ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

REITERATED HOMOGENIZATION OF

HYPERBOLIC-PARABOLIC EQUATIONS IN DOMAINS WITH TINY HOLES

HERMANN DOUANLA, ERICK TETSADJIO

Abstract. This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic.

Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of the same type as the problem with oscillating coefficients.

1. Introduction

In porous media with tiny holes, we study the asymptotic behaviour (asε→0) of the solution to the following problem with rapidly oscillating coefficients:

ρx
ε^{2}

∂^{2}uε

∂t^{2} +βx
ε, t

ε^{2}
∂uε

∂t −div A x

ε, x
ε^{2}

∇uε

=f in Ω^{ε}×(0, T),
Ax

ε, x
ε^{2}

∇uε·ν_{ε}= 0 on (∂Ω^{ε}\∂Ω)×(0, T),
uε= 0 on (∂Ω^{ε}∩∂Ω)×(0, T),

uε(x,0) =u^{0}(x) in Ω^{ε},
ρ(x

ε^{2})∂u_{ε}

∂t (x,0) =ρ^{1}^{2}(x

ε^{2})v^{0}(x) in Ω^{ε},

(1.1)

where Ω is a bounded domain in R^{N} (N ≥ 3) locally located on one side of its
Lipschitz boundary ∂Ω, f ∈ L^{2}(0, T;L^{2}(Ω)), u^{0} ∈ H_{0}^{1}(Ω), v^{0} ∈L^{2}(Ω), T > 0 is
a fixed real number representing the final time of the process and Ω^{ε}is a domain
with periodically distributed tiny holes. The coefficientsρ,β and the matrixAare
periodic. A detailed description of the domain Ω^{ε} and precise assumptions on the
coefficients are given in the next section.

Equations of the form (1.1) are usually called hyperbolic-parabolic equations (H- P equations) and appears when modelling wave processes arising for instance, in heat theory (ρ= 0 andβ 6= 0), theories of hydrodynamics, electricity, magnetism, light, sound and in elasticity theory (ρ 6= 0 and β 6= 0) (see e.g., [17, 18]). It is also well known [2, 20] that equations of the form (1.1) model the process of small longitudinal linear elastic vibration in a thin inhomogeneous rod, in this case,

2010Mathematics Subject Classification. 35B27, 76M50, 35L20.

Key words and phrases. Hyperbolic-parabolic equation; perforated domain; tiny holes;

multi-scale convergence.

c

2017 Texas State University.

Submitted October 26, 2016. Published February 27, 2017.

1

ρ6= 0 is the linear density of the rod, β =β(y)6= 0 the dissipation coefficient, A the Young’s modulus,f the distribution of the density of an external force directed along the rod anduε the displacement function.

The homogenization problem for H-P equations was first studied by Bensoussan, Lions and Papanicolau [3] in a fixed domain by means of compactness arguments and Tatar’s test function method. Bakhvalov and Panasenko [2] considered the same problem and utilized the formal asymptotic expansion method combined with maximum principles to prove homogenization results.

To the best of our knowledge, Migorski [11] was the first to address the homoge-
nization problem for H-P equations in perforated domains. In a domain perforated
with holes of sizeε, he considered aY-periodic matrixAand assumed some strong
convergence hypotheses onρ^{ε}andβ^{ε}to prove a homogenization theorem by means
of the test function method. Timofte [21] considered the same problem as Migorski
but withρ^{ε} =ε and a non-linear source term. Yang and Zhao [23] addressed the
same problem as Migorski by means of the periodic unfolding method. It is worth
pointing out that none of the just mentioned works falls within the framework of
reiterated homogenization and those in perforated domains deal with holes of size
ε.

In the situations whereρ6= 0 andβ = 0, orρ= 0 andβ6= 0 there are numerous works that are indeed related to the homogenization problem for H-P equations. In this direction we quote [4, 5, 6, 7, 9, 10, 12, 13, 14, 22] and references therein. We also mention that Nnang [16] has studied the deterministic homogenization problem for weakly damped nonlinear H-P equations in a fixed domain withρ= 1.

In this work, the matrixAoscillates on two scales and our domain is perforated
with tiny holes of size ε^{2} so that our work falls within the scope of reiterated
homogenization. Moreover, we have a time dependent functionβ^{ε}and we utilised
Nguetseng’s two scale convergence method [15]. A passage to the limit (asε→0)
yields a macroscopic problem which is of the same type as theε-problem: an H-P
equation.

This article is organized as follows. Section 2 deals with the geometric setting of the problem and detailed assumptions on the data. In Section 3 some estimates and compactness results are proven. In Section 4, we recall the basics of the multi- scale convergence theory and formulate a suitable version of its main compactness theorem to be used in the proof of our main result. We also proved some preliminary convergence results. In the fourth section our main result is formulated and proved.

2. Setting of the problem

Let us recall here the setting for the perforated domain Ω^{ε}(see e.g., the pioneer-
ing work on homogenization of differential equations in perforated domains [4]). Let
Z = (0,1)^{N} be the unit cube in R^{N} and let Θ⊂Z be a compact set in R^{N} with
a smooth boundary∂Θ, a non-empty interior and such that the Lebesgue measure
of the setZ\Θ is different from zero. For ε >0, we set

t^{ε}={k∈Z:ε^{2}(k+ Θ)⊂Ω}, Θ^{ε}=∪_{k∈t}εε^{2}(k+ Θ)
and we define the porous medium as:

Ω^{ε}= Ω\Θ^{ε}.

It appears by construction thatt^{ε} is finite since Ω is bounded. Hence Θ^{ε} is closed
and Ω^{ε} is open. One can observe that Ω^{ε} represents the subregion of Ω obtained

from Ω by removing a finite number of periodically distributed holes {ε^{2}(k+ Θ) :
k∈t^{ε}} of sizeε^{2}. In theε-problem (1.1),νεis the outward unit normal to Ω^{ε}on

∂Ω^{ε}\∂Ω. If we setZ^{∗}=Z\Θ and denote byχG the characteristic function of the
setG, the perforated domain Ω^{ε} can also be defined by

Ω^{ε}=

x∈Ω : χZ^{∗}(x
ε^{2}) = 1 .
Hence

χΩ^{ε}(x) =χZ^{∗}(x

ε^{2}) (x∈Ω).

For further needs we introduce the Hilbert space

Vε={u∈H^{1}(Ω^{ε}) :u= 0 on∂Ω}

endowed with the gradient norm

kukV_{ε} =k∇uk_{(L}2(Ω^{ε}))^{N} (u∈Vε).

We now state the assumptions on the data. The ε-problem (1.1) is constrained as follows:

(A1) Uniform ellipticity. The matrix A(y, z) = (aij(y, z))_{1≤i,j≤N} ∈ L^{∞}(R^{N} ×
R^{N})^{N}^{×N} is real, symmetric and there exists a positive constant Λ>0 such
that

kaijk_{L}^{∞}_{(}_{R}N×R^{N})≤Λ for 1≤i, j≤N,

N

X

ij=1

aij(y, z)ζiζj≥Λ^{−1}|ζ|^{2}for a.e. (y, z)∈R^{N} ×R^{N} and allζ∈R^{N}.
(A2) Positivity ofρandβ. The functionsρ(z)∈ C^{1}(R^{N}) andβ(y, τ)∈L^{∞}(R^{N}×

R) satisfy

ρ(z)≥0 inR^{N},

β(y, τ)≥α >0 a.e. inR^{N} ×R. (2.1)
(A3) Periodicity. LetY = (0,1)^{N}, Z= (0,1)^{N} andT = (0,1). We assume that
the functionβisY× T-periodic and that for any 1≤i, j≤N, the function
a_{ij} isY×Z-periodic. We also assume that the functionρisZ-periodic and
further satisfy

MZ^{∗}(ρ) =
Z

Z^{∗}

ρ(z)dz >0.

The main result of this article reads as follows (the matrixAbappearing therein is defined later).

Theorem 2.1. Assume that hypotheses (A1)–(A3) hold and letu_{ε} (ε > 0) be the
unique solution to (1.1). Then as ε→0 we have

uε→u0 inL^{2}(ΩT),
whereu0∈L^{2}(0, T;H_{0}^{1}(Ω)) is the unique solution to

Z

Z^{∗}

ρ(z)dz∂^{2}u0

∂t^{2} +Z 1
0

Z

Y

β(y, τ)dy dτ∂u0

∂t − 1

|Z^{∗}|div Aˆ∇xu0

=f(x, t) inΩ×(0, T),

u_{0}= 0 on∂Ω×(0, T),
u_{0}(x,0) =u^{0}(x) in Ω,

Z

Z^{∗}

ρ(z)dz∂u0

∂t (x,0) =Z

Z^{∗}

pρ(z)dz

v^{0}(x) in Ω.

Unless otherwise specified, vector spaces throughout are considered overR, and
scalar functions are assumed to take real values. The numerical spaceR^{N} and its
open sets are provided with the Lebesgue measure denoted bydx=dx1...dxN. The
usual gradient operator will be denoted by ∇. Throughout, C denotes a generic
constant independent ofεthat can change from one line to the next. We will use
the following notation. The centered dot stands for the Euclidean scalar product
inR^{N} while the absolute value or modulus is denoted by| · |.

LetF(R^{m}), (m≥3 integer) be a given function space and letU be a bounded
domain inR^{m}. The Lebesgue measure ofU is denoted by|U|and the mean value
of a functionv overU is denoted and defined by

MU(v) = 1

|U|

Z

U

v(x)dx.

We denote by Fper(U) the space of functions in Floc(R^{m}) (when it makes sense)
that areU-periodic, and by F_{#}(U) the space of those functionsv ∈F_{per}(U) with
R

Uv(y)dy= 0.

The letterE denotes throughout a family of strictly positive real numbers (0<

ε < 1) admitting 0 as accumulation point while a fundamental sequence is any
ordinary sequence of real numbers 0< ε_{n}<1, such thatε_{n}→0 asn→+∞. The
time derivatives ^{∂u}_{∂t} and ^{∂}_{∂t}^{2}^{u}2 are sometimes denoted byu^{0} andu^{00}, respectively. For
ε >0 the functionsx7→χZ^{∗}(_{ε}^{x}2),x7→ρ(_{ε}^{x}2), (x, t)7→β(^{x}_{ε},_{ε}^{t}2) and x7→A(^{x}_{ε},_{ε}^{x}2)
are sometimes denoted byχ^{ε}_{Z}∗,ρ^{ε},β^{ε} andA^{ε}, respectively.

3. Estimates and compactness results

We recall that [3, Theorem 1.1] for anyε >0 the evolution problem (1.1) admits a unique solutionuεthat satisfies

uε∈L^{∞}(0, T;Vε)∩L^{2}(0, T;V_{ε}^{0}),
u^{0}_{ε}∈L^{2}(0, T;L^{2}(Ω^{ε})), √

ρ^{ε}u^{0}_{ε}∈L^{∞}(0, T;L^{2}(Ω^{ε})),
ρ^{ε}u^{00}_{ε} ∈L^{2}(0, T;V_{ε}^{0})

uε(0) =u^{0}, ρ^{ε}u^{0}_{ε}(0) =√
ρ^{ε}v^{0}.

Proposition 3.1. Under hypotheses(A1)–(A3), the following estimates hold:

kuεkL^{∞}(0,T;V_{ε})≤C, (3.1)
k√

ρ^{ε}u^{0}_{ε}k_{L}^{∞}_{(0,T;L}2(Ω^{ε}))≤C, (3.2)
ku^{0}_{ε}k_{L}2(0,T;L^{2}(Ω^{ε}))≤C, (3.3)
kρ^{ε}u^{00}_{ε}k_{L}2(0,T;V_{ε}^{0})≤C, (3.4)
whereC is a positive constant which does not depend on ε.

Proof. We follow [3]. Let t∈[0, T]. We multiply the first equation of (1.1) byu^{0}_{ε}
and integrate over Ω^{ε}to get

Z

Ω^{ε}

h ρx

ε^{2}

u^{00}_{ε}u^{0}_{ε}+βx
ε, t

ε^{2}

(u^{0}_{ε})^{2}−u^{0}_{ε}div A(x
ε, x

ε^{2})∇uε

i dx=

Z

Ω^{ε}

f u^{0}_{ε}dx.

Which is also written 1 2 Z

Ω^{ε}

ρ x
ε^{2}

[(u^{0}_{ε})^{2}]^{0}dx+
Z

Ω^{ε}

β x ε, t

ε^{2}

(u^{0}_{ε})^{2}dx

− Z

Ω^{ε}

u^{0}_{ε}div A(x
ε, x

ε^{2})∇uε

dx

= Z

Ω^{ε}

f u^{0}_{ε}dx.

(3.5)

But

1 2

Z

Ω^{ε}

ρ(x

ε^{2})[(u^{0}_{ε})^{2}]^{0}dx= 1
2

d dt

Z

Ω^{ε}

ρ(x
ε^{2})

(u^{0}_{ε})^{2}
dx

= 1 2

d dt(ρ(x

ε^{2})u^{0}_{ε}, u^{0}_{ε})_{L}2(Ω^{ε})

and

Z

Ω^{ε}

β x ε, t

ε^{2}

(u^{0}_{ε})^{2}dx= β^{ε}u^{0}_{ε}, u^{0}_{ε}

L^{2}(Ω^{ε}),
so that, on settingR

Ω^{ε}A^{ε}∇uε∇u^{0}_{ε}dx:=A^{ε}(u_{ε}, u^{0}_{ε}), the Green formula

− Z

Ω^{ε}

u^{0}_{ε}div(A(x
ε, x

ε^{2})∇uε)dx=
Z

Ω^{ε}

A^{ε}∇uε∇u^{0}_{ε}dx
and the following consequence of the symmetry hypothesis onA,

− Z

Ω^{ε}

u^{0}_{ε}div A(x
ε, x

ε^{2})∇uε
dx= 1

2 d

dtA^{ε}(uε, uε)
allow us to rewrite (3.5) as follows

1 2

d

dt(ρ^{ε}u^{0}_{ε}, u^{0}_{ε})L^{2}(Ω^{ε})+ (β^{ε}u^{0}_{ε}, u^{0}_{ε})L^{2}(Ω^{ε})+1
2

d

dtA^{ε}(uε, uε) = (f, u^{0}_{ε})L^{2}(Ω^{ε}). (3.6)
We now integrate (3.6) on [0, t] and obtain

1

2(ρ^{ε}u^{0}_{ε}(t), u^{0}_{ε}(t))_{L}2(Ω^{ε})−1

2(ρ^{ε}u^{0}_{ε}(0), u^{0}_{ε}(0))_{L}2(Ω^{ε})+1

2A^{ε}(u_{ε}(t), u_{ε}(t))

−1

2A^{ε}(uε(0), uε(0)) +
Z t

0

β^{ε}u^{0}_{ε}(s), u^{0}_{ε}(s)

L^{2}(Ω^{ε})ds

= Z t

0

(f(s), u^{0}_{ε}(s))_{L}2(Ω^{ε})ds.

Using the initial conditions, we obtain 1

2k√

ρ^{ε}u^{0}_{ε}(t)k^{2}_{L}2(Ω^{ε})+1

2A^{ε}(uε(t), uε(t)) +
Z t

0

(β^{ε}u^{0}_{ε}(s), u^{0}_{ε}(s))_{L}2(Ω^{ε})ds

= 1

2A^{ε}(u^{0}, u^{0}) +1

2kv^{0}k^{2}_{L}2(Ω^{ε})+
Z t

0

(f(s), u^{0}_{ε}(s))L^{2}(Ω^{ε})ds.

Using the positivity of β, the boundedness and ellipticity hypotheses on A, the Cauchy-Schwartz and Young’s inequalities, one readily arrives at

k√

ρ^{ε}u^{0}_{ε}(t)k^{2}_{L}2(Ω^{ε})+ 1

Λkuε(t)k^{2}_{V}

ε+ 2α Z t

0

ku^{0}_{ε}(s)k^{2}_{L}2(Ω^{ε})ds

≤Λk∇u^{0}k^{2}_{L}2(Ω)^{N} +kv^{0}k^{2}_{L}2(Ω)+α
Z t

0

ku^{0}_{ε}(s)k^{2}_{L}2(Ω^{ε})ds+ 2
α

Z t

0

kf(s)k^{2}_{L}2(Ω)ds,

which implies (3.1)-(3.3) as easily seen from k√

ρ^{ε}u^{0}_{ε}(t)k^{2}_{L}2(Ω^{ε})+kuε(t)k^{2}_{V}_{ε}+
Z t

0

ku^{0}_{ε}(s)k^{2}_{L}2(Ω^{ε})ds

≤C
ku^{0}k^{2}_{H}1

0(Ω)+kv^{0}k^{2}_{L}2(Ω)+kfk^{2}_{L}2(0,T;L^{2}(Ω))

.

(3.7)

We use the main equation in (1.1) to deduce that

kρ^{ε}u^{00}_{ε}kL^{2}(0,T;V_{ε}^{0})=k −β^{ε}u^{0}_{ε}+ div(A^{ε}∇uε) +fkL^{2}(0,T;V_{ε}^{0})≤C,

which completes the proof.

Since solutions of (1.1) are defined on Ω^{ε}_{T} = (0, T)×Ω^{ε} but not on Ω_{T} =
(0, T)×Ω, we introduce a family of extension operators so that the sequence of
extensions to Ω of solutions to (1.1) belongs to a fixed space in which we can study
its asymptotic behaviour. The following result is a classical extension property
[4, 11].

Proposition 3.2. For any ε > 0, there exists a bounded linear operator Pε such that

Pε∈ L(L^{2}(0, T;Vε); L^{2}(0, T;H_{0}^{1}(Ω)))∩ L(L^{2}(0, T;L^{2}(Ω^{ε})); L^{2}(0, T;L^{2}(Ω)))
and

Pεu=u a.e. inΩ^{ε}_{T}, (3.8)

P_{ε}u^{0}= (P_{ε}u)^{0} a.e. inΩ^{ε}_{T}, (3.9)
kPεukL^{2}(0,T;L^{2}(Ω))≤CkukL^{2}(0,T;L^{2}(Ω^{ε})), (3.10)
kPεuk_{L}2(0,T;H_{0}^{1}(Ω))≤Ckuk_{L}2(0,T;Vε). (3.11)
An immediate consequence of Proposition 3.1 and Proposition 3.2 is the following
estimates that will be useful in the sequel.

Proposition 3.3. Let ε > 0 and let u_{ε} be the solution to (1.1). There exists a
constant C >0 independent of εsuch that

kP_{ε}u_{ε}k_{L}2(0,T;H^{1}_{0}(Ω))≤C, (3.12)
k(Pεuε)^{0}kL^{2}(0,T;L^{2}(Ω))≤C, (3.13)
k(Pεuε)^{0}k_{L}2(0,T;H^{−1}(Ω))≤C. (3.14)
We are now in a position to formulate our first compactness result.

Theorem 3.4. The sequence(Pεuε)ε>0 is relatively compact in L^{2}(0, T;L^{2}(Ω)).

Proof. It is a consequence of proposition 3.3 and a classical embedding result. We define

W ={u∈L^{2}(0, T;H_{0}^{1}(Ω)) :u^{0}∈L^{2}(0, T;H^{−1}(Ω))}

and endow it with the norm

kukW =kuk_{L}2(0,T;H^{1}_{0}(Ω))+kukL^{2}(0,T;H^{−1}(Ω)) u∈W.

It is well known from Aubin-Lions’ lemma that the injectionW bL^{2}(0, T;L^{2}(Ω))
is compact. The proof is complete since the sequence (Pεuε)ε>0 is bounded in W

as seen from (3.12)-(3.14).

4. Multiscale convergence and preliminary results

In this section, we recall the definition and main compactness theorem of the multi-scale convergence theory [1, 15]. We also adapt some existing results in this method to our framework. We eventually prove some preliminary convergence results needed in the homogenization process of problem (1.1).

4.1. Multiscale convergence method.

Definition 4.1. A sequence (u_{ε})_{ε∈E} ⊂L^{2}(Ω_{T}) is said to weakly multi-scale con-
verge towards u0 ∈L^{2}(ΩT ×Y ×Z× T) (denoteduε

−−−−→w−ms u0) inL^{2}(ΩT), if as
ε→0,

Z

Ω_{T}

u_{ε}(x, t)ϕ(x, t,x
ε, x

ε^{2}, t
ε^{2})dx dt

→ Z Z Z Z

Ω_{T}×Y×Z×T

u0(x, t, y, z, τ)ϕ(x, t, y, z, τ)dx dt dy dz dτ

(4.1)

for allϕ∈L^{2}(ΩT;Cper(Y ×Z× T)).

A sequence (uε)_{ε∈E} ⊂L^{2}(ΩT) is said to strongly multi-scale converge towards
u0∈L^{2}(ΩT×Y×Z× T) (denoteduε

−−−−→s−ms u0) inL^{2}(ΩT), if it weakly multi-scale
converges tou0in L^{2}(ΩT ×Y ×Z× T) and further satisfies

kuεk_{L}2(ΩT)→ ku0k_{L}2(ΩT×Y×Z×T) asε→0.

Remark 4.2. (i) Let u ∈ L^{2}(ΩT;Cper(Y ×Z × T)) and define for ε ∈ E,
u^{ε}: ΩT →Rby

u^{ε}(x, t) =u x, t,x
ε, x

ε^{2}, t
ε^{2}

for (x, t)∈ΩT.

Then u^{ε} −−−−→^{w−ms} u and u^{ε} −−−−→^{s−ms} u in L^{2}(ΩT) as ε → 0. We also have
u^{ε}→ueinL^{2}(ΩT) -weak asε→0, with

u(x, t) =e Z Z Z

Y×Z×T

u(·,·, y, z, τ)dy dz dτ.

(ii) Let u ∈ C(ΩT;L^{∞}_{per}(Y ×Z× T)) and define u^{ε} like in (i) above. Then
u^{ε w}−−−−→^{−ms} uinL^{2}(ΩT) asε→0.

(iii) If (u_{ε})_{ε∈E} ⊂L^{2}(Ω_{T}) andu_{0}∈L^{2}(Ω_{T}×Y×Z× T) are such thatu_{ε}−−−−→^{w−ms}
u0in L^{2}(ΩT), then (4.1) still holds forϕ∈ C(ΩT;L^{∞}_{per}(Y ×Z× T)).

(iv) SinceχΩ^{ε}(x) =χZ^{∗}(_{ε}^{x}2) for almost everyx∈Ω and anyε∈E, we deduce
from (ii) above that, asε→0,χΩ^{ε}

−−−−→w−ms χZ^{∗} in L^{2}(Ω).

The following two theorems are the backbone of the multi-scale convergence method [1, 15].

Theorem 4.3. Any bounded sequence inL^{2}(Ω_{T})admits a weakly multi-scale con-
vergent subsequence.

Theorem 4.4. Let (uε)_{ε∈E} be a bounded sequence in L^{2}(0, T;H_{0}^{1}(Ω)), E being a
fundamental sequence. There exist a subsequence still denoted by (uε)_{ε∈E} and a
triplet(u0, u1, u2) in the space

L^{2}(0, T;H_{0}^{1}(Ω))×L^{2}(Ω_{T};L^{2}(T;H_{per}^{1} (Y)))×L^{2}(Ω_{T};L^{2}(Y × T;H_{per}^{1} (Z)))

such that, asε→0,

uε→u0 inL^{2}(0, T;H_{0}^{1}(Ω))-weak (4.2)

∂u_{ε}

∂xi

−−−−→w−ms ∂u_{0}

∂xi

+∂u_{1}

∂yi

+∂u_{2}

∂zi

inL^{2}(Ω_{T}) (1≤i≤N). (4.3)
Remark 4.5. In theorem 4.4, the functionsu1 andu2 are unique up to additive
functions of variablesx, t, τ andx, t, y, τ, respectively. It is therefore crucial to fix
the choice of u1 andu2 in accordance with our needs. To formulate the version of
theorem 4.4 we will use, we introduce the space

H_{#ρ}^{1} (Z^{∗}) =

u∈H_{per}^{1} (Z) :
Z

Z^{∗}

ρ(z)u(z)dz= 0 and its dense subspace

C_{#ρ}^{∞}(Z^{∗}) =

u∈ C^{∞}_{per}(Z) :
Z

Z^{∗}

ρ(z)u(z)dz= 0 .

Theorem 4.6. Let (u_{ε})_{ε∈E} be a bounded sequence in L^{2}(0, T;H_{0}^{1}(Ω)), E being a
fundamental sequence. There exist a subsequence still denoted by (u_{ε})_{ε∈E} and a
triplet(u0, u1, u2) in the space

L^{2}(0, T;H_{0}^{1}(Ω))×L^{2}(ΩT;L^{2}(T;H_{#}^{1}(Y)))×L^{2}(ΩT;L^{2}(Y × T;H_{#ρ}^{1} (Z^{∗})))
such that, asε→0,

uε→u0 inL^{2}(0, T;H_{0}^{1}(Ω))-weak (4.4)

∂u_{ε}

∂xi

−−−−→w−ms ∂u_{0}

∂xi

+∂u_{1}

∂yi

+∂u_{2}

∂zi

inL^{2}(Ω_{T}) (1≤j≤N). (4.5)
4.2. Preliminary results. Before formulating some preliminary convergence re-
sults needed later, we recall some results on periodic distributions (see e.g., [7,
19]). As above, let L^{2}_{#ρ}(Z^{∗}) denotes the space of functions u ∈ L^{2}_{per}(Z) with
R

Z^{∗}ρ(z)u(z)dz= 0, and consider the following Gelfand triple
H_{#ρ}^{1} (Z^{∗})⊂L^{2}_{#ρ}(Z^{∗})⊂(H_{#ρ}^{1} (Z^{∗}))^{0}.

If u∈ L^{2}_{#ρ}(Z^{∗}) and v ∈H_{#ρ}^{1} (Z^{∗}), we have [u, v] = (u, v) where [·,·] denotes the
duality pairing between (H_{#ρ}^{1} (Z^{∗}))^{0} and H_{#ρ}^{1} (Z^{∗}) while (·,·) denotes the scalar
product in L^{2}_{#ρ}(Z^{∗}). The topological dual of L^{2}(Y × T;H_{#ρ}^{1} (Z^{∗})) is L^{2}(Y ×
T; (H_{#ρ}^{1} (Z^{∗}))^{0}) andC_{per}^{∞}(Y)⊗ C_{per}^{∞}(T)⊗ C_{#ρ}^{∞}(Z^{∗}) is dense inL^{2}(Y × T;H_{#ρ}^{1} (Z^{∗})).

Proposition 4.7. Let u∈ D_{per}^{0} (Y × T ×Z)and assume that u is continuous on
C_{per}^{∞}(Y)⊗ C_{per}^{∞}(T)⊗ C_{#ρ}^{∞}(Z^{∗})endowed with theL^{2}_{per}(Y × T;H_{#ρ}^{1} (Z^{∗}))-norm. Then
u∈L^{2}_{per}(Y × T; (H_{#ρ}^{1} (Z^{∗}))^{0}), and further

hu, ϕi= Z 1

0

Z

Y

[u(y, τ), ϕ(y, τ,·)]dy dτ

for all ϕ ∈ C_{per}^{∞}(Y)⊗ C_{per}^{∞}(T)⊗ C_{#ρ}^{∞}(Z^{∗}), where h·,·i denotes the duality pairing
between D_{per}^{0} (Y × T ×Z) andC_{per}^{∞}(Y × T ×Z).

Proposition 4.8. Let V =

u∈L^{2}(Y × T;H_{#ρ}^{1} (Z^{∗})) :ρχ_{Z}^{∗}∂^{2}u

∂τ^{2} ∈L^{2}(Y × T; (H_{#ρ}^{1} (Z^{∗}))^{0}) .

(i) The spaceV is a reflexive Banach space when endowed with the norm
kuk=kuk_{L}2(Y×T;H_{#ρ}^{1} (Z^{∗}))+

ρχZ^{∗}

∂^{2}u

∂τ^{2}

_{L}_{2}_{(Y}_{×T}_{;(H}1

#ρ(Z^{∗}))^{0}) (u∈V).

(ii) It holds that Z 1

0

Z

Y

ρχZ^{∗}

∂^{2}u

∂τ^{2}, v

dy dτ = Z 1

0

Z

Y

u, ρχZ^{∗}

∂^{2}v

∂τ^{2}

dy dτ for all u, v∈V.

We can now formulate the main result of this section.

Theorem 4.9. Let (uε)ε∈E be the sequence of solution to (1.1), E being a funda-
mental sequence. There exist a subsequenceE^{0} ofEand a triplet(u0, u1, u2)in the
space

L^{2}(0, T;H_{0}^{1}(Ω))×L^{2}(Ω_{T};H_{#}^{1}(Y))×L^{2}(Ω_{T};L^{2}(Y × T;H_{#ρ}^{1} (Z^{∗})))
such that, asE^{0}3ε→0,

Pεuε→u0 inL^{2}(ΩT), (4.6)

∂(P_{ε}u_{ε})

∂t

−−−−→w−ms ∂u_{0}

∂t in L^{2}(ΩT), (4.7)

∂(Pεuε)

∂x_{i}

−−−−→w−ms ∂u0

∂x_{i} +∂u1

∂y_{i} +∂u2

∂z_{i} inL^{2}(ΩT) (1≤i≤N). (4.8)
The proof of Theorem 4.9 requires two preliminary results and is therefore post-
poned.

Lemma 4.10. LetE, E^{0},(uε)ε∈E and the triplet(u0, u1, u2)be as in Theorem 4.9.

It holds that

ε→0lim
1
ε^{2}

Z

ΩT

uε(x, t)ρ(x

ε^{2})χZ^{∗}(x

ε^{2})ϕ x, t,x
ε, x

ε^{2}, t
ε^{2}

dx dt

= Z Z Z Z

Ω_{T}×Y×Z×T

u_{2}(x, t, y, z, τ)ρ(z)χ_{Z}∗(z)ϕ(x, t, y, z, τ)dx dt dy dz dτ
for allϕ∈ D(ΩT)⊗ C_{per}^{∞}(Y)⊗ C^{∞}_{per}(Z)⊗ C_{per}^{∞}(T)such that

Z

Z

χ_{Z}^{∗}ρ(z)ϕ(z)dz= 0 for all(x, t, y, τ)∈Ω_{T} ×Y × T.
Proof. Letϕ∈ D(Ω_{T})⊗ C^{∞}_{per}(Y)⊗ C_{per}^{∞}(Z)⊗ C^{∞}_{per}(T) with R

Zχ_{Z}^{∗}ρ(z)ϕ(z)dz= 0,
we deduce from the Fredholm alternative the existence of a unique w∈ D(ΩT)⊗
C_{per}^{∞}(Y)⊗H_{#ρ}^{1} (Z^{∗})⊗ C_{per}^{∞}(T) such that

∆_{z}w=ϕρχ_{Z}^{∗} inZ,

w(x, t, y, τ)∈H_{#ρ}^{1} (Z^{∗}) for all (x, t, y, τ)∈Ω_{T} ×Y × T. (4.9)
But the restriction to Z^{∗} of the function w defined by (4.9) belongs to C^{3}(Z^{∗}) so
that we have

div(∇zw)^{ε}= (div∇zw)^{ε}+1

ε(divy∇zw)^{ε}+ 1

ε^{2}(∆zw)^{ε} in Ω^{ε}_{T},
and therefore

1
ε^{2}

Z

Ω_{T}

u_{ε}(x, t)ρ(x

ε^{2})χ_{Z}^{∗}(x

ε^{2})ϕ x, t,x
ε, x

ε^{2}, t
ε^{2}

dx dt

=− Z

ΩT

∇uε·χ^{ε}_{Z}∗(∇zw)^{ε}dx dt−
Z

ΩT

uεχ^{ε}_{Z}∗(div∇zw)^{ε}dx dt

−1 ε

Z

Ω_{T}

u_{ε}χ^{ε}_{Z}∗(div_{y}∇zw)^{ε}dx dt.

As,E^{0}3ε→0, (4.8) and (iii) of Remark 4.2 reveal that the first term in the right
hand side of this equality converges to

− Z Z Z Z

Ω_{T}×Y×Z×T

(∇xu0+∇yu1+∇zu2)·χZ^{∗}(∇zw)dx dt dy dz dτ

= Z Z Z Z

ΩT×Y×Z×T

u2χZ^{∗}(∆zw)dx dt dy dz dτ,

while the second one converges to zero. As regards the third term, since the test
function therein, (divy∇zw)^{ε}, depends on the zvariable, its limit cannot be com-
puted as usual like in [8, Theorem 2.3] even if its mean value overY is zero. This
requires some further investigation. From

div(∇yw)^{ε}= (div∇yw)^{ε}+1

ε(divy∇yw)^{ε}+ 1

ε^{2}(divz∇yw)^{ε} in Ω^{ε}_{T}
and

(divz∇yw)^{ε}= (divy∇zw)^{ε} in Ω^{ε}_{T},
it follows that

−1 ε

Z

Ω_{T}

u_{ε}χ^{ε}_{Z}∗(div_{y}∇zw)^{ε}dx dt

=ε Z

ΩT

uεχ^{ε}_{Z}∗(div∇yw)^{ε}dx dt+
Z

ΩT

uεχ^{ε}_{Z}∗(∆yw)^{ε}dx dt
+ε

Z

Ω_{T}

∇u_{ε}·χ^{ε}_{Z}∗(∇_{y}w)^{ε}dx dt.

Therefore,

−1 ε

Z

Ω_{T}

uεχ^{ε}_{Z}∗(divy∇zw)^{ε}dx dt

→ Z Z Z Z

ΩT×Y×Z×T

u0χZ^{∗}(∆yw)dx dt dy dz dτ= 0

asE^{0} 3ε→0. The proof is complete.

Lemma 4.11. LetE, E^{0},(uε)_{ε∈E} and the triplet(u0, u1, u2)be as in Theorem 4.9.

It holds that

ε→0lim 1 ε Z

Ω_{T}

u_{ε}(x, t)ρ(x

ε^{2})χ_{Z}^{∗}(x

ε^{2})ϕ x, t,x
ε, x

ε^{2}, t
ε^{2}

dx dt

= Z Z Z Z

ΩT×Y×Z×T

u1(x, t, y, z, τ)ρ(z)χZ^{∗}(z)ϕ(x, t, y, z, τ)dx dt dy dz dτ
for allϕ∈ D(ΩT)⊗ C_{per}^{∞}(Y)⊗ C^{∞}_{per}(Z)⊗ C_{per}^{∞}(T)such that

Z

Y

ϕ(y)dy= 0 for all(x, t, z, τ)∈ΩT×Z× T.

Proof. Let ϕ ∈ D(ΩT)⊗ C_{per}^{∞}(Y)⊗ C_{per}^{∞}(Z)⊗ C_{per}^{∞}(T) with R

Y ϕ(y)dy = 0 and
considerw∈ D(ΩT)⊗ C^{∞}_{per}(Y)⊗ C_{per}^{1} (Z)⊗ C^{∞}_{per}(T) such that

∆yw=ϕρ inY,

w(x, t, z, τ)∈ C^{∞}_{#}(Y) for all (x, t, z, τ)∈Ω_{T}×Z× T.
Recalling that

div(∇yw)^{ε}= (div∇yw)^{ε}+1

ε(div_{y}∇yw)^{ε}+ 1

ε^{2}(div_{z}∇yw)^{ε} in Ω_{T},
the following holds

1 ε

Z

Ω_{T}

u_{ε}(x, t)ρ(x

ε^{2})χ_{Z}^{∗}(x

ε^{2})ϕ x, t,x
ε, x

ε^{2}, t
ε^{2}

dx dt

=1 ε

Z

Ω_{T}

uε(x, t)χZ^{∗}(x

ε^{2})(∆yw) x, t,x
ε, x

ε^{2}, t
ε^{2}

dx dt

=− Z

ΩT

∇uε·χ^{ε}_{Z}∗(∇yw)^{ε}−
Z

ΩT

uεχ^{ε}_{Z}∗(div∇yw)^{ε}dx dt

− 1
ε^{2}

Z

Ω_{T}

uεχ^{ε}_{Z}∗(divz∇yw)^{ε}dx dt.

(4.10)

AsR

Zdiv_{z}(∇_{y}w)dz= 0 we follow the lines of reasoning in the proof of Lemma 4.10
to compute the limit of the last term in (4.10). We find that as E^{0} 3 ε→ 0 the
right-hand side of (4.10) converges to

Z Z Z Z

Ω_{T}×Y×Z×T

χZ^{∗}

h−(∇xu0+∇yu1+∇zu2)·(∇yw)−u0(div∇yw)

−u2(divz∇yw)i

dx dt dy dz dτ

= Z Z Z Z

ΩT×Y×Z×T

u1χZ^{∗}(∆yw)dx dt dy dz dτ

= Z Z Z Z

Ω_{T}×Y×Z×T

u_{1}χ_{Z}^{∗}ϕρ dx dt dy dz dτ,

and the proof is complete.

Proof of Theorem 4.9. According to Proposition 3.3, Theorem 3.4 and Theorem 4.6,
it remains to prove (4.7) and to justify that the functionu1 in the triplet given by
Theorem 4.6 actually belongs to L^{2}(Ω_{T};H_{#}^{1}(Y)), i.e., u_{1} does not depend on the
variable τ. We start with the fact that u_{1} ∈ L^{2}(Ω_{T};H_{#}^{1}(Y)). To prove this, let
ψ∈ D(ΩT)⊗ C_{#}^{∞}(Y)⊗ C_{per}^{∞}(T) and consider the functionψ^{ε}∈ D(ΩT) defined by

ψ^{ε}(x, t) =ε^{3}ψ x, t,x
ε, t

ε^{2}

, (x, t)∈Ω_{T}.
Usingψ^{ε} as a test function in problem (1.1), we obtain

Z

ΩT

ρ^{ε}χ^{ε}_{Z}∗(Pεuε)∂^{2}ψ^{ε}

∂t^{2} dx dt+
Z

ΩT

β^{ε}χ^{ε}_{Z}∗

∂(Pεuε)

∂t ψ^{ε}dx dt
+

Z

Ω_{T}

A^{ε}∇(P_{ε}u_{ε})·χ^{ε}_{Z}∗∇ψ^{ε}dx dt=
Z

Ω_{T}

f χ^{ε}_{Z}∗ψ^{ε}dx dt.

LettingE^{0}3ε→0 in this equation, the term in the right-hand side and the second
and third terms on the left hand side obviously converge to zero, so that

lim

E^{0}3ε→0

Z

Ω_{T}

ρ^{ε}χ_{Ω}ε(P_{ε}u_{ε})∂^{2}ψ^{ε}

∂t^{2} dx dt= 0. (4.11)
However,

∂^{2}ψ^{ε}

∂t^{2} =ε^{3}∂^{2}ψ

∂t^{2} + 2ε∂^{2}ψ

∂t∂τ +1 ε

∂^{2}ψ

∂τ^{2}. (4.12)

Substituting (4.12) in (4.11) we realize that lim

E^{0}3ε→0

1 ε

Z

ΩT

(Pεuε)(x, t)ρ(x

ε^{2})χZ^{∗}(x
ε^{2})∂^{2}ψ

∂τ^{2}(x, t,x
ε, t

ε^{2})dx dt= 0.

Using Lemma 4.11, this is equivalent to Z Z Z Z

ΩT×Y×Z×T

u1(x, t, y, τ)ρ(z)χZ^{∗}(z)∂^{2}ψ

∂τ^{2}(x, t, y, τ)dx dt dy dz dτ = 0,
which, by takingψ=ψ1⊗ψ2⊗ψ3withψ1∈ D(ΩT),ψ2∈ C_{#}^{∞}(Y) andψ3∈ C^{∞}_{per}(T),
also writes

Z

Z^{∗}

ρ(z)dz Z Z

ΩT×Y

ψ1(x, t)ψ2(y) Z

T

u1(x, t, y, τ)∂^{2}ψ3

∂τ^{2} (τ)dτ

dx dt dy= 0.

The hypothesisMZ^{∗}(ρ)>0 and the arbitrariness ofψ1 andψ2yields
Z 1

0

u_{1}(x, t, y, τ)∂^{2}ψ_{3}

∂τ^{2} (τ)dτ = 0 for allψ_{3}∈ C_{per}^{∞}(T).

Taking in particularψ3(τ) =e^{−2iπpτ} (p∈Z\ {0}), we obtain
Z 1

0

u1(x, t, y, τ)e^{−2iπpτ}dτ = 0 for allp∈Z\ {0}. (4.13)
The Fourier series expansion of the periodic functionτ7→u1(x, t, y, τ) writes

u1(x, t, y, τ) =X

p∈Z

Cpe^{2iπpτ} whereCp=
Z 1

0

u1(x, t, y, τ)e^{−2iπpτ}dτ.

However, (4.13) implies thatCp= 0 for allp∈Z\ {0}, so thatu1(x, t, y, τ) =C0= R1

0 u1(x, t, y, τ)dτ. This proves that the functionu1 is independent ofτ.

We now prove (4.7). It follows from (3.13) and Theorem 4.3 that there exists
w∈L^{2}(ΩT ×Y ×Z× T) such that, as E^{0}3ε→0,

∂(P_{ε}u_{ε})

∂t

−−−−→w−ms w inL^{2}(Ω_{T}). (4.14)
Since (4.6) implies that ^{∂(P}_{∂t}^{ε}^{u}^{ε}^{)} → ^{∂u}_{∂t}^{0} weakly in D^{0}(Ω_{T}) as E^{0} 3 ε → 0, while
(4.14) implies ^{∂(P}_{∂t}^{ε}^{u}^{ε}^{)} −−−−→ M^{w−ms} _{Y}_{×Z×T}(w) weakly in D^{0}(Ω_{T}) asE^{0} 3ε→0 it is
sufficient to prove that the function w does not depend on the variables y, z and
τ to conclude that w = ^{∂u}_{∂t}^{0} in L^{2}(Ω_{T}). Firstly, we prove that the function w
does not depend on the variable z. Letθ∈ D(ΩT),ϕ∈ C_{per}^{∞}(Y), ψ∈ C_{per}^{∞}(Z) and
ϑ∈ C_{per}^{∞}(T) and definew^{ε}(x, t) =θ(x, t)ϕ(^{x}_{ε})ψ(_{ε}^{x}2)ϑ(_{ε}^{t}2) forε∈E^{0}and (x, t)∈ΩT.
Passing to the limit asE^{0}3ε→0 in the equality

−ε^{2}D ∂

∂xj

∂(Pεuε)

∂t

, w^{ε}E

L^{2}(0,T;H^{−1}(Ω)),L^{2}(0,T;H_{0}^{1}(Ω))

= Z

ΩT

ϑ t
ε^{2}

∂(Pεuε)

∂t h

ε^{2} ∂θ

∂x_{j}ϕ^{ε}ψ^{ε}+θϕ^{ε} ∂ψ

∂z_{j}
^{ε}

+εθ ∂ϕ

∂y_{j}
^{ε}

ψ^{ε}i
dx dt,
we obtain (keep in mind that (3.13) implies the boundedness in L^{2}(0, T;H^{−1}(Ω))
of the first term in the duality bracket just above)

0 = Z

Ω_{T}×Y×Z×T

w(x, t, y, z, τ)θ(x, t)ϕ(y)∂ψ

∂z_{j}(z)ϑ(τ)dx dt dy dz dτ
which by the arbitrariness ofθ, ϕandϑimplies

Z

Z

w(x, t, y, z, τ)∂ψ

∂zj

(z)dz= 0 for all (x, t, y, τ)∈ΩT ×Y × T,

which proves thatwdoes not depend on the variablez. Similarly, one easily proves
thatwdoes not depend onyby passing to the limit in the following equality (where
w^{ε}(x, t) =θ(x, t)ϕ(^{x}_{ε})ϑ(_{ε}^{t}2) forε∈E^{0} and (x, t)∈ΩT)

−εD ∂

∂x_{j}(∂(Pεuε)

∂t

, w^{ε}E

L^{2}(0,T;H^{−1}(Ω)),L^{2}(0,T;H_{0}^{1}(Ω))

= Z

Ω_{T}

ϑ( t

ε^{2})∂(Pεuε)

∂t h

ε∂θ

∂xj

ϕ^{ε}+θ(∂ϕ

∂yj

)^{ε}i
dx dt.

As for the independence ofu1from the variableτ, we have the following equality,
wherew^{ε}(x, t) =θ(x, t)ϑ(_{ε}^{t}2) forε∈E^{0} and (x, t)∈ΩT

−ε^{2}
ρ^{ε}∂^{2}u_{ε}

∂t^{2} , w^{ε}

L^{2}(0,T;V_{ε}^{0}), L^{2}(0,T;V_{ε})

= Z

ΩT

ρ(x

ε^{2})χZ^{∗}(x

ε^{2})∂(Pεuε)

∂t h

ε^{2}∂θ

∂tϑ^{ε}+θ ∂ϑ

∂τ
^{ε}i

dx dt,
which, after a limit passage asE^{0}3ε→0 (keeping (3.4) in mind) leads to

0 =M_{Z}^{∗}(ρ)
Z 1

0

w(x, t, τ)∂ϑ

∂τdτ for all (x, t)∈Ω_{T}.

HoweverMZ^{∗}(ρ)>0 and the proof is complete.

Remark 4.12. To capture all the microscopic and mesoscopic behaviours of the phenomenon modelled by problem (1.1), one must take test functions of the form

ψ_{ε}(x, t) =ψ_{0}(x, t) +εψ_{1}
x, t,x

ε, t
ε^{2}

+ε^{2}ψ_{2}
x, t,x

ε, x
ε^{2}, t

ε^{2}

,

with ψ0 ∈ D(ΩT),ψ1 ∈ D(ΩT)⊗ C_{#}^{∞}(Y)⊗ C_{per}^{∞}(T) andψ2 ∈ D(ΩT)⊗ C_{per}^{∞}(Y)⊗
C_{#ρ}^{∞}(Z^{∗})⊗ C_{per}^{∞}(T). Theorem 4.9 informs us that the functionu1 does not depend
on the variable τ so that in the homogenization process of problem (1.1) we can
instead use test functions of the form

ψε(x, t) =ψ0(x, t) +εψ1 x, t,x ε

+ε^{2}ψ2 x, t,x
ε, x

ε^{2}, t
ε^{2}

, (4.15)

whereψ1∈ D(ΩT)⊗ C_{#}^{∞}(Y) andψ0, ψ2 are as above.

5. Homogenization process

In this section, we pass to the limit in the limit in the variational formulation of problem (1.1) and formulate the microscopic problem, the mesoscopic problem and the macroscopic problem, successively.