabla_{x}v_{1}(t,x,z)\bigg{]}\\\
&-d_{1}\,|Z|\,\partial_{z}^{2}v_{1}(t,x,z)+|Z|\sum_{j=1}^{M}b_{1,j}\,v_{1}(t,x,z)\,v_{j}(t,x,z)\\\
&=\displaystyle\int_{\Gamma}c_{1}(x,z)\,(u_{1}(t,x,{y},z)-v_{1}(t,x,z))_{+}\,d\sigma(y)\,\,\,\;\;\;\text{
in}\,\,[0,T]\times\Omega\end{split}$ (4.46)

$[A\,\nabla_{x}v_{1}(t,x,z)]\cdot\nu=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{
on}\,\,[0,T]\times\Gamma_{L}$ (4.47)

where $A$ is a matrix with constant coefficients defined by

$A_{ij}=\displaystyle\int_{Z}(\nabla_{y}w_{i}+\hat{e}_{i})\cdot(\nabla_{y}w_{j}+\hat{e}_{j})\,d{y}.$

Let us now rewrite Eq. (2.1) as follows:

$\begin{split}&\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}u_{1}^{\epsilon}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\psi_{1}\,dt\,dx\,dz+\epsilon^{2}\,D_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\nabla_{x}u_{1}^{\epsilon}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\nabla_{x}\psi_{1}\,dt\,dx\,dz\\\
&+{\tilde{D}}_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{z}u_{1}^{\epsilon}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\partial_{z}\psi_{1}\,dt\,dx\,dz+\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}-v_{1}^{\epsilon})_{+}\,\psi_{1}\,dt\,d\sigma_{\epsilon}\\\
&=-\displaystyle\int_{0}^{T}\int_{\Omega}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,u_{1}^{\epsilon}\,u_{j}^{\epsilon}\bigg{]}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\psi_{1}\,\,dt\,dx\,dz+\displaystyle\int_{0}^{T}\int_{\Omega}f^{\epsilon}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\psi_{1}\,\,dt\,dx\,dz\end{split}$ (4.48)

where ${\tilde{\chi}}(\frac{\displaystyle x}{\displaystyle\epsilon})$ is the
characteristic function of $\Pi_{\epsilon}$. If we choose as test function:

$\psi_{1}:=\tilde{\psi}_{1}\bigg{(}t,x,\frac{x}{\epsilon},z\bigg{)}$

where $\tilde{\psi}_{1}\in
C^{1}([0,T]\times\overline{\Omega};C_{\\#}^{\infty}(Y))$, Eq. (4.48) reads:

$\begin{split}&\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}u_{1}^{\epsilon}(t,x,z)\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\tilde{\psi}_{1}(t,x,\frac{x}{\epsilon},z)\,dt\,dx\,dz\\\
&+\epsilon^{2}\,D_{1}\,\displaystyle\int_{0}^{T}\int_{\Omega}\nabla_{x}u_{1}^{\epsilon}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\bigg{[}\nabla_{x}\tilde{\psi}_{1}(t,x,\frac{x}{\epsilon},z)+\frac{1}{\epsilon}\nabla_{y}\tilde{\psi}_{1}(t,x,\frac{x}{\epsilon},z)\bigg{]}\,dt\,dx\,dz\\\
&+{\tilde{D}}_{1}\,\,\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{z}u_{1}^{\epsilon}(t,x,z)\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,\partial_{z}\tilde{\psi}_{1}(t,x,\frac{x}{\epsilon},z)\,dt\,dx\,dz\\\
&+\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}c_{1}(x,z)\,(u_{1}^{\epsilon}(t,x,z)-v_{1}^{\epsilon}(t,x,z))_{+}\,{\tilde{\psi}}_{1}(t,x,\frac{x}{\epsilon},z)\,\,dt\,d\sigma_{\epsilon}(x,z)\\\
&=-\displaystyle\int_{0}^{T}\int_{\Omega}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,u_{1}^{\epsilon}(t,x,z)\,u_{j}^{\epsilon}(t,x,z)\bigg{]}\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,{\tilde{\psi}}_{1}(t,x,\frac{x}{\epsilon},z)\,dt\,dx\,dz\\\
&+\displaystyle\int_{0}^{T}\int_{\Omega}f^{\epsilon}(t,x,z)\,{\tilde{\chi}}(\frac{\displaystyle
x}{\displaystyle\epsilon})\,{\tilde{\psi}}_{1}(t,x,\frac{x}{\epsilon},z)\,dt\,dx\,dz\end{split}$
(4.49)

Passing to the two-scale limit we obtain:

$\begin{split}&\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{X}\partial_{t}u_{1}(t,x,{y},z)\,\tilde{\psi}_{1}(t,x,{y},z)\,d{y}\\\
&+D_{1}\,\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{X}\nabla_{y}u_{1}(t,x,{y},z)\,\nabla_{y}\tilde{\psi}_{1}(t,x,{y},z)\,d{y}\\\
&+\tilde{D}_{1}\,\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{X}\partial_{z}u_{1}(t,x,{y},z)\,\partial_{z}\tilde{\psi}_{1}(t,x,{y},z)\,d{y}\\\
&+\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{\Gamma}c_{1}(x,z)\,(u_{1}(t,x,{y},z)-v_{1}(t,x,z))_{+}\,\tilde{\psi}_{1}(t,x,{y},z)\,d\sigma({y})\\\
&=-\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{X}\bigg{[}\sum_{j=1}^{M}a_{1,j}\,u_{1}(t,x,{y},z)\,u_{j}(t,x,{y},z)\bigg{]}\,\tilde{\psi}_{1}(t,x,{y},z)\,d{y}\\\
&+\displaystyle\int_{0}^{T}\,dt\int_{\Omega}\,dx\,dz\int_{X}f(t,x,{y},z)\,\tilde{\psi}_{1}(t,x,{y},z)\,d{y}\end{split}$
(4.50)

An integration by parts shows that Eq. (4.50) is a variational formulation
associated with the following homogenized system:

$\begin{split}&\frac{\displaystyle\partial u_{1}}{\displaystyle\partial
t}(t,x,{y},z)-D_{1}\,\triangle_{y}u_{1}(t,x,{y},z)-\tilde{D}_{1}\,\partial_{z}^{2}u_{1}(t,x,{y},z)\\\
&=-\sum_{j=1}^{M}a_{1,j}\,u_{1}(t,x,{y},z)\,u_{j}(t,x,{y},z)+f(t,x,{y},z),\;t>0,\;(x,z)\in\Omega,\;{y}\in
X\end{split}$ (4.51) $\displaystyle D_{1}\,\nabla_{y}u_{1}(t,x,{y},z)\cdot\nu$
$\displaystyle=-c_{1}(x,z)\,(u_{1}(t,x,{y},z)-v_{1}(t,x,z))_{+},\;t>0,\;(x,z)\in\Omega,\;\;{y}\in\Gamma$
(4.52) $\partial_{z}u_{1}(t,x,{y},z)=0,\,\,\,\qquad
t>0,\;\;(x,z)\in\overline{D}\times\\{0,L\\},\;\;{y}\in X$ (4.53)

To conclude, by continuity, we have that

$u_{1}(t=0,x,{y},z)=U_{1}(x,{y},z)\,\,\,\qquad\;\;\;\;\;(x,z)\in\Omega,\;\;{y}\in
X$ (4.54)

The proof in the case $1<m\leq M$ is achieved by applying exactly the same
arguments considered when $m=1$.

## Acknowledgements

S. L. is supported by GNFM of INdAM, Italy.

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## Appendix A

In the following, we generalize the interpolation-trace inequality from
Ladyzenskaja [29](Eq. (2.21), p. 69):

$\|u\|^{2}_{L^{2}(\Gamma)}\leq C\|\nabla
u\|_{L^{2}(\Omega)}\,\|u\|_{L^{2}(\Omega)}$ (A.1)

which is valid for any function $u(x)\in H^{1}(\Omega)$ with:

$\displaystyle\int_{\Omega}u(x)\,dx=0$ (A.2)

where $\Gamma$ is an $(n-1)$-dimensional boundary of an $n$-dimensional domain
$\Omega$.

Let us consider now functions $u(x)\in H^{1}(\Omega)$ which do not satisfy
(A.2). In this case, one can define

$u_{0}:=\frac{1}{|\Omega|}\displaystyle\int_{\Omega}u(x)\,dx$ (A.3)

and apply (A.1) to $(u-u_{0})$. Hence,

$\displaystyle\displaystyle\int_{\Gamma}|u(x)|^{2}\,d\sigma(x)\leq
2\bigg{[}\|u-u_{0}\|^{2}_{L^{2}(\Gamma)}+\|u_{0}\|^{2}_{L^{2}(\Gamma)}\bigg{]}$
$\displaystyle\leq
2\,C\,\|\nabla(u-u_{0})\|_{L^{2}(\Omega)}\|u-u_{0}\|_{L^{2}(\Omega)}+\frac{\displaystyle
2|\Gamma|}{\displaystyle|\Omega,|^{2}}\bigg{|}\displaystyle\int_{\Omega}u(x)\,dx\bigg{|}^{2}$
$\displaystyle\leq 2\,C\bigg{[}\eta\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\eta^{-1}\|u-u_{0}\|^{2}_{L^{2}(\Omega)}\bigg{]}+\frac{\displaystyle
2|\Gamma|}{\displaystyle|\Omega|^{2}}\bigg{|}\displaystyle\int_{\Omega}u(x)\,dx\bigg{|}^{2}$
(A.4)

where the Young inequality has been used with $\eta$ being a small constant.
By exploiting the Minkowski and Hölder inequalities, respectively, to estimate
the terms:

$\|u-u_{0}\|^{2}_{L^{2}(\Omega)}\leq
2\bigg{[}\|u\|^{2}_{L^{2}(\Omega)}+\|u_{0}\|^{2}_{L^{2}(\Omega)}\bigg{]}$
(A.5)

$\bigg{|}\displaystyle\int_{\Omega}u(x)\,dx\bigg{|}^{2}\leq|\Omega|\,\|u\|^{2}_{L^{2}(\Omega)}$
(A.6)

Eq. (Appendix A) reads

$\|u\|^{2}_{L^{2}(\Gamma)}\leq 2\,C\,\eta\|\nabla
u\|^{2}_{L^{2}(\Omega)}+\bigg{[}8\,C\,\eta^{-1}+\frac{2|\Gamma|}{|\Omega|}\bigg{]}\|u\|^{2}_{L^{2}(\Omega)}$
(A.7)

## Appendix B

######  Theorem B.1.

(Aubin-Lions-Simon)

Let $B_{0}\subset B_{1}\subset B_{2}$ be three Banach spaces. We assume that
the embedding of $B_{1}$ in $B_{2}$ is continuous and that the embedding of
$B_{0}$ in $B_{1}$ is compact. Let $p,r$ such that $1\leq p,r\leq+\infty$. For
$T>0$, we define

$E_{p,r}=\\{v\in L^{p}(]0,T[,B_{0}),\partial_{t}v\in L^{r}(]0,T[,B_{2})\\}.$

(i) If $p<+\infty$, $E_{p,r}$ is compactly embedded in $L^{p}(]0,T[,B_{1})$.

(ii) If $p=+\infty$ and if $r>1$, $E_{p,r}$ is compactly embedded in
$C^{0}([0,T],B_{1})$.

######  Theorem B.2.

(Lower-semicontinuity of the norm)

Let $E$ and $F$ be Banach spaces and $F^{\prime}$ be the dual space of $F$.

(i) Let $\\{x_{n}\\}$ be a sequence weakly convergent to $x$ in $E$. Then, the
norm on $E$ is lower semi-continuous with respect to the weak convergence,
i.e.

$\|x\|_{E}\leq\lim\inf_{n\rightarrow\infty}\|x_{n}\|_{E}$ (B.1)

(ii) Let $\\{x_{n}\\}$ be a sequence weakly∗ convergent to $x$ in
$F^{\prime}$. Then, the norm on $F^{\prime}$ is lower semi-continuous with
respect to the weak∗ convergence, i.e.

$\|x\|_{F^{\prime}}\leq\lim\inf_{n\rightarrow\infty}\|x_{n}\|_{F^{\prime}}$
(B.2)

## Appendix C

######  Definition C.1.

A sequence of functions $v^{\epsilon}$ in $L^{2}([0,T]\times\Omega)$ two-scale
converges to $v_{0}\in L^{2}([0,T]\times\Omega\times Y)$ if

$\lim_{\epsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}v^{\epsilon}(t,x)\,\phi\bigg{(}t,x,\frac{x}{\epsilon}\bigg{)}\,dt\,dx=\int_{0}^{T}\int_{\Omega}\int_{Y}v_{0}(t,x,y)\,\phi(t,x,y)\,dt\,dx\,dy$
(C.1)

for all $\phi\in C^{1}([0,T]\times\overline{\Omega};C_{\\#}^{\infty}(Y))$.

######  Theorem C.3.

(Compactness theorem) If $v^{\epsilon}$ is a bounded sequence in
$L^{2}([0,T]\times\Omega)$, then there exists a function $v_{0}(t,x,y)$ in
$L^{2}([0,T]\times\Omega\times Y)$ such that, up to a subsequence,
$v^{\epsilon}$ two-scale converges to $v_{0}$.

######  Theorem C.4.

Let $v^{\epsilon}$ be a sequence of functions in $L^{2}([0,T]\times\Omega)$
which two-scale converges to a limit $v_{0}\in L^{2}([0,T]\times\Omega\times
Y)$. Suppose, furthermore, that

$\lim_{\epsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}|v^{\epsilon}(t,x)|^{2}\,dt\,dx=\int_{0}^{T}\int_{\Omega}\int_{Y}|v_{0}(t,x,y)|^{2}\,dt\,dx\,dy$
(C.2)

Then, for any sequence $w^{\epsilon}$ in $L^{2}([0,T]\times\Omega)$ that two-
scale converges to a limit $w_{0}\in L^{2}([0,T]\times\Omega\times Y)$, we
have

$\begin{split}\lim_{\epsilon\rightarrow
0}\int_{0}^{T}\int_{\Omega}v^{\epsilon}(t,x)\,&w^{\epsilon}(t,x)\,\phi\bigg{(}t,x,\frac{x}{\epsilon}\bigg{)}\,dt\,dx\\\
&=\int_{0}^{T}\int_{\Omega}\int_{Y}v_{0}(t,x,y)\,w_{0}(t,x,y)\,\phi(t,x,y)\,dt\,dx\,dy\end{split}$
(C.3)

for all $\phi\in C^{1}([0,T]\times\overline{\Omega};C_{\\#}^{\infty}(Y))$.

In the following, we identify $H^{1}(\Omega)=W^{1,2}(\Omega)$, where the
Sobolev space $W^{1,p}(\Omega)$ is defined by

$W^{1,p}(\Omega)=\bigg{\\{}v|v\in L^{p}(\Omega),\frac{\partial v}{\partial
x_{i}}\in L^{p}(\Omega),i=1,2,3\bigg{\\}}$

and we denote by $H^{1}_{\\#}(Y)$ the closure of $C^{\infty}_{\\#}(Y)$ for the
$H^{1}$-norm.

######  Theorem C.5.

Let $v^{\epsilon}$ be a bounded sequence in $L^{2}([0,T];H^{1}(\Omega))$ that
converges weakly to a limit $v(t,x)$ in $L^{2}([0,T];H^{1}(\Omega))$. Then,
$v^{\epsilon}$ two-scale converges to $v(t,x)$, and there exists a function
$v_{1}(t,x,y)$ in $L^{2}([0,T]\times\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$ such
that, up to a subsequence, $\nabla v^{\epsilon}$ two-scale converges to
$\nabla_{x}v(t,x)+\nabla_{y}v_{1}(t,x,y)$.

######  Theorem C.6.

Let $v^{\epsilon}$ and $\epsilon\nabla v^{\epsilon}$ be two bounded sequences
in $L^{2}([0,T]\times\Omega)$. Then, there exists a function $v_{1}(t,x,y)$ in
$L^{2}([0,T]\times\Omega;H^{1}_{\\#}(Y)/\mathbb{R})$ such that, up to a
subsequence, $v^{\epsilon}$ and $\epsilon\nabla v^{\epsilon}$ two-scale
converge to $v_{1}(t,x,y)$ and $\nabla_{y}v_{1}(t,x,y)$, respectively.

######  Theorem C.7.

Let $v^{\epsilon}$ be a sequence in $L^{2}([0,T]\times\Gamma_{\epsilon})$ such
that

$\epsilon\,\int_{0}^{T}\int_{\Gamma_{\epsilon}}|v^{\epsilon}(t,x)|^{2}\,dt\,d\sigma_{\epsilon}(x)\leq
C$ (C.4)

where $C$ is a positive constant, independent of $\epsilon$. There exist a
subsequence (still denoted by $\epsilon$) and a two-scale limit
$v_{0}(t,x,y)\in L^{2}([0,T]\times\Omega;L^{2}(\Gamma))$ such that
$v^{\epsilon}(t,x)$ two-scale converges to $v_{0}(t,x,y)$ in the sense that

$\lim_{\epsilon\rightarrow
0}\,\epsilon\,\displaystyle\int_{0}^{T}\int_{\Gamma_{\epsilon}}v^{\epsilon}(t,x)\,\phi\bigg{(}t,x,\frac{x}{\epsilon}\bigg{)}\,dt\,d\sigma_{\epsilon}(x)=\displaystyle\int_{0}^{T}\int_{\Omega}\int_{\Gamma}v_{0}(t,x,y)\,\phi(t,x,y)\,dt\,dx\,d\sigma(y)$
(C.5)

for any function $\phi\in
C^{1}([0,T]\times\overline{\Omega};C_{\\#}^{\infty}(Y))$.