# Time-Dependent Global Attractor for a Class of Nonclassical Parabolic Equations

Authors: Fang-hong Zhang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Based on the recent theory of time-dependent global attractors in the works of Conti et al. (2013)  and di Plinio et al. (2011), we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations.


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## Body

## 1. Introduction

LetΩ be a bounded set of R3 with smooth boundary ∂Ω. For any τ∈R, we consider the following nonclassical equation:
(1)ɛ(t)ut-Δut-Δu+f(u)=g(x),inΩ×[τ,∞),u=0,on⁡∂Ω,u(x,τ)=a(x),x∈Ω,
where a:Ω→R are assigned data and ɛ(t) is a decreasing bounded function satisfying
(2)lim⁡t→+∞ɛ(t)=0,
and L>0 is such that
(3)sup⁡t∈R|ɛ(t)|≤L.The nonlinearityf∈C2(R), with f(0)=0, is assumed to satisfy the inequality
(4)|f′′(s)|≤c(1+|u|),forsomec≥0,
along with the dissipation condition
(5)lim⁡inf⁡|s|→∞f(s)s>-λ1,
where λ1 is the first eigenvalue of -Δ in H01(Ω) and g(x)∈L2(Ω).The classical reaction diffusion equation has a long history in mathematical physics and appears in many mathematical models. It arises in several bead mark problems of hydrodynamics and heat transfer theory, such as heat transfer, as well as in solid-fluid in Hradionconfigurations and, of course, in standard situations mass diffusion and flow through porous media [1, 2]. In 1980, Aifantis in [1] pointed out that the classical diffusion equation does not suffice to describe transport in media with two temperatures or two diffusions as well as in cases where the diffusions substances behave as a viscous fluid. It turns out that new terms appear in the classical diffusion equation when such effects are considered. In particular, the mixed spatiotemporal derivative Δut consistently appears in several generalized reaction-diffusion models [1, 3–6] and this is our motivation for studying (1) which contains, in addition, the nonlinear term f(u) and inhomogeneous term g(x), along with the time-dependent parameter ɛ(t). The presence of the Δut has some important consequences on the character of the solution of the partial differential equation under consideration. For example, the classical reaction diffusion equation has smoothing effect; for example, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for (1), if the initial data u0 belongs to H01(Ω), then the solution u(t,x) with u(0,x)=u0 is always in H01(Ω) and has no higher regularity because of the term -Δut.In the case whenɛ(t)=ɛ is a positive constant, the asymptotic behavior of solutions to (1) has been extensively studied by several authors in [2, 7–16] and the references therein. In the general case of time dependence, that is, ɛ=ɛ(t), the longtime behavior of the nonclassical equation has not been considered so far. In this paper, we borrow some ideas from the following previous contributors: Conti et al. in [17] who introduced the theory of time-dependent global attractors and apply the theory to the wave equations; Caraballo et al. who introduced a one-parameter family of Banach spaces in the context of cocycles for nonautonomous and random dynamical systems in [18] as well as time-dependent spaces [19] in the context of stochastic partial differential equations; and Flandoli and Schmalfuss in [20] who introduced a family of metric spaces depending on a parameter and applied it to the stochastic form of Navier-Stokes equations. In this paper, based on the recent theory of time-dependent global attractors of Conti et al. [17] and di Plinio et al. [21], we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations as described by (1).The paper is organized as follows. In Section2, we present some preliminaries, establish the necessary notation and functions spaces to be used in the subsequent analysis, and give some useful lemmas. In Section 3, we prove the existence of time-dependent global attractors for the nonclassical parabolic equation and its regularity. Our main results are Theorems 14 and 16.


## 2. Preliminaries

In this section, we introduce some notations and definitions, along with a lemma.We setH=L2(Ω), with inner product 〈·,·〉 and norm ∥·∥. For 0≤σ≤2, we define the hierarchy of (compactly) nested Hilbert spaces:
(6)Hσ=dom⁡(Aσ/2),〈w,v〉σ=〈Aσ/2w,Aσ/2v〉,∥w∥σ=∥Aσ/2w∥.
Then, for t∈R and 0≤σ≤2, we introduce the time-dependent spaces Htσ=Hσ+1 endowed with the time-dependent product norms:
(7)∥u∥Htσ2=ɛ(t)∥u∥σ2+∥u∥σ+12.
The symbol σ is always omitted whenever zero. In particular, the time-dependent phase space where we settle the problem is
(8)Ht=H1with∥u∥Ht2=ɛ(t)∥u∥2+∥u∥12.
Then, we have the compact embeddings:
(9)Htσ⊂Ht,0<σ≤2,
with injection constants independent of t∈R.Note that the spacesHt are all the same as linear spaces, and the norms ∥z∥Ht2 and ∥·∥Hτ2 are equivalent for any fixed t, τ∈R.According to (5), we have the following lemma.Lemma 1.
The following inequalities hold for some0<ν<1 and c1≥0:
(10)2〈F(u),1〉≥-(1-ν)∥u∥12-c1,〈f(u),u〉≥-(1-ν)∥u∥12-c1,∀u∈H1.


## 3. Existence of the Time-Dependent Global Attractor

### 3.1. Well-Posedness

For anyτ∈R, we rewrite the problem (1) as
(11)ɛ(t)ut+Aut+Au+f(u)=g(x),t>τ,u(x,τ)=a.Using the Galerkin approximation method, we can obtain the following result concerning the existence and uniqueness of solutions; see, for example, [2, 7, 8, 15, 16].Lemma 2.
Under the assumptions of (2)–(5), for any a∈Hτ, there is a unique solution u of (1), on any interval [τ,t] with t≥τ,
(12)u∈C([τ,t];H1).
Furthermore, for i=1,2, let uτi∈Hτ be two initial conditions such that ∥uτi∥Hτ≤R and denote by ui the corresponding solutions to the problem of (11). Then the following estimates hold as follows:
(13)∥u1(t)-u2(t)∥Ht2≤eK(t-τ)∥uτ1-uτ2∥Ht2,t≥τ,
for some constant K=K(R)≥0.According to Lemma2 above, the family of maps with t≥τ∈R(14)U(t,τ):Hτ⟶Ht,
acting as
(15)uτ⟶u(t)=U(t,τ)zτ,
where u is the unique solution of (11) with initial time τ and initial condition uτ=Hτ, defines a strongly continuous process on the family {Ht}t∈R.


### 3.2. Time-Dependent Absorbing Set

Definition 3.
A time-dependent absorbing set for the processU(t,τ) is a uniformly bounded family B={Bt}t∈R with the following property: for every R≥0 there exists θe=θe(R)≥0 such that
(16)τ≤t-θe⟹U(t,τ)Bτ(R)⊂Bt.Lemma 4.
Under the assumptions of (2)–(5), for uτ∈Hτ, t≥τ, let U(t,τ)uτ be the solution of (1); then, there exist positive constants δ, Cν,∥g∥ and an increasing positive function Q such that
(17)∥U(t,τ)uτ∥Ht≤Q(∥uτ∥Hτ)e-δ(t-τ)+Cδ,ν,∥g∥,∀t≥τ.Proof.
Multiplying (11) by u, we obtain
(18)ddt(ɛ∥u∥2+2∥∇u∥2)+2∥∇u∥2-ɛ′∥u∥2+2〈f(u),u〉-2〈u,g〉=0.
Noting that ɛ′<0 and using (3) and Young and Poincaré inequalities, for δ>0 small, we infer that
(19)ddt(ɛ∥u∥2+2∥∇u∥2)+δ(ɛ∥u∥2+∥∇u∥2)≤Cδ,ν,∥g∥.
By the Gronwall lemma, we have
(20)∥u(t)∥Ht2≤Ce-δ(t-τ)∥u(τ)∥Ht2+Cδ,ν,∥g∥.
This completes the proof.Lemma 5 (time-dependent absorbing set).
Under the assumptions of (2)–(5), there exists a constant R1>0, such that the family B={Bt(R1)}t∈R is a time-dependent absorbing set for U(t,τ).Proof.
From the proof of Lemma4, for u∈Bτ(R), there exists θe≥0, provided that t-τ≥θe,
(21)∥U(t,τ)uτ∥Ht≤R0.
This concludes the proof of the existence of the time-dependent absorbing set.
We can assume that the time-dependent absorbing setBt=Bt(R1) is positively invariant (namely, U(t,τ)Bτ⊂Bt for all t≥τ). Indeed, calling θe the entering time of Bt such that
(22)U(t,τ)Bτ⊂Bt,∀τ≤t-θe,
we can substitute Bt with the invariant absorbing family:
(23)⋃τ≤t-θe‍U(t,τ)Bτ⊂Bt.


### 3.3. Time-Dependent Global Attractor

As introduced in [17], for t∈R, let Xt be a family of normed spaces; we consider the collection
(24)K={R={Kt}t∈R:Kt⊂Xtcompact,hRpullbackattracting}.
When K≠Ø we say that the process is asymptotically compact.Definition 6.
One calls a time-dependent global attractor the smallest element ofK; that is, the family A={At}t∈R∈K such that
(25)At⊂Kt,∀t∈R,
for any element R={Kt}t∈R∈K.Theorem 7 (see [17]).
IfU(t,τ) is asymptotically compact, then the time-dependent attractor A exists and coincides with the set A={At}t∈R. In particular, it is unique.According to Definition6, the existence of the time-dependent global attractor will be proved by a direct application of the abstract Theorem 7. Precisely, in order to show that the process is asymptotically compact, we will exhibit a pullback attracting family of compact sets. To this aim, the strategy classically consists in finding a suitable decomposition of the process in the sum of a decaying part and of a compact one.


#### 3.3.1. The First Decomposition of the System Equations

For the nonlinearityf, following [11, 15, 17], we decompose f as follows:
(26)f=f0+f1,
where f0,f1∈C2(R) satisfy, for some c≥0,
(27)|f0′′(u)|≤c(1+|u|),∀u∈R,(28)f0(u)u≥0,∀u∈R,(29)f0(0)=f0′(0)=0,(30)|f1′(u)|≤c,∀u∈R.Noting thatB={Bt(R1)}t∈R is a time-dependent absorbing set for U(t,τ)uτ, then for each initial data uτ∈Bτ(R1), we decompose U(t,τ) as
(31)U(t,τ)uτ=U1(t,τ)uτ+U2(t,τ)uτ,
where v=U1(t,τ)uτ and w=U2(t,τ)uτ solve the following equations, respectively:
(32)ɛ(t)vt+Avt+Av+f0(v)=0,U1(τ,τ)=uτ,(33)ɛ(t)wt+Awt+Aw+f(u)-f0(v)=g,U2(τ,τ)=0.Lemma 8.
Under assumptions of (2)–(5), (27)–(30), there exists δ=δ(B)>0 such that
(34)∥U1(t,τ)uτ∥Ht≤C3e-δ(t-τ),∀t≥τ.Proof.
Multiplying (32) by v, we obtain
(35)ddt(ɛ∥v∥2+∥∇v∥2)-ɛ′∥v∥2+2∥∇v∥2+2〈f(v),v〉=0.
Using (28) and noting that ɛ′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(36)ddt(ɛ∥v∥2+∥∇v∥2)+δ(ɛ∥v∥2+∥∇v∥2)≤0.
By the Gronwall lemma, we complete the proof.Remark 9.
From Lemmas4 and 8, we have the uniform bound
(37)sup⁡t≥τ[∥U(t,τ)∥Ht+∥U1(t,τ)∥Ht+∥U2(t,τ)∥Ht]≤C.Lemma 10.
Under the assumptions of (2)–(5), (27)–(30), there exists M=M(B)>0 such that
(38)sup⁡t≥τ∥U2(t,τ)uτ∥Ht1/3≤M.Proof.
Multiplying (33) by A(1/3)w, we obtain
(39)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+2∥A2/3w∥2+2〈f(u)-f0(v)-g,A1/3w〉=ɛ′∥A1/6w∥2.
In view of Remark 9 and the growth of f and f0, using the embedding H1⊂L6(Ω), we have
(40)2|〈f(u),A1/3w〉|≤c∫Ω‍(1+|u|3)|A1/3w|≤c(1+∥u∥L63)∥A1/3w∥≤c∥A1/3w∥≤14∥A2/3w∥2+C,(41)2|〈f0(v)+g,A1/3w〉|≤c∥A1/3w∥≤14∥A2/3w∥2+C∥g∥.
Noting that ε′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(42)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+δ(ɛ∥A1/6w∥2+∥A2/3w∥2)≤C.
By the Gronwall lemma, we complete the proof.Remark 11.
From Lemma10, we immediately have the following regularity result: At is bounded in Ht1/3 (with a bound independent of t).Theorem 12 (see [17]).
IfU(t,τ) is a T-closed process for some T>0, which possesses a time-dependent global attractor A, then A is invariant.Remark 13 (see [17]).
If the processU(t,τ) is closed, it is T-closed, for any T>0. Note that if the process U(t,τ) is a continuous (or even norm-to-weak continuous) map for all t≥τ, then the process is closed.Theorem 14 (existence of the time-dependent global attractor).
Under the assumptions of (2)–(5), the process U(t,τ):Hτ→Ht generated by problem (1) admits an invariant time-dependent global attractor A={At}t∈R.According to Lemma10, we consider the family K={Kt}t∈R, where
(43)Kt={u∈Ht1/3:∥u∥Ht1/3≤M},where  Kt is compact by the compact embedding Ht1/3⊂Ht; besides, since the injection constants are independent of t, K is uniformly bounded. Hence, according to Lemmas 5, 8, and 10, K is pullback attracting, and the process U(t,τ) is asymptotically compact, which proves the existence of the unique time-dependent global attractor. In order to state the invariance of the time-dependent global attractor, due to the strong continuity of the process U(t,τ) stated in Lemma 2, according to Remark 13, the process U(t,τ) is closed, and it is T-closed, for some T>0; then by Theorem 12, we know that the time-dependent global attractor A is invariant.


### 3.4. Regularity of the Time-Dependent Global Attractor

#### 3.4.1. The Second Decomposition of the System Equations

We fixτ∈R and each initial data uτ∈Aτ, decomposing U(t,τ) as
(44)U(t,τ)uτ=U3(t,τ)uτ+U4(t,τ)uτ,
where v=U3(t,τ)uτ and w=U4(t,τ)uτ solve the following equations, respectively:
(45)ɛ(t)vt+Avt+Av=0,U3(τ,τ)=uτ,(46)ɛ(t)wt+Awt+Aw+f(u)=g,U4(τ,τ)=0.
As a particular case of Lemma 8, we learn that
(47)∥U3(t,τ)uτ∥Ht≤C4e-δ(t-τ),∀t≥τ.Lemma 15.
Under assumptions of (2)–(5), for some M=M(A)>0, one has the uniform bound
(48)sup⁡t≥τ∥U4(t,τ)uτ∥Ht1≤M.Proof.
Multiplying (46) by Awt+Aw, we obtain
(49)ddt(ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉)+2ɛ∥A1/2wt∥2+2∥Awt∥2+2∥Aw∥2+2〈f(u),Aw〉-2〈g,Aw〉+2〈f(u),Awt〉=ɛ′∥A1/2w∥2.
We denote
(50)M(t)=ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉,
noting that
(51)c∥w∥Ht12-C6≤M(t)≤c∥w∥Ht12+C7.
Noting that (2) and using Young and Poincaré inequalities, for δ>0 small, we infer
(52)ddtM(t)+δM(t)+2ɛ∥A1/2wt∥2+2∥Awt∥2≤-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉-2〈f(u),Awt〉+ɛ′∥A1/2w∥2.
Denoting byC>0 a generic constant depending on the size of At in Ht1/3, using the invariance of the attractor, we find
(53)∥U(t,τ)u∥Ht1/3≤C.
Exploiting the embeddings H1/3⊂L18/7, H4/3⊂L18, we get
(54)∥f(u)∥1≤∥f′(u)∥L9∥A1/2u∥L18/7≤C(1+∥u∥L18)≤C;
this yields
(55)-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉≤Cδ,∥g∥∥Aw∥≤2∥Aw∥2+C,-2〈f(u),Awt〉≤C∥Awt∥≤2∥Awt∥2+C;
noting that ɛ′<0, we infer
(56)ddtM(t)+δM(t)≤C.
By the Gronwall lemma, we can get (48) immediately.
This completes the proof.Therefore, we have the following regularity result.Theorem 16 (regularity of the time-dependent global attractor).
Under the assumptions of (2)–(5), the time-dependent global attractor A={At}t∈R, At is bounded in Ht1, with a bound independent of t.In fact, we define(57)Et={u∈Ht1:∥u∥Ht1≤M1};
according to inequalities (47) and (48), for all t∈R, we have
(58)lim⁡τ→-∞dist⁡(U(t,τ)Aτ,Et)=0,
where dist⁡ denotes the Hausdorff semidistance in Ht; that is,
(59)dist⁡(B,C)=sup⁡x∈BdistHt(x,C)=sup⁡x∈Binf⁡y∈C∥x-y∥Ht.From Theorem14, we know that the time-dependent global attractor A={At}t∈R is invariant; this means that
(60)dist⁡(At,Et)=0.Hence,At⊂Et¯=Et; that is, At is bounded in Ht1, with a bound independent of t.


## 3.1. Well-Posedness

For anyτ∈R, we rewrite the problem (1) as
(11)ɛ(t)ut+Aut+Au+f(u)=g(x),t>τ,u(x,τ)=a.Using the Galerkin approximation method, we can obtain the following result concerning the existence and uniqueness of solutions; see, for example, [2, 7, 8, 15, 16].Lemma 2.
Under the assumptions of (2)–(5), for any a∈Hτ, there is a unique solution u of (1), on any interval [τ,t] with t≥τ,
(12)u∈C([τ,t];H1).
Furthermore, for i=1,2, let uτi∈Hτ be two initial conditions such that ∥uτi∥Hτ≤R and denote by ui the corresponding solutions to the problem of (11). Then the following estimates hold as follows:
(13)∥u1(t)-u2(t)∥Ht2≤eK(t-τ)∥uτ1-uτ2∥Ht2,t≥τ,
for some constant K=K(R)≥0.According to Lemma2 above, the family of maps with t≥τ∈R(14)U(t,τ):Hτ⟶Ht,
acting as
(15)uτ⟶u(t)=U(t,τ)zτ,
where u is the unique solution of (11) with initial time τ and initial condition uτ=Hτ, defines a strongly continuous process on the family {Ht}t∈R.


## 3.2. Time-Dependent Absorbing Set

Definition 3.
A time-dependent absorbing set for the processU(t,τ) is a uniformly bounded family B={Bt}t∈R with the following property: for every R≥0 there exists θe=θe(R)≥0 such that
(16)τ≤t-θe⟹U(t,τ)Bτ(R)⊂Bt.Lemma 4.
Under the assumptions of (2)–(5), for uτ∈Hτ, t≥τ, let U(t,τ)uτ be the solution of (1); then, there exist positive constants δ, Cν,∥g∥ and an increasing positive function Q such that
(17)∥U(t,τ)uτ∥Ht≤Q(∥uτ∥Hτ)e-δ(t-τ)+Cδ,ν,∥g∥,∀t≥τ.Proof.
Multiplying (11) by u, we obtain
(18)ddt(ɛ∥u∥2+2∥∇u∥2)+2∥∇u∥2-ɛ′∥u∥2+2〈f(u),u〉-2〈u,g〉=0.
Noting that ɛ′<0 and using (3) and Young and Poincaré inequalities, for δ>0 small, we infer that
(19)ddt(ɛ∥u∥2+2∥∇u∥2)+δ(ɛ∥u∥2+∥∇u∥2)≤Cδ,ν,∥g∥.
By the Gronwall lemma, we have
(20)∥u(t)∥Ht2≤Ce-δ(t-τ)∥u(τ)∥Ht2+Cδ,ν,∥g∥.
This completes the proof.Lemma 5 (time-dependent absorbing set).
Under the assumptions of (2)–(5), there exists a constant R1>0, such that the family B={Bt(R1)}t∈R is a time-dependent absorbing set for U(t,τ).Proof.
From the proof of Lemma4, for u∈Bτ(R), there exists θe≥0, provided that t-τ≥θe,
(21)∥U(t,τ)uτ∥Ht≤R0.
This concludes the proof of the existence of the time-dependent absorbing set.
We can assume that the time-dependent absorbing setBt=Bt(R1) is positively invariant (namely, U(t,τ)Bτ⊂Bt for all t≥τ). Indeed, calling θe the entering time of Bt such that
(22)U(t,τ)Bτ⊂Bt,∀τ≤t-θe,
we can substitute Bt with the invariant absorbing family:
(23)⋃τ≤t-θe‍U(t,τ)Bτ⊂Bt.


## 3.3. Time-Dependent Global Attractor

As introduced in [17], for t∈R, let Xt be a family of normed spaces; we consider the collection
(24)K={R={Kt}t∈R:Kt⊂Xtcompact,hRpullbackattracting}.
When K≠Ø we say that the process is asymptotically compact.Definition 6.
One calls a time-dependent global attractor the smallest element ofK; that is, the family A={At}t∈R∈K such that
(25)At⊂Kt,∀t∈R,
for any element R={Kt}t∈R∈K.Theorem 7 (see [17]).
IfU(t,τ) is asymptotically compact, then the time-dependent attractor A exists and coincides with the set A={At}t∈R. In particular, it is unique.According to Definition6, the existence of the time-dependent global attractor will be proved by a direct application of the abstract Theorem 7. Precisely, in order to show that the process is asymptotically compact, we will exhibit a pullback attracting family of compact sets. To this aim, the strategy classically consists in finding a suitable decomposition of the process in the sum of a decaying part and of a compact one.


### 3.3.1. The First Decomposition of the System Equations

For the nonlinearityf, following [11, 15, 17], we decompose f as follows:
(26)f=f0+f1,
where f0,f1∈C2(R) satisfy, for some c≥0,
(27)|f0′′(u)|≤c(1+|u|),∀u∈R,(28)f0(u)u≥0,∀u∈R,(29)f0(0)=f0′(0)=0,(30)|f1′(u)|≤c,∀u∈R.Noting thatB={Bt(R1)}t∈R is a time-dependent absorbing set for U(t,τ)uτ, then for each initial data uτ∈Bτ(R1), we decompose U(t,τ) as
(31)U(t,τ)uτ=U1(t,τ)uτ+U2(t,τ)uτ,
where v=U1(t,τ)uτ and w=U2(t,τ)uτ solve the following equations, respectively:
(32)ɛ(t)vt+Avt+Av+f0(v)=0,U1(τ,τ)=uτ,(33)ɛ(t)wt+Awt+Aw+f(u)-f0(v)=g,U2(τ,τ)=0.Lemma 8.
Under assumptions of (2)–(5), (27)–(30), there exists δ=δ(B)>0 such that
(34)∥U1(t,τ)uτ∥Ht≤C3e-δ(t-τ),∀t≥τ.Proof.
Multiplying (32) by v, we obtain
(35)ddt(ɛ∥v∥2+∥∇v∥2)-ɛ′∥v∥2+2∥∇v∥2+2〈f(v),v〉=0.
Using (28) and noting that ɛ′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(36)ddt(ɛ∥v∥2+∥∇v∥2)+δ(ɛ∥v∥2+∥∇v∥2)≤0.
By the Gronwall lemma, we complete the proof.Remark 9.
From Lemmas4 and 8, we have the uniform bound
(37)sup⁡t≥τ[∥U(t,τ)∥Ht+∥U1(t,τ)∥Ht+∥U2(t,τ)∥Ht]≤C.Lemma 10.
Under the assumptions of (2)–(5), (27)–(30), there exists M=M(B)>0 such that
(38)sup⁡t≥τ∥U2(t,τ)uτ∥Ht1/3≤M.Proof.
Multiplying (33) by A(1/3)w, we obtain
(39)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+2∥A2/3w∥2+2〈f(u)-f0(v)-g,A1/3w〉=ɛ′∥A1/6w∥2.
In view of Remark 9 and the growth of f and f0, using the embedding H1⊂L6(Ω), we have
(40)2|〈f(u),A1/3w〉|≤c∫Ω‍(1+|u|3)|A1/3w|≤c(1+∥u∥L63)∥A1/3w∥≤c∥A1/3w∥≤14∥A2/3w∥2+C,(41)2|〈f0(v)+g,A1/3w〉|≤c∥A1/3w∥≤14∥A2/3w∥2+C∥g∥.
Noting that ε′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(42)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+δ(ɛ∥A1/6w∥2+∥A2/3w∥2)≤C.
By the Gronwall lemma, we complete the proof.Remark 11.
From Lemma10, we immediately have the following regularity result: At is bounded in Ht1/3 (with a bound independent of t).Theorem 12 (see [17]).
IfU(t,τ) is a T-closed process for some T>0, which possesses a time-dependent global attractor A, then A is invariant.Remark 13 (see [17]).
If the processU(t,τ) is closed, it is T-closed, for any T>0. Note that if the process U(t,τ) is a continuous (or even norm-to-weak continuous) map for all t≥τ, then the process is closed.Theorem 14 (existence of the time-dependent global attractor).
Under the assumptions of (2)–(5), the process U(t,τ):Hτ→Ht generated by problem (1) admits an invariant time-dependent global attractor A={At}t∈R.According to Lemma10, we consider the family K={Kt}t∈R, where
(43)Kt={u∈Ht1/3:∥u∥Ht1/3≤M},where  Kt is compact by the compact embedding Ht1/3⊂Ht; besides, since the injection constants are independent of t, K is uniformly bounded. Hence, according to Lemmas 5, 8, and 10, K is pullback attracting, and the process U(t,τ) is asymptotically compact, which proves the existence of the unique time-dependent global attractor. In order to state the invariance of the time-dependent global attractor, due to the strong continuity of the process U(t,τ) stated in Lemma 2, according to Remark 13, the process U(t,τ) is closed, and it is T-closed, for some T>0; then by Theorem 12, we know that the time-dependent global attractor A is invariant.


## 3.3.1. The First Decomposition of the System Equations

For the nonlinearityf, following [11, 15, 17], we decompose f as follows:
(26)f=f0+f1,
where f0,f1∈C2(R) satisfy, for some c≥0,
(27)|f0′′(u)|≤c(1+|u|),∀u∈R,(28)f0(u)u≥0,∀u∈R,(29)f0(0)=f0′(0)=0,(30)|f1′(u)|≤c,∀u∈R.Noting thatB={Bt(R1)}t∈R is a time-dependent absorbing set for U(t,τ)uτ, then for each initial data uτ∈Bτ(R1), we decompose U(t,τ) as
(31)U(t,τ)uτ=U1(t,τ)uτ+U2(t,τ)uτ,
where v=U1(t,τ)uτ and w=U2(t,τ)uτ solve the following equations, respectively:
(32)ɛ(t)vt+Avt+Av+f0(v)=0,U1(τ,τ)=uτ,(33)ɛ(t)wt+Awt+Aw+f(u)-f0(v)=g,U2(τ,τ)=0.Lemma 8.
Under assumptions of (2)–(5), (27)–(30), there exists δ=δ(B)>0 such that
(34)∥U1(t,τ)uτ∥Ht≤C3e-δ(t-τ),∀t≥τ.Proof.
Multiplying (32) by v, we obtain
(35)ddt(ɛ∥v∥2+∥∇v∥2)-ɛ′∥v∥2+2∥∇v∥2+2〈f(v),v〉=0.
Using (28) and noting that ɛ′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(36)ddt(ɛ∥v∥2+∥∇v∥2)+δ(ɛ∥v∥2+∥∇v∥2)≤0.
By the Gronwall lemma, we complete the proof.Remark 9.
From Lemmas4 and 8, we have the uniform bound
(37)sup⁡t≥τ[∥U(t,τ)∥Ht+∥U1(t,τ)∥Ht+∥U2(t,τ)∥Ht]≤C.Lemma 10.
Under the assumptions of (2)–(5), (27)–(30), there exists M=M(B)>0 such that
(38)sup⁡t≥τ∥U2(t,τ)uτ∥Ht1/3≤M.Proof.
Multiplying (33) by A(1/3)w, we obtain
(39)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+2∥A2/3w∥2+2〈f(u)-f0(v)-g,A1/3w〉=ɛ′∥A1/6w∥2.
In view of Remark 9 and the growth of f and f0, using the embedding H1⊂L6(Ω), we have
(40)2|〈f(u),A1/3w〉|≤c∫Ω‍(1+|u|3)|A1/3w|≤c(1+∥u∥L63)∥A1/3w∥≤c∥A1/3w∥≤14∥A2/3w∥2+C,(41)2|〈f0(v)+g,A1/3w〉|≤c∥A1/3w∥≤14∥A2/3w∥2+C∥g∥.
Noting that ε′<0, by Young and Poincaré inequalities, for δ>0 small, we infer
(42)ddt(ɛ∥A1/6w∥2+∥A2/3w∥2)+δ(ɛ∥A1/6w∥2+∥A2/3w∥2)≤C.
By the Gronwall lemma, we complete the proof.Remark 11.
From Lemma10, we immediately have the following regularity result: At is bounded in Ht1/3 (with a bound independent of t).Theorem 12 (see [17]).
IfU(t,τ) is a T-closed process for some T>0, which possesses a time-dependent global attractor A, then A is invariant.Remark 13 (see [17]).
If the processU(t,τ) is closed, it is T-closed, for any T>0. Note that if the process U(t,τ) is a continuous (or even norm-to-weak continuous) map for all t≥τ, then the process is closed.Theorem 14 (existence of the time-dependent global attractor).
Under the assumptions of (2)–(5), the process U(t,τ):Hτ→Ht generated by problem (1) admits an invariant time-dependent global attractor A={At}t∈R.According to Lemma10, we consider the family K={Kt}t∈R, where
(43)Kt={u∈Ht1/3:∥u∥Ht1/3≤M},where  Kt is compact by the compact embedding Ht1/3⊂Ht; besides, since the injection constants are independent of t, K is uniformly bounded. Hence, according to Lemmas 5, 8, and 10, K is pullback attracting, and the process U(t,τ) is asymptotically compact, which proves the existence of the unique time-dependent global attractor. In order to state the invariance of the time-dependent global attractor, due to the strong continuity of the process U(t,τ) stated in Lemma 2, according to Remark 13, the process U(t,τ) is closed, and it is T-closed, for some T>0; then by Theorem 12, we know that the time-dependent global attractor A is invariant.


## 3.4. Regularity of the Time-Dependent Global Attractor

### 3.4.1. The Second Decomposition of the System Equations

We fixτ∈R and each initial data uτ∈Aτ, decomposing U(t,τ) as
(44)U(t,τ)uτ=U3(t,τ)uτ+U4(t,τ)uτ,
where v=U3(t,τ)uτ and w=U4(t,τ)uτ solve the following equations, respectively:
(45)ɛ(t)vt+Avt+Av=0,U3(τ,τ)=uτ,(46)ɛ(t)wt+Awt+Aw+f(u)=g,U4(τ,τ)=0.
As a particular case of Lemma 8, we learn that
(47)∥U3(t,τ)uτ∥Ht≤C4e-δ(t-τ),∀t≥τ.Lemma 15.
Under assumptions of (2)–(5), for some M=M(A)>0, one has the uniform bound
(48)sup⁡t≥τ∥U4(t,τ)uτ∥Ht1≤M.Proof.
Multiplying (46) by Awt+Aw, we obtain
(49)ddt(ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉)+2ɛ∥A1/2wt∥2+2∥Awt∥2+2∥Aw∥2+2〈f(u),Aw〉-2〈g,Aw〉+2〈f(u),Awt〉=ɛ′∥A1/2w∥2.
We denote
(50)M(t)=ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉,
noting that
(51)c∥w∥Ht12-C6≤M(t)≤c∥w∥Ht12+C7.
Noting that (2) and using Young and Poincaré inequalities, for δ>0 small, we infer
(52)ddtM(t)+δM(t)+2ɛ∥A1/2wt∥2+2∥Awt∥2≤-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉-2〈f(u),Awt〉+ɛ′∥A1/2w∥2.
Denoting byC>0 a generic constant depending on the size of At in Ht1/3, using the invariance of the attractor, we find
(53)∥U(t,τ)u∥Ht1/3≤C.
Exploiting the embeddings H1/3⊂L18/7, H4/3⊂L18, we get
(54)∥f(u)∥1≤∥f′(u)∥L9∥A1/2u∥L18/7≤C(1+∥u∥L18)≤C;
this yields
(55)-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉≤Cδ,∥g∥∥Aw∥≤2∥Aw∥2+C,-2〈f(u),Awt〉≤C∥Awt∥≤2∥Awt∥2+C;
noting that ɛ′<0, we infer
(56)ddtM(t)+δM(t)≤C.
By the Gronwall lemma, we can get (48) immediately.
This completes the proof.Therefore, we have the following regularity result.Theorem 16 (regularity of the time-dependent global attractor).
Under the assumptions of (2)–(5), the time-dependent global attractor A={At}t∈R, At is bounded in Ht1, with a bound independent of t.In fact, we define(57)Et={u∈Ht1:∥u∥Ht1≤M1};
according to inequalities (47) and (48), for all t∈R, we have
(58)lim⁡τ→-∞dist⁡(U(t,τ)Aτ,Et)=0,
where dist⁡ denotes the Hausdorff semidistance in Ht; that is,
(59)dist⁡(B,C)=sup⁡x∈BdistHt(x,C)=sup⁡x∈Binf⁡y∈C∥x-y∥Ht.From Theorem14, we know that the time-dependent global attractor A={At}t∈R is invariant; this means that
(60)dist⁡(At,Et)=0.Hence,At⊂Et¯=Et; that is, At is bounded in Ht1, with a bound independent of t.


## 3.4.1. The Second Decomposition of the System Equations

We fixτ∈R and each initial data uτ∈Aτ, decomposing U(t,τ) as
(44)U(t,τ)uτ=U3(t,τ)uτ+U4(t,τ)uτ,
where v=U3(t,τ)uτ and w=U4(t,τ)uτ solve the following equations, respectively:
(45)ɛ(t)vt+Avt+Av=0,U3(τ,τ)=uτ,(46)ɛ(t)wt+Awt+Aw+f(u)=g,U4(τ,τ)=0.
As a particular case of Lemma 8, we learn that
(47)∥U3(t,τ)uτ∥Ht≤C4e-δ(t-τ),∀t≥τ.Lemma 15.
Under assumptions of (2)–(5), for some M=M(A)>0, one has the uniform bound
(48)sup⁡t≥τ∥U4(t,τ)uτ∥Ht1≤M.Proof.
Multiplying (46) by Awt+Aw, we obtain
(49)ddt(ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉)+2ɛ∥A1/2wt∥2+2∥Awt∥2+2∥Aw∥2+2〈f(u),Aw〉-2〈g,Aw〉+2〈f(u),Awt〉=ɛ′∥A1/2w∥2.
We denote
(50)M(t)=ɛ∥A1/2w∥2+2∥Aw∥2-2〈g,Aw〉,
noting that
(51)c∥w∥Ht12-C6≤M(t)≤c∥w∥Ht12+C7.
Noting that (2) and using Young and Poincaré inequalities, for δ>0 small, we infer
(52)ddtM(t)+δM(t)+2ɛ∥A1/2wt∥2+2∥Awt∥2≤-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉-2〈f(u),Awt〉+ɛ′∥A1/2w∥2.
Denoting byC>0 a generic constant depending on the size of At in Ht1/3, using the invariance of the attractor, we find
(53)∥U(t,τ)u∥Ht1/3≤C.
Exploiting the embeddings H1/3⊂L18/7, H4/3⊂L18, we get
(54)∥f(u)∥1≤∥f′(u)∥L9∥A1/2u∥L18/7≤C(1+∥u∥L18)≤C;
this yields
(55)-2〈f(u),Aw〉-(2+2δ)〈g,Aw〉≤Cδ,∥g∥∥Aw∥≤2∥Aw∥2+C,-2〈f(u),Awt〉≤C∥Awt∥≤2∥Awt∥2+C;
noting that ɛ′<0, we infer
(56)ddtM(t)+δM(t)≤C.
By the Gronwall lemma, we can get (48) immediately.
This completes the proof.Therefore, we have the following regularity result.Theorem 16 (regularity of the time-dependent global attractor).
Under the assumptions of (2)–(5), the time-dependent global attractor A={At}t∈R, At is bounded in Ht1, with a bound independent of t.In fact, we define(57)Et={u∈Ht1:∥u∥Ht1≤M1};
according to inequalities (47) and (48), for all t∈R, we have
(58)lim⁡τ→-∞dist⁡(U(t,τ)Aτ,Et)=0,
where dist⁡ denotes the Hausdorff semidistance in Ht; that is,
(59)dist⁡(B,C)=sup⁡x∈BdistHt(x,C)=sup⁡x∈Binf⁡y∈C∥x-y∥Ht.From Theorem14, we know that the time-dependent global attractor A={At}t∈R is invariant; this means that
(60)dist⁡(At,Et)=0.Hence,At⊂Et¯=Et; that is, At is bounded in Ht1, with a bound independent of t.



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*Source: 748321-2014-08-28.xml*