# Quantitative Homogenization of Attractors of Non-Newtonian Filtration Equations

Authors: Emil Novruzov
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

For a rapidly spatially oscillating nonlinearityg we compare solutions uϵ of non-Newtonian filtration equation ∂tβ(uϵ)-D(|Duϵ|p-2Duϵ+φ(uϵ)Duϵ)+g(x,x/ϵ,uϵ)=f(x,x/ϵ) with solutions u0 of the homogenized, spatially averaged equation
∂tβ(u0)-D(|Du0|p-2Du0+φ(u0)Du0)+g0(x,u0)=f0(x). Based on an ε-independent a priori estimate, we prove that ||β(uϵ)-β(u0)||L1(Ω)≤Cϵeρt uniformly for all t≥0. Besides, we give explicit estimate for the distance between the nonhomogenized Aϵ and the homogenized A0 attractors in terms of the parameter ϵ; precisely, we show fractional-order semicontinuity of the global attractors for ϵ↘0:distL1(Ω)(Aϵ,A0)≤Cϵγ.


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## Body

## 1. Introduction

This paper is devoted to the study of nonlinear parabolic equations related to thep-Laplacian operator. The problems related to such type of equation arise in many applications in the fields of mechanics, physics, and biology (non-Newtonian fluids, gas flow in porous media, spread of biological populations, etc.).For example, in the study of the water in filtration through porous media, Darcy’s linear relation(*)V=-K(τ)ωx
satisfactorily describes flow conditions provided that the velocities are small. Here V represents the velocity of the water, τ is the volumetric water (moisture) content, K(τ) is  the hydraulic conductivity, and ω is the total potential, which can be expressed as the sum of a hydrostatic potential ψ(τ) and a gravitational potential z: ω(τ)=ψ(τ)+z. However, from the physical point of view, (*) fails to describe the flow for large velocities. To get a more accurate description of the flow in this case, several nonlinear versions of (*) have been proposed. One of these versions is (1.1)Vα=-K(τ)ωx
which leads us to the equation(**)ut-div{|u|m|∇u|p-2∇u}=f(x,t,u).
For the first time, nonlinear relationship instead of Darcy`s relation are suggested by Dupuit and Forchheimer. This modification, known as Darcy-Forchheimer's relation, has led to much research from experimental, theoretical, and numerical points of view. For example, Darcy-Forchheimer equations are widely used in reservoir engineering and other subsurface applications.One-dimensional (by spatial variable) variant of (**) with p∈[3/2,2) and m>p-1 arises in the study of a turbulent flow of a gas in a one-dimensional porous medium. This phenomenon was first described by Leibenson in [1]. One of the first papers devoted to the existence problem for such type of equation was an initiating paper by Raviart [2]. In [3], the author investigated the regularity problem for more general case (which includes Leibenson's equation after changing |u|σu=v) of non-Newtonian equation as(1.2)∂tβ(v)+divB⃗(x,t,v,∇v)+B0(x,t,v,∇v)=0
with some restriction on the functions B⃗  and B0. Earlier, in [4] authors have investigated existence and the stabilization of solutions (as t↗∞) of nonlinear parabolic equation of  mentioned type describing certain models related to turbulent flows. Also note that there is an extensive literature devoted to the existence, regularity, and the large-time behavior of solutions of (1.2) under various conditions on functions β,B⃗, and B0 (see [2–10] and references therein).In this paper we show that, under natural assumptions on the termsf and g, the longtime behavior of solutions of the equation(1.3)∂tβ(u0)-D(|Du0|p-2Du0+φ(u0)Du0)+g0(x,u0)=f0(x)
can be described in terms of the global attractor Aϵ of the associated dynamical system related to the equation(1.4)∂tβ(uϵ)-D(|Duϵ|p-2Duϵ+φ(uϵ)Duϵ)+g(x,xϵ,uϵ)=f(x,xϵ),
where functions g(x,x/ϵ,uϵ) and f(x,x/ϵ) are constructed according to some ideas previously presented in [11], where authors carry out a quantitative comparison with the averaged or homogenized equations, in particular for quasiperiodic inhomogeneities with Diophantine frequencies (see [11, pages 176–180]). Note that a quantitative homogenization aims at determining a specific rate of convergence. Method of the construction of the homogenized equation allows us to assert that ||β(uϵ)-β(u0)||L1(Ω)≤Cϵeρt (Theorem 3.1 below) uniformly for all t≥0. As we mentioned above, this result requires g and f to depend quasiperiodically on the rapid space variable z=x/ϵ. At the same time, being interested in quantitative strong convergence not only of individual trajectories but also of global attractors, we also show that global attractors Aϵ tend to attractor A0 in a suitable sense providing fractional-order semicontinuity of the global attractors for ϵ↘0. On a related note, the author would like to mention several results on homogenization of the attractor for nonlinear parabolic equations.In [12] the Cauchy problem for parabolic equations on Riemannian manifolds with complicated microstructure has been considered and a connection between global attractors of the initial problem of the homogenized one has been established. The asymptotical behavior of the global attractor of the boundary value problem for semilinear equation(1.5)utϵ-Δuϵ+f(uϵ)=hϵ(x)
was investigated in [13] and it was shown that this tended in a suitable sense to the finite-dimensional weak global attractor of some system of a parabolic p.d.e. coupled with an o.d.e.Also it is necessary to note the papers in [14–16], where the media properties are assumed to be oscillatory, focusing on the homogenization of the attractor for semilinear parabolic equations (see [14]) and systems (see [15]), and of the quasilinear parabolic equations (see [16]).Note that previous approaches on homogenization of attractors are limited to certain types of nonlinear equations. In particular, they assume the main part of the equation to be linear or monotone. The consideration of equation (1.4) is a first step in the investigation of the more general equation(1.6)∂tβ(uϵ)-D(a(xϵ)(|Duϵ|p-2+φ(uϵ))Duϵ)+g(x,xϵ,uϵ)=f(x,xϵ),
where the media properties are assumed to be oscillatory. This often arises in porous media flows [17]. We also note the paper in [18] where authors deal with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero-order term and show that the family of solutions of the studied problem converges in law.Our work is inspired, on one hand, by results of the studies in [7, 8] which are devoted to the existence and regularity of the attractor and, on the other hand, by the paper in [11] that is related to quantitative homogenization of the global attractor for reaction-diffusion systems. For proof of the existence we use sketch of the proof of corresponding result; however, compared to the studies in [7, 8] we consider the equation with an additional nonlinear term φ(s). Note that this term prevents us from applying the result from the paper in [4] which is often used to prove the uniqueness of the solution. Here we use a technique (see [19]) which is based on the fact that the interval [a,b] may be divided into subintervals where the sign of the “difference” u(x,t)-v(x,t) (u(x,t) and v(x,t) are solutions with the same initial condition) does not change for fixed t. Also note that the results of the paper in [11] cannot be directly applied to prove homogenization of the attractor for (1.4) because of nonlinear terms β and φ.Our paper is organized as follows. In Section2 we consider the Dirichlet problem(1.7)∂tβ(u)-D(|Du|p-2Du+φ(u)Du)+g(x,u)=f(x),(1.8)u|Γ=0,Γ=∂Ω×(0,T),(1.9)u(x,0)=u0(x),
where D=∂x,Ω=(a,b) under the following hypotheses(H1)
u0(x) and β(u0) are in L2(Ω),(H2)
β(ζ) is an increasing locally Lipschitzian function from R to R, with β(0)=0.(H3)
g(·,ζ)ζ⩽c1|ζ|k+c2,∂ζg(·,ζ)>-λ, (without loss of generality we suppose that g(·,0)=0) and there exist positive constants c3 and c4 such that sign(ζ)g(x,ζ)≥c3|β(ζ)|q-1-c4 for a. e. (x,t)∈Ω×R and q>2.(H4)
φ(ζ) is a function from the space C1 such that φ(ζ)≥0,φ(0)=0,  and, for each M and |ζ|≤M, |φ′(ζ)|≤(|ζ|l+c)(β′(ζ))α almost everywhere, for some l,c≥0 and α≥1.(H5)
f(x)∈L∞(Ω),p≥4.Under the mentioned condition we show the existence and uniqueness of the solution and the existence of theL2-global attractor.Further, in order to show more regularity of the solution we suppose the following condition.(H6)
For eachM and |ζ|≤M,|g(x,ζ)-f(x)|≤(|ζ|l1+c)(β′(ζ))α1 almost everywhere, for some l1,c≥0 and α1≥1.Finally in Section3, we derive, under conditions (H1)–(H5), the following estimate:(1.10)∥β(uϵ)-β(u0)∥L1(Ω)≤Cϵeρt.Also, under additional condition ong we prove fractional-order semicontinuity of the global attractors for ϵ↘0, namely, the validity of the estimation distL1(Ω)(Aϵ,A0)≤Cϵγ, where Aϵ and A0 are global attractors of the semigroups generated by the problems (1.4), (1.8), (1.9) and (1.3), (1.8), (1.9), respectively.


## 2. Existence and Uniqueness

First, we suppose thatϵ in equation (1.4) is a constant. This leads us to the problem (1.7), (1.8), (1.9) where g(x,u) and f(x) are denoted as g(x,x/ϵ,uϵ) and f(x,x/ϵ) correspondingly.We use the standard regularization of the problem (1.7), (1.8), (1.9):(2.1)∂tβη(u)-D((|Du|p-2+φ(u)+η)Du)+g(x,u)=f(x),(1.8)u|Γ=0,Γ=∂Ω×(0,T),(1.9)u(x,0)=u0(x),
where the sequence βη∈C1(R) is such that βη(0)=0,βη→β in Cloc(R), Mη>βη′>η, and  |βη|≤|β|.Let(u0η)η>0 be a sequence in C0∞(Ω) such that u0η→u0 almost everywhere in Ω and ||u0η||L2(Ω),||βη(u0η)||L2(Ω)≤c, with constant c>0. To show the solvability of the problem (1.8), (1.9), (2.1) we use the result for the classical solvability in the large given by Ladyžhenskaya et al. (see [20, Theorem 4.1(5.2), ch. VI]).Our proof is based on a priori estimates and is similar to that in [7]. First, using the properties of φ and βη, and then multiplying the equation by |βη(uη)|kβη(uη), we deduce, analogously to [7, 8], the following lemma.Lemma 2.1.
Under the hypotheses(H1)–(H5) for any η∈(0,1) the following estimates hold: ||uη||L∞(τ,T;L∞(Ω))⩽c(τ,T),(2.2)||βη(uη)||L∞(0,T;L2(Ω))∩Lq(QT)⩽c(T),||uη||Lp(0,T;W01,p(Ω))⩽c(T).Proof.
Multiplying equation (2.1) by |βη(uη)|kβη(uη) and integrating by parts, we get
(2.3)1k+2ddt∫Ω|βη(uη)|k+2dx+(k+1)∫Ω(|Duη|p-2+φ(uη)+η)|βη(uη)|kβη′(uη)×|Duη|2dx+∫Ωg(x,uη)|βη(uη)|kβη(uη)dx=∫Ωf(x)|βη(uη)|kβη(uη)dx.
By virtue of the positivity of theβη′(uη) and φ(uη) we have
(2.4)1k+2ddt∫Ω|βη(uη)|k+2dx+∫Ωg(x,uη)|βη(uη)|kβη(uη)dx≤∫Ωf(x)|βη(uη)|kβη(uη)dx.
By conditions(H2) and (H3) we derive
(2.5)1k+2ddt∫Ω|βη(uη)|k+2dx+c3∫Ω|βη(uη)|k+qdx≤∫Ωf(x)|βη(uη)|kβη(uη)dx+c4∫Ω|βη(uη)|kβη(uη)dx.
Consequently, from(H5) we obtain
(2.6)1k+2ddt∫Ω|βη(uη)|k+2dx+c3∫Ω|βη(uη)|k+qdx≤(c(f)+c4)∫Ω|βη(uη)|k+1dx.
Further, using Holder's inequality on both sides of the latter inequality, we deduce that there exist two constantsα0>0 and λ0>0 such that
(2.7)ddt||βη(uη)||Lk+2(Ω)+λ0||βη(uη)||Lk+2(Ω)q-1≤α0,
which implies from Ghidaglia's lemma [9] that
(2.8)∥βn(uη)∥Lk+2(Ω)≤(α0λ0)1/(q-1)+1(λ0(q-2)t)1/(q-2)=c(t).
Ask→∞ in (2.8), and for t≥τ>0, we have
(2.9)||βη(uη)(t)||L∞(Ω)≤c(τ),
which implies that
(2.10)||uη(t)||L∞(Ω)≤max(βη-1(c(τ)),|βη-1(-c(τ))|)=δη.
Sinceβη converges to β in Cloc(ℝ), then the sequence δη is bounded in ℝ as η→+∞. Thus δη is bounded by max(βη-1(c(τ)),|βη-1(-c(τ))|), which is finite. Whence ||uη||L∞(τ,T;L∞(Ω))⩽c(τ).
On the other hand, takingk=0 in (2.3), using Holder’s inequality, and integrating over [0,T], we obtain the second estimate of the statement of Lemma 2.1 as
(2.11)||βη(uη)||L∞(0,T;L2(Ω))∩Lq(QT)⩽c(T).
Further, in order to prove the latter estimate of Lemma2.1, we multiply (2.1) by uη and integrate over Qt:
(2.12)1k+2∫0t∫Ω∂tβη(uη)uηdxdθ+(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ+∫0t∫Ωg(x,uη)uηdxdθ=∫0t∫Ωf(x)uηdxdθ.
Therefore,(2.13)1k+2∫0t∫Ω∂t(βη(uη)uη)dxdθ+(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ+∫0t∫Ωg(x,uη)uηdxdθ=∫0t∫Ωf(x)uηdxdθ+1k+2∫0t∫Ωβη(uη)∂tuηdxdθ.
Hence,(2.14)1k+2∫Ωβη(uη(t))uη(t)dx-1k+2∫Ωβη(u0η)u0ηdx+(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ+∫0t∫Ωg(x,uη)uηdxdθ=∫0t∫Ωf(x)uηdxdθ+1k+2∫0t∫ΩdBη(uη)dτdxdθ,
where Bη(s)=∫0sβη(θ)dθ.
Now, taking into account thatβη(0)=0 and βη′>η>0, we conclude that Bη(s)≤βη(s)s. Thus,
(2.15)1k+2∫Ωβη(uη)(t)uη(t)dx-1k+2∫Ωβη(u0η)u0ηdx+(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ+∫0t∫Ωg(x,uη)uηdxdθ≤∫0t∫Ωf(x)uηdxdθ+1k+2∫ΩBη(uη(t))dx-1k+2∫ΩBη(u0η)dx≤∫0t∫Ωf(x)uηdxdθ+1k+2∫Ωβη(uη)(t)uη(t)dx,
or
(2.16)(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ+∫0t∫Ωg(x,uη)uηdxdθ≤∫0t∫Ωf(x)uηdxdθ+1k+2∫Ωβη(u0η)u0ηdx.
From condition∂ζg>-λ we obtain that g(·,ζ)>-λζ and, consequently,
(2.17)(k+1)∫0t∫Ω(|Duη|p-2+φ(uη)+η)|Duη|2dxdθ≤∫0t∫Ω|uη|2(λ+1)dxdθ+∫0t∫Ω|f|2dxdθ+1k+2∫Ω|βη(u0η)|2dx+1k+2∫Ω|u0η|2dx.
Therefore,(2.18)∫0T∫Ω|Duη|pdxdθ≤c(T).
Thereby assertion follows.Corollary 2.2.
Under condition of Lemma2.1 there exists c such that
(2.19)1τ∫tt+τ∫Ω|Duη|pdxdτ≤cfor arbitraryτ.(It is sufficient to carry out the proof of Lemma2.1 using integration on the interval (t,t+τ) instead of (0,t) and taking into account (2.8).)Lemma 2.3.
Under the hypotheses(H1)–(H5) there exist constants ci such that for any η∈(0,1) the following estimates hold: ||uη||L∞(τ,T;W01,p(Ω))⩽c(τ,T),(2.20)∫τT∫Ω(βη′(uη))2|∂tuη|2dxdθ≤c(τ,T),∫tt+τ∫Ωβη′(uη)|∂tuη|2dxdθ≤c(τ)(t⩾τ).Proof.
Multiplying (2.1) by ∂tuη and integrating over [s,τ+t], where τ≤t≤s, we get
(2.21)∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ-∫st+τ∫ΩD((|Duη|p-2+φ(uη)+η)Duη)∂tuηdxdθ+∫st+τ∫Ωg(x,uη)∂tuηdxdθ=∫st+τ∫Ωf(x)∂tuηdxdθ.
Then, using integration by parts, we derive
(2.22)∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+∫st+τ∫Ω((|Duη|p-2+φ(uη)+η)Duη)D∂tuηdxdθ+∫st+τ∫Ω∂tG(x,u)dxdθ=∫st+τ∫Ω∂t(f(x)uη)dxdθ,
where G(·,s)=∫0sg(·,ζ)dζ.
Hence, using condition(H4), we get
(2.23)∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+∫st+τddθ(1p+2∫Ω|Duη(θ)|pdx+12∫Ωφ(uη(θ))|Duη(θ)|2dx+12∫Ωη|Duη(θ)|2dx+∫ΩG(x,uη(θ))dx-∫Ωf(x)uη(θ))dθ=12∫st+r∫Ωφ′(uη(θ))×|Duη(θ)|2∂tuηdxdθ≤12∫st+r∫Ω(|uη|l+c)(β′(s))α|Duη|2|∂tuη|dxdθ.
Further, taking into account thatβη(uη) is uniformly bounded by η (uη∈[-δ,δ], where δ is the bound in the proof of Lemma 2.1), it is possible to choose βη so that βη′≤L, where L is the Lipschitz constant of β(ζ) on [-δ,δ]. Therefore,
(2.24)12∫st+τ∫Ω(|uη|l+c)(β′(s))α|Duη|2|∂tuη|dxdθ≤ϵ2∫st+τ∫Ω(βη′(uη))α|∂tuη|2dxdθ+12ϵ∫st+τ∫Ω(|uη|l+c)2|Duη|4dxdθ≤ϵ2Lα-1∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+12ϵ∫st+τ∫Ω(|uη|l+c)2|Duη|4dxdθ.
Consequently,(2.25)∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+∫st+τddθ(1p+2∫Ω|Duη(θ)|pdx+12∫Ωφ(uη(θ))|Duη(θ)|2dx+12∫Ωη|Duη(θ)|2dx+∫ΩG(x,uη(θ))dx-∫Ωf(x)uη(θ)dx)dθ≤ϵ2Lα-1∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+12ϵ∫st+τ∫Ω(|uη|l+c)2(|Duη|p+1)dxdθ.
Now, using Lemma2.1 and Corollary 2.2, we easily deduce
(2.26)12∫st+τ∫Ωβη′(uη)|∂tuη|2dxdθ+1p+2∫Ω|Duη|p(t+τ)dx+∫Ωφ(uη)|Duη|2(t+τ)dx+12∫Ωη|Duη|2(t+τ)dx+∫ΩG(x,uη(t+τ))dx-∫Ωf(x)uη(t+τ)dx≤1p+2∫Ω|Duη|p(s)dx+∫Ωφ(uη)|Duη|2(s)dx+12∫Ωη|Duη|2(s)dx+∫ΩG(x,uη(s))dx-∫Ωf(x)uη(s)dx+12ϵcτ.
After integrating over[t,t+τ], we derive
(2.27)τ1p+2∫Ω|Duη|p(t+τ)dx+τ∫Ωφ(uη)|Duη|2(t+τ)dx+12τ∫Ωη|Duη|2(t+τ)dx≤1p+2∫tt+τ∫Ω|Duη|p(s)dxds+∫tt+τ∫Ωφ(uη)|Duη|2(s)dxds+12∫tt+τ∫Ωη|Duη|2(s)dxds+∫tt+τ∫ΩG(x,uη(s))dxds+∫tt+τ∫Ωf(x)uη(s)dxds-τ∫Ωf(x)uη(t+τ)dx-τ∫ΩG(x,uη(t+τ))dx+c(τ)τ.
Since, by virtue of conditions imposed on the functiong,-(λ/2)|ζ|2<G(·,ζ)≤c|ζ|k+(λ/2)|ζ|2+c, then
(2.28)1p+2∫Ω|Duη|p(t+τ)dx+∫Ωφ(uη)|Duη|2(t+τ)dx≤1τ(1p+2∫tt+τ∫Ω|Duη|p(s)dxds+∫tt+τ∫Ωφ(uη)|Duη|2(s)dxds+12∫tt+τ∫Ωη|Duη|2(s)dxds+∫tt+τ∫Ω(c|uη|k+(λ2)|uη|2+c)dxds+∫tt+τ∫Ωf(x)uη(s)dxds)+∫Ω(c|uη|k+(λ2)|uη|2+c)(t)dx+∫Ωf(x)uη(t)dx+c(τ)≤1τ(1p+2∫tt+τ∫Ω|Duη|p(s)dxds+12ɛ∫tt+τ∫Ω|Duη|4(s)dxds+12ɛ-1∫tt+τ∫Ωφ2(uη(s))dxds+12∫tt+τ∫Ωη|Duη|2(s)dxds+∫tt+τ∫Ω(c|uη(s)|k+(λ2)|uη(s)|2+c)dxds+∫tt+τ∫Ω|f(x)uη(s)|dxds)+c(τ).
As we noted above,βη′≤L on [-δ,δ]. Hence,
(2.29)∫tt+τ∫Ωφ2(uη)dxds≤∫tt+τ∫Ω(∫0uηφ′(s)ds)2dxds≤∫tt+τ∫Ω||uη||L∞(τ,T;L∞(Ω))(||uη||L∞(τ,T;L∞(Ω))l+c)Lαdxds.
Thus, we obtain that(2.30)1p∫Ω|Duη|p(t+τ)dx+∫Ωφ(uη)|Duη|2(t+τ)dx≤c(τ)
for t≥τ.
Now returning to (2.23), we have
(2.31)∫tt+τ∫Ωβη′(uη)|∂tuη|2dxdθ≤c(τ).
Also, choosingβη so that βη′≤L, we derive
(2.32)∫τT∫Ω(βη′(uη))2|∂tuη|2dxdθ≤c(τ,T).
Thus, the lemma is proved.Passage to the Limit in (2.1).
Analogously to [7, 8], by estimates from Lemmas 2.1 and 2.3, we deduce that there exist that a subsequence uη such thatuη→u weakly in Lp(0,T;W01,p(Ω)) and a.e.,βη(uη)→β(u) strongly in C([0,T];L2(Ω)),∂tβη(uη)→∂tβ(u) weakly in L2(Q),D((|Duη|p-2Duη)→χ weakly in Lp′(0,T;W-1,p′(Ω)).
Further, it is easy to see that(2.33)g(x,uη)→g(x,u)weaklyinL2(Q).Indeed, we know thatuη→u  a.e. Besides, by virtue of the embedding ℜ(Q)⊂C(Q), where ℜ(Q) is a domain of the solvability, the operator g:ℜ(Q)→L2(Q) generated by the expression g(x,u) is bounded. Hence, by the continuity ofg  we obtain that g(x,uη)→g(x,u)a.e. By applying a known lemma from [10, (1.1, Lemma 1.3)]  we can conclude that(2.34)g(x,uη)→g(x,u)weaklyinL2(Q).Arguing similarly, by help of the lemma from [10, (1.1, Lemma 1.3)] and conditions imposed on φ(ζ), we obtain(2.35)φ(uη)Duη→φ(u)DuweaklyinL2(Q),(2.36)φ(uη)Duη→φ(u)DuweaklyinL2(Q),D(φ(uη)Duη)→D(φ(u)Du)weaklyinL2(0,T;W-1,2(Ω)).Observe now that from (2.35), in view of the known theorem,(2.37)∫Qφ(uη)|Duη|2dxdt=||φ(u)Du||L2(Q)≤limn↗∞inf||φ(uη)Duη||L2(Q)=∫Qφ(uη)|Duη|2dxdt.Consequently, using standard monotonicity argument [10], we derive thatχ=D(|Du|p-2Du).Therefore, the following theorems hold.Theorem 2.4.
Under hypotheses(H1)–(H5) the problem (1.7), (1.8), (1.9) has a weak solution u(x,t) such that u∈Lp(0,T;W01,p(Ω))∩L∞(0,T;L2(Ω))∩L∞(τ,T;W01,p(Ω)), for all τ>0 and β(u)∈C([0,T];L2(Ω))∩Lq(Q).Now, we prove that the solution of the problem is unique.Lemma 2.5.
Suppose that the assumptions of Theorem2.4 are fulfilled and there exists a constant c such that the function ζ→g(x,ζ)+cβ(ζ) is increasing for a.e.x∈Ω. Then the solution of the problem (1.7), (1.8), (1.9) is unique.Proof.
Letu(x,t) and v(x,t) be two solutions of the problem (1.7), (1.8), (1.9) with the same initial condition: u(x,0)=v(x,0)=u0(x). Consider the “difference” u(x,t0)-v(x,t0), where t0 is an arbitrary point from (0,T). The above “difference” is a continuously differentiable function for almost all t>0, because by virtue of estimation ∫τT∫Ω(βη′(uη))2|∂tuη|2dxdt≤c2(τ,T), using the equation, we can conclude that |Du|p-2Du∈L2(τ,T;W01,2(Ω)) and consequently Du(x,t)∈C(Ω¯) for almost all t. We choose t0 such that u(x,t0)∈C1(Ω¯). Hence, the interval [a,b] may be divided into the subintervals where sign of “difference” u(x,t0)-v(x,t0) does not change. Let (x1,x2) be an interval such that u(x,t0)-v(x,t0)>0 and u(xi,t)=v(xi,t)(i=1,2¯). Then from (1.7) we obtain that (2.38)∫x1x2∂t(β(u)-β(v))dx-∫x1x2D(|Du|p-2Du-|Dv|p-2Dv)dx-∫x1x2D(φ(u)Du-φ(v)Dv)dx+∫x1x2(g(x,u)-g(x,v))dx=0.
By applying Newton-Leibniz formula we have
(2.39)[ddt∫x1x2(β(u)-β(v))dx-|Du|p-2Du(x2,t)+|Dv|p-2Dv(x2,t)+|Du|p-2Du(x1,t)-|Dv|p-2Dv(x1,t)-φ(u)Du(x2,t)+φ(v)Dv(x2,t)+φ(u)Du(x1,t)-φ(v)Dv(x1,t)+∫x1x2g(x,u)+cβ(u)-g(x,v)-cβ(v)dx-∫x1x2cβ(u)-cβ(v)dx]t=t0=0.
SinceDu(x1,t0)≥Dv(x1,t0),Du(x2,t0)≤Dv(x2,t0),u(xi,t)=v(xi,t)(i=1,2¯),  it follows that
(2.40)ddt∫x1x2(β(u)-β(v))dx|t=t0<c∫x1x2(β(u)-β(v))dx|t=t0.
Note that the functions ψ(t)=∫x1x2((β(u)-β(v))dx and ψ̃(t)=∫x1x2|β(u)-β(v)|dx are absolutely continuous by virtue of the inclusion ∂tβ(u)∈L2(τ,T,L2(Ω)). Hence, these functions possess a derivate for almost all t. Besides, it is obvious that ψ̃(t)≥ψ(t), and if β(u(x,t′))-β(v(x,t′))⩾0 for x∈(x1,x2),then, ψ̃(t′)=ψ(t′). Consequently,(2.41)ddt∫x1x2|β(u)-β(v)|dx|t=t′≤ddt∫x1x2(β(u)-β(v))dx|t=t′
for almost all t′. From here, without loss of generality, we can assume that
(2.42)ddt∫x1x2|β(u)-β(v)|dx|t=t0≤ddt∫x1x2(β(u)-β(v))dx|t=t0.
It follows that(2.43)ddt∫x1x2|β(u)-β(v)|dx|t=t0<c∫x1x2|β(u)-β(v)|dx|t=t0.
The same estimation holds for an arbitrary interval on which  u(x,t0)-v(x,t0) does not change its sign. Summing up similar inequalities over subintervals, we get
(2.44)ddt∫ab|β(u)-β(v)|dx|t=t0<c∫ab|β(u)-β(v)|dx|t=t0
almost everywhere. In view of integrability of both sides of the latter inequality (β(u)∈W1,2(τ,T;L2(Ω))) we have
(2.45)∫t′t′′ddt∫ab|β(u)-β(v)|dxdt<c∫t′t′′∫ab|β(u)-β(v)|dxdt,
where t′′>t′>τ, or
(2.46)∫ab|β(u)-β(v)|dx|t=t′′<∫ab|β(u)-β(v)|dx|t=t′+c∫t′t′′∫ab|β(u)-β(v)|dxdt.
Thus, by Gronwall's lemma,(2.47)∫ab|β(u)-β(v)|dx|t=t′′≤ec(t′′-t′)∫ab|β(u)-β(v)|dx|t=t′.
Now, taking into account thatβ(u)∈C(0,T;L2(Ω)) and u(x,0)=v(x,0)=u0(x), we obtain u(x,t)=v(x,t), which finishes the proof of uniqueness.Remark 2.6.
Note that the condition on the functionζ→g(x,ζ)+cβ(ζ) from the lemma can be changed to the following.
(H2)′ The function β-1(s) is a locally Lipschitzian function from R to R (it follows directly from the proof).
In this case, we can exclude the condition(H4) of Theorem 2.4 and conditions (H4) and (H6) of Theorem 2.8 below.So, analogously to the corresponding result from [8], we obtain that problem (1.7), (1.8), (1.9) generates a continuous semigroup St:L2(Ω)→L2(Ω)Stu0=u(t,·) and the following theorem holds.Theorem 2.7.
Assume that assumptions of Lemma2.5 are satisfied. Then the semigroup St associated with the boundary value problem (1.7), (1.8), (1.9) possesses a maximal attractor A which is bounded in W01,p(Ω)∩L∞(Ω), compact, and connected in L2(Ω).(For the concepts of absorbing sets and global attractors used here, we refer the reader to [9]).Under an additional condition we can obtain more regularity foru(x,t).Theorem 2.8.
Assume thatu0(x)∈W1,∞(Ω),β(u0)∈L∞(Ω), and the conditions (H2)–(H6) are fulfilled. Then Du∈L∞(Q).Proof.
We prove this fact by multiplying (2.1) on expression |Duη|σ∂tuη, taking into account of the arbitrarity of σ(2.48)∫0T∫Ωβη′(uη)|∂tuη|2|Duη|σdxdt-∫0T∫ΩD(|Duη|p-2Duη)∂tuη|Duη|σdxdt-η∫0T∫ΩD2uη∂tuη|Duη|σdxdt-∫0T∫ΩD(φ(uη)Duη)∂tuη|Duη|σdxdt+∫0T∫Ω(g(x,uη)-f(x))∂tuη|Duη|σdxdt=0.
Let us estimate the terms of (2.48) separately.
Integrating by parts the second integral of (2.48), we get
(2.49)-∫0T∫ΩD(|Duη|p-2Duη)∂tuη|Duη|σdxdt=-(p-1)∫0T∫Ω|Duη|p-2D2uη∂tuη|Duη|σdxdt=-(p-1)∫0T∫Ω|Duη|p+σ-2D2uη∂tuηdxdt=-p-1p+σ-1∫0T∫ΩD(|Duη|p+σ-2Duη)∂tuηdxdt=p-1p+σ-1∫0T∫Ω|Duη|p+σ-2DuηD∂tuηdxdt=p-1(p+σ-1)(p+σ)∫0Tddt∫Ω|Duη|p+σdxdt.
Arguing similarly, we obtain(2.50)-η∫0T∫ΩD2uη∂tuη|Duη|σdxdt=-ησ+1∫0T∫ΩD(|Duη|σDuη)∂tuηdxdt=ησ+1∫0T∫Ω|Duη|σDuηD∂tuηdxdt=η(σ+1)(σ+2)∫0Tddt∫Ω|Duη|σ+2dxdt.
A series simple calculations give us the estimate of the third term of (2.48):
(2.51)-∫0T∫ΩD(φ(uη)Duη)∂tuη|Duη|σdxdt=1σ+2∫0T∫Ωφ(uη)∂t|Duη|σ+2dxdt+σσ+1∫0T∫Ωφ(uη)∂tuηD(|Duη|σDuη)dxdt=1σ+2∫0T∫Ωφ(uη)∂t|Duη|σ+2dxdt-σσ+1∫0T∫Ωφ(uη)D∂tuη|Duη|σDuηdxdt-σσ+1∫0T∫Ωφ′(uη)∂tuη|Duη|σ+2dxdt=-1(σ+1)(σ+2)∫0T∫Ωφ′(uη)∂tuη|Duη|σ+2dxdt+1(σ+1)(σ+2)∫0T∫Ω∂t(φ(uη)|Duη|σ+2)dxdt-σσ+1∫0T∫Ωφ′(uη)∂tuη|Duη|σ+2dxdt=-σ+1σ+2∫0T∫Ωφ′(uη)∂tuη|Duη|σ+2dxdt+1(σ+1)(σ+2)∫0T∫Ω∂t(φ(uη)|Duη|σ+2)dxdt≥-σ+1σ+2∫0T∫Ω(|uη|l1+c)(βη′(uη))α1|∂tuη||Duη|σ+2dxdt+1(σ+1)(σ+2)×∫0T∫Ω∂t(φ(uη)|Duη|σ+2)dxdt≥-(σ+1)ϵ(σ+2)2∫0T∫Ω(βη′(uη))α1|∂tuη|2|Duη|σdxdt-(σ+1)2ϵ(σ+2)∫0T∫Ω(|uη|l1+c)2|Duη|σ+4dxdt+1(σ+1)(σ+2)×∫0T∫Ω∂t(φ(uη)|Duη|σ+2)dxdt≥-(σ+1)ϵ(σ+2)2Lα1-1∫0T∫Ωβη′(uη)|∂tuη|2|Duη|σdxdt-(σ+1)2ϵ(σ+2)∫0T∫Ω(|uη|l1+c)2|Duη|σ+p+2dxdt-(σ+1)2ϵ(σ+2)∫0T∫Ω(|uη|l1+c)2dxdt∫Ωφ(uη)|Duη|σ+2dx-∫Ωφ(u0)|Du0|σ+2dx.
Further, taking into account thatβη(uη) is uniformly bounded by η (uη∈[-δ,δ], where δ is the bound in the proof of Lemma 2.1), it is possible to choose βη so that βη′≤L, where L is the Lipschitz constant of β(ζ) on [-δ,δ]. Therefore,
(2.52)|∫0T∫Ω(g(x,uη)-f(x))∂tuη|Duη|σdxdt|≤|∫0T∫Ω(|uη|l1+c)(βη′(uη))α1|∂tuη||Duη|σdxdt|≤ϵ2∫0T∫Ω(βη′(uη))α1|∂tuη|2|Duη|σdxdt+12ϵ∫0T∫Ω(|uη|l1+c)2|Duη|σdxdt≤ϵ2Lα1-1∫0T∫Ωβη′(uη)|∂tuη|2|Duη|σdxdt+12ϵ∫0T∫Ω(|uη|l1+c)2|Duη|σdxdt.
Also note that ifβ(u0)∈L∞(Ω) then using the same arguments as in proof of (2.8) we conclude that ||uη||L∞(QT)⩽c. Thus, combining (2.49)–(2.52) and choosing ϵ sufficiently small, we rewrite (2.48) in the form
(2.53)∫Ω|Duη|p+σdx≤c1∫0T∫Ω|Duη|p+σdxdt+c2T+c3∫Ω|Du0|p+σdx.
Applying Gronwall 's lemma, we get(2.54)∫Ω|Duη|p+σdx⩽(c2T+c3∫Ω|Du0|p+σdx)ec1T,
where the constant ci does not depend on σ. Consequently,
(2.55)(1|Ω|∫Ω|Duη|p+σdx)1/(p+σ)⩽(c3|Ω|∫Ω|Du0|p+σdx+c2|Ω|T)1/(p+σ)ec1T/(p+σ)⩽ec1T/(p+σ)(c3|Ω|∫Ω|Du0|p+σdx)1/(p+σ)+ec1T/(p+σ)(c2|Ω|T)1/(p+σ).
By lettingσ tend to +∞, we get necessary estimation. Theorem is proved.


## 3. Quantitative Homogenization

As we mentioned earlier, our goal is to compare the global behavior of solutionsuϵ(x,t) of (1.4) for ϵ→0 with solutions u=u0(x,t) of the homogenized equation(3.1)∂tβ(u)-D(|Du|p-2Du+φ(u)Du)+g0(x,u)=f0(x),
where (3.1) and (1.4) are supplied with the same initial data uϵ(x,0)=u0(x,0)=u0(x), and homogenized nonlinearity g0 and inhomogeneity f0 are defined according to assumption from [11, pages 172-174].We suppose that the functiong(x,z,ζ) has the following structure:(3.2)g(x,z,ζ)=∑j=1Mbj(x,z)gj(ζ),
where gj∈C1 is supposed to satisfy a condition (H3).For allj=1, we suppose that bj(x,z) are bounded:(3.3)|bj(x,z)|≤C,
and the average b0j(x) of bj(x,x/ϵ) exist in Lw*∞(Ω), for ϵ→0:(3.4)〈bj(x,xϵ),v(x)〉→ϵ→0〈b0j(x),v(x)〉,
for v(x)∈L1(Ω), where 〈·,·〉 indicates duality.We also suppose that(3.5)〈f(x,xϵ),v(x)〉→ϵ→0〈f0(x),v(x)〉,
for any v(x)∈L2(Ω).The equation∂tβ(u0)-D((|Du0|p-2Du0+φ(u0)Du0)+g0(x,u0)=f0(x) is called the homogenization of equation if(3.6)g0(x,u0)=∑j=1Mb0j(x)gj(u0).Denotingb̃j(x,z)=bj(x,z)-b0j(x), for x∈Ω,z∈R, we assume that there exist functions Bj(x,z) which are uniformly bounded for all x∈Ω,z∈R given as(3.7)|Bj(x,z)|≤C
and which represent b̃j such that b̃j(x,z)=∂zBj(x,z).  With respect to the x-derivatives we assume ϵ-independent L1-bound(3.8)||∂xBj(·,·ϵ)||L1(Ω)≤C,
uniformly for all j=1,…,M. Here ∂x is a partial’s derivatives with respect to the first argument x of the function Bj(x,z).Analogously, we denotef̃(x,z)=f(x,z)-f0(x) and require the existence of a function J(x,z) such that f̃(x,z) admits a divergence representation f̃(x,z)=∂zJ(x,z).Besides, we assume bounds(3.9)|J(x,z)|≤C,||∂xJ(·,·ϵ)||L1(Ω)≤C.Note that sufficient conditions which guarantee the existence of divergence representations forb̃j(x,z) and f̃(x,z) by help of Bj(x,z) and J(x,z), respectively, are established in [11, Theorem 3.2, pages 176-180].The following theorem holds.Theorem 3.1.
Letg(x,z,w) and f(x,z) satisfy conditions (3.2)–(3.9) and let assumptions of Lemma 2.5 be fulfilled. Then there exists a positive constant ρ such that the solutions uϵ(x,t) and u0(u,x) of the respective problems (1.4), (1.8), (1.9) and (1.3), (1.8), (1.9) with equal initial data u0(x)∈L2(Ω) satisfy the quantitative homogenization estimation
(3.10)||β(uϵ)-β(u0)||L1(Ω)≤Cϵeρtuniformly for0≤t<∞.Proof.
Consider the “difference”uϵ(x,t0)-u0(x,t0), where t0 is an arbitrary point from (0,T). The above “difference” is a continuously differentiable function, in virtue of Lemma 2.3. Hence, the interval [a,b] may be divided into the subintervals where sign of  the “difference” uϵ(x,t0)-u0(x,t0) does not change. Let (x1,x2) be an interval such that uϵ(x,t0)-u0(x,t0)>0 and uϵ(xi,t0)=u0(xi,t0)(i=1,2¯). Then from equations (1.3) and (1.4) we obtain that (3.11)ddt∫x1x2β(uϵ)-β(u0)dx-∫x1x2D(|Duϵ|p-2Duϵ-|Du0|p-2Du0)dx-∫x1x2D(φ(uϵ)Duϵ-φ(u0)Du0)dx+∫x1x2(g0(x,uϵ)-g0(x,u0))dx=-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx+∫x1x2(f(x,xϵ)-f0(x))dx.
By applying Newton-Leibniz formula we have(3.12)[ddt∫x1x2(β(uϵ)-β(u0))dx-|Duϵ|p-2Duϵ(x2,t)+|Du0|p-2Du0(x2,t)+|Duϵ|p-2Duϵ(x1,t)-|Du0|p-2Du0(x1,t)+φ(uϵ)Duϵ(x2,t)-φ(u0)Du0(x2,t)-φ(uϵ)Duϵ(x1,t)+φ(u0)Du0(x1,t)+∫x1x2(g0(x,uϵ)-g0(x,u0))dx+∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx-∫x1x2(f(x,xϵ)-f0(x))dxddt]t=t0=0.
SinceDuϵ(x1,t0)≥Du0(x1,t0),Duϵ(x2,t0)≤Du0(x2,t0), and uϵ(xi,t)=u0(xi,t)(i=1,2¯), it follows that
(3.13)ddt∫x1x2(β(uϵ)-β(u0))dx|t=t0<-∫x1x2(g0(x,uϵ)-g0(x,u0))dx-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx+∫x1x2(f(x,xϵ)-f0(x))dx|t=t0.
Note that(3.14)-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx|t=t0=-∑j=1M∫x1x2(bj(x,xϵ)gj(uϵ)-b0j(x)gj(uϵ))dx|t=t0=-∑j=1M∫x1x2b̃j(x,xϵ)gj(uϵ)dx|t=t0.
Obviously,(3.15)∂zBj(x,xϵ)=ϵddxBj(x,xϵ)-ϵ∂xBj(x,xϵ),
where, as we mentioned above, ∂x indicate partial derivatives with respect to the first argument x of the function Bj(x,z).
Consequently,(3.16)-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx|t=t0=-∑j=1M∫x1x2(∂zBj(x,xϵ)gj(uϵ))dx|t=t0=-ϵ∑j=1M∫x1x2ddxBj(x,xϵ)gj(uϵ)dx+ϵ∑j=1M∫x1x2∂xBj(x,xϵ)gj(uϵ)dx|t=t0.
Therefore, in view of condition(H3) and (3.7)
(3.17)-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx|t=t0≤ϵ∑j=1M∫x1x2|Bj(x,xϵ)||∂xgj(uϵ)|dx+ϵ∑j=1M||∂xBj(.,.ϵ)||L1(x1,x2)||gj(uϵ)||L∞(x1,x2)|t=t0≤Cϵ∑j=1M∫x1x2|∂xgj(uϵ)|dx+ϵ∑j=1M∥∂xBj(·,·ϵ)∥L1(x1,x2)∥gj(uϵ)∥L∞(x1,x2)|t=t0.
Observe now that the approximate solutionuηϵ is bounded in the space L∞(τ,T;W01,p(Ω)) uniformly with respect to ϵ, because in the proof of this Lemma 2.3 we use the same constant ci and  λ (condition (H3)) to estimate every gj(ω) from (3.2), and from condition (3.5) and (H5) (using known theorem on boundedness of the weakly convergence sequence in normed space) we can conclude that |f(x,x/ϵ)|≤C. Thus, ||uϵ||L∞(τ,T;W01,p(Ω))≤C. Consequently, ∑j=1M∫x1x2|∂xgj(uϵ)|dx≤C and (3.17) becomes
(3.18)-∫x1x2(g(x,xϵ,uϵ)-g0(x,uϵ))dx|t=t0≤Cϵ+Cϵ∑j=1M∥∂xBj(·,·ϵ)∥L1(x1,x2)|t=t0.
Analogously, by (3.9), using the same arguments as in proof of (3.18), we deduce that
(3.19)∫x1x2(f(x,xϵ)-f0(x))dx≤Cϵ.
Now, bearing in mind that the functionζ→g0(x,ζ)+cβ(ζ) is increasing and combining (3.18) and (3.19), we rewrite (3.13) in the form
(3.20)ddt∫x1x2(β(uϵ)-β(u0))dx|t=t0<C∫x1x2|β(uϵ)-β(u0)|dx|t=t0+Cϵ+Cϵ∑j=1M||∂xBj(·,·ϵ)||L1(x1,x2)|t=t0.
Further, note that the functionsψ(t)=∫x1x2((β(uϵ)-β(u0))dx and ψ̃(t)=∫x1x2|β(uϵ)-β(u0)|dx are absolutely continuous (β(uϵ)t∈L2(τ,T,L2(Ω)). Besides, it is obvious that ψ̃(t)≥ψ(t) and ψ̃(t0)=ψ(t0). Consequently,
(3.21)ddt∫x1x2|β(uϵ)-β(u0)|dx|t=t0≤ddt∫x1x2(β(uϵ)-β(u0))dx|t=t0≤C∫x1x2|β(uϵ)-β(u0)|dx|t=t0+Cϵ+Cϵ∑j=1M∥∂xBj(·,·ϵ)∥L1(x1,x2)|t=t0.
Hence, by condition (3.8)
(3.22)ddt∫x1x2|β(uϵ)-β(u0)|dx|t=t0≤C∫x1x2|β(uϵ)-β(u0)|dx|t=t0+Cϵ.
The same estimation holds for an arbitrary interval on whichuϵ(x,t)-u0(u,x) does not change its sign. Summing up similar inequalities over subintervals, we get
(3.23)ddt∫ab|β(uϵ)-β(u0)|dx|t=t0<C∫ab|β(uϵ)-β(u0)|dx|t=t0+Cϵ.
Consequently,(3.24)∫ab|β(uϵ)-β(u0)|dx|t=t′′≤Cϵeρ(t′′-t′)(t′′>t′>τ).
Thus, taking into account thatβ(uϵ)∈C(0,T;L2(Ω)),(3.25)∫ab|β(uϵ)-β(u0)|dx|t≤Cϵeρt.
Thereby, assertion follows.Remark 3.2.
It is easy to see that the condition on the functionζ→g(x,ζ)+cβ(ζ) in Lemma 2.5, which we use in Theorem 3.1, can be changed to the (H2)′. In this case we can exclude condition (H4).Now, note that if∑j=1Mb0j(x)<c̃<0 and condition (H2)′ is fulfilled then we derive, using simple reasoning, the following exponential attraction with exponential rate ν>0: distL1(Ω)(St0u0;A0)⩽ce-νt.Indeed, we know that solution of the corresponding stationary problem belongs to the attractor, that is, it belongs toA0. Denoting this solution by v, we will use the same arguments as in proof of Lemma 2.5.Definingu=St0u0, we have(3.26)ddt∫x1x2β(u)-β(v)dx-∫x1x2D(|Du|p-2Du-|Dv|p-2Dv)dx-∫x1x2D(φ(u)Du-φ(v)Dv)dx+∫x1x2(g0(x,u)-g0(x,v))dx=0,
where x1 and x2 are such that β(u)-β(v)≥0 on the interval (x1,x2). Hence,(3.27)ddt∫x1x2β(u)-β(v)dx-∫x1x2D(|Du|p-2Du-|Dv|p-2Dv)dx-∫x1x2D(φ(u)Du-φ(v)Dv)dx+∑j=1Mb0j(x)∫x1x2(gj(u)-gj(v))dx=0.
Bearing into mind that gj(w) is supposed to satisfy condition (H3) (∂ζgj>-λ), we have(3.28)ddt∫x1x2β(u)-β(v)dx≤-λc̃∫x1x2(u-v)dx.
Using (H2)′, we derive(3.29)ddt∫x1x2β(u)-β(v)dx≤-λc̃∫x1x2(β(u)-β(v))dx.
Hence, from(2.42)(3.30)ddt∫x1x2|β(u)-β(v)|dx≤-λc̃∫x1x2|β(u)-β(v)|dx.
Further, arguing similarly to the proof of Lemma 2.5, we arrive to(3.31)ddt∫ab|β(u)-β(v)|dx<-λc̃∫ab|β(u)-β(v)|dx,
or(3.32)∫ab|β(u)-β(v)|dx<c(u0,v)e-λc̃t.Consequently, by condition(H2)′(3.33)∫x1x2|u-v|dx≤L1∫x1x2|β(u)-β(v)|dx≤L1c(u0,v)e-λc̃t=ce-νt.Thus, we obtain thatdistL1(Ω)(St0u0;A0)⩽ce-νt, where ν=λc̃,L1  is the Lipschitz constant.Hence, using Remarks2.6 and 3.2, with the help of Lemma 4.1 of the study in [11], we obtain the estimate for the distance between the nonhomogenized Aϵ and the homogenized A0 attractors in terms of the parameter ϵdistL1(Ω)(Aϵ,A0)≤Cϵγ. So, the following theorem holds.Theorem 3.3.
Letg(x,z,w) and f(x,z) satisfy conditions (3.2)–(3.9), and assumptions (H1)–(H3),(H2)′, and (H5) are fulfilled. Also suppose that  ∑j=1Mb0j(x)<c̃<0. Then the global attractors Aϵ of the problem (1.4), (1.8), (1.9) satisfy an upper semicontinuity distance estimate of the form distL1(Ω)(Aϵ,A0)≤Cϵγ.



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*Source: 173408-2010-06-29.xml*