Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

Shane Cooper / Anton Savostianov
Published Online: 2019-08-20 | DOI: https://doi.org/10.1515/anona-2020-0024

Abstract

Homogenisation of global 𝓐ε and exponential 𝓜ε attractors for the damped semi-linear anisotropic wave equation t2uε+ytuεdivaxεuε+f(uε)=g, on a bounded domain Ω ⊂ ℝ3, is performed. Order-sharp estimates between trajectories uε(t) and their homogenised trajectories u0(t) are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator divaxε and its homogenised limit div (ah∇). Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts 𝓐0 and 𝓜0 are established. These results imply error estimates of the form distX(𝓐ε, 𝓐0) ≤ ϰ and distXs(Mε,M0)Cεϰ in the spaces X = L2(Ω) × H–1(Ω) and X = (Cβ(Ω))2. In the natural energy space 𝓔 := H01(Ω) × L2(Ω), error estimates dist𝓔(𝓐ε, Tε 𝓐0) ≤ Cεϰ and distEs(Mε,TεM0)Cεϰ are established where Tε is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.

Keywords: dampedwave equation; global attractor; exponential attractor; homogenisation; homogenization; error estimates

MSC 2010: 35B40; 35B45; 35L70; 35B27

Introduction

In this article we consider the following damped semi-linear wave equation in a bounded smooth domain Ω ⊂ ℝ3 with rapidly oscillating coefficients:

t2uε+ytuεdivaxεuε+f(uε)=g(x),xΩ,t0,(uε,tuε)|t=0=ξ,uε|Ω=0.(0.1)

Such equations appear, for example, in the context of non-linear ascoustic oscillations in periodic composite media (see for example [1]).

For fixed ε > 0, the long-time behaviour of uε has been intensively studied in many works under various assumptions on the non-linearity f and force g. In the context of dissipative PDEs the long-time dynamics can be studied in terms of global attractors. Intuitively speaking, the global attractor is a compact subset of the infinite-dimensional phase space which attracts all trajectories that originate from bounded regions of phase space. Therefore, the global attractor is in some sense a ‘much smaller’ subset of phase space that characterises the long-time dynamics of the system (see for example [2, 3, 4, 5, 6, 7]).

It is well-known that for suitable assumptions on the non-linearity (cf. [2, 7]) that problem (0.1) possesses a global attractor 𝓐ε and an important question to ask, from the point of view of applications, is about the asymptotic structure, with respect to ε, of the global attractor 𝓐ε in the limit of small ε. Asymptotics for global attractors have been studied, in the context of reaction diffusion equations and the damped wave equation, with respect to ‘lower-order’ rapid spatial oscillations in the dampening, non-linearity and/or forces g (see [8, 9, 10, 11]). Yet surprisingly, to the best knowledge of the authors, little or no work has been performed on the asymptotics of attractors for hyperbolic dissipative systems with ‘higher-order’ rapid spatial oscillations such as in (0.1). We mention here the works [12] that perform a quantitative analysis of the asymptotics of global attractors in the context of reaction diffusion equations. We also mention the works [13, 14] that determine the limit-behaviour of global attractors, in the context of reaction-diffusion and hyperbolic equations, for a particular choice of rapidly oscillating coefficients that degenerate in the limit of small period. Aside from the very limited amount of work done on the asymptotics of global attractors for dissipative PDEs with rapid oscillations, no work has been done on the asymptotics of exponential attractors. This article is dedicated to performing these studies for problems of the form (0.1).

In this article we aim to study the long-time behaviour of trajectories uε to (0.1), for small parameter ε, from the point of view of homogenisation theory. In homogenisation theory, the mapping

Aεu:=diva(ε)u,

for periodic uniformly elliptic and bounded coefficients a(⋅), is well-known to converge (in an appropriate sense) in the limit of small ε to

A0u:=div(ahu),

where ah is the ‘effective’ or ‘homogenised’ constant-coefficient matrix associated to a(⋅) (see for example [15] and references therein). As such, it is natural to compare the long-time dynamics of uε to the long-time dynamics of u0 the solution to homogenised problem

t2u0+ytu0divahu0+f(u0)=g(x),xΩ,t0,(u0,tu0)|t=0=ξ,u0|Ω=0.(0.2)

Homogenisation theory has been studied intensively since the 1970’s and amongst the extensive works we focus on works related to quantitative estimates of the form

Aε1A01L(L2(Ω))Cε,(0.3)

where the mappings have been equipped with appropriate boundary conditions. Such (sharp) order-ε results, that are now standard, has been proved by various authors using various techniques (see the monograph [16] for a review of some of these techniques). We mention here the results of particular interest to our article; in the case of bounded domain with Dirichlet or Neumann boundary conditions the order-sharp estimates were proved for the first time in [17, 18] and utilised the (order-sharp) estimate proved in [19, 20] for the whole space (and periodic torus).

While some work has been done to provide order-sharp operator estimates for individual trajectories in the parabolic (cf. [15, 16]) or hyperbolic settings (for smooth enough initial data) (cf. [21, 22, 23, 24]), no work is done on providing order-sharp operator estimates for attractors in dissipative PDEs.

Our first main result is the following estimate1 between the global attractors 𝓐ε and 𝓐0, associated to problem (0.1) and (0.2) respectively, in the energy spaces 𝓔–1 := L2(Ω) × H–1(Ω) and (Cβ(Ω))2 (see Theorem 4.3 and Corollary 4.1):

distE1(Aε,A0)CAε1A01L(L2(Ω))ϰ,dist(Cβ(Ω¯))2(Aε,A0)CAε1A01L(L2(Ω))θϰ,(0.4)

for some ϰ, θ ∈(0, 1). Upon combining this result with the operator estimate (0.3) gives the desired error estimates between global attractors.

The above inequality is new in the homogenisation theory of attractors. Moreover, this result is important from the general perspective as it establishes the upper semi-continuity of global attractors of the damped wave equation in terms of the elliptic part of the PDE. Indeed, in the proof of this result we do not use the asymptotic structure in ε of Sε(t) in terms of S0(t). The arguments are purely operator-theoretic in nature and only require that the elliptic operator is self-adjoint and boundedly invertible (see Section 4). In particular, if Aε and A0 were positive elliptic operators A = div(a∇) and B = div(b∇) for two different matrices a and b, the above continuity result still holds. Additionally, the same can be said for different boundary conditions: one can replace Dirichlet boundary conditions with other types of boundary conditions under the sole requirement that A = div(a∇) defines a self-adjoint operator in L2(Ω) (see Section 7 for details).

Let us say a few words on the method of proof of (0.4). This result is essentially proved by establishing the following (sharp) estimate between trajectories uε(t) and u0(t) for initial data in 𝓐ε (Theorem 4.2):

uε(t)u0(t)L2(Ω)+tuε(t)tu0(t)H1(Ω)MeKtAε1A01L(L2(Ω)),t0.(0.5)

Then, to prove (0.4), we combine this novel estimate with the exponential attraction property of 𝓐0 which is known to hold ‘generically’ on an open dense subset of forces g:

σ>0suchthatforeveryboundedsetBEthefollowingestimateholds:distE(S0(t)B,A0)M(BE)eσt,t0.

Notice that estimate (0.5) is optimal; indeed, upon substituting the right-hand side with ε we arrive at the expected order-sharp estimates in ε (just as in the elliptic case (0.3)).

Aside from (0.4), a natural question to ask is if we can compare the global attractors in the energy space 𝓔 := H01(Ω) × L2(Ω). In general estimates of the form (0.4) are not to be expected in 𝓔 and this is due to the fact that, on the level of asymptotic expansions, the trajectories ∇uε(t) are not close to ∇u0(t) but instead are close to

Jεu0(t,x):=u0(t,x)+εi=13Ni(xε)xiu0(t,x).

Here Ni are the solutions to the so-called auxiliary cell problem (see Section 1). Indeed, in Homogenisation theory it is known that (0.3) does not generally hold in H1(Ω) but rather the following ‘corrector’ estimate

Aε1gJεA01gH1(Ω)CεgL2(Ω),

holds (cf. the above citations on error estimates in homogenisation of elliptic systems). For this reason, we introduce the notion of correction to attractors:

Tεξ:=(Jεξ1,ξ2),ξ=(ξ1,ξ2)A0,

and our next main result is the following corrector estimate (Theorem 5.3):

distE(Aε,TεA0)Cεϰ.(0.6)

To the best of our knowledge, in all previous works, no corrector estimates were provided in the homogenisation of attractors. To prove this result we naturally aim to establish an inequality of the form:

uε(t)Jεu0(t)H01(Ω)MeKtε,t0,(0.7)

for initial data ξ ∈ 𝓐ε. It turns out that for such initial data the trajectory u0(t) does not contain enough regularity for such a result to hold. This issue is due to the hyperbolic nature of the problem and does not appear, for example, in the context of parabolic equations. To overcome this issue we introduce specially prepared initial data ξ0 for the trajectory u0 as follows: ξ01H01(Ω) is the solution to

div(ahξ01)=div(a(ε)ξ1) inΩ.

Then, for such a choice of initial data, we readily establish inequality (0.7) (Theorem 5.2 and Corollary 5.1) and consequently prove (0.6). Such initial data was originally introduced in [25] in the homogenisation (without error estimates) of the linear wave equation.

An important question from the point of view of applications is whether or not the estimates (0.4), (0.6) hold in the symmetric Hausdorff distance

dists(Aε,A0)=max{dist(Aε,A0),dist(A0,Aε)}.

To prove this one would need to show that for sufficiently small ε the global attractor 𝓐ε is in fact (generically) an exponential attractor with exponent, and set of generic forces, independent of ε. Such a result seems reasonable from the perspective of considering 𝓐ε to be an ‘appropriate’ perturbation of the global attractor 𝓐0 and applying the theory of regular attractors, see for example [2, 26]. Such a result has yet to be established and we intend to carry out this study in future work.

That being said, it is known that, in general, global attractors are not continuous (in the symmetric Hausdorff distance) under perturbations and that the rate of attraction can be arbitrarily slow. For this reason the theory of exponential attractors was developed; such exponential attractors are known to be stable under perturbations and attract bounded sets exponentially fast in time. Importantly, exponential attractors also occupy ‘small’ subsets of phase space in the sense that they have finite fractal dimension, cf. [27, 28, 29, 30].

Motivated by the above discussion, and the desire for estimates in the symmetric Hausdorff distance, we also study the relationship between exponential attractors associated to problems (0.1) and (0.2). In fact we construct exponential attractors 𝓜ε and 𝓜0 whose (finite) fractal dimension and exponents of attraction are independent of ε, and we determine the following analogues of (0.4) and (0.6) in the symmetric distance (Theorem 6.1, Corollary 6.1 and Theorem 6.3):

distE1s(Mε,M0)CAε1A01L(L2(Ω))ϰ,dist(Cβ(Ω¯))2s(Mε,M0)CAε1A01L(L2(Ω))θϰ,distEs(Mε,TεM0)Cεϰ.(0.8)

To establish the last inequality above we developed further (in Theorem 6.4) the known abstract construction of exponential attractors of semi-groups to include the case of semi-groups that admit asymptotic expansions (i.e. ‘corrections’ such as 𝓣ε).

We end the introduction with some words on the structure of this article. In Section 1, we formulate precise assumptions on the non-linearity f and the elliptic part of (0.1), (0.2). Also, we recall relevant known well-posedness results as well as results on the existence of global attractors associated with (0.1), (0.2). For the reader’s convenience, details on the corresponding attractor theory is provided in Appendix A. In Section 2, for the dynamical systems generated by problems (0.1), (0.2), we establish existence and smoothness results for an attracting set (which contains the global attractors). These results will be crucial in justifying error estimates between anisotropic and homogenised attractors. In Section 3, we establish the convergence, in the limit of ε → 0, of the anisotropic global attractor 𝓐ε to the homogenised attractor 𝓐0 in the spaces 𝓔–1 and (Cβ(Ω))2. In Section 4, we derive the central (order-sharp) estimate (0.5) on the difference between trajectories uε(t) and u0(t) of the corresponding anisotropic and homogenised problems. Then, based on this, we demonstrate the quantitative estimates (0.4) on the distance between global attractors 𝓐ε and 𝓐0. Estimate (0.6) between the global attractor 𝓐ε and first-order correction Tε 𝓐0 in the energy space 𝓔 is proved in Section 5. Section 6 is devoted to exponential attractors 𝓜ε, 𝓜0 associated with problems (0.1), (0.2) and consists of two parts. In Subsection 6.1, existence of the exponential attractors is proved and estimates (0.8) in 𝓔–1 and (Cβ(Ω))2 are obtained. The results in this section rely on a variant of a standard abstract result on the construction of exponential attractors; this construction is included in Appendix B. In Subsection 6.2, we compare the distance between the exponential attractor 𝓜ε and the first-order correction Tε𝓜0 in the energy space 𝓔. Subsection 6.2 rests on a new abstract theorem, presented in Appendix C, which compares the distance between exponential attractors which admit correction. We discuss, and prove the corresponding results for the cases of Neumann and periodic boundary conditions in Section 7. Some refinements of the results obtained in Sections 2-4 related to boundary corrections in homogenisation theory are the subject of Appendix D.

Notations

We document here notations frequently used throughout the article. The L2(Ω) inner product is given by (u, v) := ∫Ω u(x)v(x) dx, with norm denoted by ∥u∥ := (u, u)1/2 for u, vL2(Ω). We frequently consider initial data in the energy spaces 𝓔–1 := L2(Ω) × H–1(Ω), and 𝓔 := H01(Ω) × L2(Ω). These spaces are equipped with norms whose squares are given as |ξE12:=ξ12+ξ2H1(Ω)2andξE2:=ξ12+ξ22 for admissible pairs2 ξ = (ξ1, ξ2). For any function z(t) we set ξz(t) to be the pair (z(t), tz(t)) where tz denotes the distributional (time) derivative. For a Banach space E, BE(0, r) denotes the ball centered at 0 of radius r in E; the symbol [⋅]E denotes the closure in E; the one-sided and symmetric Hausdorff distances between two sets A, BE are respectively defined as distE(A, B) := supaA infbBabE and distEs(A, B) := max {distE(A, B), distE(B, A)}. The standard Euclidean basis is denoted by {ek}k=13.

1 Preliminaries

Throughout the article, unless stated otherwise, we adopt the convention that M and K denote generic constants whose precise value may vary from line to line.

For a given matrix a()={aij()}i,j=13 we denote by ah={aijh}i,j=13 the homogenised matrix corresponding to a(⋅) whose constant coefficients are given by the formula

aijh:=Q(aij(y)+k=13aik(y)ykNj(y))dy.

Here Ni, i ∈ {1, 2, 3}, is the solution to the so-called cell problem:

divy(a(y)yNi(y))=divy(a(y)ei),yQ=[0,1)3,QNi(y)dy=0,Ni(+ej)=Ni()j{1,2,3}.(1.1)

It is well-known that if a(⋅) is symmetric, bounded and uniformly elliptic, then so is ah with the exact same bounds (see for example [15, Section 1]). Furthermore, as ah is constant it is clearly periodic. Consequently, both problem (0.1) and (0.2) are problems of the form

t2u+ytudivau+f(u)=g(x),xΩ,t0,(u,tu)|t=0=ξ,u|Ω=0,(1.2)

with the same generic assumptions on coefficients, forces and non-linearity; we collect these assumptions together here:

LetΩR3beaboundedsmoothdomain,gL2(Ω),a()={aij()}i,j=13satisfyingaijL(R3),aij=aji,aij(+ek)=aij(),i,j,k{1,2,3},&ν|η|2a(y)η.ην1|η|2,ν>0,yR3,ηR3;andfC2(R)satisfyingf(s)sK1,f(s)K2,|f(s)|K3(1+|s|),f(0)=0,sR,(H1)

for some positive constants ν, Ki.

Remark 1.1

We note that above assumptions on f imply the following bounds which are important in obtaining dissipative estimates.

  1. There exists K4 > 0 and K5 > 0 such that |f′(s)| ≤ K4(1 + |s|2), |f(s) | ≤ K5(1 + |s|3), s ∈ ℝ.

  2. The anti-derivative3 F(s) = 0s f(τ) satisfies K22s2F(s)f(s)s+K22s2,sR.

  3. For all μ > 0 there exists Kμ > 0 such that F(s) ≥ –Kμμs2, s ∈ ℝ.

Also note that the assumption f(0) = 0 is, in fact, not a restriction since f(0) always can be included into the forcing term g.

We begin with some basic existence, continuity and dissipative estimate results. Particular attention is paid to the dependence of these results on the matrix a, assuming that the other variables (Ω and f) are fixed. As these results are standard we shall omit the proofs, commenting here that they are easily argued by the techniques employed in Appendix A.

Theorem 1.1

Assume (H1). Then, for any initial data ξ ∈ 𝓔, problem (1.2) possesses a unique energy solution u with ξuC(ℝ+; 𝓔). Moreover, the following dissipative estimate is valid:

ξu(t)E2+ttu(τ)2dτM(ξE)eβt+M(g),t0,(1.3)

for some non-decreasing function M and constant β > 0 that depend only on ν.

A consequence of the dissipative estimate (1.3), growth restrictions on f, and uniform ellipticity of a(⋅) we have the following continuous dependence on initial data.

Corollary 1.1

Let u1 and u2 be two energy solutions to problem (1.2) with initial data ξ1, ξ2 ∈ 𝓔 respectively. Then the following estimate

ξu1(t)ξu2(t)EMeKtξ1ξ2E,t0,

holds for some constant M > 0 and K = K(∥ξ1𝓔, ∥ξ2𝓔, ∥g∥, ν).

Additionally, we have the following continuous dependence in 𝓔–1.

Corollary 1.2

Let u1 and u2 be two energy solutions to problem (1.2) with initial data ξ1, ξ2 ∈ 𝓔 respectively. Then the following estimate

ξu1(t)ξu2(t)E1MeKtξ1ξ2E1,t0,

holds for some constant M > 0 and K = K(∥ξ1𝓔, ∥ξ2𝓔, ∥g∥, ν).

We now proceed to study the long-time behaviour of solutions u from the point of view of infinite-dimensional dynamical systems. In particular the problem (1.2) defines a dynamical system (𝓔, S(t)) by

S(t):EE,S(t)ξ=ξu(t),(1.4)

where u is a solution to the problem (1.2) with initial data ξ. The limit behaviour of a dissipative dynamical system as time goes to +∞ can be described in terms of a so-called global attractor. Let us briefly recall its definition (see [2, 3, 5, 7]).

Definition 1.1

Let S(t) : 𝓔 → 𝓔 be a semi-group acting on a Banach space 𝓔. Then a set 𝓐 is called a global attractor for the dynamical system (𝓔, S(t)) if it possesses the following properties:

  1. The set 𝓐 is compact in 𝓔;

  2. The set 𝓐 is strictly invariant:

    S(t)A=A,t0;

  3. The set 𝓐 uniformly attracts every bounded set B of 𝓔, that is

    limt+distE(S(t)B,A)=0.

One can show that if a global attractor exists then it is unique. Also, the following description of the global attractor in terms of bounded trajectories is known (see e. g. [2, 3]):

A={ξ0E:ξ(t)L(R;E),ξ(0)=ξ0,S(t)ξ(s)=ξ(t+s),sR,t0}.(1.5)

Now, the dissipative estimate (1.3) implies the existence of a bounded positively invariant absorbing set 𝓑 ⊂ 𝓔 (which depends only on ν):

S(t)BB,t0.(1.6)

To prove that a global attractor exists for problem (1.2) we utilise the following classical result ([2, 3, 5, 7]).

Theorem 1.2

A dynamical system (𝓔, S(t)) possesses a global attractor 𝓐 in 𝓔 if the following conditions hold:

  1. The dynamical system (𝓔, S(t)) is asymptotically compact: there exists a compact set 𝒦 ⊂ 𝓔 such that

    limt+distE(S(t)B,K)=0,forallboundedsetsBE;

  2. For each t ≥ 0 the operators S(t) : 𝓔 → 𝓔 are continuous.

    Under such conditions, it follows that 𝓐 not only exists but also 𝓐 ⊂ 𝒦.

Note that Corollary 1.1 implies that the evolution operator S(t), given by (1.4), has continuous dependence on the initial data. Let us focus on the existence of a compact attracting set.

Introducing the space

E1:={ξ=(ξ1,ξ2)E|div(aξ1)L2(Ω),ξ2H01(Ω)},ξE12:=div(aξ1)2+ξ22,(1.7)

we have the following known result that states there exists an attracting ball in 𝓔1.

Theorem 1.3

Assume (H1), and let S(t) be the semi-group defined by (1.4). Then, there exists a ball in 𝓔1 that attracts the set 𝓑, from (1.6), in 𝓔. More precisely, the inequality

distE(S(t)B,BE1(0,R))Meβt,t0,

holds for some positive constants R, M and β that depend only on ν.

The proof of Theorem 1.3 is presented for the reader’s convenience in Appendix A and is based on a splitting of trajectory u, into the smooth and contractive parts, that was developed in [31].

Consequently, as 𝓔1 is compact in 𝓔 we see from Theorem 1.3 that 𝒦 = B𝓔1(0, R) is a compact attracting set and, by Theorem 1.2, there exists a global attractor. That is, the following result holds.

Theorem 1.4

Assume (H1). Then, the dynamical system (𝓔, S(t)) given by (1.4) possesses a global attractor 𝓐 ⊂ 𝓔1 such that:

AE1M(g),A=K|t=0,(1.8)

where 𝓚 is the set of bounded energy solutions to problem (1.2) defined for all t ∈ ℝ, cf. (1.5).

2 Smoothness of the global attractor

Above we demonstrated that the global attractor 𝓐 is a bounded subset of 𝓔1. We shall now establish some additional regularity of 𝓐. These results will be used later on to derive homogenisation error estimates.

We are going to show that 𝓐 is contained in the more regular set

E2:={ξE1|(div(aξ1)+g)H01(Ω) and div(aξ2)L2(Ω)},ξE22:=div(aξ1)+gH01(Ω)2+div(aξ1)2+div(aξ2)2,

and that 𝓐 is bounded in the following sense: ∥𝓐∥𝓔2M.

To this end, we shall show that B𝓔1(0, R) is exponentially attracted, in 𝓔, to some ‘ball’ 4

BE2(0,R1):={ξE2|ξE2R1}.

Then by utilising the so-called transitivity property of exponential attraction we establish that 𝓑 (from (1.6)) is attracted to B𝓔2(0, R1) exponentially in 𝓔 and, therefore, we will show that 𝓐 is bounded in 𝓔2.

Let us begin with the following theorem which provides a useful dissipative estimate for problem (0.2) with initial data in 𝓔1 (see (1.7)).

Theorem 2.1

Assume (H1). Then for any initial data ξ ∈ 𝓔1 the energy solution u to problem (1.2) is such that ξuL(ℝ+; 𝓔1), and the following dissipative estimate is valid:

t2u(t)+ξu(t)E1M(ξE1)eβt+M(g),t0,

for some non-decreasing function M and constant β > 0 that depend only on ν.

Since this result is standard we omit the proof. We only remark here that, by differentiating the first equation of (1.2) in time, one first obtains a dissipative estimate for ∥ξtu(t)∥𝓔 which readily implies the uniform bound on ∥div(au)(t)∥.

Remark 2.1

Note that by elliptic regularity we have the inequality

uCα(Ω¯)Cdiv(au),C=C(ν)>0,(2.1)

for sufficiently small α = α(ν) and admissible u. Here Cα(Ω) is the Hölder space of order α:

Cα(Ω¯)={uC(Ω¯):supx,yΩ¯,xy|u(x)u(y)||xy|α<},uCα(Ω¯):=maxxΩ¯|u(x)|+supx,yΩ¯,xy|u(x)u(y)||xy|α.

Thus, we have a dissipative estimate for u, given by Theorem 2.1, in the Cα(Ω) norm.

Consider GH01(Ω) such that – div(aG) = gL2(Ω), and, for initial data ξB𝓔1(0, R), the decomposition of the solution u to (1.2) as follows: u = v + w where

t2v+ytvdiv(av)=0,xΩ,t0,ξv|t=0=(ξ1G,ξ2),v|Ω=0,(2.2)

and

t2w+ytwdiv(aw)=f(u)+g,xΩ,t0,ξw|t=0=(G,0),w|Ω=0.(2.3)

It is clear from standard linear estimates (e.g. Theorem 1.1 for f = g = 0) that

ξv(t)EeβtM(g),t0,(2.4)

for some constant β > 0 and non-decreasing function M that depend only on ν. Additionally, we have the following lemma on the regularity of w.

Lemma 2.1

Assume (H1), ξB𝓔1(0, R) and w solves (2.3). Then

div(aw)(t)+gH01(Ω)+div(atw)(t)M(g),t0,

for some non-decreasing function M that depends only on ν.

Proof

By differentiating the first equation of (2.3) in time and by our choice of initial data (G, 0) we find that p := tw solves

t2p+ytpdiv(ap)=f(u)tu=:G1,xΩ,t0,ξp|t=0=(0,f(ξ1)),p|Ω=0.(2.5)

Moreover, q := tp solves

t2q+ytqdiv(aq)=f(u)|tu|2f(u)t2u=:G2,xΩ,t0,ξq|t=0=(f(ξ1),yf(ξ1)f(ξ1)ξ2),q|Ω=0.

By the dissipative estimate in 𝓔1 (cf. Theorem 2.1 and Remark 2.1) we find that

tu(t)+u(t)Cα(Ω¯)M(g),t0.

This inequality and the conditions on the non-linearity f (see (H1)) imply that

ξp(0)E+G1L(R+;L2(Ω))M(g);ξq(0)E+G2L(R+;L2(Ω))M(g).

Therefore, using the dissipative estimate in 𝓔 ((1.3)) we conclude

p(t)+tp(t)M(g),&q(t)+tq(t)M(g),t0.

Returning back to p = tw, we rewrite (2.5) to find

div(atw)(t)=G1(t)+ytp(t)+tq(t)M(g),t0.

Rewriting the first equation in (2.3), and using cubic growth of f (see Remark 1.1.a) gives

div(aw)(t)+gH01(Ω)=q(t)+yp(t)+f(u(t))H01(Ω)M(g),t0.

Hence, the desired result holds and the proof is complete.□

Combining (2.4) and Lemma 2.1 produces the following result.

Corollary 2.1

Assume (H1) and let S(t) be the semi-group defined by (1.4). Then, there exists aballin 𝓔2 that attracts B𝓔1(0, R) in 𝓔. More precisely, the inequality

distE(S(t)BE1(0,R),BE2(0,R1))Meβt,t0,

holds for some positive constants R1, M and β that depend only on ν.

Let us now recall the so-called transitivity property of exponential attraction (cf. [30, Theorem 5.1] for a proof):

Theorem 2.2

Let E be a Banach space, S(t) a semi-group acting on E, and E1 be a positively invariant subset of E, i.e. S(t) E1E1 for all t ≥ 0, such that

S(t)ξ1S(t)ξ2EM0eK0tξ1ξ2E,ξ1,ξ2E1,

for some constants M0, K0 > 0. Furthermore, assume that there exist subsets E2E1 and E3E such that

distE(S(t)E1,E2)M1eβ1t,distE(S(t)E2,E3)M2eβ2t,t0,

for some M1, M2, β1 > 0 and β2 > 0. Then

distE(S(t)E1,E3)Meβt,t0,

for M = M0 M1 + M2 and β=β1β2K0+β1+β2.

Note that Theorem 1.1 (in particular (1.6)), Theorem 1.3 and Corollary 2.1 imply that the assumptions of the above theorem hold for E = 𝓔, E1 = 𝓑, E2 = B𝓔1(0, R) and E3 = B𝓔2(0, R1). Therefore, we see that B𝓔2(0, R1) attracts the positively invariant absorbing set 𝓑 and, therefore, bounded sets in 𝓔. That is the following result holds.

Theorem 2.3

Assume (H1), S(t) given by (1.4) and B𝓔2(0, R1) given by Corollary 2.1. Then, for every bounded B in 𝓔 the following assertion

distE(S(t)B,BE2(0,R1))M(BE)eβt,t0,

holds for some non-decreasing M and β > 0 that depend only on ν.

We are now ready to prove that the global attractor is bounded in 𝓔2.

Theorem 2.4

Assume (H1) and let 𝓐 be the global attractor of the dynamical system (𝓔, S(t)) given by (1.4). Then

AE2M(g),(2.6)

for some non-decreasing M that depends only on ν.

Remark 2.2

Note that (2.6) implies the following estimate

A(Cα(Ω¯))2M(g),(2.7)

for a non-decreasing function M that depends only on ν and the exponent α from Remark 2.1.

Proof of Thoerem 2.4

The proof follows from the strict invariance of the global attractor (property 2. of Definition 1.1) and Theorem 2.3. Indeed, for an arbitrary δ-neighbourhood 𝓞δ(B𝓔2(0, R1)) of B𝓔2(0, R1) in 𝓔, one has

A=S(t)AOδ(BE2(0,R1)),

for some t = t(δ). Therefore 𝓐 ⊂ [B𝓔2(0, R1)]𝓔 and it remains to note that, since B𝓔2(0, R) is closed in 𝓔, the identity [B𝓔2(0, R1)]𝓔 = B𝓔2(0, R1) holds.□

We end this section with one more result which will be useful later.

Theorem 2.5

Assume (H1). Then, for any initial data ξ ∈ 𝓔2, the energy solution u to problem (1.2) is such that ξuL(ℝ+; 𝓔2) and the following dissipative estimate is valid:

t3u(t)+t2u(t)+ξu(t)E2M(ξE2)eβt+M(g),t0,(2.8)

for some non-decreasing function M and constant β > 0 that depend only on ν > 0.

The proof is very close to the proof of Lemma 2.1 and for this reason is omitted. We only remark that, since 𝓔2 ⊂ 𝓔1 and the dissipative estimate in 𝓔1 is already known, we see that the quantity ∥u(t)∥L(Ω) is bounded. Thus, basically, one applies linear dissipative estimates to the equations for p and q in the proof of Lemma 2.1 with the appropriately changed initial data.

3 Homogenisation and convergence of global attractors

Let us now consider the dynamical systems Sε(t) and S0(t) generated by problems (0.1) and (0.2) respectively. In Theorem 2.4 we established that Sε (respect. S0) has a global attractor 𝓐ε (respect. 𝓐0). Moreover, Theorem 2.4 informs us that 𝓐ε is a, uniformly in ε, bounded subset of Eε2 and 𝓐0 is a bounded subset of E02, where

Eε2:={ξ(H01(Ω))2|(div(a(ε)ξ1)+g)H01(Ω),div(a(ε)ξ2)L2(Ω)},ξEε22:=div(a(ε)ξ1)+gH01(Ω)2+div(a(ε)ξ1)2+div(a(ε)ξ2)2,(3.1)

and

E02:={ξ(H01(Ω))2|(div(ahξ1)+g)H01(Ω),div(ahξ2)L2(Ω)},ξE022:=div(ahξ1)+gH01(Ω)2+div(ahξ1)2+div(ahξ2)2.(3.2)

Remark 3.1

We note that, by elliptic regularity (see Remark 2.1), the global attractors 𝓐ε are uniformly in ε bounded subsets of Eε2 ∩ (Cα(Ω))2. Additionally for 𝓐0, as ah is constant, we can readily deduce that 𝓐0 is a bounded subset of E02 ∩ (H2(Ω))2. That is, the inequalities

AεEε2+Aε(Cα(Ω¯))2M(g),&A0E02+A0(H2(Ω))2M(g),

hold for some non-decreasing function M independent of ε.

The main result of this section is the following theorem which establishes convergence of the global attractors 𝓐ε to the global attractor 𝓐0 in the one-sided Hausdorff distance.

Theorem 3.1

The global attractor 𝓐ε of the problem (0.1) converges to the global attractor 𝓐0 of the homogenised problem (0.2) in the following sense

limε0dist(Cβ(Ω¯))2(Aε,A0)=0,

for any 0 ≤ β < α where α is given in Remark 3.1.

To prove Theorem 3.1 we shall use the following classical homogenisation theorem for elliptic PDEs (see for example [15, Section 1]).

Theorem 3.2

(Homogenisation theorem) Let Ω ⊂ ℝ3 be a bounded smooth domain, a(⋅) a positive bounded periodic matrix and εn → 0 as n → ∞. Then for any sequence gnH–1(Ω) that strongly converges to g in H–1(Ω) we have that unH01(Ω) the weak solution of

div(a(xεn)un)=gn,

weakly converges in H01(Ω) to u0 the weak solution of

div(ahu0)=g.

Remark 3.2

In general, one cannot expect strong convergence of un to u0 in H01(Ω) since this would imply that the homogenised matrix ah is simply the averageQ a(y)dy. Clearly this formula for the homogenised matrix is, in general, not true and it is known that the equality ah = ∫Q a(y)dy holds if, and only if, divy a = 0 in weak sense.

A consequence of the above observation is that, in general, we can not expect convergence of the attractors 𝓐ε to 𝓐0 in the strong topology of 𝓔. To obtain such convergence results a correction to 𝓐0 needs to be made, see Section 5 for further information.

Proof of Theorem 3.1

Fix an arbitrary sequence εn → 0 and ξn ∈ 𝓐εn. To prove the result it is sufficient to show that there exists ξ0 ∈ 𝓐0 such that ξn converges, up to some subsequence, to ξ0 in (Cβ(Ω))2 as n → ∞.

For each n ∈ ℕ, we denote by un ∈ 𝓚εn the bounded (for all time) in 𝓔 solution of (0.1) that satisfies ξun(0) = ξn. Now, 𝓐ε is a (uniformly in ε) bounded subset of (H01(Ω) ∩ Cα(Ω))2 (see Remark 3.1). Moreover, it is well-known that Cα(Ω) is compactly embedded in Cβ(Ω), for any 0 ≤ β < α. Therefore, up to some discarded subsequence,

ξnconverges strongly in(Cβ(Ω¯))2to someξ0(H01(Ω)Cβ(Ω¯))2.(3.3)

It remains to prove that ξ0 ∈ 𝓐0, and this is established if we demonstrate that ξ0 = ξu0(0) for some bounded (for all time) in 𝓔 solution u0 to (0.2). The remainder of the proof is to establish the existence of such a u0. In what follows convergence is meant up to an appropriately discarded subsequence.

By Remark 3.1 and the strict invariance of 𝓐ε (property 2 of Definition (1.1)) there exists M > 0 such that

un(t)+div(a(xεn)un)(t)+un(t)Cα(Ω¯)+tun(t)+div(a(xεn)tun)(t)+tun(t)Cα(Ω¯)M,(3.4)

for all n ∈ ℕ and all t ∈ ℝ.

Let us fix z ∈ ℤ. Using (3.4) we find

un is bounded in W1:={wL([z,z+2];H01(Ω))|twL([z,z+2];L2(Ω))}.

Similarly, since (cf. (0.1))

t2un=ytun+div(a(xεn)un)f(un)+g,(3.5)

assertion (3.4) and the cubic growth condition of f (Remark 1.1(a)) imply that

tun is bounded in W1.

Furthermore, differentiating (3.5) in t gives

t3un=yt2un+div(a(xεn)tun)f(un)tun.

This equation, along with (3.4), the boundedness of tun in W1 and growth assumption on f imply that

t2un is bounded in {wL([z,z+2];L2(Ω))|twL([z,z+2];H1(Ω))}.

Therefore, since the embeddings H01(Ω) ⊂ L2(Ω) and L2(Ω) ⊂ H–1(Ω) are compact, by Aubin-Lions lemma we deduce that

unustrongly inC([z,z+2];L2(Ω))asn;tuntustrongly inC([z,z+2];L2(Ω))asn;t2unt2ustrongly inC([z,z+2];H1(Ω))asn.(3.6)

Let us demonstrate that u solves (0.2) on the time interval [z, z + 2]. To this end we are going to pass to the limit in

div(a(xεn)un)=t2unytunf(un)+g=:hn.(3.7)

Due to (3.6) we know that

hn(t)t2u(t)ytu(t)f(u(t))+g strongly in H1(Ω) for all t[z,z+2].

Therefore, by an application of the homogenisation theorem (Theorem 3.2), we conclude, that for every t ∈ [z, z + 2], un(t) weakly converges in H01(Ω) to the solution u0(t) of the homogenised problem

div(ahu0(t))=t2u(t)ytu(t)f(u(t))+g.

It follows from (3.6) and the weak convergence un(t) ⇀ u0(t) in H01(Ω) that u(t) = u0(t) for all t ∈ [z, z + 2]. Consequently, from this identity and the above equation, we see that u0 (weakly) solves

t2u0+ytu0div(ahu0)+f(u0)=g,t[z,z+2].

Let us argue that the above equation holds for all time. Indeed, by a Cantor diagonalisation argument we see that the convergences (3.6) can be taken to hold for all z ∈ ℤ. Then, by noting that any ϕC0(ℝ; C0(Ω)) can be represented as a finite sum of smooth functions whose individual supports (w.r.t to time) are in some [z, z + 2], we deduce that u0 weakly solves the homogenised equation (0.2). Hence, u0 is a bounded in 𝓔 solution to (0.2) for all time.

It remains to show that ξu0(0) = (u0(0), tu0(0)) equals ξ0. On the one hand, from (3.3) we see that ξn converges strongly to ξ0 in (L2(Ω))2. On the other hand, by (3.6) (for z = 0) ξn = (un(0), tun(0)) converges strongly to (u0(0), tu0(0)) in (L2(Ω))2. Hence, (u0(0), tu0(0)) = ξ0 and the proof is complete.□

4 Rate of convergence to the homogenised global attractor

We shall begin with recalling an important result on error estimates in homogenisation theory of elliptic PDEs. Recall, for fixed ε > 0, the mappings

Aεu:=div(a(ε)u),&A0u:=div(ahu).(4.1)

Theorem 4.1

(Theorem 3.1, [16]). Let Ω ⊂ ℝ3 be a bounded smooth domain, symmetric periodic matrix a(⋅) satisfying uniform ellipticity and boundedness assumptions, Aε and A0 given by (4.1) and gL2(Ω). Let also uε, u0H01(Ω) solve the problems

Aεuε=g,inΩ,uε|Ω=0,&A0u0=g,inΩ,u0|Ω=0.

Then, the following estimate

uεu0Cεg,(4.2)

holds for some constant C = C(ν, Ω).

Remark 4.1

Note that inequality (4.2) is equivalent to the following operator estimate on resolvents:

Aε1A01L(L2(Ω))Cε.

In what follows we wish to compare properties of the semi-groups associated to (0.1) and (0.2) via estimates in terms of ε. In fact, we shall provide stronger estimates in terms of the difference Aε1A01L(L2(Ω)). The mentioned ε estimates then immediately follow by Remark 4.1.

Our first important result is the following continuity estimate.

Theorem 4.2

Let Eε2 be the set (3.1), R > 0. Then, for all ξBEε2(0,R)={ξEε2,ξEε2R}, the inequality

Sε(t)ξS0(t)ξE1MeKtAε1A01L(L2(Ω)),t0,(4.3)

holds for some non-decreasing functions M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Proof of Theorem 4.2

Let us fix ξ, set ξuε(t) := Sε(t)ξ, ξu0(t) := S0(t)ξ, and define rε := uεu0. Then, rε solves

t2rε+ytrε+A0rε=A0uεAεuε+f(u0)f(uε),xΩ,t0,ξrε|t=0=0,rε|Ω=0.(4.4)

By testing the first equation in (4.4) with A01 trε we deduce that

ddt(12(trε,A01trε)+12rε2)+y(trε,A01trε)=(A0uεAεuε,A01trε)+(f(u0)f(uε),A01trε).(4.5)

We compute

(A0uεAεuε,A01trε)=(A0uε,A01trε)(Aεuε,A01trε)=(uε,trε)(Aεuε,A01trε)=(Aεuε,Aε1trε)(Aεuε,A01trε)=(Aεuε,(Aε1A01)trε).

Furthermore,

(Aεuε,(Aε1A01)trε)=ddt(Aεuε,(Aε1A01)rε)(Aεtuε,(Aε1A01)rε).

Therefore, we can rewrite (4.5) as

ddtΛ+y(trε,A01trε)=(Aεtuε,(Aε1A01)rε)+(f(u0)f(uε),A01trε),(4.6)

for

Λ(t):=12(trε(t),A01trε(t))+12rε(t)2(Aεuε(t),(Aε1A01)rε(t)),t0.

We now aim to bound the right-hand-side of (4.6) in terms of Aε1A01L(L2(Ω))2 and Λ, then subsequently apply Gronwall’s inequality and the following standard estimate

νϕH1(Ω)2(ϕ,A01ϕ)ν1ϕH1(Ω)2,ϕH1(Ω)(4.7)

to deduce the desired result.

To this end, let us first estimate the non-linear term. Using the growth restriction on f′ (see Remark 1.1a) and Hölder’s inequality (for exponents (p1, p2, p3) = (3, 2, 6)) we compute

(f(uε)f(u0),A01trε)M((1+|uε|2+|u0|2)|rε|,|A01trε|)M1+|uε|2+|u0|2L3(Ω)rεA01trεL6(Ω).(4.8)

Then, by the Sobolev embedding L6(Ω) ⊂ H1(Ω), the fact that uε and u0 are bounded in 𝓔 (see dissipative estimate (1.3)) and (4.7) we compute

(f(uε)f(u0),A01trε)MrεA01trεH1(Ω)MrεtrεH1(Ω)Mrε(trε,A01trε)12M1(12rε2+12(trε,A01trε)),

for some positive M1. By utilising the above inequality in (4.6) we infer that

ddtΛ(2M1AεuεAεtuε,(Aε1A01)rε)2M1(Aεuε,(Aε1A01)rε)++M1(12rε2+12(trε,A01trε)).

Now, by the dissipative estimate in Eε2 (Theorem 2.5) we have the following uniform bounds in t and ε:

Aεuε(t)+Aεtuε(t)M,t0,ε>0,(4.9)

which we use along with the Cauchy-Schwarz inequality to compute

(2M1AεuεAεtuε,(Aε1A01)rε)MAε1A01L(L2(Ω))2+M12rε2.

By collecting the above inequalities together we deduce that

ddtΛMAε1A01L(L2(Ω))2+2M1Λ.

Consequently, by applying Gronwall’s inequality and the initial data ξrε|t=0 = 0 we have

12(trε(t),A01trε(t))+12rε(t)2(Aεuε(t),(Aε1A01)rε(t))e2M1tMM1Aε1A01L(L2(Ω))2,t0.

Now, we compute

(Aεuε,(Aε1A01)rε)AεuεAε1A01L(L2(Ω))rεAεuε2Aε1A01L(L2(Ω))2+14rε2.

Hence, the above two inequalities along with (4.7) and (4.9) demonstrate (4.3) and the proof is complete.□

Along with Theorem 4.2, to prove error estimates on the distance between global attractors we need the following exponential attraction property of 𝓐0:

thereexistsaconstantσ>0suchthatforeveryboundedsetBEtheestimatedistE(S0(t)B,A0)M(BE)eσt,t0,holdsforsomenondecreasingfunctionM.(H2)

It is known that, for problem (0.2), the property (H2) is a generic assumption in the sense that it holds for an open dense subset of forces gL2(Ω) (cf. [2]).

We are now ready to formulate and prove our main result of this section.

Theorem 4.3

Assume (H1) and (H2). Let 𝓐ε and 𝓐0 be the global attractors of the dynamical systems (𝓔, Sε(t)) and (𝓔, S0(t)) corresponding to the problems (0.1) and (0.2). Then the following estimate

distE1(Aε,A0)MAε1A01L(L2(Ω))ϰ,ϰ=σ(K+σ),(4.10)

holds. Here, K is as in Theorem 4.2, σ as in (H2), and M = M(∥g∥) is a non-decreasing function independent of ε.

Proof

The assertion follows from the already obtained estimate (4.3) and the exponential attraction property (H2). Indeed, let ξεAεBEε2(0,R1) be arbitrary. Then due to (2.6) there exists a complete bounded trajectory ξuε(t) ∈ 𝓚ε, such that ξuε(0) = ξε. Let us fix an arbitrary T ≥ 0 and consider ξT,ε = ξuε(–T) ∈ 𝓐ε. By Theorem 4.2 we deduce

ξεS0(T)ξT,εE1MκeKT,for κ=Aε1A01L(L2(Ω)).

for some M and K which are independent of ε and ξε ∈ 𝓐ε. On the other hand, due to exponential attraction (H2) we have

distE1(S0(T)ξT,ε,A0)MeσT.

Therefore, using the triangle inequality, we derive

distE1(ξε,A0)M(κeKT+eσT).(4.11)

We recall that T ≥ 0 is arbitrary and therefore we choose T that minimizes the right hand side of (4.11). For example, taking T = T(ε) such that κeKT = eσT yields

distE1(ξε,A0)2MAε1A01L(L2(Ω))ϰ,ϰ=σ(K+σ),

and since ξε ∈ 𝓐ε is arbitrary we obtain the desired inequality (4.10).□

To complement the convergence result in Theorem 3.1, we have the following error estimates.

Corollary 4.1

Assume (H1) and (H2). Let α > 0 be given by Remark 2.1, ϰ as in Theorem 4.3 and 0 ≤ β < α. Then the inequality

dist(Cβ(Ω¯))2(Aε,A0)MAε1A01L(L2(Ω))θϰ,θ=αβ2+α,

for some non-decreasing function M = M(∥g∥) which is independent of ε.

Proof

The corollary follows directly from the uniform boundedness of 𝓐ε and 𝓐0 in (Cα(Ω))2 (Remark 3.1), the estimate on the distance between attractors in 𝓔–1 (cf. (4.10)) and the interpolation inequalities

uL(Ω)CuH1(Ω)ϑuCα(Ω¯)1ϑ,uH1(Ω)Cα(Ω¯),where ϑ=α2+α,uCβ(Ω¯)2uCα(Ω¯)β/αuL(Ω)(1β/α),uCα(Ω¯).

5 Approximation of global attractors with error estimates in the energy space 𝓔

In addition to the obtained estimates in Section 4 on the distance in 𝓔–1 we would like to obtain estimates in the energy space 𝓔. Note that we can not expect, in general, convergence of the global attractors in the strong topology of 𝓔, cf. Remark 3.2. As in the elliptic case, estimates in H1(Ω)-norm require involving the correction εi Ni(ε) xiu0 of homogenised trajectories u0. To this end, we introduce the ‘correction’ operator 𝓣ε : H2(Ω) → H1(Ω) given by

Jεw(x):=w(x)+εi=13Nixεxiw(x),xΩ.(5.1)

Here, Ni, i ∈ {1, 2, 3}, are the solutions to the cell problem (1.1).

Now, it is known that Ni, i = 1, 2, 3, are multipliers in H1(Ω) (see [32, Section 13] and [33, Proposition 9.3]); in particular the following non-trivial estimate holds (see [16, Section 3]): there exists C = C(ν, Ω) such that

Ω|yNi(xε)u(x)|2dxCΩ(|u(x)|2+ε2|u(x)|2)dx,uH1(Ω).

Consequently, the following inequality

JεwC(w+εwH2(Ω)),wH2(Ω),(5.2)

holds for some C > 0 independent of ε and w. Indeed, this follows from the above multiplier estimate and the fact NiL(Q) (by elliptic regularity).

Now, we are ready to present the well-known corrector estimate result in elliptic homogenisation theory which improves the L2-estimate given in Theorem 4.1 to H1-norm.

Theorem 5.1

(Theorem 3.1, [16]). Let Ω ⊂ ℝ3 be a bounded smooth domain, periodic matrix a(⋅) satisfying uniform elliptic and boundedness assumptions, Aε and A0 given by (4.1) and gL2(Ω). Let also uε, u0H01(Ω) solve the problems

Aεuε=g,inΩ,uε|Ω=0,&A0u0=g,inΩ,u0|Ω=0.

Then, the following estimate

uεJεu0H01(Ω)Cεg,(5.3)

holds for some constant C = C(ν, Ω).

Remark 5.1

Note that inequality (5.3) is equivalent to the following operator estimate:

Aε1gJεA01gH01(Ω)Cεg,gL2(Ω).

As in Theorem 4.3, we would like to compare the distance between Sε(t) ξ, for ξEε2, to some trajectory for S0 but this time in the energy space 𝓔. However, here the trajectory S0(t) ξ is not a suitable candidate as it does not have the sufficient regularity needed to apply the above corrector estimates. To overcome this difficulty we carefully choose our initial data for the homogenised problem (0.2).

More precisely, let us recall the spaces Eε2,E02 given in (3.1), (3.2), and introduce the bounded linear operator Πε:Eε2E02 given by

Πε(ξ1,ξ2):=(ξ01,ξ02), where the termξ0iH2(Ω)H01(Ω),i=1,2,satisfiesdiv(ahξ0i)=div(aεξi).(5.4)

The operator Πε has the following nice properties.

Lemma 5.1

The operator Πε:Eε2E02 is a bijection that satisfies:

ΠεξE02=ξEε2,ξEε2;(5.5)

Πεξξ(L2(Ω))2Aε1A01L(L2(Ω))ξEε2,ξEε2.(5.6)

Proof

The bijective property and equality (5.5) directly follow from the definitions of Eε2,E02 and the identity Πε (ξ1, ξ2) = (A01 Aεξ1, A01 Aεξ2). Inequality (5.6) follows from the identity

A01Aεξiξi=(A01Aε1)Aεξi.

We now compare Sε(t)ξ with S0(t)Πεξ in 𝓔 for ξEε2. The following result is the direct analogue of Theorem 4.2 when one replaces the initial data ξ by Πεξ in problem (0.2).

Theorem 5.2

Let Eε2 be the set (3.1). Then, for every ξBEε2(0, R), the following inequalities

Sε(t)ξS0(t)ΠεξE1MeKtAε1A01L(L2(Ω)),t0,(5.7)

tSε(t)ξtS0(t)ΠεξE1MeKtAε1A01L(L2(Ω))1/2,t0,(5.8)

hold for some non-decreasing functions M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Proof

First note that inequality (5.7) is a consequence of the Lipschitz continuity of S0 in 𝓔–1 (Corollary 1.2), Lemma 5.1 and (4.3). Indeed,

Sε(t)ξS0(t)ΠεξE1Sε(t)ξS0(t)ξE1+S0(t)ξS0(t)ΠεξE1Sε(t)ξS0(t)ξE1+MeKtξΠεξE1MeKtAε1A01L(L2(Ω)).

It remains to prove (5.8).

Set ξuε(t) := Sε(t)ξ, ξu0(t) := S0(t)Πεξ. We begin by noting the following uniform bounds in t and ε:

t2uε+uε+Aεtuε+t2u0+u0M.(5.9)

Indeed, these bounds are a consequence of identity Πε BEε2(0, R) = BE02(0, R) and the dissipative estimates for uε and u0 in Eε2 and E02 respectively (Theorem 2.5 for a = a(ε) and a = ah respectively).

Now, the difference rε := uεu0 solves

t2rε=ytrε+A0u0Aεuε+f(u0)f(uε),xΩ,t0,ξrε|t=0=ξΠεξ,rε|Ω=0.(5.10)

Note that by the definition of Πε, (5.4), we have

ξtrε|t=0=(ξ2ξ02,y(ξ02ξ2)+f(ξ01)f(ξ1)).

Upon handling the non-linearity as in (4.8), and utilising Lemma 5.1 we conclude that

ξtrε(0)E1CAε1A01L(L2(Ω)).(5.11)

Now, by differentiating the first equation in (5.10) in time (and then adding A0trε to both sides) we find that qε := trε solves

t2qε+ytqε+A0qε=A0tuεAεtuε+f(u0)tu0f(uε)tuε,xΩ,t0,ξqε|t=0=ξtrε(0),qε|Ω=0.

Testing the first equation in the above problem with A01tqε gives

ddt(12(tqε,A01tqε)+12qε2)+y(tqε,A01tqε)=(A0tuεAεtuε,A01tqε)+(f(u0)tu0f(uε)tuε,A01tqε).

We aim to prove the inequality

ddtΛMeKtAε1A01L(L2(Ω))+MΛ,Λ:=12(tqε,A01tqε)+12qε2(5.12)

for some M and K independent of ε and ξ0, which subsequently implies the desired result via an application of Gronwall’s inequality and (5.11). As usual, we shall utilise the H–1-norm equivalence given by (4.7).

So it remains to prove (5.12). By arguing as in Theorem 4.2, we utilise the identity tqε = t2 uεt2 u0 and uniform bounds (5.9) to compute

|(A0tuεAεtuε,A01tqε)|=|(Aεtuε,(Aε1A01)tqε)|AεtuεAε1A01L(L2(Ω))tqεMAε1A01L(L2(Ω)).(5.13)

Let us now handle the non-linear term. We compute

(f(u0)tu0f(uε)tuε,A01tqε)=(f(u0)qε,A01tqε)+((f(u0)f(uε))tuε,A01tqε)=:I1+I2.

The arguments to bound I1 and I2 will use the uniform bounds on uε and u0 given by (5.9).

By the growth condition on f and the H–1-norm equivalence (4.7), we compute

|I1|=|(f(u0)qε,A01tqε)|M((1+|u0|2)|qε|,|A01tqε|)M1+|u0|2L3(Ω)qεA01tqεL6(Ω)MqεtqεH1(Ω)M(12qε2+12(tqε,A01tqε)).

Additionally, by Hölder’s inequality (for exponents (p1, p2, p3, p4) = (6, 2, 6, 6)) we compute

|I2|=|((f(u0)f(uε))tuε,A01tqε)|M((1+|u0|+|uε|)|rε||tuε|,|A01tqε|)M1+|u0|+|uε|L6(Ω)rεtuεL6(Ω)A01tqεL6(Ω)M(12rε2+12(tqε,A01tqε)).

The above assertion and (5.7) imply

|I2|C(e2KtAε1A01L(L2(Ω))2+12(tqε,A01tqε)).

Combining the above calculations leads to the inequality (5.12). The proof is complete.□

The following estimate is an immediate consequence of Theorem 5.2 and standard elliptic theory.

Corollary 5.1

Let Eε2 be the set (3.1), ξBEε2(0, R) and set ξuε(t) := Sε(t)ξ, ξu0(t) := S0(t)Πεξ. Let 𝓣ε be given by (5.1). Then, the following inequality

uε(t)Jεu0(t)H1(Ω)MeKtε,t0,(5.14)

holds for some non-decreasing M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Proof

Note that uεH01(Ω) satisfies the equation

Aεuε=t2uεytuεf(uε)+g=:Fε(t),t0,

and u0H01(Ω) satisfies

A0u0=t2u0ytu0f(u0)+g=:F0(t),t0.

Since ξBEε2(0, R) then by (5.5) we have Πε ξBE02(0, R) and the dissipative estimate in E02 (Theorem 2.5 for a = ah) gives F0L(ℝ+; L2(Ω)). Let us introduce the intermediate function ε = ε(t) ∈ H01(Ω) the solution to

Aεu~ε=F0(t),t0.

Then, by Theorem 5.1 we have

u~ε(t)Jεu0(t)H01(Ω)CεF0(t),t0,

and, since Aε1 is uniformly bounded in 𝓛(H–1(Ω), H01(Ω)), we have

uε(t)u~ε(t)H01(Ω)CFε(t)F0(t)H1(Ω),t0.

Therefore, by the triangle inequality, we have

uε(t)Jεu0(t)H01(Ω)C(εF0L(R+;L2(Ω))+Fε(t)F0(t)H1(Ω)),t0.(5.15)

Now, upon estimating the non-linear term as in the proof of Theorem 5.2, along with utilising Remark 4.1 and Theorem 5.2, we readily deduce that

Fε(t)F0(t)H1(Ω)MeKtε,t0.

The above inequality along with (5.15) imply the desired result and the proof is complete.□

Let us now provide estimates on the distance in the energy space. As in Corollary 5.1 this requires adding an appropriate correction to the attractor 𝓐0. To this end, we introduce the corrector Tε : E02 → (L2(Ω))2 which maps the pair ξ = (ξ1, ξ2) to the pair

Tεξ=(Jεξ1,ξ2).(5.16)

By (5.2), we readily deduce the following inequality: there exists a constant C > 0, independent of ε, such that the inequality

distEs(TεA,TεB)C(distEs(A,B)+εdistE02s(A,B)),A,BE02,(5.17)

holds.

By inequality (5.8) and Corollary 5.1 we have shown the following result.

Corollary 5.2

Let Eε2 be the set (3.1), ξBEε2(0, R) and set ξuε(t) := Sε(t)ξ, ξu0(t) := S0(t)Πεξ. Then, the inequality

Sε(t)ξTεS0(t)ΠεξEMeKtε,

holds for some non-decreasing M = M(R, ∥g∥) and K = K(R, ∥ g∥) independent of ε.

The following estimate on the global attractors in 𝓔 holds.

Theorem 5.3

Assume (H1) and (H2). Let 𝓐ε and 𝓐0 the global attractors of problems (0.1) and (0.2) respectively, and let Tε be given by (5.16). Then, the following estimate

distE(Aε,TεA0)Mεϰ,

holds for some M = M(∥g∥) which is independent of ε. Here ϰ is as in Theorem 4.3.

Proof

The method of proof follows along the same lines as the argument for Theorem 4.3 and so we shall only sketch it here.

For ξε ∈ 𝓐ε and T > 0, consider ξT,ε ∈ 𝓐ε that satisfies Sε(T)ξT,ε = ξ0. Then, by Corollary 5.2 we have

ξεTεS0(T)ΠεξT,εEMeKTε.

Furthermore, by (5.17) we have

distE(TεS0(T)ΠεξT,ε,TεA0)C(distE(S0(T)ΠεξT,ε,A0)+εdistE02(S0(T)ΠεξT,ε,A0)).

Now, to control the second term on the above right we use the fact that Πε 𝓐ε and 𝓐0 are bounded subsets of E02 (see Remark 3.1 and inequality (5.5)) and that we have a dissipative estimate for S0(t) on E02 (see Theorem 2.5). Consequently, we compute

distE(ξε,TεA0)distE(ξε,TεS0(T)ΠεξT,ε)+distE(TεS0(T)ΠεξT,ε,TεA0)M1eKTε+M2distE(S0(T)ΠεξT,ε,A0),

and the remainder of the proof utilises the exponential attraction property of 𝓐0, as in Theorem 4.3.□

Remark 5.2

  1. The appearance of ε in (5.3) is a well-known consequence of the fact that the correction 𝓣εu0 does not approximate well the function uε in a ε-neighbourhood of the boundary. In particular, the reduced power of ε appears in the estimate due to the fact that 𝓣εu0 does not satisfy the Dirichlet boundary conditions and a ‘boundary correction’ is needed. In general, the explicit ε-dependence (i.e. leading-order asymptotics) of this boundary correction is not known.

  2. In certain situations, such as when Ω is the whole space or a torus (see Remark 7.2), there is no need for the boundary correction and, consequently, the error estimate (5.3) is order ε. In such situations we expect order εϰ in our estimate on the distance between global attractors in 𝓔 (Theorem 5.3). As it stands, our argument does not provide such an estimate and this is because the power in the right-hand side of (5.8) is not optimal. This is consciously done to avoid unnecessary complications and we provide an argument in Appendix D that gives the expected power.

  3. Let us return to Remark 3.2. In this case it is interesting to note that estimate (5.3) is order ε. This is simply because the cell solutions Ni are trivial (Ni ≡ 0) and there is no need for boundary corrections; indeed, this can be readily seen by noting that the right-hand-side in problem (1.1) is zero in this situation. Consequently 𝓣ε = I and (under the refinement in Appendix D) we have the following improvement of Theorem 5.3:

    distE(Aε,A0)Mεϰ.

6 Exponential attractors: existence, homogenisation and convergence rates

Let us recall the definition of an exponential attractor for a dynamical system.

Definition 6.1

Let S(t) : 𝓔 → 𝓔, t ≥ 0, be a semi-group acting on a Banach space 𝓔. Then a set 𝓜 is called an exponential attractor for the dynamical system (𝓔, S(t)) if it possesses the following properties:

  1. The set 𝓜 is compact in 𝓔 with finite fractal (box-counting) dimension dimf(𝓜, 𝓔);

  2. The set 𝓜 is positively invariant:

    S(t)MM,t0;

  3. The set 𝓜 exponentially attracts every bounded set B of 𝓔, that is

    distE(S(t)B,M)M(BE)eσt,t0,

    for some non-decreasing M and constant σ > 0.

6.1 Existence of exponential attractors and continuity in 𝓔−1

Let us present our main result for this subsection.

Theorem 6.1

Assume (H1). Then, the dynamical systems (𝓔, Sε(t)), ε > 0 and (𝓔, S0(t)) generated by problems (0.1) and (0.2) respectively possess exponential attractors 𝓜ε, 𝓜0(H01(Ω))2 such that the following properties hold:

  1. div(a(ε)ξ1)+gH01(Ω)+div(a(ε)ξ2)+ξ(Cα(Ω¯))2M(g), for all ξ = (ξ1, ξ2) ∈ 𝓜ε;

  2. div(ahξ1)+gH01(Ω)+div(ahξ2)+ξ(H2(Ω))2M(g), for all ξ = (ξ1, ξ2) ∈ 𝓜0;

  3. For every bounded set B ⊂ 𝓔 one has

    distE(Sε(t)B,Mε)+distE(S0(t)B,M0)M(BE)eσt,t0;

  4. dimf(Mε,E)+dimf(M0,E)D;

  5. distE1s(Mε,M0)MAε1A01L(L2(Ω))ϰ.

Here α is the same as in Remark 2.1 and the constants M > 0, σ > 0, 0 < ϰ < 1 and D ≥ 0 are independent of ε.

Corollary 6.1

Assume (H1). Let α > 0 be given by Remark 2.1, ϰ as in Theorem 6.1 and 0 ≤ β < α. Then the inequality

dist(Cβ(Ω¯))2s(Mε,M0)MAε1A01L(L2(Ω))θϰ,θ=αβ2+α,

for some non-decreasing function M = M(∥g∥) which is independent of ε.

The remainder of the section is dedicated to the proof of Theorem 6.1. First, we recall a variation of an abstract result which establishes the existence of an exponential attractor 𝓜ε, for a parameter-dependent family of semi-groups Sε, whose characteristics are independent of ε (see Appendix B, [29, Theorem 2.10] and [30, Section 3, Theorem 3.1]).

Theorem 6.2

Let 𝓔 be a Banach space and Eε1, ε ≥ 0, be a family of Banach spaces compactly embedded into 𝓔 uniformly in the following sense:

  1. There exists c0 independent of ε ≥ 0 such that ξEc0ξEε1 for all ξEε1;

  2. For all μ > 0, r > 0 there exists a finite cover of BEε1(0, r) consisting of balls radius of μ in 𝓔 with centers 𝓤ε(μ, r) ⊂ BEε1(0, δr), for some δrr, satisfying

    cardUε(μ,r)N(μ,r),

    for some finite N(μ, r) independent of ε.

Let us consider, for each ε ≥ 0, a map defined on 𝓔 such that

Sε:O(Bε)Bε,O(Bε):=Bε+r[0,1]rUε(14K,1)Uε(1K,R),

where the set BεBEε1(0, R) is closed in 𝓔. Furthermore, we assume Sε satisfies the following properties:

  1. for every ξ1 and ξ2 from 𝓞(Bε), the difference Sεξ1Sεξ2 can be represented in the form:

    Sεξ1Sεξ2=vε+wε,withvεE12ξ1ξ2E,wεEε1Kξ1ξ2E,(6.1)

    for K > 0 independent of ε.

  2. Furthermore, there exists a Banach space 𝓔−1 ⊃ 𝓔 such that

    ξE1c1ξE,ξE;S0ξ1S0ξ2E1Lξ1ξ2E1,ξ1O(Bε),ξ2O(B0),

    for constants c−1 and L > 0.

Then, for every ε ≥ 0, the discrete dynamical system (Bε, Sε) possesses an exponential attractor 𝓜ε ⊂ 𝓞(Bε). The exponent of attraction σ > 0 is independent of ε ≥ 0 and dimf(𝓜ε, 𝓔) ≤ D for some positive D independent of ε (see Definition 6.1). Moreover

distE-1s(Mε,M0)C(supξO(Bε)SεξS0ξE-1+distE-1s(Bε,B0)distE-1s(Uε(14K,1),U0(14K,1))+distE-1s(Uε(1K,R),U0(1K,R)))ϰ,(6.2)

where the constants C > 0 and ϰ = ϰ(c0, L, K, δ1) are independent of ε.

The proof of Theorem 6.2 is postponed to Appendix B.

We now move on to the proof of Theorem 6.1. As in the usual way, we first construct exponential attractors for the discrete dynamical systems with maps Sε := Sε(T), S0 := S0(T), for large enough T > 0. Then by a standard procedure, clarified below, one arrives at exponential attractors for the continuous dynamical systems (𝓔, Sε(t)), t ≥ 0.

Proof of Theorem 6.1

  • Step 1

    Construction of discrete exponential attractors. Recall the maps Aε and A0 given by (4.1). Let 𝓔 = H01(Ω) × L2(Ω), 𝓔−1 = L2(Ω) × H−1(Ω), and let Eε1andE01 be given by (1.7) for a(⋅) = a(ε) and a(⋅) = ah respectively). Then property (i) is an immediate consequence of the uniform ellipticity of a(⋅) and Poincaré’s inequality.

  • Proof of (ii)

    We shall provide an explicit construction for the covers. Moreover, it will be important later that we produce a cover such that

    Uε(μ,r)Eε2BEε1(0,δr),&distE1s(Uε(μ,r),U0(μ,r))CrAε1A01L(L2(Ω)),(6.3)

    for some Cr > 0 independent of ε ≥ 0.

    For this reason we seek a cover of BEε1(0, r) in the form

    i=1N(μ,r)BE(ξiε,μ),for ξiε=(Aε1(pi+g),qiε)Eε2.

    To ensure ξ are in Eε2 we see that (pi, q) should belong to (H01(Ω))2 with Aε qi εL2(Ω).

    We now proceed with the construction of such a cover. As L2(Ω) × H01(Ω) is compactly embedded in H−1(Ω) × L2(Ω) then, for each μ̂ > 0, there exist finitely many (pi, qi0), i = 1, …, N(μ̂, r), such that

    BL2(Ω)×H01(Ω)((g,0),r)i=1N(μ^,r)BH1(Ω)×L2(Ω)((pi,qi0),μ^),(pi,qi0)BL2(Ω)×H01(Ω)((g,0),r).

    Additionally, due to density arguments, we can suppose

    (pi,qi0)H01(Ω)×H2(Ω).

    Moreover, as the eigenfunctions of Aε form an orthonormal basis for L2(Ω) we can find q such that Aε qi εL2(Ω) and

    qiεqi0min{μ^,Aε1A01L(L2(Ω))},i=1,,N(μ^,r).(6.4)

    Therefore, we have the covering

    BL2(Ω)×H01(Ω)((g,0),r)i=1N(μ^,r)BH1(Ω)×L2(Ω)((pi,qiε),2μ^),ε0.

    Now, for fixed ξBEε1(0, r) we readily deduce from the ellipticity of a that

    (ξ1Aε1(pi+g))ν1Aεξ1pigH1(Ω).

    Furthermore, it is clear that (Aεξ1g,ξ2)BL2(Ω)×H01(Ω)((g,0),r). Consequently, one can readily check that

    BEε1(0,r)i=1N(μ^,r)BE((Aε1(pi+g),qiε),2(1ν1)μ^).

    Additionally, since q are obtained by truncating qi0 with respect to the eigenfunctions of Aε, we compute

    qiε2ν1(Aεqiε,qiε)ν1(Aεqi0,qi0)ν2qi02,

    and so we deduce that

    (Aε1(pi+g),qiε)BEε1(0,(1ν1)r).

    Hence, upon setting μ^=12(1ν1)μ, we see that the centers

    Uε(μ,r):={(Aε1(pi+g),qiε)|i=1,,N(12(1ν1)μ,r)},ε0,(6.5)

    satisfy (ii) for δr = (1 ∨ ν−1) r. Also the additional desired properties (6.3) hold.

    Construction of Bε and Sε. We set Bε := BEε2(0, R2) to be the absorbing ball provided by Theorem 2.5 for 𝓔2 = Eε2, and a(⋅) = aε in the case ε > 0 and a(⋅) ≡ ah for ε = 0. The radius R2 is independent of ε and clearly Bε is closed in 𝓔.

    Since Bε is an absorbing set in Eε2 and, by (6.3), 𝓞(Bε) is a subset of Eε2, we can choose T1 large enough (and independent of ε) such that Sε := Sε(T), ε ≥ 0, satisfies

    Sε:O(Bε)Bε,O(Bε)=Bε+r[0,1rUε(14K,1)Uε(1K,R).

    Let us verify properties (1) and (2) of Sε.

  • Proof of (1)

    For ξi ∈ 𝓞(Bε) ⊂ Eε2, i = 1, 2, let ξui(t) = Sε(t) ξi. Consider the splitting ui = vi + wi given by (2.2)-(2.3), and set v = v1v2 and w = w1w2.

    As the equation for v is linear then obviously the inequality

    v(T2)E12ξ1ξ2E,

    holds for large enough time T2 (independent of ε).

    From (2.3) we find that w solves

    t2w+ytwdiv(aw)=f(u2)f(u1),xΩ,t0,ξw|t=0=(0,0),w|Ω=0,(6.6)

    for a = a(ε) or aah. Moreover, p = t w solves

    t2p+ytpdiv(ap)=f(u2)tu2f(u1)tu1,xΩ,t0,ξp|t=0=(0,f(ξ21)f(ξ11)),p|Ω=0.(6.7)

    Using the fact that our initial data is from Eε2 we conclude that ui, t ui are bounded in L(Ω) uniformly in ε. Then upon testing the first equation in (6.7) with t p, rewriting the subsequent right-hand-side in the form

    (f(u2)(tu2tu1),tp)+((f(u2)f(u1))tu1,tp),

    we obtain via standard arguments, and the Lipschitz continuity of Sε(t) in 𝓔 (Corollary 1.1), the uniform estimate

    tp(t)+p(t)MeKtξ1ξ2E,t0.

    Consequently, we use p = t w and (6.6) to conclude

    ξw(t)Eε1MeKtξ1ξ2E,t0,

    for some positive constants M and K independent of ε and ξi. Therefore, for T = max{T1, T2}, property (1) holds.

  • Proof of (2)

    This property is given by Corollary 1.2 for aah.

    Hence, the assumptions of Theorem 6.2 hold and therefore Theorem 6.1 holds for the discrete dynamical systems (Bε, Sε(T)) with discrete exponential attractors Mdε. Indeed, Theorem 6.1 (1)-(4) hold due to the choice of Bε and 𝓤ε, and (5) follows from (6.2), (6.3), Theorem 4.2, Lemma 5.1 and the fact that the map Πε : BεB0 is a bijection.

  • Step 2

    Discrete to continuous dynamics. From the discrete exponential attractors Mdε we can build exponential attractors 𝓜ε for the original dynamical systems (𝓔, Sε(t)) by the following standard construction ([5]):

    Mε:=τ[0,T]Sε(τ)Mdε,ε0.(6.8)

Indeed, the properties (1)-(4) can be easily verified due to dissipative estimate in Eε2, Lipschitz continuity with respect to initial data in 𝓔 (Corollary 1.1) on the bounded set Bε:

Sε(t)ξ1Sε(t)ξ2EMξ1ξ2E,ξ1,ξ2Bε,ε0,

and Lipschitz continuity with respect to time:

Sε(τ1)ξSε(τ2)ξEM|τ1τ2|,τ1,τ2[0,T],ξBε,ε0,

for some constant M > 0 (independent of ε). Indeed, the continuity in time follows from the uniform boundedness of Bε in the space Eε1. It remains to check the continuity property (5) for the exponential attractors 𝓜ε. This readily follows from the fact that (5) holds for the discrete exponential attractors Mdε, Theorem 4.2 and the following computation:

distE1s(Mε,M0)=distE1s(τ[0,T]Sε(τ)Mdε,τ[0,T]S0(τ)Md0)supτ[0,T]distE1s(Sε(τ)Mdε,S0(τ)Md0)supτ[0,T]distE1s(Sε(τ)Mdε,S0(τ)Mdε)+supτ[0,T]distE1s(S0(τ)Mdε,S0(τ)Md0)supτ[0,T]MeKτAε1A01L(L2(Ω))+supτ[0,T]LdistE1s(Mdε,Md0).

6.2 Continuity of exponential attractors in 𝓔

Theorem 6.1 (5) demonstrates Hélder continuity between the exponential attractors 𝓜ε and 𝓜0 in the space 𝓔−1. In this section we provide continuity results in the energy space 𝓔. Unlike in 𝓔−1, in the stronger topology of 𝓔 this requires a correction (such as in Definition 5.16) of the exponential attractor 𝓜0. More precisely, the main result of this section is the following theorem.

Theorem 6.3

Assume (H1) and let 𝓜ε, 𝓜0 be the exponential attractors constructed in Theorem 6.1. Then, the following estimate is valid:

distEs(Mε,TεM0)Mεϰ,ε>0,(6.9)

where thecorrectionoperator Tε is given by (5.16), 0 < ϰ < 1 as in Theorem 6.1 and the constant M > 0 is independent of ε.

To prove this result, we make an important development of Theorem 6.2 to provide estimates between exponential attractors which admit correction. That is we establish the following new result.

Theorem 6.4

Let assumptions of Theorem 6.2 be satisfied and 𝓜ε, 𝓜0 be the exponential attractors constructed therein. Additionally, assume that:

  • 3

    for every ε > 0 there exists a bijection Πε:Eε1E01 that satisfies

    ΠεBε=B0;

  • 4

    for every ε > 0 there exists acorrectionoperator Tε : E01 → 𝓔 which possesses the property

    Tεξ1Tεξ2ELcorξ1ξ2E+m(ε)forallξ1,ξ2O(B0);

    for some constant Lcor > 0 independent of ε and positive function m(⋅) with m(0+) = 0.

  • 5

    the maps Sε are uniformly Lipschitz continuous in 𝓔 with respect to ε > 0, that is

    Sεξ1Sεξ2ELξ1ξ2E,ξ1,ξ2O(Bε),

    with some constant L > 1 independent of ε > 0.

Then the following estimate

distEs(Mε,TεM0)C(supξO(B0)SεΠε1ξTεS0ξE+supξO(B0)TεξΠε1ξE+m(ε)distEs(Uε(14K,1),TεU0(14K,1))+distEs(Uε(1K,R),TεU0(1K,R)))ϰ,(6.10)

holds for constant C > 0 independent of ε and ϰ as in Theorem 6.2.

The proof of this result is presented in Appendix C.

Proof of Theorem 6.3

Let the sets Bε, 𝓞(Bε), ε ≥ 0, and the operator Sε = Sε(T) be as in Theorem 6.1.

We first establish, based on the abstract result Theorem 6.4, the estimate (6.9) for the discrete exponential attractors Mdε (defined in the proof of Theorem 6.1). That is we prove the following inequality:

distEs(Mdε,TεMd0)Mεϰ,ε>0,(6.11)

for some constant M > 0.

Let us check that the assumptions of Theorem 6.4 hold. Indeed, assumption (3) follows from the fact that Bε = BEε2(0, R2) (see the proof of Theorem 6.1) and Definition 5.4 of the projector Πε (where we note that Πε can be trivially extended to the map from Eε1 onto E01, preserving the bijection property). Assumption (4) holds with m(ε) = (for some constant C > 0, independent of ε) due to the multiplier estimate (5.17) and the fact that 𝓞(B0) is a bounded subset of E02 by construction. Assumption (5) is a consequence of Corollary 1.1. Hence the assumptions of Theorem 6.4 hold and (6.10) holds for the discrete exponential attractors Mdε and Md0.

Let us now estimate the terms on the right-hand side of (6.10) in terms of ε. Since Πε:Eε2E02 is bijective and preserves the norm (5.1), and since 𝓞(B0) ⊂ Eε2 is bounded, we see that Πε1O(B0)Eε2 = O(B0)E02; that is the set Πε1O(B0) is bounded in Eε2. Therefore, this observation and Corollary 5.2 imply that

supξO(B0)SεΠε1ξTεS0ξE=supξΠε1O(B0)SεξTεS0ΠεξEMε,(6.12)

for some M > 0 independent of ε > 0. Also from the identity

JεA01Aεww=(JεA01Aε1)Aεw,(6.13)

and Remark 5.1 we deduce that

supξO(B0)TεξΠε1ξE=supξΠε1O(B0)TεΠεξξEMε,(6.14)

for some constant M > 0 independent of ε > 0. It remains to compare the distance between the covers present in the right-hand side of (6.10). To this end, we notice that if ξiε:=(Aε1(pi+g),qiε)Uε(μ,r), then

ξiεTεξi0=((Aε1JεA01)(pi+g),qiεqi0),ε>0.(6.15)

Consequently, due to Remark 5.1 and the properties of q (see (6.4)) one can see that

distEs(Uε(μ,r),TεU0(μ,r))Crε,(6.16)

for some constant Cr > 0 independent of ε, μ. Upon collecting the above estimates we derive (6.11).

It remains to establish (6.9) for the exponential attractors 𝓜ε. It is sufficient to show that

distEs(Mε,TεM0)LdistEs(Mdε,TεMd0)+LsupξΠε1O(B0)TεΠεξξE++supτ[0,T]supξΠε1O(B0)Sε(τ)ξTεS0(τ)ΠεξE.(6.17)

Indeed, since ϰ < 1, the above inequality, (6.11), (6.14) and Corollary 5.2 implies (6.9).

Let us demonstrate (6.17):

distEs(Mε,TεM0)=distEs(τ[0,T]Sε(τ)Mdε,τ[0,T]TεS0(τ)Md0)supτ[0,T]distEs(Sε(τ)Mdε,TεS0(τ)Md0)supτ[0,T]distEs(Sε(τ)Mdε,Sε(τ)Πε1Md0)+supτ[0,T]distEs(Sε(τ)Πε1Md0,TεS0(τ)Md0)LdistEs(Mdε,Πε1Md0)+supτ[0,T]distEs(Sε(τ)Πε1Md0,TεS0(τ)Md0)LdistEs(Mdε,TεMd0)+LdistEs(TεMd0,Πε1Md0)+supτ[0,T]distEs(Sε(τ)Πε1Md0,TεS0(τ)Md0)LdistEs(Mdε,TεMd0)+LsupξO(B0)TεξΠε1ξE+supτ[0,T]supξO(B0)Sε(τ)Πε1ξTεS0(τ)ξELdistEs(Mdε,TεMd0)+LsupξΠε1O(B0)TεΠεξξE+supτ[0,T]supξΠε1O(B0)Sε(τ)ξTεS0(τ)ΠεξE.

Hence the theorem is proved. □

7 The case of different boundary conditions

In this section we are going to show that the analogues of the obtained homogenisation error estimates for the global and exponential attractors still hold if we change the Dirichlet boundary conditions to be either Neumann or periodic.

Let Ω ⊂ ℝ3 be a smooth bounded domain and 𝓗1 := H1(Ω) or Ω be a three-dimensional torus 𝕋3 := [0, )3, > 0, with

H1:={uH1(Ω)|u(x+ek)=u(x),k{1,2,3}}.

In both cases we endow 𝓗1 with the norm

uH12:=u2+u2,uH1.

For the maps Aε be given by (4.1), ε ≥ 0, we consider the problem

t2uε+ytuε+(Aε+1)uε+f(uε)=g(x),xΩ,t0,(uε,tuε)|t=0=ξ,(7.1)

endowed with either Neumann

a(ε)uεn|Ω=0,ε>0,ahu0n|Ω=0,ε=0,(N)

or periodic

uε(x+ek)=uε(x),uε(x+ek)=uε(x),k{1,2,3},ε0,(P)

boundary conditions.

It is well-known that problem (7.1) with either boundary conditions (N) or (P) is well-posed in the energy space 𝓔 := 𝓗1 × L2(Ω) and, therefore, defines a dynamical system (𝓔, Sε(t)) where

Sε(t)ξ:=ξuε(t),t0,

for uε(t) the unique solution of the corresponding problem with initial data ξ.

Moreover, is well-known that Aε + 1 : 𝓓(Aε + 1) ⊂ L2(Ω) → L2(Ω) is self-adjoint, where

D(Aε+1)={uH1|AεuL2(Ω),a(ε)un|Ω=0},ε>0,{uH1|A0uL2(Ω),ahun|Ω=0},ε=0,

for condition (N) or

D(Aε+1)={uH1|AεuL2(Ω),u(x+ek)=u(x),k{1,2,3}},ε0,

for condition (P). Setting

Eε2:={ξ(D(Aε+1))2|(Aεξ1g)H1},ξEε22:=Aεξ1gH12+(Aε+1)ξ12+(Aε+1)ξ22,ε0,(7.2)

it is straightforward to see from Appendix A and Sections 2-6 that the following theorem holds.

Theorem 7.1

Assume (H1). Then, for every ε ≥ 0, the dynamical systems (𝓔, Sε(t)) generated by problem (7.1) with boundary conditions (N) or (P) possesses a global attractor 𝓐ε, and exponential attractor 𝓜ε, of finite fractal dimension such that:

AεMεEε2,AεEε2MεEε2M(g),Aε=Kε|t=0,distE(Sε(t)B,Mε)eσtM(BE),t0,forallboundedBE,dimf(Aε,E)dimf(Mε,E)D,

where the constants σ, D > 0 and non-decreasing function M are independent of ε. Here 𝓚ε is the set of all bounded energy solutions to problem (7.1), with (N) or (P), defined for all t ∈ ℝ.

Let us now discuss error estimates between the anisotropic and homogenised attractors. It is known that the main homogenisation results, Theorems 4.1 and 5.1, remain valid for the case of Neumann and periodic boundary conditions.

Theorem 7.2

([16]) Let Ω ⊂ ℝ3 be a bounded smooth domain or three-dimensional torus 𝕋3, ε > 0, periodic matrix a(⋅) satisfying uniform ellipticity and boundedness assumptions, Aε and A0 given by (4.1) and gL2(Ω). Let also uε ∈ 𝓓(Aε + 1), u0 ∈ 𝓓(A0 + 1), solve the equations

(Aε+1)uε=ginΩ,(A0+1)u0=ginΩ.

Then, the following estimates

uεu0Cεg,(7.3)

uεJεu0H1Cεg,(7.4)

hold for some constant C = C(ν, Ω). Here the operator 𝓣ε is given in (5.1).

Remark 7.1

Note that inequalities (7.3) and (7.4) are equivalent to the following operator estimates:

(Aε+1)1(A0+1)1L(L2(Ω))Cε,(Aε+1)1gJε(A0+1)1gH1Cεg,gL2(Ω).

Remark 7.2

In the case of periodic boundary conditions (P), where Q = [0, 1)3 and Ω = [0, )3, if ε ∈ ℕ then for w ∈ 𝓓(A0 + 1) the corrector 𝓣ε w belongs to 𝓗1. In this setting it is well-known that one can improve the bound in (7.4) from ε to ε. Consequently, as discussed in Remark 5.2, for this case we can replace ε with ε in the relevant results below.

Let us also define the energy space of order − 1:

E1:=L2(Ω)×(H1),

where (𝓗1)* stands for the dual space of 𝓗1.

We now draw the reader’s attention to the fact that the key theorems (Theorems 4.2 and 5.2) on the distance between trajectories in 𝓔−1 are in terms of resolvents of the operator Aε, ε ≥ 0. The key point to note is that the proofs of these results essentially rely on the fact Aε is self-adjoint and (uniformly in ε) bounded and positive. Since the operator Aε + 1, for Neumann (N) or periodic (P) boundary conditions, also possesses these properties one can see that analogues of Theorems 4.2-5.2 readily hold (after appropriately changing the projector Πε). Namely, upon defining Πε:Eε2E02, for Eε2 given by (7.2), as follows

Πε(ξ1,ξ2):=(ξ01,ξ02), where the termξ0iD(A0+1),i=1,2,satisfies(A0+1)ξ0i=(Aε+1)ξi,(7.5)

we have the following result.

Theorem 7.3

Let Eε2 be given by (7.2) and Sε(t) be the solution operator to the problem (7.1) with Neumann (N) or periodic (P) boundary conditions. Then, for all ξEε2,ξEε2R, R > 0, the inequalities

Sε(t)ξS0(t)ξE1+Sε(t)ξS0(t)ΠεξE1MeKt(Aε+1)1(A0+1)1L(L2(Ω)),tSε(t)ξtS0(t)ΠεξE1MeKt(Aε+1)1(A0+1)1L(L2(Ω))1/2,Sε(t)ξTεS0(t)ΠεξEMeKtε,t0,

hold for some non-decreasing functions M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Based on Theorem 7.3 and arguing along the same lines as in Sections 4 - 6 we obtain the following theorem on the comparison of distances between anisotropic and homogenised attractors in terms of ε.

Theorem 7.4

Assume (H1) and (H2). Let 𝓐ε, 𝓜ε, ε ≥ 0 be attractors corresponding to problem (7.1), with Neuman (N) or periodic (P) boundary conditions, provided by Theorem 7.1. Let also α > 0 be such an exponent that (Aε + 1)−1 ∈ 𝓛(L2(Ω), Cα(Ω)) and 0 ≤ β < α. Then, the following estimates

distE1(Aε,A0)Mεϰ,distE(Aε,TεA0)Mεϰ,dist(Cβ(Ω¯))2(Aε,A0)Mεθϰ,distE1s(Mε,M0)Mεϰ,distEs(Mε,TεM0)Mεϰ,dist(Cβ(Ω¯))2s(Mε,M0)Mεθϰ,

hold for some non-decreasing M = M(∥g∥) and constants ϰ ∈ (0, 1), θ = αβ2+α independent of ε. Here Tε is thecorrectionoperator defined by (5.16).

A Proof of Theorem 1.3

To prove Theorem 1.3 we perform a splitting of the solution u = v + w to the problem (1.2) into asymptotically contractive and compact parts. This form of splitting was intoduced in [31].

Let us consider

t2v+ytvdiv(av)+Lv+f(u)f(w)=0,xΩ,t0,ξv|t=0=ξu(0),v|Ω=0,(A.1)

and

t2w+ytwdiv(aw)+Lw+f(w)=Lu+g,xΩ,t0,ξw|t=0=0,w|Ω=0,(A.2)

where the fixed constant L > 0 is specified below.

Recall that 𝓑 denotes a positive invariant absorbing set of the semigroup (𝓔, S(t)) (see (1.6)). Similar to Theorem 1.1 we have the following result.

Lemma A.1

Assume (H1), ξu(0) ∈ 𝓑, L > 0 be an arbitrary constant and w solve the equation (A.2). Then the estimate

ξw(t)EML(BE),t0,

holds for some non-decreasing function ML that depends only on ν and L.

The proof of Lemma A.1 follows from the multiplication of the first equation in (A.2) by t w + κ w with sufficiently small κ > 0 and the fact that we already know that ∥ξu(t)∥𝓔M(∥𝓑∥𝓔) for all t ≥ 0 (due to the dissipative estimate (1.3)).

Lemma A.2

Assume (H1), ξu(0) ∈ 𝓑, L > 0 be an arbitrary constant and w solve (A.2). Then, for every μ > 0 the estimate

sttw(τ)2dτμ(ts)+ML(BE)μ,ts0,(A.3)

holds for some non-decreasing function ML that depends only on ν and L.

Proof

Multiplying the equation (A.2) by t w, integrating in Ω and using Lemma A.1 we obtain

ddtΛ+ytw2=L(tu,w)yμ+y1L2ML(BE)μtu2,(A.4)

where

Λ=12(tw2+(aw,w)+Lw2)+(F(w),1)L(u,w)(g,w).

From the dissipative estimate (1.3) and positive invariance (1.6) we see that

sttu(τ)2dτM(BE),ts0.(A.5)

Integrating (A.4) in time from s to t, using Lemma A.1 and (A.5) we derive the desired inequality (A.3) for some new function ML. □

Before continuing, let us recall the following modified Gronwall’s lemma.

Lemma A.3

(Modified Gronwall’s Lemma [31]). Let Λ : ℝ+ → ℝ+ be an absolutely continuous function satisfying

ddtΛ(t)+2μΛ(t)h(t)Λ(t)+k,

where μ > 0, k ≥ 0 and st h(τ) μ(ts) + m, for all ts ≥ 0 and some m ≥ 0. Then

Λ(t)Λ(0)emeμt+kemμ,t0.

We are now ready to show that v exponentially goes to 0 in the energy space 𝓔.

Proposition A.1

Assume (H1) and ξu(0) ∈ 𝓑. Then, for sufficiently large constant L = L(y, ν, f), the estimate

ξv(t)EML(BE)eβt,t0,

holds for some non-decreasing function ML and constant β > 0 that depend only on ν and L.

Proof

Fix κ > 0 to be specified below. Multiplying equation (A.1) by t v + κ v in L2(Ω) we find (after some algebraic manipulation) that

ddtΛ+(yκ)tv2+κ((av,v)+Lv2+(f(u)f(w),v))=(f(u)f(w),twv)12(f(u)tu,|v|2),(A.6)

for

Λ:=12(tv2+(av,v)+Lv2)+κ(tv,v)+κy2v2+(f(u)f(w),v)12(f(u),|v|2).(A.7)

Now by the lower bound on f′ (see (H1)) we compute

Lv2+(f(u)f(w),v)=Lv2+(01f(λu+(1λ)w)dλ,|v|2)(LK2)v2.

Thus, for L > K2, (A.6) implies

ddtΛ+(yκ)tv2+κ(av,v)(f(u)f(w),twv)12(f(u)tu,|v|2).(A.8)

We shall establish below, for sufficiently large L, the equivalence

CνΛ12tv2+12(av,v)2Λ.(A.9)

as well as the inequalities

(f(u)f(w),twv)κ4(av,v)+ML(BE)tw2Λ,(A.10)

12(f(u),tu|v|2)κ4(av,v)+M(BE)tu2Λ,(A.11)

Consequently, for 0 < κ < y /2, inequalities (A.8)-(A.11) imply

ddtΛ+CνκΛhΛ, for h(t)=ML(BE)(tw(t)2+tu(t)2).

This inequality, Lemma A.2 (with small enough μ) and (A.5) show that the assumptions of the Modified Gronwall’s Lemma (Lemma A.3) hold. Whence

Λ(t)M(B)Λ(0)e12Cνκt,t0.

From (A.9), and the fact ξv(0) = ξu(0), we prove the desired result. Therefore, to complete the proof it remains to establish (A.9)-(A.11).

Let us prove (A.9). We shall prove the upper bound, as the argument for the lower bound is similar. For κ ∈ (0, y /2), utilising the dissipative estimate for u (1.3) and the bounds on f′ (see (H1) and Remark 1.1.a) we compute

Λ14tv2+12((av,v)+Lv2)+κ(y2κ)v2+(01f(λu+(1λ)w)dλ,|v|2)12(f(u),|v|2)14tv2+12((av,v)+Lv2)K2v2K42(1+|u|2,|v|2)14tv2+12((av,v)+Lv2)(K2+K42)v2K42uL4(Ω)2v1/2vL6(Ω)3/214tv2+14(av,v)+(L2K2K42M(BE))v2.

Then for large enough L, we deduce Λ14tv2+14(av,v) and the upper bound in (A.9) holds.

To prove (A.10) and (A.11), we use dissipative bounds on u and w (Lemma A.1) plus the growth assumption on f″ to establish

(f(u)f(w),twv)K5(1+|u|+|w|,|tw||v|2)K51+|u|+|w|L6(Ω)tw|v|2L3(Ω)ML(BE)twv2κν4v2+ML(BE)tw2v2,

and

12(f(u),tu|v|2)M(BE)tuvL6(Ω)2κν4v2+M(BE)tu2v2.

Then the desired inequalities follow by invoking the ellipticity of a and the now established (A.9). The proof is complete. □

To complete the proof of Theorem 1.3 it remains to prove that ξw is a bounded trajectory in 𝓔1, this is the subject of the next result.

Proposition A.2

Assume (H1) and ξu(0) ∈ 𝓑. Then, for sufficiently large constant L = L(y, ν, f), the inequality

div(aw)(t)+tw(t)+t2w(t)ML(BE),t0,

holds for some non-decreasing function ML that depends only on ν and L.

Proof

Let us set q := t w, then q solves

t2q+ytqdiv(aq)+Lq+f(w)q=Ltu,xΩ,t0,ξq|t=0=(0,Lu(0)+g),q|Ω=0.

Multiplying the first equation above by t q + κ q and integrating in Ω we find

ddtΛ+(y2κ)tq2+κ(tq2+(aq,q)+Lq2+(f(w),|q|2))=L(tu,tq)+κL(tu,tw)+12(f(w)tw,|q|2),(A.12)

for

Λ:=12(tq2+(aq,q)+Lq2+(f(w),|q|2))+κ(tq,q)+κy2q2.

The identity (A.12) can be rewritten in the form

ddtΛ+(y2κ)tq2+2κΛ=2κ2(tw,tq)+κ2ytw2++L(tu,tq)+κL(tu,tw)+12(f(w),tw|q|2)=:H.(A.13)

Arguing in a similar manner as in the proof of (A.9) we have

CνξqE2Λ(A.14)

for some Cν, as long as L = L( y, ν, f) is large enough. Using the growth condition of f″ (see (H1)), the dissipative estimate for u (1.3), energy estimate for w (Lemma A.1) and arguing as in the proof of (A.10), the right-hand side H(t) can be estimated as follows:

HML(BE)δ+δξqE2+ML(BE)δtw2ξqE2,(A.15)

for any δ > 0. Choosing 0 < κ < y2, δ small, and collecting (A.13), (A.14), (A.15) we derive

ddtΛ+κΛML(BE)+ML(BE)tw2Λ.

Consequently, using Lemma A.2 (with small enough μ) and applying the modified Gronwall’s lemma we determine that

tw(t)+t2w(t)ML(BE),t0.(A.16)

It now readily follows that

div(aw)ML(BE),t0.

Indeed, by rewriting equation (A.2) in the form

div(aw)=t2wytwLwf(w)+Lu+g=:H,xΩ,t0,

then due to Theorem 1.1, Lemma A.1 and (A.16) we see that ∥H(t)∥ ≤ ML(∥𝓑∥𝓔). Hence, the proof is complete. □

B Proof of Theorem 6.2

The proof of Theorem 6.2 is an adaptation of a construction for exponential attractors presented in [29, Theorem 2.10]. The difference here is one needs to keep track on the parameter dependence of all the sets used in the construction and incorporate the fact we compare the symmetric distance in a topology different to that in which the exponential attractors are constructed. For the reader’s convenience we shall provide the details here.

B.1 Construction of the exponential attractors

Let us introduce notations for the ‘starting’ cover 𝓤ε(1K, R) and the ‘model’ cover 𝓤ε(14K, 1):

V0(ε):=Uε(1K,R),U(ε):=Uε(14K,1)={ξiε}i=1N,ε0,

where N0 := card𝓥0(ε) = N(1K,R)andN:=N(14K,1) are, by assumption, independent of ε ≥ 0.

We shall begin with constructing a family of sets 𝓥k(ε), k ∈ ℕ, that satisfy5

Vk(ε)O(Bε),Sε(k)BεξVk(ε)BE(ξ,1K34k),kN,ε0.(B.1)

Note that, by the assumptions of Theorem 6.2, the above property holds for k = 0. We now assume that the set 𝓥k(ε) exists, for some fixed k, and are going to construct from it the set 𝓥k+1(ε). From (B.1) it follows that

Sε(k+1)BεξVk(ε)SεBE(ξ,1K34k),ε0.

Let us consider an element SεζSεBE(ξ,1K34k) for some ξ ∈ 𝓥k(ε). Due to the splitting (6.1) we have

SεζSεξ=vε+wε,vεE12K34k,wεEε134k,ε0.

Therefore, by using the model cover 𝓤(ε) of BEε1(0, 1), we see that

wεBEε1(0,34k)i=1NBE34kξiε,14K34k.

Since Sε ζ = Sεξ + vε + wε we deduce that

Sε(k+1)BεξVk(ε)i=1NBESεξ+34kξiε,1K34k+1,ε0.

As ξi,εEε1δ1 we conclude that (B.1) holds for

Vk+1(ε):=SεVk(ε)+34kU(ε)O(Bε),kZ+,ε0.(B.2)

Now, it is straightforward to verify the following properties of 𝓥k(ε):

cardVk(ε)=N0Nk,distE(Sε(k)Bε,Vk(ε))1K34k,distEs(Vk+1(ε),SεVk(ε))c0δ134k,kN,ε0.(B.3)

Based on the sets 𝓥k(ε) we construct the sets Ek(ε) ⊂ 𝓞(Bε):

E1(ε):=V1(ε),Ek+1(ε):=Vk+1(ε)SεEk(ε),kN,ε0,(B.4)

that clearly satisfy

cardEk(ε)kN0Nk,SεEk(ε)Ek+1(ε),distE(Sε(k)Bε,Ek(ε))<1K34k,kN,ε0.(B.5)

We shall now demonstrate that the sets

Mε:=M^εE,M^ε:=k=1Ek(ε),ε0,(B.6)

are exponential attractors for the discrete dynamical systems (Bε, Sε). To this end we use the following result.

Lemma B.1

Let the assumptions of Theorem 6.2 hold and the sets Ek(ε), k ∈ ℕ, ε ≥ 0, be given by (B.4). Then, there exist constants M1 = M1(c0, K, δ1) > 0 and ω = ω(c0, K, δ1) ∈ (0, 1) (both independent of ε) such that for all ε ≥ 0 we have

distE(Ek(ε),Sε(n)Bε)M134ωk,forallnN,kN:knω.

The proof of this lemma, basically, repeats the proof of Lemma 2.3 from [29], so we omit the proof.

Now, we are ready to verify that the constructed sets 𝓜ε satisfy Definition 6.1. The positive invariance and the uniform exponential attraction property (with σ = ln43)

distE(Sε(k)Bε,Mε)1K34k,kN,ε0,(B.7)

follow directly from (B.5)2, (B.5)3 and (B.6). From the construction it also follows that 𝓜ε ⊂ 𝓞(Bε) and thus 𝓜ε is compact in 𝓔 for every ε ≥ 0. Let us check that dimf(𝓜ε, 𝓔) ≤ D uniformly with respect to ε ≥ 0. To this end we need to estimate the minimal number Nr(𝓜ε, 𝓔) of open balls with radius r > 0 in 𝓔 needed to cover 𝓜ε. Note that, since the cover is open, Nr(𝓜ε, 𝓔) = Nr(𝓜̂ε, 𝓔). We argue that for any r > 0 there exist kr ∈ ℕ and nr ∈ ℕ (independent of ε) such that

distE(k=kr+1Ek(ε),Vnr(ε))<r,ε0.(B.8)

Indeed, let kr and nr be parameters, then by the triangle inequality we have

distE(k=kr+1Ek(ε),Vnr(ε))distE(k=kr+1Ek(ε),Sε(nr)Bε)+distESε(nr)Bε,Vnr(ε),ε0.

Using (B.3)2 and taking nr6 1ln(4/3)ln2rK0+1 we obtain

distESε(nr)Bε,Vnr(ε)<r2,ε0.

Also applying Lemma B.1 for any kr ∈ ℕ such that krnrω, we find that

distE(k=kr+1Ek(ε),Sε(nr)Bε)M134ωkrM134nr<r2,ε0,

if nr1ln(4/3)ln2M1r0+1. Therefore (B.8) is valid for nr and kr of the form

nr=1ln(4/3)ln1r0+C1(c0,K,δ1),kr=1ωln(4/3)ln1r0+C2(c0,K,ω,δ1).

Using the control on the number of elements for 𝓥k(ε) and Ek(ε), (B.6) and (B.8) we can estimate Nr(𝓜̂ε, 𝓔) as follows

Nr(M^ε,E)k=1krcardEk(ε)+cardVnr(ε)k=1krkN0Nk+N0Nnr(kr2+1)N0Nkr.

This estimate readily yields

dimf(Mε,E):=lim supr+0lnNr(Mε,E)ln1rlnNωln(4/3)=:D,ε0.(B.9)

B.2 Estimate on the symmetric distance

Derivation of the estimate on the symmetric distance distE1s(Mε,M0) relies on the following result.

Lemma B.2

Let the assumptions of Theorem 6.2 hold and the sets Ek(ε), k ∈ ℕ, ε ≥ 0, be given by (B.4). Then for all k ∈ ℕ and ε ≥ 0 the following estimate

distE1s(Ek(ε),Ek(0))MLk(supξO(Bε)SεξS0ξE1+distE1s(Uε(14K,1),U0(14K,1))+distE1s(Uε(1K,R),U0(1K,R)),(B.10)

holds for some constant M = M(L) independent of ε and k.

Proof

Fix ε ≥ 0.

  • Step 1

    We first establish (B.10) for the sets 𝓥k(ε), 𝓥k(0). To this end it is convenient to introduce the notations

    dk:=distE1s(Vk(ε),Vk(0)),kZ+,d^0:=distE1s(U(ε),U(0));s0:=supξO(Bε)SεξS0ξE1.

    It is sufficient to establish that the following recurrent chain of inequalities

    dk+1s0+d^0+Ldk,kZ+.(B.11)

    Indeed, upon iterating these inequalities one finds

    dkLk+11L1(s0+d^0+d0),kZ+.(B.12)

    Let us prove (B.11). Note that, from the construction of 𝓥k(ε) (B.2), we readily have the following inequalities

    distE1s(Vk+1(ε),Vk+1(0))distE1s(SεVk(ε),S0Vk(0))+d^0,kZ+.(B.13)

    Let us now verify the inequality

    distE1s(SεA,S0C)s0+LdistE1s(A,C), for all AO(Bε),CO(B0).(B.14)

    Fixing arbitrary aA, cC and using Lipschitz continuity of S0 in 𝓔−1 we obtain

    SεaS0cE1SεaS0aE1+S0aS0cE1SεaS0aE1+LacE1s0+LacE1.

    Consequently (B.14) holds. Hence, upon combining (B.13) with (B.14), we deduce (B.11) and step 1 is complete.

  • Step 2

    We claim that the sets Ek(ε), Ek(0) satisfy the same inequality as in (B.12), namely

    distE1s(Ek(ε),Ek(0))Lk+11L1(s0+d0+d^0),kN.(B.15)

Since E1(ε) = 𝓥1(ε) for all ε ≥ 0, the above inequality is true for k = 1. Assume (B.15) holds for k = m and let us verify it for k = m + 1. It is straightforward to check that for any A1, A2 ⊂ 𝓞(Bε), C1, C2 ⊂ 𝓞(B0) the following inequality

distE1s(A1A2,C1C2)distE1s(A1,C1)distE1s(A2,C2)(B.16)

holds. Therefore, due to (B.4), it is enough to show that

distE1s(SεEm(ε),S0Em(0))(s0+d0+d^0)Lm+21L1.

This inequality is a direct consequence of (B.14) and the induction assumption. Indeed, we compute

distE1s(SεEm(ε),S0Em(0))s0+LdistE1s(Em(ε),Em(0))(s0+d0+d^0)1+LLm+11L1=(s0+d0+d^0)Lm+21L1,

as required. Hence, inequality (B.15) yields the desired result with M(L)=LL1.

We proceed to the proof of the estimate (6.2) on the distance distE1s(Mε,M0). We fix ε ≥ 0 and set

d~:=supξO(Bε)SεξS0ξE1+distE1s(Bε,B0)+distE1s(Uε(14K,1),U0(14K,1))+distE1s(Uε(1K,R),U0(1K,R)).(B.17)

In fact, we will only demonstrate how to obtain the estimate (6.2) for dist𝓔−1(𝓜ε, 𝓜0) as the other side (dist𝓔−1(𝓜0, 𝓜ε)) can be done similarly. Let k ∈ ℕ be arbitrary and fix ξεEk(ε). Due to the just proved Lemma B.2 we have

distE1(ξε,M0)distE1(ξε,Ek(0))MLkd~,kN,ε0.(B.18)

On the other hand, we will show below that

distE1(ξε,M0)M(d~Lnω+34n),for all nN,kN:knω.(B.19)

Using (B.18) for knω and (B.19) we deduce that

distE1(ξε,M0)Md~Lnω+34n,(B.20)

for some M = M(c0, c−1, K, L, δ1) which is independent of ε. Optimizing n in the above inequality, for example taking n=ωωln(4/3)+Llnωln(4/3)d~lnL0, we conclude the desired estimate (6.2) with ϰ=ωln(4/3)ωln(4/3)+ln(L).

It remains to prove (B.19). By the triangle inequality we have

distE-1(ξε,M0)distE-1(ξε,Sε(n)Bε)+distE-1(Sε(n)Bε,S0(n)B0)+distE-1(S0(n)B0,M0).(B.21)

Let us estimate each of the terms on the right hand side of (B.21) separately. Using Lemma B.1 and considering knω we obtain

distE1(ξε,Sε(n)Bε)M134n,for all nN,kN:knω.(B.22)

Iterating (B.14) we find

distE1(Sε(n)Bε,S0(n)B0)(s0+distE1s(Bε,B0))Ln+1L1d~Ln+1L1.(B.23)

Finally, due to the continuous embedding 𝓔 ⊂ 𝓔−1 (assumption (2)) and the exponential attraction property of 𝓜0 (B.7) we see that

distE1(S0(n)B0,M0)c1distE(S0(n)B0,M0)c11K34n,nN.(B.24)

Hence (B.19) holds and the proof is complete.

C Proof of Theorem 6.4

Derivation of the estimate on the symmetric distance with correction relies on the following interesting modification of Lemma B.2.

Lemma C.1

Let the assumptions of Theorem 6.4 hold and the sets Ek(ε), k ∈ ℕ, ε ≥ 0, be given by (B.4). Then for all k ∈ ℕ and ε ≥ 0 the following estimate

distEs(Ek(ε),TεEk(0))MLk(supξO(B0)SεΠε1ξTεS0ξE+supξO(B0)TεξΠε1ξE+distEs(Uε(14K,1),TεU0(14K,1))+distEs(Uε(1K,R),TεU0(1K,R))),(C.1)

holds with some constant M = M(L) which is independent of ε and k.

Proof

We follow the strategy of Lemma B.2 and fix ε ≥ 0.

We first derive an estimate on the distance between 𝓥k(ε) and Tε 𝓥k(0). Let us introduce the notations

dk:=distEs(Vk(ε),TεVk(0)),kZ+,d^0:=distEs(U(ε),TεU(0));s0:=supξO(B0)SεΠε1ξTεS0ξE+LsupξO(B0)Πε1ξTεξE.

We are going to verify the recurrent chain of inequalities

dk+1s0+d^0+Ldk,kZ+.(C.2)

From the construction of 𝓥k(ε) (B.2) we see

distEs(Vk+1(ε),TεVk+1(0))distEs(SεVk(ε),TεS0Vk(0))+d^0,kZ+.(C.3)

We now argue that

distEs(SεA,TεS0C)s0+LdistEs(A,TεC),for all AO(Bε),CO(B0).(C.4)

Indeed, fixing aA, cC and using the uniform (with respect to ε > 0) Lipschitz continuity of Sε in 𝓔 (assumption (5) of Theorem 6.4) we compute

SεaTεS0cESεaSεΠε1cE+SεΠε1cTεS0cELaΠε1cE+SεΠε1cTεS0cELaTεcE+LTεcΠε1cE+SεΠε1cTεS0cE.

The above inequality, obviously, implies (C.4). Combining (C.3) and (C.4) we establish the recurrent inequalities (C.2) which yield

dk(s0+d0+d^0)Lk+11L1,kZ+.

To derive the estimate (C.1) on the distance distEs(Ek(ε),TεEk(0)) we simply argue as in Step 2 of Lemma B.2. □

We are ready to prove the theorem. We fix ε ≥ 0 and set

d~:=supξO(B0)SεΠε1ξTεS0ξE+supξO(B0)TεξΠε1ξE+distEs(Uε(14K,1),TεU0(14K,1))+distEs(Uε(1K,R),TεU0(1K,R)).

As in the proof of Theorem 6.2 we will only consider dist𝓔(𝓜ε, Tε𝓜0) as the other side can argued in a similar manner. Let k ∈ ℕ and ξεEk(ε) be fixed. Then according to Lemma C.1 we have

distE(ξε,TεM0)distE(ξε,TεEk(0))MLkd~,kN,ε0.(C.5)

On the other hand we deduce below that

distE(ξε,TεM0)M(d~Lnω+34n)+m(ε),knω,nN,(C.6)

for ω given in Lemma B.1. The estimate (C.5) for knω together with (C.6) implies

distE(ξε,TεM0)M((d~+m(ε))Lnω+34n),(C.7)

for some M = M(c0, K, L, Lcor, δ1) which is independent of ε. Optimizing n in the above inequality provides the desired result.

It remains to prove (C.6). By the triangle inequality we deduce that

distE(ξε,TεM0)distE(ξε,Sε(n)Bε)+distE(Sε(n)Bε,TεS0(n)ΠεBε)+distE(TεS0(n)ΠεBε,TεM0).(C.8)

The first term on the right hand side of (C.8) can be controlled by Lemma B.1 for knω:

distE(ξε,Sε(n)Bε)M134n.(C.9)

By the identity Πε Bε = B0 (assumption (3) of Theorem 6.4) and iterations of (C.4) we estimate the second term on the right hand side of (C.8):

distE(Sε(n)Bε,TεS0(n)ΠεBε)=distE(Sε(n)Bε,TεS0(n)B0)s0Ln1L1d~LL1Ln.(C.10)

The last term on the right hand side of (C.8) can be estimated using Πε Bε = B0 and the property of Tε (assumption (4) of Theorem 6.4) and the exponential attraction property of 𝓜0:

distE(TεS0(n)ΠεBε,TεM0)=distE(TεS0(n)B0,TεM0)LcordistE(S0(n)B0,M0)+m(ε)Lcor1K34n+m(ε).(C.11)

Hence (C.6) follows from (C.8)-(C.11) and the theorem is proved.

D On the refinement of inequality (5.8)

Let us begin by noting that in Section 5 we were actually in the position to prove the following improvement of inequality (5.8) (in Theorem 5.2).

Proposition D.1

For every ξBEε2(0,R) the inequality

tSε(t)ξtS0(t)ΠεξE1MeKtAε1A01L(L2(Ω))2/3,t0,(D.1)

holds for some non-decreasing functions M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Proof

The proof of this result follows along the same lines as in the proof of Theorem 5.2 except for the following minor alterations:

  1. In the uniform bounds (5.9) (due to Theorem 2.5) we actually have

    t2uεH01(Ω)2+t2u0H01(Ω)2M.

  2. From (1) we can see that qε = t uεt u0 satisfies the bound

    tqεtqεH1(Ω)1/2tqεH01(Ω)1/2MtqεH1(Ω)1/2,

    and so we can improve (5.13) as follows:

    |(A0tuεAεtuε,A01tqε)|=|(Aεtuε,(Aε1A01)tqε)|AεtuεAε1A01L(L2(Ω))tqεMAε1A01L(L2(Ω))tqεH1(Ω)1/2M(34Aε1A01L(L2(Ω))4/3+14tqεH1(Ω)2).

  3. From (2) we can replace (5.12) with

    ddtΛM1eKtAε1A01L(L2(Ω))4/3+M2Λ,Λ:=12(tqε,A01tqε)+12qε2,

    which then leads to the desired result. □

In order to further improve (5.8) (or rather (D.1)), and achieve the optimal bound with power one, we intend to argue as in the proof of Theorem 4.2. For this reason, we require additional regularity on the initial data ξ. In particular, we shall show that it is sufficient for ξEε2 to be such that the solution uε to (0.1) (with initial data ξ) satisfies

Aεt2uεM,t0.

Then, we shall demonstrate that this additional regularity is ‘natural’ in the sense that the global attractor 𝓐ε possesses such smoothness under the additional mild assumption on the non-linearity f:

fC3(R),|f(s)|K6,sR.(H3)

Let us introduce the mapping

Au:=div(au),

recall

E2={ξ(H01(Ω))2|(Aξ1g)H01(Ω) and Aξ2L2(Ω)},ξE22=Aξ1gH01(Ω)2+Aξ12+Aξ22,

and introduce

E3:={ξE2|A(Aξ1+f(ξ1)g)L2(Ω) and Aξ2H01(Ω)},ξE32:=A(Aξ1+f(ξ1)g)2+Aξ22+ξE22.

Our first result is that a dissipative estimate holds in 𝓔3.

Theorem D.1

Assume (H1) and (H2). Then for any initial data ξ ∈ 𝓔3 the energy solution u to problem (1.2) is such that ξuL(ℝ+; 𝓔3) and the following dissipative estimate is valid:

t4u(t)+t3u(t)H01(Ω)+At2u(t)+ξu(t)E3M(ξE3)eβt+M(g),t0,

for some non-decreasing function M and constant β > 0 that depend only on ν > 0.

Proof

We begin by noting that since ξ ∈ 𝓔2 then, by the dissipative estimate in 𝓔2 (Theorem 2.5), ξu(t) := S(t) ξ satisfies

t3u(t)+t2u(t)+ξu(t)E2M(ξE2)eβt+M(g),t0.(D.2)

In particular, we have

u(t)Cα(Ω¯)+tu(t)Cα(Ω¯)M(ξE2)eβt+M(g),t0,(D.3)

where α is given in Remark 2.1.

Now upon differentiating (1.2), in time, three times we deduce that r(t):=t3u(t) solves the equation

t2r+ytr+Ar=f(u)(tu)33f(u)tut2uf(u)t3u=:F(t),t0,

with initial data

r(0)=y2ξ2+y(Aξ1+f(ξ1)g)Aξ2f(ξ1)ξ2,

and

tr(0)=yr(0)+yAξ2+A(Aξ1+f(ξ1)g)f(ξ1)(ξ2)2+f(ξ1)(yξ2+Aξ1+f(ξ1)g).

Now by (D.2) and (D.3) we readily deduce that FL(ℝ+; L2(Ω)). Additionally, since ξ ∈ 𝓔3 we see that r(0) ∈ H01(Ω) and t r(0) ∈ L2(Ω), i.e. (r(0), t r(0)) ∈ 𝓔. Consequently, by standard linear dissipative estimates for r, we find

t4u(t)+t3u(t)H01(Ω)M(ξE3)eβt+M(g),t0,(D.4)

for some M that depends only on ν.

Now, the remaining claims are proven by differentiating (1.2) once to get

Atu(t)H01(Ω)M(ξE3)eβt+M(g),t0.

Then differentiating (1.2) one more time to get

At2u(t)M(ξE3)eβt+M(g),t0,

and finally re-arranging (1.2) to get

A(Au(t)+f(u(t))g)M(ξE3)eβt+M(g),t0.

Equipped with Theorem 2.1 we are ready to prove the desired improvement of (D.1). Namely, upon setting Eε3 to be 𝓔3 for the case a=a(ε)andBEε3(0,R):={ξEε3|ξEε3R}, the following result holds.

Theorem D.2

Assume (H1) and (H2). Then, for every ξBEε3(0, R), the following inequality

tSε(t)ξtS0(t)ΠεξE1MeKtAε1A01L(L2(Ω)),t0,

holds for some non-decreasing functions M = M(R, ∥g∥) and K = K(R, ∥g∥) which are independent of ε > 0.

Proof

The argument is similar to that in Theorem 4.2 so we shall just outline the main ideas.

Set ξuε(t) := Sε(t)ξ, ξu0(t) := S0(t)Πεξ and recall ξ0 = Πε ξ. Then by the dissipative estimates for ξuε in Eε3 (Theorem D.1) and ξu0 in E02 (Theorem 2.5) we have the following uniform bounds in t and ε:

uεH01(Ω)+Aεtuε+Aεt2uε+u0H01(Ω)M.

The difference qε := t uεt u0 solves

t2qε+ytqε+A0qε=A0tuεAεtuε+f(u0)tu0f(uε)tuε,xΩ,t0,ξqε|t=0=(ξ2ξ02,y(ξ02ξ2)+f(ξ01)f(ξ1)),qε|Ω=0,

and we have

ξqε|t=0E1CAε1A01L(L2(Ω)).

After testing the first equation in the above problem with A01tqε and some algebra (similar to that in Theorem 4.2) we deduce that

ddtΛ(Aεt2uε,(Aε1A01)qε)+(f(u0)tu0f(uε)tuε,A01tqε), where Λ:=12qε2+12(tqε,A01tqε)(Aεtuε,(Aε1A01)qε).

Now in the proof of Theorem 5.2 we showed that

|(f(u0)tu0f(uε)tuε,A01tqε)|M1(eKtAε1A01L(L2(Ω))2+12qε2+12(tqε,A01tqε)).

Therefore

ddtΛ(2M1AεtuεAεt2uε,(Aε1A01)qε)2M1(Aεtuε,(Aε1A01)qε)+M1(eKtAε1A01L(L2(Ω))2+12qε2+12(tqε,A01tqε)),

and since

|(2M1AεtuεAεt2uε,(Aε1A01)qε)|CAε1A01L(L2(Ω)2+M112qε2,

we find

ddtΛ2M1Λ+CeKtAε1A01L(L2(Ω))2,

from which the desired result follows. □

We finish this section with the following result on the smoothness of the global attractor.

Theorem D.3

Assume (H1) and (H2), and let 𝓐 be the global attractor of the dynamical system (𝓔, S(t)) given by (1.4). Then

AE3M(g),

for some non-decreasing M that depends only on ν.

Indeed this result can be proved by arguing as in Section 2 for the following splitting: for initial data ξB𝓔2(0, R1) we consider HH01(Ω) that satisfies

div(aH)=f(ξ1)ξ2L2(Ω),

and GH01(Ω) that satisfies

div(aG)=gf(ξ1)yHL2(Ω).

Then, we decompose the solution u to (1.2) as u = v + w where

t2v+ytvdiv(av)=0,xΩ,t0,ξv|t=0=(ξ1G,ξ2H),v|Ω=0,

and

t2w+ytwdiv(aw)=f(u)+g,xΩ,t0,ξw|t=0=(G,H),w|Ω=0.

The main points to highlight are that we can argue as in the proof of Theorem 2.1 (to produce an analogue of Lemma 2.1) and establish that

distE(S(t)BE2(0,R1),BE3(0,R2))Meβt,t0,

holds for some positive constants R2, M and β that depend only on ν. Then, we use the transitivity of exponential attraction (Theorem 2.2) and Corollary 2.1 to deduce that B𝓔3(0, R2) attracts bounded sets in 𝓔:

distE(S(t)B,BE3(0,R2))M(BE)eβt,t0.

This finally allows us to argue as in the proof of Theorem 4.2 to prove Theorem D.3.

Consequently, the improved regularity of the attractor (Theorem D.3) allows us to apply, when appropriate, the improved inequality (Theorem D.2) in obtaining error estimates in homogenisation (cf. Remark 5.2).

Acknowledgement

The authors would like to thank Sergey Zelik for helpful discussions related to this work. S. Cooper was supported by the EPSRC grant EP/M017281/1 (“Operator asymptotics, a new approach to length-scale interactions in metamaterials”).

References

  • [1]

    E. Sánchez-Palencia. Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes in Physics. Springer-Verlag, Berlin-New York, 1980. Google Scholar

  • [2]

    A. V. Babin and M. I. Vishik. Attractors of evolution equations, volume 25 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. Google Scholar

  • [3]

    V. Chepyzhov and M. Vishik. Attractors for equations of mathematical physics, volume 49 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2002. Google Scholar

  • [4]

    J. K. Hale. Asymptotic behavior of dissipative systems, volume 25 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1988. Google Scholar

  • [5]

    A. Miranville and S. Zelik. Attractors for dissipative partial differential equations in bounded and unbounded domains. In Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., pages 103–200. Elsevier/North-Holland, Amsterdam, 2008. Google Scholar

  • [6]

    G. R. Sell and Y. You. Dynamics of evolutionary equations, volume 143 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002. Google Scholar

  • [7]

    R. Temam. Infinite-dimensional dynamical systems in mechanics and physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997. Google Scholar

  • [8]

    G. Chechkin, V. Chepyzhov, and L. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete Contin. Dyn. Syst. Ser. B, 23(3):1133–1154, 2018. Web of ScienceGoogle Scholar

  • [9]

    V. Chepyzhov, M. Conti, and V. Pata. Averaging of equations of viscoelasticity with singularly oscillating external forces. J. Math. Pures Appl. (9), 108(6):841–868, 2017. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    B. Fiedler and M. Vishik. Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies. Adv. Differential Equations, 6(11):1377–1408, 2001. Google Scholar

  • [11]

    M. Vishik and B. Fidler. Quantative averaging of global attractors of hyperbolic wave equations with rapidly oscillating coefficients. Uspekhi Mat. Nauk, 57(4(346)):75–94, 2002. Google Scholar

  • [12]

    M. Efendiev and S. Zelik. Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization. Ann. Inst. H. Poincaré Anal. Non Linéaire, 19(6):961–989, 2002. CrossrefGoogle Scholar

  • [13]

    I. D. Chueshov and L. S. Pankratov. Upper semicontinuity of attractors of semilinear parabolic equations with asymptotically degenerating coefficients. Mat. Fiz. Anal. Geom., 6(1-2):158–181, 1999. Google Scholar

  • [14]

    L. S. Pankratov and I. D. Chueshov. Averaging of attractors of nonlinear hyperbolic equations with asymptotically degenerate coefficients. Mat. Sb., 190(9):99–126, 1999. Google Scholar

  • [15]

    V. V. Jikov, S. M. Kozlov, and O. A. Oleinik. Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif’yan]. Google Scholar

  • [16]

    V. V. Zhikov and S. E. Pastukhova. On operator estimates in homogenization theory. Uspekhi Mat. Nauk, 71(3(429)):27–122, 2016. Google Scholar

  • [17]

    T. Suslina. Homogenization of the Dirichlet problem for elliptic systems: L2-operator error estimates. Mathematika, 59(2):463–476, 2013. Web of ScienceCrossrefGoogle Scholar

  • [18]

    T. Suslina. Homogenization of the Neumann problem for elliptic systems with periodic coefficients. SIAM J. Math. Anal., 45(6):3453–3493, 2013. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    M. Birman and T. Suslina. Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics. In Alexander A. Borichev and Nikolai K. Nikolski, editors, Systems, Approximation, Singular Integral Operators, and Related Topics, pages 71–107, Basel, 2001. Birkhäuser Basel. Google Scholar

  • [20]

    M. Birman and T. Suslina. Periodic second-order differential operators. Threshold properties and averaging. Algebra i Analiz, 15(5):1–108, 2003. Google Scholar

  • [21]

    M. Birman and T. Suslina. Operator error estimates for the averaging of nonstationary periodic equations. Algebra i Analiz, 20(6):30–107, 2008. Google Scholar

  • [22]

    M. Dorodnyi and T. Suslina. Spectral approach to homogenization of hyperbolic equations with periodic coefficients. J. Differential Equations, 264(12):7463–7522, 2018. CrossrefWeb of ScienceGoogle Scholar

  • [23]

    T. Suslina. Spectral approach to homogenization of nonstationary Schrödinger-type equations. J. Math. Anal. Appl., 446(2):1466–1523, 2017. CrossrefGoogle Scholar

  • [24]

    Yu. Meshkova. On operator error estimates for homogenization of hyperbolic systems with periodic coefficients. arXiv: 1705.02531, 2018. Google Scholar

  • [25]

    S. Brahim-Otsmane, G. A. Francfort, and F. Murat. Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. (9), 71(3):197–231, 1992. Google Scholar

  • [26]

    M. I. Vishik, S. V. Zelik, and V. V. Chepyzhov. Regular attractors and their nonautonomous perturbations. Mat. Sb., 204(1):3–46, 2013. Google Scholar

  • [27]

    A. Eden, C. Foias, B. Nicolaenko, and R. Temam. Exponential attractors for dissipative evolution equations, volume 37 of RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar

  • [28]

    M. Efendiev, A. Miranville, and S. Zelik. Exponential attractors for a nonlinear reaction-diffusion system in R3. C. R. Acad. Sci. Paris Sér. I Math., 330(8):713–718, 2000. Google Scholar

  • [29]

    M. Efendiev, S. Zelik, and A. Miranville. Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. Proc. Roy. Soc. Edinburgh Sect. A, 135(4):703–730, 2005. CrossrefGoogle Scholar

  • [30]

    P. Fabrie, C. Galusinski, A. Miranville, and S. Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin. Dyn. Syst., 10(1-2):211–238, 2004. Partial differential equations and applications. Google Scholar

  • [31]

    V. Pata and S. Zelik. A remark on the damped wave equation. Commun. Pure Appl. Anal., 5(3):609–614, 2006. Google Scholar

  • [32]

    V. Maz’ya and T. Shaposhnikova. Theory of Sobolev multipliers, volume 337 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. With applications to differential and integral operators. Google Scholar

  • [33]

    T. Suslina. Homogenization of a periodic parabolic Cauchy problem in the Sobolev space H1(ℝd). Math. Model. Nat. Phenom., 5(4):390–447, 2010. Web of ScienceCrossrefGoogle Scholar

Footnotes

  • 1

    Here distX(A, B) denotes the one-sided Hausdorff metric between sets A and B in the strong topology of X. 

  • 2

    Here we adopt the common clash of notation for (⋅, ⋅) to mean both an inner product and represent a pair in a product space. It will be clear from the context which meaning is appropriate. 

  • 3

    The upper-bound follows from noting that g(s)=0sf(r)drf(s)sK22s2 attains its maximum at s = 0. 

  • 4

    Note that the convex functional ∥⋅∥𝓔2 is not a norm and the set 𝓔2 is an affine subset of 𝓔1. 

  • 5

    Here Sε(k) denotes the kth iteration of Sε. 

  • 6

    Here ⌊c⌋ denotes the largest integer which does not exceed c ∈ ℝ. 

About the article

Received: 2018-09-16

Accepted: 2019-03-01

Published Online: 2019-08-20

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 745–787, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0024.

Export Citation

© 2020 S. Cooper and A. Savostianov, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in