# Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

Authors: Yanjun Zhang; Qiaozhen Ma
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equationsut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.


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## Body

## 1. Introduction

We study the long-time behavior of the following class of nonclassical parabolic equations:(Eε)ut-εΔut-ωΔu+f(u)=g(x),inΩ×ℝ+,u(x,0)=u0(x),u|∂Ω=0,
where Ω⊂ℝN(N≥3) is a bounded domain with smooth boundary ∂Ω, ε∈[0,1] and ω>0 are two parameters, the external force g is time independent, and the nonlinearity f satisfies some specified conditions later.Whenε=0 for the fixed constant ω(>0), equation (E0) is a usual reaction-diffusion equation, and its asymptotic behavior has been studied extensively in terms of attractors by many authors; see [1–5].For each fixedε=ε0>0, equation (Eε0) is a nonclassical reaction-diffusion equation, which arises as models to describe physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction; see [6–8] and references therein. Aifantis in [6] provided a quite, general approach for obtaining these equations. The asymptotic behavior of the solutions for this equation has been studied by many authors; see [9–16].For the fixed constantω(>0), any ε∈[0,1], and the long-time behavior of the solutions of (Eε) has been considered by some researchers; see [10, 13]. In [10] the author proved the existence of a class of attractors in H2∩H01 with initial data u0∈H2∩H01 and the upper semicontinuity of attractors in H01 under subcritical assumptions and g(x)=0 in the case of N≤3. In [13] similar results have been shown when N≥3 and g(x)∈H01(Ω).In this paper, inspired by the ideas in [17, 18] and motivated by the dynamical results in [19–22], we study the uniform (with respect to the parameter ε∈[0,1]) qualitative analysis (a priori estimates) for the solutions of the nonclassical parabolic equations (Eε) and then give some information about the relation between the solutions of (E0) and those of (Eε). Our main difficulty comes from the critical nonlinearity and the uniformness with respect to ε∈[0,1].This paper is organized as follows. In Section2, we introduce basic notations and state our main results. In Section 3, we recall some abstract results that we will use later. In Section 4, we present several dissipative estimates about the solution of (Eε) when g(x)∈H-1(Ω), which hold uniformly with respect to ε∈[0,1]. The main results are proved for g(x)∈L2(Ω) in Section 5. Moreover, in Section 6, as an application, we construct a finite dimensional exponential attractor and prove the upper semicontinuity of the global attractor obtained in Section 5.


## 2. Main Results

Before presenting our main results, we first state the basic mathematical assumptions for considering the long-time behaviors of the nonclassical parabolic equations and then introduce some notations that we will use throughout this paper.(i)
f∈𝒞1(ℝ) with f(0)=0 and satisfies the following conditions:
(1)|f′(s)|≤C0(1+|s|(N+2)/(N-2)-1),∀s∈ℝ,(2)liminf|s|→∞f(s)s>-λ1,
where C0 is a positive constant and λ1 is the first eigenvalue of -Δ on H01. The number (N+2)/(N-2)-1 is called the critical exponent. f is not compact in this case, and this is one of the essential difficulties in studying the asymptotic regularity.(ii)
Assumption on the parametersε∈[0,1] and ω>0. From the work in [18, 19], we know that a very large damping has the effect of freezing the system, if the damping acts only on the velocity ut, and this prevents the squeezing of the component u. Therefore, the most dissipative situation occurs in between, that is, for a certain damping ε*, which depends on the other coefficient of the equation. Therefore, in our frame, we choose ω>1 such that 1/ω<ε as ε∈[0,1] in order to obtain the uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions of (Eε).(iii)
A=-Δ with domain D(A)=H2∩H01, and consider the family of Hilbert space D(As/2), s∈ℝ with the standard inner products and norms, respectively,
(3)〈·,·〉D(As/2)=〈As/2·,As/2·〉,∥·∥D(As/2)=∥As/2·∥.
In particular, 〈·,·〉 and ∥·∥ mean the L2(Ω) inner product and norm, respectively.(iv)
ℋs=D(A1+s/2), s∈[0,1] with the usual norm
(4)∥u∥ℋs2=∥A(1+s)/2u∥2.
In particular, we denote ℋ=ℋ0=H01(Ω) and ∥·∥ℋ=∥·∥H01.(v)
For eachu∈ℋ, we define ∥·∥ℋεs (ε,s∈[0,1]) as
(5)∥u∥ℋεs2=∥As/2u∥2+ε∥A(s+1)/2u∥2
and define ℋεs as
(6)ℋεs=cl∥·∥ℋε(ℋ1).
Then (ℋεs,∥·∥ℋε) is a Banach Space for every ε,s∈[0,1].The global well-posedness of solutions and its asymptotic behavior for(Eε0) have been studied extensively under assumptions (1)-(2) by many authors in [9–14] and references therein in fact note that (ℋε00,∥·∥ℋε00)≅(ℋ,∥·∥ℋ) for each fixed ε0.The main results of this paper are the following asymptotic regularity.Theorem 1.
Under assumptions (1), (2), and ω>1, there exist a positive constant ν, a bounded (in ℋ1) subset 𝔹⊂ℋ1, and a continuous increasing function Q(·):[0,∞)→[0,∞) such that, for any bounded (in  ℋ) subset B0⊂ℋ,
(7)∀ε∈[0,1],distℋ(Sε(t)B0,𝔹)≤Q(∥B∥ℋ)e-νt∀ε∈[0,1],distℋ(Sε(t)B0,𝔹)≤====∀t≥0,
where 𝔹,ν, and Q(·) are all independent of ε, and {Sε(t)}t≥0 is the semigroup generated by (Eε) in ℋε0.This result says that asymptotically, for each(Eε), the solutions are exponentially approaching a more regular fixed subset 𝔹 uniformly (with respect to ∈[0,1]) for ω>1. Moreover, it implies the following results.(1)
For eachε∈[0,1], {Sε(t)}t≥0 has a global attractor 𝒜ε in ℋ, and
(8)⋃ε∈[0,1]‍𝒜ε⊂clℋ1(𝔹).(2)
Based on Theorem1, applying the abstract result devised in [23, 24], for each ε∈[0,1] we can prove the existence of a finite dimensional exponential attractor ℰε in ℋ. Moreover, our attraction is uniform (with respect to ε∈[0,1]) under the ℋ-norm (not only with the ℋε0-norm); see Lemma 19.(3)
Since the global attractor𝒜ε⊂ℰε, it also implies that the fractal dimension of the global attractor 𝒜ε is finite. Moreover, in line with Theorem 1, we prove the upper semicontinuity of 𝒜ε at ε=0; see Lemma 20.For the proof of Theorem1, the main difficulty comes from the critical nonlinearity and the uniformness with respect to ε∈[0,1].Hereafter, we will also use the following notation: denote by𝒥 the space of continuous increasing functions J:[0,∞)→[0,∞) and by 𝒟 the space of continuous decreasing functions β:[0,∞)→[0,∞) such that β(∞)<1. Moreover, C, Ci, and ci are the generic constants, and Q(·), Qi(·)∈𝒥 are generic functions, which are all independent of ε; otherwise we will point out clearly.


## 3. Preliminaries

In this section, we recall some results used in the main part of the paper.The first result comes from [17], which will be used to prove the asymptotic regularity for the case g(x)∈L2(Ω).Lemma 2 (see [17]).
LetX and V be two Banach spaces and {T(t)}t≥0 a C0-semigroup on X with a bounded absorbing set B-⊂X. For every x∈B-, assume that there exist two solution operators Vx(t) on X and Ux(t) on V satisfying the following properties.(i)
For any two vectorsy∈X and z∈V satisfying y+z=x,
(9)T(t)x=Vx(T)y+Ux(t)z∀t≥0.(ii)
There existsα∈𝒟 such that
(10)supx∈B-∥Vx(t)y∥X≤α(t)∥y∥X,∀y∈B-.(iii)
There areβ∈𝒟 and J∈𝒥 such that
(11)supx∈B-∥Ux(t)z∥V≤β(t)∥z∥V+J(t),∀z∈V.Then, there exist positive constants ρ, K, and ϑ such that
(12)distX(T(t)B-,BV(ρ))≤Ke-ϑt,∀t≥0,
where BV(ρ)={z∈V:∥z∥V≤ρ}.Next, we recall a criterion for the upper semicontinuity of attractors.Lemma 3 (see [25, 26]).
Let{Tλ(t)}t≥0 be a family of semigroups defined on the Banach space X, and for each λ∈Λ, let {Tλ(t)}t≥0 have a global attractor 𝒜λ. Assume further that λ0 is a nonisolated point of Λ and that there exist s>0, t0>0, and a compact set K⊂X such that
(13)⋃λ∈𝒩λ(λ0,s)‍𝒜λ⊂K,(14)ifλn⟶λ0andxn⟶x0(xn∈𝒜asn≠0),ifλn⟶λ0andthenTλn(t0)xn⟶Tλ0(t0)x0.
Then the global attractors 𝒜λ are upper semicontinuous on Λ at λ=λ0; that is,
(15)limΛ∋λ→λ0distX(𝒜λ,𝒜λ0)=0.Lemma 4 (see [27]).
LetΦ be an absolutely continuous positive function on ℝ+, which satisfies for some ϵ>0 the differential inequality
(16)ddtΦ(t)+2ϵΦ(t)≤h1(t)Φ(t)+h2(t),
for almost every t∈ℝ+, where h1 and h2 are functions on ℝ+ such that
(17)∫τt‍|h1(y)|≤m1(1+(t-τ)μ),∀t≥τ≥0,
for some m1≥0 and μ∈[0,1], and
(18)supt≥0∫tt+1‍|h2(y)|dy≤m2,
for some m2≥0. Then
(19)Φ(t)≤ρ1Φ(0)e-ϵt+ρ2,∀t∈ℝ+,
for some ρ1=ρ1(m1,μ)≥1 and
(20)ρ2=ρ1m2eϵ1-e-ϵ.For the proof, we refer the reader to [27, Lemma2.2].A standard Gronwall-type lemma will also be needed.Lemma 5.
LetΨ be an absolutely continuous positive function on ℝ+, which satisfies for some Ψ>0 the differential inequality
(21)ddtΨ(t)+ϵΦ(t)≤κe-ιtΨ(t)+J(t)
for some ϵ,κ,ι>0 and some J∈𝒥. Then,
(22)Ψ(t)≤eκ/ιe-ϵtΨ(0)+ϵ-1eκ/ιJ(t).


## 4. Uniformly Decaying Estimates inℋ

In this section, we always assume that (1), (2), andω>1 such that 1/ω<ε as ε∈[0,1] hold and g(x) only belongs to H-1(Ω), so all results in this section certainly hold for the case g(x)∈L2(Ω).The main purpose of this section is to deduce some dissipative estimates about the semigroups{Sε(t)}t≥0(ε∈[0,1]) associated with (Eε) in ℋ. Here, using the method in [19, 20, 22] for a strongly damped wave equation and a semilinear second order evolution equation, we will show that the radius of the absorbing set of {Sε(t)}t≥0 associated with (Eε) in ℋ can be chosen to be independent of ε∈[0,1].Lemma 6.
There exists a positive constantM, which depends only on ω, ∥g∥H-1, and coefficients of (1)-(2), satisfying that for any ε∈[0,1] and any bounded (in ℋε0) subset B∈ℋε0, there is a tB=t(∥B∥ℋε0)>0 (which depends only on the bound of ∥B∥ℋε0) such that
(23)∥Sε(t)u∥ℋ≤M,∀t≥tB,∀u∈B,
where both tB and M are independent of ε∈[0,1].Proof.
Throughout the proof, the generic constantsC,Cj(j=1,2,…) are independent of ε. For clarity, we separate the proof into three claims.
Claim  1. There exists an M1 which depends on ω, |Ω|, ∥g∥H-1 (but independent of B and ε) such that
(24)∀ε∈[0,1],∥Sε(t)B∥ℋε02≤M1,ast≥T1B,
where T1B=T1(∥B∥ℋε0) depends on ∥B∥ℋε0 but not on ε.
Multiplying(Eε) by u, we have
(25)ddt(∥u∥2+ε∥∇u∥2)+2ω∥∇u∥2+2〈f(u),u〉=2〈g,u〉.
By virtue of (2), we conclude that there exists 0<λ<λ1, C1>0 such that
(26)2〈f(u),u〉=2∫Ω‍f(u)udx≥-2λ∥u∥2-2C1.
At the same time, by the Hölder inequality, we get(27)2〈g,u〉≤1ω∥g∥H-12+ω∥∇u∥2.
Substituting (26) and (27) into (25) and noticing 1/ω<ε as ω>1, we obtain
(28)ddt(∥u∥2+ε∥∇u∥2)+ε0(∥u∥2+ε∥∇u∥2)≤ζ(∥u∥2+ε∥∇u∥2)+C(1+∥g∥H-12),
where ζ=max{2λ-ε0,ε0} and ε0 is a small positive constant such that λ>ε0/2.
And then applying Lemma4 to above inequality, it follows that
(29)∥u∥2+ε∥∇u∥2≤ρ1[∥u(0)∥2+ε∥∇u(0)∥2]e-(ε0/2)t+ρ2,
where ρ1=ρ1(ζ), ρ2=ρ1C∥g∥H-12eε0/2/(1-e-(ε0/2)).
Then, Claim 1 follows from (29) immediately.
Claim  2. There exists an M2 which depends on ω, |Ω|,  and ∥g∥H-1 (but is independent of B and ε) such that
(30)∀ε∈[0,1],∫T1B∞‍∥∇u(s)∥ds≤M2,
where T1B is given in Claim 1.
Noting (25) and taking λ=(1/2)λ1 in (26), it yields
(31)ddt(∥u∥2+ε∥∇u∥2)+(ω-1)∥∇u∥2≤C(1+∥g∥H-12).
Then, for anyt≥T1B, integrating (31) over [T1B,t] and using Claim  1, we can complete this claim immediately.
Claim  3. Multiplying (Eε) by ut, we find
(32)ddt(∥∇u∥2+2ω∫Ω‍F(u)dx-2ω〈g,u〉)+2ω∥ut∥2+2εω∥∇ut∥2=0;
furthermore,
(33)ddt(∥∇u∥2+2ω∫Ω‍F(u)dx-2ω〈g,u〉)≤0.
Then, from assumptions (1)-(2), Claim 1, and using Hölder inequality, there holds
(34)∥∇u∥2+2ω∫Ω‍F(u)dx-2ω〈g,u〉≥12∥∇u∥2-C1ω(1+∥∇u∥2N/(N-2)+∥g∥H-12)≥12∥∇u∥2-C1ω2(1+M1N/(N-2)+∥g∥H-12);(35)∥∇u∥2+2ω∫Ω‍F(u)dx-2ω〈g,u〉≤2∥∇u∥2+C1ω2(1+M1N/(N-2)+∥g∥H-12).
On the other hand, from Claim 2 we know that for eachu0∈B there is a time t0∈[T1B,T1B+1] such that
(36)∥∇u(t0)∥≤M2,
where t0 depends on u0.
Whent≥T1B+1, for each u0∈B, integrating (33) over [t0,t] and applying (34)–(36), we obtain that
(37)∥∇u∥≤4M2+2(C1+C2)ω2(1+MN/(N-2)+∥g∥H-12).
Now, takingM=4M2+(2(C1+C2)/ω2)(1+MN/(N-2)+∥g∥H-12) (is independent of ε∈[0,1]), we can complete our proof.Remark 7.
Observing that above process of proof, we can also deduce that, for anyε∈[0,1] and any B⊂ℋε0,
(38)∥Sε(t)B∥ℋε02≤Q(ω,∥B∥ℋε0),∀t≥0,
where Q(·)∈𝒥 is independent of B and ε.
Moreover, ifB is bounded in ℋ, then we can obtain
(39)ε∈[0,1],∥Sε(t)B∥ℋ2≤Cω,∥B∥ℋ,∀t≥0
for some constant Cω,∥B∥ℋ which depends on ω,  ∥B∥ℋ. Indeed, from the fact that there is a constant c1 such that ∥·∥ℋε0≤c1∥·∥ℋ for any ε∈[0,1], (39) can be obtained just by repeating the proof of Lemma 6 and taking t0=0 in (35) since B is bounded in ℋ.
On the other hand, from the proof of Claim 3 as follows, we can get further estimates aboutut(40)∀ε∈[0,1],∫T1B+1∞‍(∥ut∥2+ε∥∇ut∥2)≤M,∀ε∈[0,1],∫T1B+1∞‍==1===∀t≥0,u0∈B.Lemma 8.
There exists a positive constantM3 such that for any ε∈[0,1] and any bounded (in ℋε0) subset B∈ℋε0,
(41)∥ut(t)∥2+ε∥∇ut∥2+ω∫T1B+2t‍∥∇ut(s)∥2ds≤M3,∥ut(t)∥2+ε∥∇ut∥2+ω∫T1B+2t‍∥∇m∥2∀t≥T1B+2,
where u(t)=Sε(t)u0, u0∈B, T1B is the time given in Claim  1, and M3 only depends on ω but is independent of B and ε.Proof.
By differentiation of(Eε), we can obtain the following equation:
(42)utt-εΔutt-ωΔut+f′(u)ut=0.
Multiplying (42) by ut, we have
(43)ddt(∥ut∥2+ε∥∇ut∥2)+2ω∥∇ut∥2=-2〈f′(u)ut,ut〉.
Whent≥T1B+1, using Lemma 6, there holds
(44)|-2〈f′(u)ut,ut〉|≤C(1+∥∇u∥4/(N-2))∥∇ut∥∥∇ut∥≤C4ω(1+∥∇u∥8/(N-2))∥∇ut∥2+ω∥∇ut∥2≤CM,ω(∥ut∥2+ε∥∇ut∥2)+ω∥∇ut∥2.
So, we obtain(45)ddt(∥ut∥2+ε∥∇ut∥2)+ω∥∇ut∥2≤CM,ω(∥ut∥2+ε∥∇ut∥2).
Therefore, ast≥T1B+2, for u0∈B, integrating (45) over [T1B+2,t] and substituting (40), we can complete our proof at once.For later applications, we present some Hölder continuity of{Sε(t)}t≥0 in ℋε0.Lemma 9.
For any bounded(ℋε0) subset B⊂ℋε0, there exists a constant Cω,∥B∥ℋε0 which depends only on ω and ∥B∥ℋε0 such that
(46)∥Sε(t)u1-Sε(t)u2∥ℋε0≤eCω,∥B∥ℋε0t∥u1-u2∥ℋε0,∥Sε(t)u1-Sε(t)u2∥ℋε0≤eCω,∥B∥ℋε0∀t≥0,ui∈B;(47)∥Sε(t)u1-Sε(t)u2∥ℋε0≤eCω,∥B∥ℋε0t∥u1-u2∥ℋε012,∥Sε(t)u1-Sε(t)u2∥ℋε0=≤∀t≥T1B+2,ui∈B.Proof.
Letu1  and  u2 be the solutions of (Eε) corresponding to the initial data u1  and  u2. Then the difference u-=u1-u2 satisfies
(48)u-t-εΔu-t-ωΔu-+f(u1)-f(u2)=0,
with initial data u-(0)=u1-u2.
For (46), multiplying (48) by u-, we have
(49)ddt(∥u-∥2+ε∥∇u-∥2)+2ω∥∇u-∥2=-2〈f(u1)-f(u2),u-〉≤C∫Ω‍(1+|u1|4/(N-2)+|u2|4/(N-2))|u-||u-|dx≤C(1+∥u1∥4/(N-2)+∥u2∥L2N/(N-2)4/(N-2))∥u-∥L2N/(N-2)∥u-∥L2N/(N-2)≤CMε∥u-∥2+2ω∥u-∥2,
where we used (38). Then, when applying Gronwall lemma, we can obtain (46).
For (47), when t≥T1B+2, multiplying (48) by u- and combining with Lemma 8, we have
(50)ω∥∇u-∥2=-〈u-t,u-〉+ε〈Δu-t,u-〉-〈f(u1)-f(u1),u-〉≤M3(∥u-∥+ε∥∇u-∥)+εCM∥∇u-∥2+ω2∥∇u-∥2.
Hence, by (47) we complete the proof.Hereafter, we denote the uniformly (with respect toε∈[0,1]) bounded absorbing set obtained in Lemma 6 as B0, that is,
(51)B0={u∈ℋ:∥u∥ℋ2≤M},
and denote the time by Λ0 such that Lemmas 6 and 8 hold for B0; that is,
(52)∥Sε(t)B0∥ℋ2+∥ut(t)∥2+ε∥∇ut(t)∥2+ω∫T1B+2t‍∥∇ut(s)∥2ds≤M-=M+M3,
holds for any ε∈[0,1] and all t≥Λ0. Moreover, similar to Remark 7, noting now that B0 is bounded in ℋ, we have
(53)∀ε∈[0,1],∥Sε(t)B0∥ℋ2≤CM,∀t≥0.


## 5. Proof of the Main Results

Throughout this section, we always assume that (1), (2), and ω>1 hold for g(x)∈L2(Ω).


### 5.1. Decomposition of the Equation

For the nonlinear functionf satisfying (1)-(2), from [12, 17, 19, 22] for our situation we know that f allows the following decomposition f=f0+f1, where f0,f1∈𝒞1(ℝ) and satisfy
(54)|f0(u)|≤C|u|(N+2)/(N-2),∀u∈ℝ,(55)f0(u)u≥0,∀u∈ℝ,(56)|f1(u)|≤C(1+|u|γ)withsomeγ<N+2N-2,∀u∈ℝ,(57)liminf|u|→∞f1(u)u>-λ1.Now, decomposing the solutionSε(t)u0=u into the sum
(58)Sε(t)u0=Dε(t)u0+Kε(t)u0,
for any t≥0 and any u0∈ℋ, where Dε(t)u0=v(t) and Kε(t)u0=w(t) are the solutions of the following equations:
(59)vt-εΔvt-ωΔv+f0(v)=0inΩ×ℝ+,v(x,0)=u0,v|∂Ω=0,(60)wt-εΔwt-ωΔw+f(u)-f0(v)=g(x)inΩ×ℝ+,w(x,0)=0,w|∂Ω=0.Applying the general results in [9, 12, 14], we know that both (59) and (60) are global well-posed in ℋ, and {Dε(t)}t≥0 also forms a semigroup.Moreover, as in Section4, we can deduce a similar estimate for {Dε(t)}t≥0 in ℋ, and so {Kε(t)}t≥0. There exist constants CM (M is given in Lemma 6) and Λ1 such that for any ε∈[0,1] and any u0∈B0,
(61)∥Dε(t)B0∥ℋ2+∥vt(t)∥2+ε∥∇vt∥2+ω∫T1B+2t‍∥∇vt(s)∥2ds≤M-,∀t≥Λ1,(62)∀ε∈[0,1],∥Dε(t)B0∥ℋ2+∥Kε(t)B0∥ℋ2≤CM,∀ε∈[0,1],∥Dε(t)B0∥ℋ2+∥Kε(t)B0∥ℋ2∀t≥0.


### 5.2. The First A Priori Estimate

We begin with the decay estimates for the solution of (59).Lemma 10.
There exists a constantk>0 and Q(·)∈𝒥 such that
(63)∥Dε(t)B0∥ℋ2≤CM-,ωQ(∥B0∥ℋ)e-kt,======∀t≥0andanyε∈[0,1],
where both k and Q(·) are independent of ε∈[0,1].Proof.
Multiplying (59) by v, we have
(64)ddt(∥v∥2+ε∥∇v∥2)+2ω∥∇v∥2+2〈f0(v),v〉=0.
By means of (55), it follows that (d/dt)(∥v∥2+ε∥∇v∥2)≤0.
Therefore, there existsk1>0 such that
(65)∥v(t)∥2+ε∥∇v(t)∥2≤∥v(0)∥2+ε∥∇v(0)∥2≤Q(∥B0∥ℋ)e-k1t,
for all t≥0 and any ε∈[0,1].
As a result, we multiply (59) by v and obtain
(66)ω∥∇v∥2≤∥vt∥∥v∥+ε∥∇vt∥∥∇v∥-〈f0(v),v〉.
Then integrating with (55), (61), (62), and (65), we conclude
(67)ω∥∇v∥2≤M-(∥v∥+ε∥∇v∥)≤M-Q(∥B0∥ℋ)e-k1t.
Thus, using the following Lemma 11 with (67), allows us to complete our proof by taking k=k1/2 and some increasing function Q(·).Lemma 11.
Let{S(t)}t≥0 be a continuous semigroup on the Banach space X, satisfying
(68)∥S(t)B∥X≤Q1(∥B∥X)e-μt,∀t≥t0,∥{S(t)B:t≥0}∥X≤Q2(∥B∥X).
Then(69)∥S(t)B∥X≤Q3(∥B∥X)e-μt,∀t≥0.Its proof is obvious and we omit it here.The next estimate is about the solution of (60).Lemma 12.
For every (given)T>0 and any ε∈[0,1], there is a positive constant J which only depends on T,ω,∥g∥,  and  ∥u0∥ℋ such that the solutions of (60) satisfy
(70)∥Kε(t)B0∥ℋεσ2≤J,
where both J are independent of ε∈[0,1], and σ=min{1/4,(N+2-(N-2)γ)/2}.Proof.
Multiplying (60) by Aσw(t) and integrating over Ω, Then the proof is completely similar to that in [12, Lemma3.4], so, we omit it.Based on Lemmas10 and 12, following the idea in Zelik [21], we can now decompose u(t) as follows.Lemma 13.
Letu(t) be the solution of (Eε) corresponding to the initial data u0∈B0. Then, for any η>0, we can decompose u(t)=Sε(t)u0 as
(71)u(t)=v1(t)+w1(t),∀t≥0,
where v1(t) and w1(t) satisfy the following estimates:
(72)∫st‍∥∇v1(τ)∥2dτ≤η(t-s)+Cη,∀t≥s≥0,∥A(1+σ)/2w1(t)∥2≤Kη,∀t≥0,
with the constants Cη and Kη depending on η,ω,∥B0∥ℋ, and ∥g∥, but both independent of ε∈[0,1].Proof.
The proof is completely similar to that of [12, Lemma 4.6] and [22, Lemma5.4], since the estimates in Lemmas 10 and 12 hold uniformly with respect to ε∈[0,1].Note that in the above decomposition in Lemma13, we can require further that v1(t) satisfies the following: there is a constant M5 which depends only on ω, ∥u0∥ℋ such that
(73)∥∇v1(t)∥2≤Q(∥B0∥ℋ):=M4,∀t≥0,u0∈B0.


### 5.3. The Second A Priori Estimate

The main purpose of this subsection is to deduce some uniformly asymptotic (with respect toε∈[0,1] and t) the a priori estimates about the solution of (Eε).Lemma 14.
There exists positive constantsν-, R->0, and Q(·)∈𝒥 such that for each ε∈[0,1], there is a subset B-ε⊂ℋε1 satisfying
(74)∥B-ε∥ℋε12=supu∈B-ε{∥∇u∥2+ε∥Δu∥2}≤R-,
and the exponential attraction
(75)distℋε0(Sε(t)B0,B-ε)≤Q1(∥B0∥ℋ)e-ν-t,∀t≥0,
where all ν-,R-, and Q1(·) are independent of ε∈[0,1], and distℋε0(·,·) denotes the Hausdorff semidistance with respect to the ℋε0-norm.Proof.
It is convenient to separate our proof into three steps. We emphasize, especially, that all the generic constants in the proof are independent ofε∈[0,1].
Step  1. We first claim that (recall σ=min{1/4,(N+2-(N-2)γ)/2}): ∃νσ, Rσ>0 and Qσ(·)∈𝒥 such that for each ε∈[0,1], there is a subset B-σ,ε⊂ℋεσ satisfying
(76)∥B-σ,ε∥ℋεσ2=supu∈B-ε{∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2}≤Rσ
and the exponential attraction
(77)distℋε0(Sε(t)B0,B-σ,ε)≤Qσ(∥B0∥ℋ)e-νσt,∀t≥0.
We will apply Lemma2 with X=ℋε0 and V=ℋεσ (note that B0⊂ℋ⊂ℋε0 for any ε∈[0,1]). From (54), we can write
(78)f0(s)=sφ(s)with|φ(s)|≤C|s|4/(N-2).
For anyx∈B0 and y∈ℋε0, z∈ℋεσ satisfying x=y+z, we decompose the solution of (Eε) as Sε(t)(x)=Vxε(t)y+Uxε(t)z, where
(79)Vxε(t)y=v-(t),Uxε(t)z=w-(t),
which uniquely solves the following equations, respectively:
(80)v-t-εΔv-t-ωΔv-=h1,v-(x,0)=y,v-|∂Ω=0,(81)w-t-εΔw-t-ωΔw-=h2,w-(x,0)=z,w-|∂Ω=0,
with h1=-v-φ(v) and h2=g(x)-f(u)+v-φ(v), and v(t) is the solution of (59) corresponding to the initial data x.
For (80), from (54), (56), (78), and Lemmas 10 and 12, we can directly calculate that
(82)∥h1∥L2N/(N+2)≤C∥∇v-∥∥∇v∥4/(N-2)≤CM-,ω,Ne-k′t∥∇v-∥,
where k′=(2/(N-2))k, k is given in Lemma 10.
Multiplyingv- by (80), we have
(83)ddt(∥v-∥2+ε∥∇v-∥2)+2ω∥∇v-∥2=2〈h1,v-〉≤2∥h1∥L2N/(N+2)∥v-∥L2N/(N-2)≤CM-,ω,Ne-k′t∥∇v-∥2≤CM,ω,Ne-2k′tε∥∇v-∥2+ω∥∇v-∥2.
Furthermore, using the similar estimates of Lemma6, we get
(84)ddt(∥v-∥2+ε∥∇v-∥2)+ε1(∥v-∥2+ε∥∇v-∥2)≤2CM-,ω,Ne-2k′t(∥v-∥2+ε∥∇v-∥2),
where ε1 is a small positive constant such that ε1≤min{CM-,ω,Ne-2k′t,λ1ω} for all t≥0.
And then applying Lemma5 to above inequality, there holds
(85)∥Vxε(t)y∥ℋε02≤eCM,ω,N/k′e-ε1t∥y∥ℋε02.
For (81), since
(86)h2=g(x)-f(u)+f(v)-f1(v)+wφ(v)-w-φ(v),
then
(87)|h2|≤|g|+C|w|(1+|u|4/(N-2)+|v|4/(N-2))|h2|≤+C|w-||v|4/(N-2)+C(1+|v|γ).
Using Hölder inequality we get(88)∥h2∥L2N/(N+2(1-σ))≤2C|Ω|(2-2σ)/2N∥g∥+2C∥w∥L2N/(N-2(1+σ))×(1+∥u∥L2N/(N-2)4/(N-2)+∥v∥L2N/(N-2)4/(N-2))+2C∥v∥L2N/(N-2)4/(N-2)∥w-∥L2N/(N-2(1-σ))+2C(1+∥v∥L2N/(N-2)4/(N-2))≤C|Ω|∥g∥2+C∥A(1+σ)/2w∥×(1+∥∇u∥4/(N-2)+∥∇v∥4/(N-2))+C∥∇v∥4/(N-2)∥A(1+σ)/2w-∥+C(1+∥∇v∥4/(N-2))≤CM-,ω,Ne-k′t∥A(1+σ)/2w-∥+CM-,∥g∥2,|Ω|J(t),∀t≥0,
where we used (53), (62), and Lemmas 10 and 12.
Hence, multiplyingAσw- by (83), we have
(89)ddt(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+2ω∥A(1+σ)/2w-∥2=2〈h2,w-〉≤2∥h2∥L2N/(N+2(1-σ))∥Aσw-∥L2N/(N-2(1-σ))≤2CM-,ω,Ne-2k′tε∥A(1+σ)/2w-∥2+2CM-,∥g∥2,|Ω|J(t)+ω∥A(1+σ)/2w-∥2.
Furthermore, we have(90)ddt(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+ε1(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)≤2CM-,ω,Ne-2k′t(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+CM-,∥g∥2,|Ω|J(t),
where ε1 is a small positive constant given in (84).
Then, using Lemma5 we obtain
(91)∥Uxε(t)z∥ℋεσ2≤eM-,ω,N/k′e-ε1t∥z∥ℋεσ2+1ε1eε1CM-,∥g∥2,|Ω|J(t).
Therefore, combining (85) and (91), we can verify that all the conditions of Lemma 2 are satisfied for the cases X=ℋε0, V=ℋεσ, and T(t)=Sε(t). Moreover, since there is a c1>0 (independent of ε) such that c1∥B0∥ℋ≥∥B0∥ℋε0 for any ε∈[0,1] and the constants in our estimates are all independent of ε; consequently, νσ, Rσ, and Qσ(·) are all independent of ε∈[0,1], and then we can deduce our claim.
Step  2. We claim that there exists a constant R-σ>0 which depends only on Rσ such that
(92)∀ε∈[0,1],∥Sε(t)B-σ,ε∥ℋεσ2≤R-σ,∀t≥0.
Multiplying(Eε) by Aσu(t), we only need to note the following:
(93)12ddt(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)+ω∥A(1+σ)/2u∥2=〈g,Aσu〉-〈f(u),Aσu〉.
First, sinceσ<1, we have σ<(1+σ)/2 and then
(94)|〈g,Aσu(t)〉|≤∥g∥∥Aσu∥≤∥g∥∥A(1+σ)/2u∥≤4ω∥g∥2+ω16∥A(1+σ)/2u∥2,(95)〈f(u),Aσu〉≤C∫Ω‍(1+|u|(N+2)/(N-2))|Aσu|dx≤4ω|Ω|2+ω16∥A(1+σ)/2u∥2+C∫Ω‍|u|4/(N-2)|u||Aσu|dx,
while
(96)∫Ω‍|u|4/(N-2)|u||Aσu|dx≤∫Ω‍(|v1|4/(N-2)+|w1|4/(N-2))|u||Aσu|dx,∫Ω‍|v1|4/(N-2)|u||Aσu|dx≤C∥∇v1∥4/(N-2)∥A(1+σ)/2∥2≤ω16∥A(1+σ)/2u∥2+CM4∥∇v1∥∥A(1+σ)/2u∥2,
where we used (73).
Moreover, sinceσ≤1/4, we have 2≤2N(N-2)/(N(N-4-2σ)+4(1+3σ))≤2N/(N-2-2σ) and then
(97)∫Ω‍|w1|4/(N-2)|u||Aσu|dx≤C∥∇w1∥L2N/(N-2-2σ)4/(N-2)∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))∥Aσu∥≤C∥A(1+σ)/2w1∥4/(N-2)×∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))∥A(1+σ)/2u∥≤CKη4/(N-2)∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))×∥A(1+σ)/2u∥≤CKη4/(N-2)(CM+ω32CKη4/(N-2)∥A(1+σ)/2u∥)×∥A(1+σ)/2u∥≤ω32∥A(1+σ)/2u∥2+CMKη4/(N-2)∥A(1+σ)/2u∥≤CMKη8/(N-2)+ω16∥A(1+σ)/2u∥2,
where Kη is given in Lemma 13.
Hence, substituting the above estimates into (93), applying the Poincaré inequality we have
(98)ddt(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)+(C-CM4∥∇v1∥2)(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)≤8ω∥g∥2+CM(1+Kη8/(N-2)).
Then using the Gronwall inequality and integrating over[0,t] (from Lemma 12), we obtain
(99)∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2≤e-∫0t‍(C-CM4∥∇v1(s)∥2)ds≤≤×(∥Aσ/2u(0)∥2+ε∥A(1+σ)/2u(0)∥2)≤≤+CM(1+Kη8/(N-2))≤≤×∫0t‍e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds≤≤+8ω∫0t‍∥g∥2e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds.
Takingη (in Lemma 13) small enough such that η<C/2CM4, we have
(100)∫0t‍e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds=∫0t‍eC(s-t)e∫st‍CM4∥∇v1(y)∥2)dyds≤∫0t‍eC(s-t)eCM4η(t-s)+CM4Cηds≤eCM4Cη∫0t‍eC(s-t)/2ds≤2eCM4CηC,(101)e-∫0t‍(C-CM4∥∇v1(s)∥2)ds≤e-Ct/2eCM4Cη.
Thus,(102)∫0t‍∥g∥2e-∫ts‍(C-CM5∥∇v1(y)∥2)dyds≤eCM4Cη1-e-(C/2)∥g∥2.
Substituting above (100) and (102) into (99), we get that for all t≥0(103)∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2≤eCM5CηRσ≤+2Cω(1+Kη8/(N-2))CeCM5Cη≤+eCM5Cη1-e-(C/2)∥g∥2:=R-σ.
Step  3.  Based on Step 1 and Step 2, applying the attraction transitivity lemma given in [28, Theorem5.1] and noticing the Holder continuity Lemma 9, we can prove our lemma by performing a standard bootstrap argument, whose proof is now simple since Step 1 makes the nonlinear term become subcritical to some extent.


### 5.4. Proof of Theorem1

Lemma14 has shown some asymptotic regularities; however, the radius of ∥B-ε∥ℋ1 depends on ε and the distances only under the ℋε0-norm.To prove Theorem1, we first give two lemmas as preliminary.Lemma 15.
There exsits a constantR1>0 such that for any bounded (in ℋε1) subset B⊂ℋε1, there exsits T1=T1(∥B∥ℋε1) such that
(104)∀ε∈[0,1],∥Sε(t)B∥ℋε12≤R1,∀t≥T1.Proof.
Multiplying(Eε) by -Δu, we find
(105)ddt(∥∇u∥2+ε∥∇u∥2)+2ω∥∇u∥2=-2〈f(u),-Δu〉+2〈g,-Δu〉,2〈g,-Δu〉≤Cω∥g∥2+ω2∥∇u∥2.
Notingu∈ℋε1⊂ℋε0, from Lemma 6, yields
(106)|-2〈f(u),-Δu〉|≤2∫Ω‍|f′(u)||∇u||∇u|dx≤C(1+∥∇u∥4/(N-2))∥Δu∥∥Δu∥≤CMε∥∇u∥2+ω2∥∇u∥2,
hence, we obtain(107)ddt(∥∇u∥2+ε∥∇u∥2)+ε2(∥∇u∥2+ε∥∇u∥2)≤2CM(∥∇u∥2+ε∥∇u∥2)+Cω∥g∥2,
where ε2 is a small, positive constant.
Similarly, with using Lemma4 we finally complete the proof.Lemma 16.
There exists a constantR2>0 such that for any bounded (in ℋε1) subset B⊂ℋε1, there is a T2=T2(∥B∥ℋε1) such that
(108)∀ε∈[0,1],∥Sε(t)B∥ℋ12≤R2,∀t≥T2.Proof.
From Lemma15, we only need to estimate that the bound of ∥Δu∥2 is independent of ε∈[0,1].
Applying Lemma15 again, we have
(109)ddt(∥∇u∥2+ε∥∇u∥2)+(2ω-2CCM-1)∥∇u∥2≤Cω∥g∥2.
Takingω=max{1,2CCM} which may provide that 1/ω<ε and 2ω-2CCM-1>0, integrating (109) on [t,t+1], and from Lemma 15, when t≥T1 we yield
(110)∫tt+1‍∥Δu(τ)∥2dτ≤Cω,R12ω-2CCM-1.
Hence, multiplying(Eε) by -Δut, we can complete our proof by applying the uniform Gronwall lemma.Now, we are ready to prove Theorem1.Proof of Theorem1.
Set(111)𝔹={u∈ℋ1:∥u∥ℋ12≤R2},
where the constant R2 comes from Lemma 16.
From Lemmas16 and 14, we know that there is a t0 such that Sε(t)B-ε⊂𝔹 (recall that B-ε is given in (78)) for all t≥t0 and any ε∈[0,1].
On the other hand, note that∃c1,c2>0 such that
(112)c1∥·∥ℋε0≤∥·∥ℋ≤c2∥·∥ℋε1.
Then, from Lemma9, there exists t1 which depends only on ∥B0∥ℋ and ∥B-ε∥ℋε1 (so only on M, R-) such that
(113)==∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-t∀ε∈[0,1],==∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-t∥u1-u2∥ℋε01/2,====∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-tx∀t≥t1,u1∈B0⊂ℋ⊂ℋε0,u2∈B-ε⊂ℋε1⊂ℋε0,(114)∀ε∈[0,1],Sε(t)B0⊂B0,∀t≥t1.
Therefore, from Lemma14, we have
(115)distℋ(Sε(t+t0+t1)B0,𝔹)≤distℋ(Sε(t+t0+t1)B0,Sε(t0+t1)B-ε)≤CM,R-,t0+t1distℋ1/2(Sε(t)B0,B-ε)≤CM,R-,t0+t1Q1(∥B0∥ℋ)e-(ν-/2)t.
Hence, noting thatt0,t1, and R- are all fixed, we can complete the proof by taking ν=ν-/2 and applying Lemma 11.


## 5.1. Decomposition of the Equation

For the nonlinear functionf satisfying (1)-(2), from [12, 17, 19, 22] for our situation we know that f allows the following decomposition f=f0+f1, where f0,f1∈𝒞1(ℝ) and satisfy
(54)|f0(u)|≤C|u|(N+2)/(N-2),∀u∈ℝ,(55)f0(u)u≥0,∀u∈ℝ,(56)|f1(u)|≤C(1+|u|γ)withsomeγ<N+2N-2,∀u∈ℝ,(57)liminf|u|→∞f1(u)u>-λ1.Now, decomposing the solutionSε(t)u0=u into the sum
(58)Sε(t)u0=Dε(t)u0+Kε(t)u0,
for any t≥0 and any u0∈ℋ, where Dε(t)u0=v(t) and Kε(t)u0=w(t) are the solutions of the following equations:
(59)vt-εΔvt-ωΔv+f0(v)=0inΩ×ℝ+,v(x,0)=u0,v|∂Ω=0,(60)wt-εΔwt-ωΔw+f(u)-f0(v)=g(x)inΩ×ℝ+,w(x,0)=0,w|∂Ω=0.Applying the general results in [9, 12, 14], we know that both (59) and (60) are global well-posed in ℋ, and {Dε(t)}t≥0 also forms a semigroup.Moreover, as in Section4, we can deduce a similar estimate for {Dε(t)}t≥0 in ℋ, and so {Kε(t)}t≥0. There exist constants CM (M is given in Lemma 6) and Λ1 such that for any ε∈[0,1] and any u0∈B0,
(61)∥Dε(t)B0∥ℋ2+∥vt(t)∥2+ε∥∇vt∥2+ω∫T1B+2t‍∥∇vt(s)∥2ds≤M-,∀t≥Λ1,(62)∀ε∈[0,1],∥Dε(t)B0∥ℋ2+∥Kε(t)B0∥ℋ2≤CM,∀ε∈[0,1],∥Dε(t)B0∥ℋ2+∥Kε(t)B0∥ℋ2∀t≥0.


## 5.2. The First A Priori Estimate

We begin with the decay estimates for the solution of (59).Lemma 10.
There exists a constantk>0 and Q(·)∈𝒥 such that
(63)∥Dε(t)B0∥ℋ2≤CM-,ωQ(∥B0∥ℋ)e-kt,======∀t≥0andanyε∈[0,1],
where both k and Q(·) are independent of ε∈[0,1].Proof.
Multiplying (59) by v, we have
(64)ddt(∥v∥2+ε∥∇v∥2)+2ω∥∇v∥2+2〈f0(v),v〉=0.
By means of (55), it follows that (d/dt)(∥v∥2+ε∥∇v∥2)≤0.
Therefore, there existsk1>0 such that
(65)∥v(t)∥2+ε∥∇v(t)∥2≤∥v(0)∥2+ε∥∇v(0)∥2≤Q(∥B0∥ℋ)e-k1t,
for all t≥0 and any ε∈[0,1].
As a result, we multiply (59) by v and obtain
(66)ω∥∇v∥2≤∥vt∥∥v∥+ε∥∇vt∥∥∇v∥-〈f0(v),v〉.
Then integrating with (55), (61), (62), and (65), we conclude
(67)ω∥∇v∥2≤M-(∥v∥+ε∥∇v∥)≤M-Q(∥B0∥ℋ)e-k1t.
Thus, using the following Lemma 11 with (67), allows us to complete our proof by taking k=k1/2 and some increasing function Q(·).Lemma 11.
Let{S(t)}t≥0 be a continuous semigroup on the Banach space X, satisfying
(68)∥S(t)B∥X≤Q1(∥B∥X)e-μt,∀t≥t0,∥{S(t)B:t≥0}∥X≤Q2(∥B∥X).
Then(69)∥S(t)B∥X≤Q3(∥B∥X)e-μt,∀t≥0.Its proof is obvious and we omit it here.The next estimate is about the solution of (60).Lemma 12.
For every (given)T>0 and any ε∈[0,1], there is a positive constant J which only depends on T,ω,∥g∥,  and  ∥u0∥ℋ such that the solutions of (60) satisfy
(70)∥Kε(t)B0∥ℋεσ2≤J,
where both J are independent of ε∈[0,1], and σ=min{1/4,(N+2-(N-2)γ)/2}.Proof.
Multiplying (60) by Aσw(t) and integrating over Ω, Then the proof is completely similar to that in [12, Lemma3.4], so, we omit it.Based on Lemmas10 and 12, following the idea in Zelik [21], we can now decompose u(t) as follows.Lemma 13.
Letu(t) be the solution of (Eε) corresponding to the initial data u0∈B0. Then, for any η>0, we can decompose u(t)=Sε(t)u0 as
(71)u(t)=v1(t)+w1(t),∀t≥0,
where v1(t) and w1(t) satisfy the following estimates:
(72)∫st‍∥∇v1(τ)∥2dτ≤η(t-s)+Cη,∀t≥s≥0,∥A(1+σ)/2w1(t)∥2≤Kη,∀t≥0,
with the constants Cη and Kη depending on η,ω,∥B0∥ℋ, and ∥g∥, but both independent of ε∈[0,1].Proof.
The proof is completely similar to that of [12, Lemma 4.6] and [22, Lemma5.4], since the estimates in Lemmas 10 and 12 hold uniformly with respect to ε∈[0,1].Note that in the above decomposition in Lemma13, we can require further that v1(t) satisfies the following: there is a constant M5 which depends only on ω, ∥u0∥ℋ such that
(73)∥∇v1(t)∥2≤Q(∥B0∥ℋ):=M4,∀t≥0,u0∈B0.


## 5.3. The Second A Priori Estimate

The main purpose of this subsection is to deduce some uniformly asymptotic (with respect toε∈[0,1] and t) the a priori estimates about the solution of (Eε).Lemma 14.
There exists positive constantsν-, R->0, and Q(·)∈𝒥 such that for each ε∈[0,1], there is a subset B-ε⊂ℋε1 satisfying
(74)∥B-ε∥ℋε12=supu∈B-ε{∥∇u∥2+ε∥Δu∥2}≤R-,
and the exponential attraction
(75)distℋε0(Sε(t)B0,B-ε)≤Q1(∥B0∥ℋ)e-ν-t,∀t≥0,
where all ν-,R-, and Q1(·) are independent of ε∈[0,1], and distℋε0(·,·) denotes the Hausdorff semidistance with respect to the ℋε0-norm.Proof.
It is convenient to separate our proof into three steps. We emphasize, especially, that all the generic constants in the proof are independent ofε∈[0,1].
Step  1. We first claim that (recall σ=min{1/4,(N+2-(N-2)γ)/2}): ∃νσ, Rσ>0 and Qσ(·)∈𝒥 such that for each ε∈[0,1], there is a subset B-σ,ε⊂ℋεσ satisfying
(76)∥B-σ,ε∥ℋεσ2=supu∈B-ε{∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2}≤Rσ
and the exponential attraction
(77)distℋε0(Sε(t)B0,B-σ,ε)≤Qσ(∥B0∥ℋ)e-νσt,∀t≥0.
We will apply Lemma2 with X=ℋε0 and V=ℋεσ (note that B0⊂ℋ⊂ℋε0 for any ε∈[0,1]). From (54), we can write
(78)f0(s)=sφ(s)with|φ(s)|≤C|s|4/(N-2).
For anyx∈B0 and y∈ℋε0, z∈ℋεσ satisfying x=y+z, we decompose the solution of (Eε) as Sε(t)(x)=Vxε(t)y+Uxε(t)z, where
(79)Vxε(t)y=v-(t),Uxε(t)z=w-(t),
which uniquely solves the following equations, respectively:
(80)v-t-εΔv-t-ωΔv-=h1,v-(x,0)=y,v-|∂Ω=0,(81)w-t-εΔw-t-ωΔw-=h2,w-(x,0)=z,w-|∂Ω=0,
with h1=-v-φ(v) and h2=g(x)-f(u)+v-φ(v), and v(t) is the solution of (59) corresponding to the initial data x.
For (80), from (54), (56), (78), and Lemmas 10 and 12, we can directly calculate that
(82)∥h1∥L2N/(N+2)≤C∥∇v-∥∥∇v∥4/(N-2)≤CM-,ω,Ne-k′t∥∇v-∥,
where k′=(2/(N-2))k, k is given in Lemma 10.
Multiplyingv- by (80), we have
(83)ddt(∥v-∥2+ε∥∇v-∥2)+2ω∥∇v-∥2=2〈h1,v-〉≤2∥h1∥L2N/(N+2)∥v-∥L2N/(N-2)≤CM-,ω,Ne-k′t∥∇v-∥2≤CM,ω,Ne-2k′tε∥∇v-∥2+ω∥∇v-∥2.
Furthermore, using the similar estimates of Lemma6, we get
(84)ddt(∥v-∥2+ε∥∇v-∥2)+ε1(∥v-∥2+ε∥∇v-∥2)≤2CM-,ω,Ne-2k′t(∥v-∥2+ε∥∇v-∥2),
where ε1 is a small positive constant such that ε1≤min{CM-,ω,Ne-2k′t,λ1ω} for all t≥0.
And then applying Lemma5 to above inequality, there holds
(85)∥Vxε(t)y∥ℋε02≤eCM,ω,N/k′e-ε1t∥y∥ℋε02.
For (81), since
(86)h2=g(x)-f(u)+f(v)-f1(v)+wφ(v)-w-φ(v),
then
(87)|h2|≤|g|+C|w|(1+|u|4/(N-2)+|v|4/(N-2))|h2|≤+C|w-||v|4/(N-2)+C(1+|v|γ).
Using Hölder inequality we get(88)∥h2∥L2N/(N+2(1-σ))≤2C|Ω|(2-2σ)/2N∥g∥+2C∥w∥L2N/(N-2(1+σ))×(1+∥u∥L2N/(N-2)4/(N-2)+∥v∥L2N/(N-2)4/(N-2))+2C∥v∥L2N/(N-2)4/(N-2)∥w-∥L2N/(N-2(1-σ))+2C(1+∥v∥L2N/(N-2)4/(N-2))≤C|Ω|∥g∥2+C∥A(1+σ)/2w∥×(1+∥∇u∥4/(N-2)+∥∇v∥4/(N-2))+C∥∇v∥4/(N-2)∥A(1+σ)/2w-∥+C(1+∥∇v∥4/(N-2))≤CM-,ω,Ne-k′t∥A(1+σ)/2w-∥+CM-,∥g∥2,|Ω|J(t),∀t≥0,
where we used (53), (62), and Lemmas 10 and 12.
Hence, multiplyingAσw- by (83), we have
(89)ddt(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+2ω∥A(1+σ)/2w-∥2=2〈h2,w-〉≤2∥h2∥L2N/(N+2(1-σ))∥Aσw-∥L2N/(N-2(1-σ))≤2CM-,ω,Ne-2k′tε∥A(1+σ)/2w-∥2+2CM-,∥g∥2,|Ω|J(t)+ω∥A(1+σ)/2w-∥2.
Furthermore, we have(90)ddt(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+ε1(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)≤2CM-,ω,Ne-2k′t(∥Aσ/2w-∥2+ε∥A(1+σ)/2w-∥2)+CM-,∥g∥2,|Ω|J(t),
where ε1 is a small positive constant given in (84).
Then, using Lemma5 we obtain
(91)∥Uxε(t)z∥ℋεσ2≤eM-,ω,N/k′e-ε1t∥z∥ℋεσ2+1ε1eε1CM-,∥g∥2,|Ω|J(t).
Therefore, combining (85) and (91), we can verify that all the conditions of Lemma 2 are satisfied for the cases X=ℋε0, V=ℋεσ, and T(t)=Sε(t). Moreover, since there is a c1>0 (independent of ε) such that c1∥B0∥ℋ≥∥B0∥ℋε0 for any ε∈[0,1] and the constants in our estimates are all independent of ε; consequently, νσ, Rσ, and Qσ(·) are all independent of ε∈[0,1], and then we can deduce our claim.
Step  2. We claim that there exists a constant R-σ>0 which depends only on Rσ such that
(92)∀ε∈[0,1],∥Sε(t)B-σ,ε∥ℋεσ2≤R-σ,∀t≥0.
Multiplying(Eε) by Aσu(t), we only need to note the following:
(93)12ddt(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)+ω∥A(1+σ)/2u∥2=〈g,Aσu〉-〈f(u),Aσu〉.
First, sinceσ<1, we have σ<(1+σ)/2 and then
(94)|〈g,Aσu(t)〉|≤∥g∥∥Aσu∥≤∥g∥∥A(1+σ)/2u∥≤4ω∥g∥2+ω16∥A(1+σ)/2u∥2,(95)〈f(u),Aσu〉≤C∫Ω‍(1+|u|(N+2)/(N-2))|Aσu|dx≤4ω|Ω|2+ω16∥A(1+σ)/2u∥2+C∫Ω‍|u|4/(N-2)|u||Aσu|dx,
while
(96)∫Ω‍|u|4/(N-2)|u||Aσu|dx≤∫Ω‍(|v1|4/(N-2)+|w1|4/(N-2))|u||Aσu|dx,∫Ω‍|v1|4/(N-2)|u||Aσu|dx≤C∥∇v1∥4/(N-2)∥A(1+σ)/2∥2≤ω16∥A(1+σ)/2u∥2+CM4∥∇v1∥∥A(1+σ)/2u∥2,
where we used (73).
Moreover, sinceσ≤1/4, we have 2≤2N(N-2)/(N(N-4-2σ)+4(1+3σ))≤2N/(N-2-2σ) and then
(97)∫Ω‍|w1|4/(N-2)|u||Aσu|dx≤C∥∇w1∥L2N/(N-2-2σ)4/(N-2)∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))∥Aσu∥≤C∥A(1+σ)/2w1∥4/(N-2)×∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))∥A(1+σ)/2u∥≤CKη4/(N-2)∥u∥L2N(N-2)/(N(N-4-2σ)+4(1+3σ))×∥A(1+σ)/2u∥≤CKη4/(N-2)(CM+ω32CKη4/(N-2)∥A(1+σ)/2u∥)×∥A(1+σ)/2u∥≤ω32∥A(1+σ)/2u∥2+CMKη4/(N-2)∥A(1+σ)/2u∥≤CMKη8/(N-2)+ω16∥A(1+σ)/2u∥2,
where Kη is given in Lemma 13.
Hence, substituting the above estimates into (93), applying the Poincaré inequality we have
(98)ddt(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)+(C-CM4∥∇v1∥2)(∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2)≤8ω∥g∥2+CM(1+Kη8/(N-2)).
Then using the Gronwall inequality and integrating over[0,t] (from Lemma 12), we obtain
(99)∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2≤e-∫0t‍(C-CM4∥∇v1(s)∥2)ds≤≤×(∥Aσ/2u(0)∥2+ε∥A(1+σ)/2u(0)∥2)≤≤+CM(1+Kη8/(N-2))≤≤×∫0t‍e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds≤≤+8ω∫0t‍∥g∥2e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds.
Takingη (in Lemma 13) small enough such that η<C/2CM4, we have
(100)∫0t‍e-∫ts‍(C-CM4∥∇v1(y)∥2)dyds=∫0t‍eC(s-t)e∫st‍CM4∥∇v1(y)∥2)dyds≤∫0t‍eC(s-t)eCM4η(t-s)+CM4Cηds≤eCM4Cη∫0t‍eC(s-t)/2ds≤2eCM4CηC,(101)e-∫0t‍(C-CM4∥∇v1(s)∥2)ds≤e-Ct/2eCM4Cη.
Thus,(102)∫0t‍∥g∥2e-∫ts‍(C-CM5∥∇v1(y)∥2)dyds≤eCM4Cη1-e-(C/2)∥g∥2.
Substituting above (100) and (102) into (99), we get that for all t≥0(103)∥Aσ/2u∥2+ε∥A(1+σ)/2u∥2≤eCM5CηRσ≤+2Cω(1+Kη8/(N-2))CeCM5Cη≤+eCM5Cη1-e-(C/2)∥g∥2:=R-σ.
Step  3.  Based on Step 1 and Step 2, applying the attraction transitivity lemma given in [28, Theorem5.1] and noticing the Holder continuity Lemma 9, we can prove our lemma by performing a standard bootstrap argument, whose proof is now simple since Step 1 makes the nonlinear term become subcritical to some extent.


## 5.4. Proof of Theorem1

Lemma14 has shown some asymptotic regularities; however, the radius of ∥B-ε∥ℋ1 depends on ε and the distances only under the ℋε0-norm.To prove Theorem1, we first give two lemmas as preliminary.Lemma 15.
There exsits a constantR1>0 such that for any bounded (in ℋε1) subset B⊂ℋε1, there exsits T1=T1(∥B∥ℋε1) such that
(104)∀ε∈[0,1],∥Sε(t)B∥ℋε12≤R1,∀t≥T1.Proof.
Multiplying(Eε) by -Δu, we find
(105)ddt(∥∇u∥2+ε∥∇u∥2)+2ω∥∇u∥2=-2〈f(u),-Δu〉+2〈g,-Δu〉,2〈g,-Δu〉≤Cω∥g∥2+ω2∥∇u∥2.
Notingu∈ℋε1⊂ℋε0, from Lemma 6, yields
(106)|-2〈f(u),-Δu〉|≤2∫Ω‍|f′(u)||∇u||∇u|dx≤C(1+∥∇u∥4/(N-2))∥Δu∥∥Δu∥≤CMε∥∇u∥2+ω2∥∇u∥2,
hence, we obtain(107)ddt(∥∇u∥2+ε∥∇u∥2)+ε2(∥∇u∥2+ε∥∇u∥2)≤2CM(∥∇u∥2+ε∥∇u∥2)+Cω∥g∥2,
where ε2 is a small, positive constant.
Similarly, with using Lemma4 we finally complete the proof.Lemma 16.
There exists a constantR2>0 such that for any bounded (in ℋε1) subset B⊂ℋε1, there is a T2=T2(∥B∥ℋε1) such that
(108)∀ε∈[0,1],∥Sε(t)B∥ℋ12≤R2,∀t≥T2.Proof.
From Lemma15, we only need to estimate that the bound of ∥Δu∥2 is independent of ε∈[0,1].
Applying Lemma15 again, we have
(109)ddt(∥∇u∥2+ε∥∇u∥2)+(2ω-2CCM-1)∥∇u∥2≤Cω∥g∥2.
Takingω=max{1,2CCM} which may provide that 1/ω<ε and 2ω-2CCM-1>0, integrating (109) on [t,t+1], and from Lemma 15, when t≥T1 we yield
(110)∫tt+1‍∥Δu(τ)∥2dτ≤Cω,R12ω-2CCM-1.
Hence, multiplying(Eε) by -Δut, we can complete our proof by applying the uniform Gronwall lemma.Now, we are ready to prove Theorem1.Proof of Theorem1.
Set(111)𝔹={u∈ℋ1:∥u∥ℋ12≤R2},
where the constant R2 comes from Lemma 16.
From Lemmas16 and 14, we know that there is a t0 such that Sε(t)B-ε⊂𝔹 (recall that B-ε is given in (78)) for all t≥t0 and any ε∈[0,1].
On the other hand, note that∃c1,c2>0 such that
(112)c1∥·∥ℋε0≤∥·∥ℋ≤c2∥·∥ℋε1.
Then, from Lemma9, there exists t1 which depends only on ∥B0∥ℋ and ∥B-ε∥ℋε1 (so only on M, R-) such that
(113)==∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-t∀ε∈[0,1],==∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-t∥u1-u2∥ℋε01/2,====∥Sε(t)u1-Sε(t)u2∥ℋ≤eCR-tx∀t≥t1,u1∈B0⊂ℋ⊂ℋε0,u2∈B-ε⊂ℋε1⊂ℋε0,(114)∀ε∈[0,1],Sε(t)B0⊂B0,∀t≥t1.
Therefore, from Lemma14, we have
(115)distℋ(Sε(t+t0+t1)B0,𝔹)≤distℋ(Sε(t+t0+t1)B0,Sε(t0+t1)B-ε)≤CM,R-,t0+t1distℋ1/2(Sε(t)B0,B-ε)≤CM,R-,t0+t1Q1(∥B0∥ℋ)e-(ν-/2)t.
Hence, noting thatt0,t1, and R- are all fixed, we can complete the proof by taking ν=ν-/2 and applying Lemma 11.


## 6. Applications of Theorem1

As for the applications of Theorem1, in this subsection, we consider the existence of finite dimensional exponential attractors and the upper semicontinuity of global attractors for problem (Eε) under assumptions (1), (2), and ω>1.


### 6.1. A Priori Estimates

For the subset𝔹 defined in (113), and from Lemmas 6 and 8 we know that there is a t𝔹 such that
(116)∀ε∈[0,1],∥ut∥2+ε∥∇ut∥2≤M3,∀t≥t𝔹,u0∈𝔹,
where u(t)=Sε(t)u0.Now, for eachε∈[0,1], define B^ε as follows:
(117)B^ε=⋃t≥t𝔹+T2‍Sε(t)𝔹,
where T2 is the time given in Lemma 16 corresponding to 𝔹. Then, for each ε∈[0,1] we have B^ε as a positive invariant under Sε(t) (i.e., Sε(t)B^ε=B^ε, for all t≥0) (from Lemma 16)
(118)∀ε∈[0,1],∥B^ε∥ℋ12≤R2.Moreover, we have the following results.Lemma 17.
Under assumptions (1), (2), and ω>1, there exists a constant 𝒯>0 such that for every ε∈[0,1], the semigroup Sε(t) satisfies the following properties: Sε(𝒯) admits a decomposition of the form
(119)Sε(𝒯)=Lε+Nε,Lε:B^ε⟶ℋε0,B^ε⟶ℋεθ,
where Lε  and  Nε satisfy the estimates
(120)∥Lεu1-Lεu2∥ℋε0≤14∥u1-u2∥ℋε0,∀u1,u2∈B^ε,∥Nεu1-Nεu2∥ℋεθ≤CR2𝒯∥u1-u2∥ℋε0,∀u1,u2∈B^ε,
where the constants CR2𝒯 are independent of ε and
(121)θ={1,N=3,4,5,6,4N-2,N>6.Proof.
For any two initial dataui∈B^ε with solution Sε(t)ui=ui (i=1,2), we decompose the difference Sε(t)u1-Sε(t)u2 as follows:
(122)Sε(t)u1-Sε(t)u2=Lε(t)(u1-u2)+Nε(t)(u1-u2),
where Lε(t)(u1-u2)=v~ solves
(123)v~t-εΔv~t-ωΔv~=0inΩ×ℝ+,v~(x,0)=u1-u2,v~|∂Ω=0,(124)w~t-εΔw~t-ωΔw~+f(u1)-f(u2)=0inΩ×ℝ+,w~(x,0)=0,w~|∂Ω=0.
Next, for clarity, we decompose the remainder proof into two steps.
Step  1. For v~(t), multiplying (123) by v~(t), we have
(125)ddt(∥v~∥2+ε∥∇v~∥2)+2μ1(∥v~∥2+ε∥∇v~∥2)≤2ωλ1(∥v~∥2+ε∥∇v~∥2),
where μ1 is a small positive constant such that μ1<ωλ1.
Using Lemma4 we can deduce that
(126)∥Lεu1-Lεu2∥ℋε02=∥v~∥ℋε02≤Q(∥B^ε∥ℋ)∥u1-u2∥ℋε02e-μ1t.
Hence, by takingT′>0 large enough, we get
(127)∥Lε(t+T′)u1-Lε(t+T′)u2∥ℋε0≤14∥u1-u2∥ℋε0,∀t≥0.
Step  2. For w~(t), multiplying (124) by Aθw~(t) (where θ is given in (121)), we obtain
(128)12ddt(∥Aθ/2w~∥2+ε∥A(1+θ)/2w~∥2)+ω∥A(1+θ)/2w~∥2=-〈f(u1)-f(u2),Aθw~〉.Case  1 (N=3,4). Then by using the embedding D(A)↪Lp(Ω) for any p≥1, we have
(129)|-〈f(u1)-f(u2),Aw~〉|≤C(1+∥u1∥L4N/(N-2)4/(N-2)+∥u2∥L4N/(N-2)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥ℋ14/(N-2)+∥u2∥ℋ14/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤CR2∥∇(u1-u2)∥∥Aw~∥≤CR2eCR2t∥u1-u2∥ℋε0∥Aw~∥≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥Aw~∥2,
where we have used (118) and (46).
Case  2 (
N
=
5,6
). Since 4N/(N-2)≤2N/(N-4) and embedding D(A)↪L2N/(N-4)(Ω), we also have
(130)|-〈f(u1)-f(u2),Aw~〉|≤C(1+∥u1∥L4N/(N-2)4/(N-2)+∥u2∥L4N/(N-2)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥L2N/(N-4)4/(N-2)+∥u2∥L2N/(N-4)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥ℋ14/(N-2)+∥u2∥ℋ14/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥Aw~∥2.
Case  3 (
N
>
6
). Noticing that embedding D(A)↪L2N/(N-4)(Ω) and (4-2θ)/2N+(N-2)/2N+(N-(1-2θ))/2N=1, we have
(131)|-〈f(u1)-f(u2),Aθw~〉|≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥A(1+θ)/2w~∥2,
therefore, for anyN≥3, we have
(132)ddt(∥Aθ/2w~∥2+ε∥A(1+θ)/2w~∥2)+ω∥A(1+θ)/2w~∥2≤2CR2,ωeCR2t∥u1-u2∥ℋε02,∀t≥0.
Hence, taking(133)𝒯=T′,Lε=Lε(𝒯),Nε=Nε(𝒯),
then, from (127) and (132), we can see that 𝒯, Lε,  and  Nε satisfy Lemma 17.Lemma 18.
Under assumptions (1), (2), and ω>1, for an arbitrary fixed time T>0 and any ε∈[0,1], the semigroup Sε(t) is Lipchitz continuous on [0,T]×B^ε in the following sense: there exists a positive constant C-T,R2 such that for any ui∈B^ε, ti∈[0,T], i=1,2,
(134)∥Sε(t1)u1-Sε(t2)u2∥ℋε0≤C-T,R2(|t1-t2|+∥u1-u2∥ℋε0).Proof.
Foru1,u2∈B^ε and t1,t2∈[0,T] we have
(135)∥Sε(t1)u1-Sε(t2)u2∥ℋε0≤∥Sε(t1)u1-Sε(t2)u1∥ℋε0+∥Sε(t2)u1-Sε(t2)u2∥ℋε0.
The second term of above inequality is handled by estimate (46). Concerning the first one
(136)∥Sε(t1)u1-Sε(t2)u1∥ℋε0=∥∫t1t2‍ddtSε(t)u1ds∥ℋε0≤|∫t1t2‍∥ddtSε(t)u1∥ℋε0ds|.
Then from (116) and (117) we can deduce
(137)∥Sε(t1)u1-Sε(t2)u1∥ℋε0≤M3|t1-t2|.
So, the proof is completed immediately.


### 6.2. Exponential Attractors

We are now ready to prove the following result about the existence of exponential attractors.Lemma 19.
Under assumptions (1), (2), and ω>1, for every ε∈[0,1], there exists a compact subset ℰ⊂ℋ1, uniformly bounded in ℋ1, which satisfies the following conditions:(i)
ℰε is semi-invariant with respect to the semigroup {Sε(t)}t≥0, that is,
(138)Sε(t)ℰε⊂ℰε,∀t≥0;(ii)
the fractal dimension ofℰε is finite, that is,
(139)dimF(ℰε,ℋ)≤Λε<∞,∀ε∈[0,1];(iii)
for eachε∈[0,1],ℰε enjoys a uniform exponential attraction property of the following form: for any bounded (inℋ) subset B⊂ℋ,
(140)distℋ(Sε(t)B,ℰε)≤Qε(∥B∥ℰ)e-ν′t,∀t≥0.Here,Λε and Qε(·) may depend on ε, but ν′ is independent of ε.Proof.
For eachε∈[0,1], we know that B^ε is invariant and compact in ℋε0. Hence, applying the abstract results established in [23, 24], from Lemmas 17 and 18 we can first construct an exponential attractor on B^ε with respect to the ℋε0-norm. Then, we can complete the proof by using the attraction transitivity lemma given in [28,Theorem5.1] from Lemma 14 and the Hölder continuity (47).


### 6.3. Upper Semicontinuity of Global Attractors

Since𝒜ε⊂ℰε, (ii) of Lemma 19 implies that the fractal dimension of the global attractor 𝒜ε is finite too. Moreover, we have the following upper semicontinuity result of 𝒜ε at ε=0.Lemma 20.
Under assumptions (1), (2), and ω>1, the global attractors {𝒜ε}ε∈[0,1] are upper semicontinuous at ε=0.                                  (141)distℋ(𝒜ε,𝒜0)⟶0asε⟶0+.Proof.
Since the global attractor𝒜ε is strictly invariant, that is, Sε(t)𝒜ε=𝒜ε for all t≥0, it is obvious to see that
(142)⋃ε∈[0,1]‍𝒜ε⊂𝔹andcompactinℋ.
Therefore, to apply Lemma3, we can take K=clℋ1(𝔹) and we only need to verify condition (14). Let ε∈[0,1] and u^=Sε(t)uε with uε∈𝒜ε; also let v^=S0(t)u0 with u0∈𝔹. Denote w^(t)=u^(t)-v^(t). Then w^ solves the following equation:
(143)w^t-ωΔw^+f(u^)-f(v^)=εΔu^t,w^(x,0)=uε-u0,w^|∂Ω=0.
Multiplying (143) by w^t, we have
(144)12ddt(ω∥∇w^∥2+2〈εΔu^,w^〉)+∥w^t∥2=-〈f(u^)-f(v^),w^t〉,(145)|-〈f(u^)-f(v^),w^t〉|≤C(1+∥u^∥ℋ14/(N-2)+∥v^∥ℋ14/(N-2))∥∇w^∥∥w^t∥≤C∥B∥ℋ1∥∇w^∥2+12∥w^t∥2,
where we used Lemma 19 and (118) and noticed the process of Step  2 of Lemma 17 for N.
On the other hand, for (145), using Poincaré inequality, we have
(146)(ω-1)∥∇w^∥2-ε2∥∇u^∥2≤ω∥∇w^∥2+2〈εΔu^,w^〉≤(ω+1λ1)∥∇w^∥2+ε2∥∇u^∥2.
So, we obtain(147)ddt(ω∥∇w^∥2+2〈εΔu^,w^〉)+∥w^t∥2≤2C∥B∥ℋ1∥∇w^∥2≤2C∥B∥ℋ1(ω∥∇w^∥2+2〈εΔu^,w^〉).
Noticing𝔹⊂ℋ1⊂ℋε1⊂ℋε0, using (24), Lemma 15, and Gronwall inequality, yields
(148)∥∇w^∥2≤1(ω-1)2C∥B∥t,ℋ1((ω+1λ1)∥uε-u0∥ℋ2+εR1)+εM1,∀t≥T1B+T1.
Hence, we know that there existst1=t1(∥𝔹∥ℋ1)≥T1B+T1 such that
(149)∥∇w^(t1+1)∥2≤1(ω-1)2Ct1,∥B∥ℋ1((ω+1λ1)∥uε-u0∥ℋ2+εR1)+M1ω-1ε,
which implies
(150)ifεn⟶0+,and𝒜εn∋un⟶u0,thenSεn(t1+1)un⟶S0(t1+1)u0.
Therefore, from (142) and (150), we can directly apply Lemma 3 to complete the proof.


## 6.1. A Priori Estimates

For the subset𝔹 defined in (113), and from Lemmas 6 and 8 we know that there is a t𝔹 such that
(116)∀ε∈[0,1],∥ut∥2+ε∥∇ut∥2≤M3,∀t≥t𝔹,u0∈𝔹,
where u(t)=Sε(t)u0.Now, for eachε∈[0,1], define B^ε as follows:
(117)B^ε=⋃t≥t𝔹+T2‍Sε(t)𝔹,
where T2 is the time given in Lemma 16 corresponding to 𝔹. Then, for each ε∈[0,1] we have B^ε as a positive invariant under Sε(t) (i.e., Sε(t)B^ε=B^ε, for all t≥0) (from Lemma 16)
(118)∀ε∈[0,1],∥B^ε∥ℋ12≤R2.Moreover, we have the following results.Lemma 17.
Under assumptions (1), (2), and ω>1, there exists a constant 𝒯>0 such that for every ε∈[0,1], the semigroup Sε(t) satisfies the following properties: Sε(𝒯) admits a decomposition of the form
(119)Sε(𝒯)=Lε+Nε,Lε:B^ε⟶ℋε0,B^ε⟶ℋεθ,
where Lε  and  Nε satisfy the estimates
(120)∥Lεu1-Lεu2∥ℋε0≤14∥u1-u2∥ℋε0,∀u1,u2∈B^ε,∥Nεu1-Nεu2∥ℋεθ≤CR2𝒯∥u1-u2∥ℋε0,∀u1,u2∈B^ε,
where the constants CR2𝒯 are independent of ε and
(121)θ={1,N=3,4,5,6,4N-2,N>6.Proof.
For any two initial dataui∈B^ε with solution Sε(t)ui=ui (i=1,2), we decompose the difference Sε(t)u1-Sε(t)u2 as follows:
(122)Sε(t)u1-Sε(t)u2=Lε(t)(u1-u2)+Nε(t)(u1-u2),
where Lε(t)(u1-u2)=v~ solves
(123)v~t-εΔv~t-ωΔv~=0inΩ×ℝ+,v~(x,0)=u1-u2,v~|∂Ω=0,(124)w~t-εΔw~t-ωΔw~+f(u1)-f(u2)=0inΩ×ℝ+,w~(x,0)=0,w~|∂Ω=0.
Next, for clarity, we decompose the remainder proof into two steps.
Step  1. For v~(t), multiplying (123) by v~(t), we have
(125)ddt(∥v~∥2+ε∥∇v~∥2)+2μ1(∥v~∥2+ε∥∇v~∥2)≤2ωλ1(∥v~∥2+ε∥∇v~∥2),
where μ1 is a small positive constant such that μ1<ωλ1.
Using Lemma4 we can deduce that
(126)∥Lεu1-Lεu2∥ℋε02=∥v~∥ℋε02≤Q(∥B^ε∥ℋ)∥u1-u2∥ℋε02e-μ1t.
Hence, by takingT′>0 large enough, we get
(127)∥Lε(t+T′)u1-Lε(t+T′)u2∥ℋε0≤14∥u1-u2∥ℋε0,∀t≥0.
Step  2. For w~(t), multiplying (124) by Aθw~(t) (where θ is given in (121)), we obtain
(128)12ddt(∥Aθ/2w~∥2+ε∥A(1+θ)/2w~∥2)+ω∥A(1+θ)/2w~∥2=-〈f(u1)-f(u2),Aθw~〉.Case  1 (N=3,4). Then by using the embedding D(A)↪Lp(Ω) for any p≥1, we have
(129)|-〈f(u1)-f(u2),Aw~〉|≤C(1+∥u1∥L4N/(N-2)4/(N-2)+∥u2∥L4N/(N-2)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥ℋ14/(N-2)+∥u2∥ℋ14/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤CR2∥∇(u1-u2)∥∥Aw~∥≤CR2eCR2t∥u1-u2∥ℋε0∥Aw~∥≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥Aw~∥2,
where we have used (118) and (46).
Case  2 (
N
=
5,6
). Since 4N/(N-2)≤2N/(N-4) and embedding D(A)↪L2N/(N-4)(Ω), we also have
(130)|-〈f(u1)-f(u2),Aw~〉|≤C(1+∥u1∥L4N/(N-2)4/(N-2)+∥u2∥L4N/(N-2)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥L2N/(N-4)4/(N-2)+∥u2∥L2N/(N-4)4/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤C(1+∥u1∥ℋ14/(N-2)+∥u2∥ℋ14/(N-2))×∥∇(u1-u2)∥∥Aw~∥≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥Aw~∥2.
Case  3 (
N
>
6
). Noticing that embedding D(A)↪L2N/(N-4)(Ω) and (4-2θ)/2N+(N-2)/2N+(N-(1-2θ))/2N=1, we have
(131)|-〈f(u1)-f(u2),Aθw~〉|≤CR2,ωeCR2t∥u1-u2∥ℋε02+ω2∥A(1+θ)/2w~∥2,
therefore, for anyN≥3, we have
(132)ddt(∥Aθ/2w~∥2+ε∥A(1+θ)/2w~∥2)+ω∥A(1+θ)/2w~∥2≤2CR2,ωeCR2t∥u1-u2∥ℋε02,∀t≥0.
Hence, taking(133)𝒯=T′,Lε=Lε(𝒯),Nε=Nε(𝒯),
then, from (127) and (132), we can see that 𝒯, Lε,  and  Nε satisfy Lemma 17.Lemma 18.
Under assumptions (1), (2), and ω>1, for an arbitrary fixed time T>0 and any ε∈[0,1], the semigroup Sε(t) is Lipchitz continuous on [0,T]×B^ε in the following sense: there exists a positive constant C-T,R2 such that for any ui∈B^ε, ti∈[0,T], i=1,2,
(134)∥Sε(t1)u1-Sε(t2)u2∥ℋε0≤C-T,R2(|t1-t2|+∥u1-u2∥ℋε0).Proof.
Foru1,u2∈B^ε and t1,t2∈[0,T] we have
(135)∥Sε(t1)u1-Sε(t2)u2∥ℋε0≤∥Sε(t1)u1-Sε(t2)u1∥ℋε0+∥Sε(t2)u1-Sε(t2)u2∥ℋε0.
The second term of above inequality is handled by estimate (46). Concerning the first one
(136)∥Sε(t1)u1-Sε(t2)u1∥ℋε0=∥∫t1t2‍ddtSε(t)u1ds∥ℋε0≤|∫t1t2‍∥ddtSε(t)u1∥ℋε0ds|.
Then from (116) and (117) we can deduce
(137)∥Sε(t1)u1-Sε(t2)u1∥ℋε0≤M3|t1-t2|.
So, the proof is completed immediately.


## 6.2. Exponential Attractors

We are now ready to prove the following result about the existence of exponential attractors.Lemma 19.
Under assumptions (1), (2), and ω>1, for every ε∈[0,1], there exists a compact subset ℰ⊂ℋ1, uniformly bounded in ℋ1, which satisfies the following conditions:(i)
ℰε is semi-invariant with respect to the semigroup {Sε(t)}t≥0, that is,
(138)Sε(t)ℰε⊂ℰε,∀t≥0;(ii)
the fractal dimension ofℰε is finite, that is,
(139)dimF(ℰε,ℋ)≤Λε<∞,∀ε∈[0,1];(iii)
for eachε∈[0,1],ℰε enjoys a uniform exponential attraction property of the following form: for any bounded (inℋ) subset B⊂ℋ,
(140)distℋ(Sε(t)B,ℰε)≤Qε(∥B∥ℰ)e-ν′t,∀t≥0.Here,Λε and Qε(·) may depend on ε, but ν′ is independent of ε.Proof.
For eachε∈[0,1], we know that B^ε is invariant and compact in ℋε0. Hence, applying the abstract results established in [23, 24], from Lemmas 17 and 18 we can first construct an exponential attractor on B^ε with respect to the ℋε0-norm. Then, we can complete the proof by using the attraction transitivity lemma given in [28,Theorem5.1] from Lemma 14 and the Hölder continuity (47).


## 6.3. Upper Semicontinuity of Global Attractors

Since𝒜ε⊂ℰε, (ii) of Lemma 19 implies that the fractal dimension of the global attractor 𝒜ε is finite too. Moreover, we have the following upper semicontinuity result of 𝒜ε at ε=0.Lemma 20.
Under assumptions (1), (2), and ω>1, the global attractors {𝒜ε}ε∈[0,1] are upper semicontinuous at ε=0.                                  (141)distℋ(𝒜ε,𝒜0)⟶0asε⟶0+.Proof.
Since the global attractor𝒜ε is strictly invariant, that is, Sε(t)𝒜ε=𝒜ε for all t≥0, it is obvious to see that
(142)⋃ε∈[0,1]‍𝒜ε⊂𝔹andcompactinℋ.
Therefore, to apply Lemma3, we can take K=clℋ1(𝔹) and we only need to verify condition (14). Let ε∈[0,1] and u^=Sε(t)uε with uε∈𝒜ε; also let v^=S0(t)u0 with u0∈𝔹. Denote w^(t)=u^(t)-v^(t). Then w^ solves the following equation:
(143)w^t-ωΔw^+f(u^)-f(v^)=εΔu^t,w^(x,0)=uε-u0,w^|∂Ω=0.
Multiplying (143) by w^t, we have
(144)12ddt(ω∥∇w^∥2+2〈εΔu^,w^〉)+∥w^t∥2=-〈f(u^)-f(v^),w^t〉,(145)|-〈f(u^)-f(v^),w^t〉|≤C(1+∥u^∥ℋ14/(N-2)+∥v^∥ℋ14/(N-2))∥∇w^∥∥w^t∥≤C∥B∥ℋ1∥∇w^∥2+12∥w^t∥2,
where we used Lemma 19 and (118) and noticed the process of Step  2 of Lemma 17 for N.
On the other hand, for (145), using Poincaré inequality, we have
(146)(ω-1)∥∇w^∥2-ε2∥∇u^∥2≤ω∥∇w^∥2+2〈εΔu^,w^〉≤(ω+1λ1)∥∇w^∥2+ε2∥∇u^∥2.
So, we obtain(147)ddt(ω∥∇w^∥2+2〈εΔu^,w^〉)+∥w^t∥2≤2C∥B∥ℋ1∥∇w^∥2≤2C∥B∥ℋ1(ω∥∇w^∥2+2〈εΔu^,w^〉).
Noticing𝔹⊂ℋ1⊂ℋε1⊂ℋε0, using (24), Lemma 15, and Gronwall inequality, yields
(148)∥∇w^∥2≤1(ω-1)2C∥B∥t,ℋ1((ω+1λ1)∥uε-u0∥ℋ2+εR1)+εM1,∀t≥T1B+T1.
Hence, we know that there existst1=t1(∥𝔹∥ℋ1)≥T1B+T1 such that
(149)∥∇w^(t1+1)∥2≤1(ω-1)2Ct1,∥B∥ℋ1((ω+1λ1)∥uε-u0∥ℋ2+εR1)+M1ω-1ε,
which implies
(150)ifεn⟶0+,and𝒜εn∋un⟶u0,thenSεn(t1+1)un⟶S0(t1+1)u0.
Therefore, from (142) and (150), we can directly apply Lemma 3 to complete the proof.



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*Source: 204270-2013-09-01.xml*