# Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains

Authors: Cung The Anh; Nguyen Duong Toan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equationut-εΔut-Δu+f(u)=g(x,t),ε∈(0,1], in a noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor 𝒜̂ε, which is upper semicontinuous at ε=0.


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## Body

## 1. Introduction

In recent years, the evolution equations on noncylindrical domains, that is, spatial domains which vary in time so their Cartesian products with the time variable are noncylindrical sets, have been investigated extensively (see, e.g., [1–3]).Much of the progress has been made for nested spatial domains which expand in time.However, the results focus mainly on formulation of the problems and existence and uniqueness theory, while the existence of attractors of such systems has been less considered, except some recent works for the reaction-diffusion equation (or the heat equation) [4, 5]. This is not really surprising since such systems are intrinsically nonautonomous even if the equations themselves contain no time-dependent terms and require the concept of a nonautonomous attractor, which has only been introduced in recent years.In this paper, we consider a class of nonautonomous nonclassical diffusion equations on bounded spatial domains which are expanding in time. First, we show how the first initial boundary value problem for these equations can be formulated as a variational problem with appropriate function spaces, and then we establish the existence and uniqueness over a finite time interval of variational solutions. Next, we show that the process of two parameter generated by such solutions has a nonautonomous pullback attractor. Finally, we study the upper semicontinuity of the obtained pullback attractor.Let{Ωt}t∈ℝ be a family of nonempty bounded open subsets of ℝN such that
(1.1)s<t⟹Ωs⊂Ωt.
From now on, we will frequently use the following notations:
(1.2)Qτ,T∶=⋃t∈(τ,T)‍Ωt×{t},Qτ:=⋃t∈(τ,∞)‍⁡Ωt×{t},∀τ∈R,∑τ,T:=⋃t∈(τ,T)‍∂Ωt×{t},∑τ:=⋃t∈(τ,∞)‍∂Ωt×{t}.
In this paper we consider the following nonautonomous equation:
(1.3)(Pϵ){ut-ɛΔut-Δu+f(u)=g(x,t)inQτ,u=0on⁡∑τ,u|=uτ(x),t=τx∈Ωτ,
where ɛ∈[0,1], the nonlinear term f and the external force g satisfy some conditions specified later on. This equation is called the nonclassical diffusion equation when ɛ>0, and when ɛ=0, it turns to be the classical reaction-diffusion equation.Nonclassical diffusion equation arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanics, and heat conduction (see, e.g., [6–9]). In the last few years, the existence and long-time behavior of solutions to nonclassical diffusion equations has attracted the attention of many mathematicians. However, to the best of our knowledge, all existing results are devoted to the study of the equation in cylindrical domains. For example, under a Sobolev growth rate of the nonlinearity f, problem (1.3) in cylindrical domains has been studied [10–13] for the autonomous case, that is the case g not depending on time t and in [14, 15] for the nonautonomous case. In this paper, we will study the existence and long-time behavior of solutions to problem (1.3) in the case of noncylindrical domains, the nonlinearity f of polynomial type satisfying some dissipativity condition, and the external force g depending on time t. It is noticed that this question for problem (1.3) in the case ɛ=0, that is, for the reaction-diffusion equation, has only been studied recently in [4, 5].In order to study problem (1.3), we make the following assumptions. (H1)
The functionf∈C1(ℝ,ℝ) satisfies that
(1.4)-β+α1|s|p≤f(s)s≤β+α2|s|p,(1.5)f′(s)≥-l,
for some p≥2, where α1,α2,β,ℓ are nonnegative constants. By (1.4), there exist nonnegative constants α~1,α~2,β~ such that
(1.6)-β~+α~1|s|p≤F(s)≤β~+α~2|s|p,
where F(u)=∫0u‍f(s)ds is the primitive of f.(H2)
The external forceg∈Lloc2(ℝN+1).(H3)
The initial datumuτ∈H01(Ωτ)∩Lp(Ωτ) is given.Since the open setΩt changes with time t, problem (1.3) is nonautonomous even when the external force g is independent of time. Thus, in order to study the long-time behavior of solutions to (1.3), we use the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of these dynamical systems (see [16] and references therein). The existence of a pullback attractor for problem (1.3) in the case ɛ=0, that is, for the classical reaction-diffusion equation, has been derived recently in [4]. In the case ɛ>0, since (1.3) contains the term -ɛΔut, this is essentially different from the classical reaction-diffusion equation. For example, the reaction-diffusion equation has some kind of “regularity”; for example, although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity, and hence we can use the compact Sobolev embeddings to obtain the existence of attractors easily. However, for problem (1.3) when ɛ>0, because of -Δut, if the initial datum uτ belongs to H01(Ωτ)∩Lp(Ωτ), the solution u(t) with the initial condition u(τ)=uτ is always in H01(Ωt)∩Lp(Ωt) and has no higher regularity, which is similar to hyperbolic equations. This brings some difficulty in establishing the existence of attractors for the nonclassical diffusion equations. Other difficulty arises since the considered domain is not cylindrical, so the standard techniques used for studying evolution equations in cylindrical domains cannot be used directly. Therefore, up to now, although there are many results on attractors for evolution equations in cylindrical domains (see, e.g., [17, 18]), little seems to be known for the equations in noncylindrical domains.In this paper, we first exploit the penalty method to prove the existence and uniqueness of a variational solution satisfying the energy equality to problem (1.3). Next, we prove the existence of a pullback attractor 𝒜^ɛ for the process associated to problem (1.3). Finally, we study the continuous dependence on ɛ of the solutions to problem (1.3), in particular we show that the solutions of the nonclassical diffusion equations converge to the solution of the classical reaction-diffusion equation as ɛ→0. Hence using an abstract result derived recently by Carvalho et al. [19] and techniques similar to the ones used in [14], we prove the upper semicontinuity of pullback attractors 𝒜^ɛ in L2(Ωt) at ɛ=0. The last result means that the pullback attractors 𝒜^ɛ of the nonclassical diffusion equations converge to the pullback attractor 𝒜^0 of the classical reaction-diffusion equations as ɛ→0, in the sense of the Hausdorff semidistance.The paper is organized as follows. In Section2, for the convenience of readers, we recall some results on the penalty method and the theory of pullback attractors. After some preliminary results in Section 2, we proceed by a penalty method to solve approximated problem, and then we also prove the existence and uniqueness of the solution to problem (1.3) in Section 3. In Section 4, a uniform estimate for the solutions is then obtained under an additional assumption of the external force g, and this will lead to the proof of existence of a pullback attractor 𝒜^ɛ in an appropriate framework. The upper semicontinuity of pullback attractors 𝒜^ɛ at ɛ=0 is investigated in Section 5. In the last section, we give some discussions and related open problems.Notations. In what follows, we will introduce some notations which are frequently used in the paper. Denote Hr:=L2(Ωr) and Vr:=H01(Ωr) for each r∈ℝ, and denote by (·,·)r and |·|r the usual inner product and associated norm in Hr and by ((·,·)) and ∥·∥r the usual gradient inner product and associated norm in Vr. For each s<t, consider Vs as a closed subspace of Vt with the functions belonging to Vs being trivially extended by zero. It follows from (1.1) that {Vt}t∈[τ,T] can be considered as a family of closed subspaces of VT for each T>τ with
(1.7)s<t⟹Vs⊂Vt.
In addition, Hr will be identified with its topological dual Hr* by means of the Riesz theorem and Vr will be considered as a subspace of Hr* with v∈Vr identified with the element fv∈Hr* defined by
(1.8)fv(h)=(v,h)r,h∈Hr.
The duality product between Vr* and Vr will be denoted by 〈·,·〉.


## 2. Preliminaries

### 2.1. Penalty Method

To study problem (1.3), for each T>τ, we consider the following auxiliary problem:
(2.1)ut-ɛΔut-Δu+f(u)=g(x,t)inQτ,T,u=0on⁡∑τ,T,u|=uτ(x),x∈Ωτ,t=τ
where τ∈ℝ, uτ:Ωτ→ℝ and g:Qτ→ℝ are given functions.The method of penalization due to Lions (see [20]) will be used to prove the existence and uniqueness of a solution to problem (2.1) satisfying an energy equality a.e. in (τ,T) and, as a consequence, the existence and uniqueness of a solution to problem (1.3) satisfying the energy equality a.e. in (τ,∞). To begin, fix T>τ and for each t∈[τ,T] denote by
(2.2)Vt⊥:={v∈VT:((v,ω))T=0,∀ω∈Vt}
the orthogonal subspace of Vt with respect the inner product in VT and by P(t)∈ℒ(VT) the orthogonal projection operator from VT onto VT⊥, which is defined as
(2.3)P(t)v∈Vt⊥,v-P(t)v∈Vt,
for each v∈VT. Finally, define P(t)=P(T) for all t>T and observe that P(T) is the zero of ℒ(VT).We will now approximateP(t) by operators which are more regular in time. Consider the family p(t;·,·) of symmetric bilinear forms on VT defined by
(2.4)p(t;v,ω):=((P(t)v,ω))T,∀v,ω∈VT,∀t≥τ.
It can be proved that the mapping [τ,∞)∋t↦p(t;v,ω)∈ℝ is measurable for all v,ω∈VT. Moreover, |p(t;v,ω)|≤∥v∥T∥ω∥T. For each integer k≥1 and each t≥τ, we define
(2.5)pk(t;v,ω):=k∫01/k‍p(t+r;v,ω)dr,∀v,ω∈VT,∀t≥τ,
and denote by Pk(t)∈ℒ(VT) the associated operator defined by
(2.6)((Pk(t)v,ω))T:=pk(t;v,ω),∀v,ω∈VT,∀t≥τ.Lemma 2.1 (see [2, 4]).
For any integer1≤h≤k, any t≥τ and every v,ω∈VT,
(2.7)pk(t;v,ω)=pk(t;ω,v),0≤ph(t;v,v)≤pk(t;v,v)≤p(t;v,v)=‖P(t)v‖T2≤‖v‖T2,pk′(t;v,v):=ddtpk(t;v,v)=k(p(t+1k;v,v)-p(t;v,v))≤0,((Pk(t)v,z))T=0,∀z∈Vt.
Moreover,   for every sequence {vk}⊂L2(τ,T;VT) weak convergent to v in L2(τ,T;VT),
(2.8)liminf⁡k→+∞∫τT‍pk(t;vk(t),vk(t))dt≥∫τT‍p(t;v(t),v(t))dt.LetJ:VT→VT* be the Riesz isomorphism defined by
(2.9)〈Jv,ω〉T:=((v,ω))T,∀v,ω∈VT,
and for each integer k≥1 and each t∈[τ,T], we denote
(2.10)Ak(t):=-Δ+kJPk(t).
Obviously, Ak(t)∈ℒ(VT,VT*),t∈[τ,T], is a family of symmetric linear operators such that the mapping t∈[τ,T]↦Ak(t)∈ℒ(VT,VT*) is measurable, bounded, and satisfies
(2.11)〈Ak(t)v,v〉T≥‖v‖T2,∀v∈VT,∀t∈[τ,T].
Let uτ∈VT be given and for each k≥1 consider the following problem:
(2.12)(uk′(t),v)T+〈Ak(t)uk(t),v〉T+ɛ〈Ak(t)uk′(t),v〉T+(f(uk(t)),v)T=(g(t),v)T,∀v∈VT,((uk(τ),v))T=((uτ,v))T.The idea of the penalty method is as follows: for eachk≥1 we first prove the existence of a solution uk to problem (2.12) (a problem in a cylindrical domain) using standard methods such as the Galerkin method, and then show that uk converges to a solution to problem (2.1) (a problem in a noncylindrical domain) in some suitable sense, and as a consequence, the existence of a solution to problem (1.3) (see Section 3 for details).


### 2.2. Pullback Attractors

Since the open setΩt changes with time t, problem (1.3) is nonautonomous even when the external force g is independent of time. Thus, in order to study the long-time behavior of solutions to (1.3), we use the theory of pullback 𝒟-attractors which is a modification of the theory in [16].Consider a processU(·,·) on a family of metric spaces {(Xt,dt);t∈ℝ}, that is, a family {U(t,τ);-∞<τ≤t<+∞} of mappings U(t,τ):Xτ→Xt such that U(τ,τ)x=x for all x∈Xτ and
(2.13)U(t,τ)=U(t,r)U(r,τ)∀τ≤r≤t.
In addition, suppose 𝒟 is a nonempty class of parameterized sets of the form 𝒟^={D(t);D(t)⊂Xt,D(t)≠∅,t∈ℝ}.Definition 2.2 (see [4]).
The processU(·,·) is said to be pullback 𝒟-asymptotically compact if the sequence {U(t,τn)xn} is relatively compact in Xt for any t∈ℝ, any 𝒟^∈𝒟, and any sequences {τn} and {xn} with τn→-∞ and xn∈D(τn).Definition 2.3 (see [4]).
A familyℬ^∈𝒟 is said to be pullback 𝒟-absorbing for the process U(·,·) if for any t∈ℝ and any 𝒟^∈𝒟, there exists τ0(t,𝒟^)≤t such that
(2.14)U(t,τ)D(τ)⊂B(t),
for all τ≤τ0(t,𝒟^).Remark 2.4.
Note that ifℬ^∈𝒟 is pullback 𝒟-absorbing for the process U(·,·) and B(t) is a compact subset of Xt for any t∈ℝ, then the process U(·,·) is pullback 𝒟-asymptotically compact.For eacht∈ℝ, let distt(D1,D2) be the Hausdorff semi-distance between nonempty subsets D1 and D2 of Xt, which is defined as
(2.15)distt(D1,D2)=sup⁡x∈D1inf⁡y∈D2dXt(x,y)forD1,D2⊂Xt.Definition 2.5 (see [4]).
The family𝒜^={A(t);A(t)⊂Xt,A(t)≠∅,t∈ℝ} is said to be a pullback 𝒟-attractor for U(·,·) if  (1)
a(t) is a compact set of Xt for all t∈ℝ,(2)
𝒜^ is pullback 𝒟-attracting, that is,
(2.16)lim⁡τ→-∞distt(U(t,τ)D(τ),A(t))=0∀D^∈D,∀t∈R,(3)
𝒜^ is invariant, that is,
(2.17)U(t,τ)A(τ)=A(t)for-∞<τ≤t<+∞.Theorem 2.6 (see [4]).
Suppose that the processU(·,·) is pullback 𝒟-asymptotically compact and that B^∈𝒟 is a family of pullback 𝒟-absorbing sets for U(·,·). Then, the family 𝒜^={A(t);t∈ℝ} defined by A(t):=Λ(B^,t),t∈ℝ, where for each D^∈𝒟 and t∈ℝ,
(2.18)Λ(D^,t):=⋂s≤t‍⋃τ≤s‍U(t,τ)D(τ)¯Xt(closureinXt)
is a pullback 𝒟-attractor for U(·,·), which in addition satisfies
(2.19)A(t)=⋃D^∈D‍Λ(D^,t)¯Xt.
Furthermore, 𝒜^ is minimal in the sense that if C^={C(t);t∈ℝ} is a family of nonempty sets such that C(t) is a closed subset of Xt and
(2.20)lim⁡τ→-∞dist⁡t(U(t,τ)B(τ),C(t))=0foranyt∈R,thenA(t)⊂C(t)∀t∈R.


### 2.3. The Upper Semicontinuity of Pullback Attractors

We now state some results on upper semicontinuity of pullback attractors, which are slight modifications of those in [19]. Because the proof is very similar to the one in [19], so we omit it here.Definition 2.7.
Let{Uɛ(·,·):ɛ∈[0,1]} be a family of evolution processes in a family of Banach spaces {Xt} with corresponding pullback 𝒟-attractors {Aɛ(t):ɛ∈[0,1],t∈ℝ}. For any bounded interval I∈ℝ, we say that {Aɛ(·)} is upper semicontinuous at ɛ=0 for t∈I if
(2.21)lim⁡sup⁡ɛ→0t∈Idistt(Aɛ(t),A0(t))=0.Theorem 2.8.
Let{Uɛ(·,·):ɛ∈[0,1]} be a family of processes with corresponding pullback 𝒟-attractors {Aɛ(·):ɛ∈[0,1]}. Then, for any bounded interval I⊂ℝ, {Aɛ(·):ɛ∈[0,1]} is upper semicontinuous at 0 for t∈I if for each t∈ℝ, for each T>0, and for each compact subset K of Xt-τ, the following conditions hold:  (i)
supτ∈[0,T]supx∈Kdistt(Uɛ(t,t-τ)x,U0(t,t-τ)x)→0asɛ→0,(ii)
⋃ɛ∈[0,1]⋃t≤t0Aɛ(t)isboundedforgivent0,(iii)
⋃0<ɛ≤1Aɛ(t)¯iscompactforeacht∈ℝ.


## 2.1. Penalty Method

To study problem (1.3), for each T>τ, we consider the following auxiliary problem:
(2.1)ut-ɛΔut-Δu+f(u)=g(x,t)inQτ,T,u=0on⁡∑τ,T,u|=uτ(x),x∈Ωτ,t=τ
where τ∈ℝ, uτ:Ωτ→ℝ and g:Qτ→ℝ are given functions.The method of penalization due to Lions (see [20]) will be used to prove the existence and uniqueness of a solution to problem (2.1) satisfying an energy equality a.e. in (τ,T) and, as a consequence, the existence and uniqueness of a solution to problem (1.3) satisfying the energy equality a.e. in (τ,∞). To begin, fix T>τ and for each t∈[τ,T] denote by
(2.2)Vt⊥:={v∈VT:((v,ω))T=0,∀ω∈Vt}
the orthogonal subspace of Vt with respect the inner product in VT and by P(t)∈ℒ(VT) the orthogonal projection operator from VT onto VT⊥, which is defined as
(2.3)P(t)v∈Vt⊥,v-P(t)v∈Vt,
for each v∈VT. Finally, define P(t)=P(T) for all t>T and observe that P(T) is the zero of ℒ(VT).We will now approximateP(t) by operators which are more regular in time. Consider the family p(t;·,·) of symmetric bilinear forms on VT defined by
(2.4)p(t;v,ω):=((P(t)v,ω))T,∀v,ω∈VT,∀t≥τ.
It can be proved that the mapping [τ,∞)∋t↦p(t;v,ω)∈ℝ is measurable for all v,ω∈VT. Moreover, |p(t;v,ω)|≤∥v∥T∥ω∥T. For each integer k≥1 and each t≥τ, we define
(2.5)pk(t;v,ω):=k∫01/k‍p(t+r;v,ω)dr,∀v,ω∈VT,∀t≥τ,
and denote by Pk(t)∈ℒ(VT) the associated operator defined by
(2.6)((Pk(t)v,ω))T:=pk(t;v,ω),∀v,ω∈VT,∀t≥τ.Lemma 2.1 (see [2, 4]).
For any integer1≤h≤k, any t≥τ and every v,ω∈VT,
(2.7)pk(t;v,ω)=pk(t;ω,v),0≤ph(t;v,v)≤pk(t;v,v)≤p(t;v,v)=‖P(t)v‖T2≤‖v‖T2,pk′(t;v,v):=ddtpk(t;v,v)=k(p(t+1k;v,v)-p(t;v,v))≤0,((Pk(t)v,z))T=0,∀z∈Vt.
Moreover,   for every sequence {vk}⊂L2(τ,T;VT) weak convergent to v in L2(τ,T;VT),
(2.8)liminf⁡k→+∞∫τT‍pk(t;vk(t),vk(t))dt≥∫τT‍p(t;v(t),v(t))dt.LetJ:VT→VT* be the Riesz isomorphism defined by
(2.9)〈Jv,ω〉T:=((v,ω))T,∀v,ω∈VT,
and for each integer k≥1 and each t∈[τ,T], we denote
(2.10)Ak(t):=-Δ+kJPk(t).
Obviously, Ak(t)∈ℒ(VT,VT*),t∈[τ,T], is a family of symmetric linear operators such that the mapping t∈[τ,T]↦Ak(t)∈ℒ(VT,VT*) is measurable, bounded, and satisfies
(2.11)〈Ak(t)v,v〉T≥‖v‖T2,∀v∈VT,∀t∈[τ,T].
Let uτ∈VT be given and for each k≥1 consider the following problem:
(2.12)(uk′(t),v)T+〈Ak(t)uk(t),v〉T+ɛ〈Ak(t)uk′(t),v〉T+(f(uk(t)),v)T=(g(t),v)T,∀v∈VT,((uk(τ),v))T=((uτ,v))T.The idea of the penalty method is as follows: for eachk≥1 we first prove the existence of a solution uk to problem (2.12) (a problem in a cylindrical domain) using standard methods such as the Galerkin method, and then show that uk converges to a solution to problem (2.1) (a problem in a noncylindrical domain) in some suitable sense, and as a consequence, the existence of a solution to problem (1.3) (see Section 3 for details).


## 2.2. Pullback Attractors

Since the open setΩt changes with time t, problem (1.3) is nonautonomous even when the external force g is independent of time. Thus, in order to study the long-time behavior of solutions to (1.3), we use the theory of pullback 𝒟-attractors which is a modification of the theory in [16].Consider a processU(·,·) on a family of metric spaces {(Xt,dt);t∈ℝ}, that is, a family {U(t,τ);-∞<τ≤t<+∞} of mappings U(t,τ):Xτ→Xt such that U(τ,τ)x=x for all x∈Xτ and
(2.13)U(t,τ)=U(t,r)U(r,τ)∀τ≤r≤t.
In addition, suppose 𝒟 is a nonempty class of parameterized sets of the form 𝒟^={D(t);D(t)⊂Xt,D(t)≠∅,t∈ℝ}.Definition 2.2 (see [4]).
The processU(·,·) is said to be pullback 𝒟-asymptotically compact if the sequence {U(t,τn)xn} is relatively compact in Xt for any t∈ℝ, any 𝒟^∈𝒟, and any sequences {τn} and {xn} with τn→-∞ and xn∈D(τn).Definition 2.3 (see [4]).
A familyℬ^∈𝒟 is said to be pullback 𝒟-absorbing for the process U(·,·) if for any t∈ℝ and any 𝒟^∈𝒟, there exists τ0(t,𝒟^)≤t such that
(2.14)U(t,τ)D(τ)⊂B(t),
for all τ≤τ0(t,𝒟^).Remark 2.4.
Note that ifℬ^∈𝒟 is pullback 𝒟-absorbing for the process U(·,·) and B(t) is a compact subset of Xt for any t∈ℝ, then the process U(·,·) is pullback 𝒟-asymptotically compact.For eacht∈ℝ, let distt(D1,D2) be the Hausdorff semi-distance between nonempty subsets D1 and D2 of Xt, which is defined as
(2.15)distt(D1,D2)=sup⁡x∈D1inf⁡y∈D2dXt(x,y)forD1,D2⊂Xt.Definition 2.5 (see [4]).
The family𝒜^={A(t);A(t)⊂Xt,A(t)≠∅,t∈ℝ} is said to be a pullback 𝒟-attractor for U(·,·) if  (1)
a(t) is a compact set of Xt for all t∈ℝ,(2)
𝒜^ is pullback 𝒟-attracting, that is,
(2.16)lim⁡τ→-∞distt(U(t,τ)D(τ),A(t))=0∀D^∈D,∀t∈R,(3)
𝒜^ is invariant, that is,
(2.17)U(t,τ)A(τ)=A(t)for-∞<τ≤t<+∞.Theorem 2.6 (see [4]).
Suppose that the processU(·,·) is pullback 𝒟-asymptotically compact and that B^∈𝒟 is a family of pullback 𝒟-absorbing sets for U(·,·). Then, the family 𝒜^={A(t);t∈ℝ} defined by A(t):=Λ(B^,t),t∈ℝ, where for each D^∈𝒟 and t∈ℝ,
(2.18)Λ(D^,t):=⋂s≤t‍⋃τ≤s‍U(t,τ)D(τ)¯Xt(closureinXt)
is a pullback 𝒟-attractor for U(·,·), which in addition satisfies
(2.19)A(t)=⋃D^∈D‍Λ(D^,t)¯Xt.
Furthermore, 𝒜^ is minimal in the sense that if C^={C(t);t∈ℝ} is a family of nonempty sets such that C(t) is a closed subset of Xt and
(2.20)lim⁡τ→-∞dist⁡t(U(t,τ)B(τ),C(t))=0foranyt∈R,thenA(t)⊂C(t)∀t∈R.


## 2.3. The Upper Semicontinuity of Pullback Attractors

We now state some results on upper semicontinuity of pullback attractors, which are slight modifications of those in [19]. Because the proof is very similar to the one in [19], so we omit it here.Definition 2.7.
Let{Uɛ(·,·):ɛ∈[0,1]} be a family of evolution processes in a family of Banach spaces {Xt} with corresponding pullback 𝒟-attractors {Aɛ(t):ɛ∈[0,1],t∈ℝ}. For any bounded interval I∈ℝ, we say that {Aɛ(·)} is upper semicontinuous at ɛ=0 for t∈I if
(2.21)lim⁡sup⁡ɛ→0t∈Idistt(Aɛ(t),A0(t))=0.Theorem 2.8.
Let{Uɛ(·,·):ɛ∈[0,1]} be a family of processes with corresponding pullback 𝒟-attractors {Aɛ(·):ɛ∈[0,1]}. Then, for any bounded interval I⊂ℝ, {Aɛ(·):ɛ∈[0,1]} is upper semicontinuous at 0 for t∈I if for each t∈ℝ, for each T>0, and for each compact subset K of Xt-τ, the following conditions hold:  (i)
supτ∈[0,T]supx∈Kdistt(Uɛ(t,t-τ)x,U0(t,t-τ)x)→0asɛ→0,(ii)
⋃ɛ∈[0,1]⋃t≤t0Aɛ(t)isboundedforgivent0,(iii)
⋃0<ɛ≤1Aɛ(t)¯iscompactforeacht∈ℝ.


## 3.   Existence and Uniqueness of Variational Solutions

For eachT>τ, denote
(3.1)Q~τ,T:=ΩT×(τ,T),Uτ,T:={Φ∈L∞(τ,T;VT)∩Lp(Q~τ,T),Φ′∈L2(τ,T;VT),Q~τ,T=ΩTΦ(τ)=Φ(T)=0,Φ(t)∈Vta.e.in(τ,T){Φ∈L∞(τ,T;VT)∩Lp(Q~τ,T),Φ′∈L2(τ,T;VT)}}.Definition 3.1.
A variational solution of (2.1) is a function u such that  (C1)
u∈L∞(τ,T,VT)∩Lp(Q~τ,T), u′∈L2(τ,T;VT),(C2)
for allΦ∈Uτ,T,
(3.2)∫τT‍[-(u(t),Φ′(t))T+((u(t),Φ(t)))T+ɛ((u′(t),Φ(t)))T+(f(u),Φ(t))T]dt=∫τT‍(g(t),Φ(t))Tdt,(C3)
u(t)∈Vt a.e. in (τ,T),(C4)
limt↓τ(t-τ)-1∫τt‍|u(r)-uτ|T2dr=0.Remark 3.2.
IfT2>T1>τ and u is a variational solution of (2.1) with T=T2, then the restriction of u to Q~τ,T1 is a variational solution of (2.1) with T=T1.DenoteQ~τ:=UT>τQ~(τ,T).Definition 3.3.
A variational solution of (1.3) is a function u:Q~τ→ℝ such that for each T>τ, its restriction to Q~τ,T is a variational solution of (2.1).To prove the uniqueness of variational solutions to problem (2.1), we need the following lemmas.Lemma 3.4 (see [4]).
Assume thatv∈L2(τ,T,VT)∩Lp(Q~τ,T) and there exist ξ∈L2(τ,T,VT*) and η∈Lp/p-1(Q~τ,T) such that
(3.3)∫τT(v(t),Φ′(t))Tdt=-∫τT〈ξ(t),Φ(t)〉Tdt-∫τT(η(t),Φ(t))Tdt,
for every function Φ∈Uτ,T.
For each0<h<T-τ, define vh by
(3.4)vh:={h-1(v(t+h)-v(t))a.e.in(τ,T-h),0a.e.in(T-h,T).
Then
(3.5)lim⁡h↓0∫τT(vh(t),ω(t))Tdt=∫τT〈ξ(t),ω(t)〉Tdt+∫τT(η(t),ω(t))Tdt,
for every function ω∈L2(τ,T;VT)∩Lp(Q~τ,T) such that ω(t)∈Vt a.e. in (τ,T).Remark 3.5.
Ifτ<T′<T and Φ∈L2(τ,T′;VT)∩Lp(ΩT×(τ,T′)), with Φ′∈L2(τ,T′;HT) satisfies Φ(τ)=Φ(T′)=0 and Φ(t)∈Vt a.e. in (τ,T′), then the trivial extension Φ~ of Φ satisfies Φ~∈Uτ,T, with (Φ~)′=Φ~′. Using the open sets Ω^t:=Ωt+T-T′,τ≤t≤T′, it is easy to see that under the conditions of (3.5), one also has
(3.6)lim⁡h↓0∫τT′-h(vh(t),ω(t))Tdt=∫τT′〈ξ(t),ω(t)〉Tdt+∫τT′(η(t),ω(t))Tdt,
for every τ≤T′≤T and every function ω∈L2(τ,T;VT)∩Lp(Q~τ,T) such that ω(t)∈Vt a.e. in (τ,T).Lemma 3.6 (see [4]).
Letvi∈L2(τ,T;VT)∩Lp(Q~τ,T),i=1,2, be two functions such that vi(t)∈Vt a.e. in (τ,T) for i=1,2. Assume that there exist ξi∈L2(τ,T;VT*),ηi∈Lp/p-1(Q~τ,T),i=1,2 such that
(3.7)∫τT‍(vi(t),Φ′(t))Tdt=-∫τT‍〈ξi(t),Φ(t)〉Tdt-∫τT‍(ηi(t),Φ(t))Tdt,i=1,2,
for every function Φ∈Uτ,T. Then,   for every pair τ≤s<t≤T of Lebesgue points of the inner product function (v1,v2)T it holds
(3.8)(v1(t),v2(t))T-(v1(s),v2(s))T=∫st‍〈ξ1(r),v2(r)〉Tdr+∫st‍〈ξ2(r),v1(r)〉Tdr+∫st‍(η1(r),v2(r))Tdr+∫st‍(η2(r),v1(r))Tdr+lim⁡h↓0h-1∫st-h‍(v1(r+h)-v1(r),v2(r+h)-v2(r))Tdr.Ifu is a variational solution of problem (2.1), then τ is the Lebesgue point of |u|T2 since the condition (C4) is satisfied. The next corollary gives an obvious consequence of (3.8).Corollary 3.7.
Ifu is a variational solution of (2.1), then for every Lebesgue point t∈(τ,T) of |u|T2 it holds
(3.9)|u(t)|T2+ɛ‖u(t)‖T2+2∫τt‍‖u(r)‖T2dr+2∫τt‍(f(u(r)),u(r))Tdr=|uτ|T2+ɛ‖uτ‖T2+2∫τt‍(g(r),u(r))Tdr+lim⁡h↓0h-1∫τt-h‍|u(r+h)-u(r)|T2dr.Proof.
Ifu is a variational solution of (2.1), then we have
(3.10)∫τT‍[-(u(t),Φ′(t))T+ɛ((∂u∂t,Φ(t)))T+((u(t),Φ(t)))T+(f(u(t)),Φ(t))T]dt=∫τT‍(g(t),Φ(t))Tdt.
Applying Lemma 3.6 with v1=v2=u, we get
(3.11)|u(t)|T2-|uτ|T2=-∫τt‍((ɛ∂u∂t+u(r),u(r)))Tdr-∫τt‍((ɛ∂u∂t+u(r),u(r)))Tdr-∫τt‍(f(u(r))-g(r),u(r))Tdr-∫τt‍(f(u(r))-g(r),u(r))Tdr+lim⁡h→0h-1∫τt-h‍|u(r+h)-u(r)|T2dr=-2∫τt‍‖u(r)‖T2dr-2∫τt‍(f(u(r)),u(r))Tdr+2∫τt‍(g(r),u(r))Tdr-ɛ‖u(t)‖T2+ɛ‖uτ‖T2+lim⁡h→0h-1∫τt-h‍|u(r+h)-u(r)|T2dr.
Hence, it implies the desired result.The aim of this section is to obtain a variational solution of (2.1) such that
(3.12)|u(t)|T2+ɛ‖u(t)‖T2+2∫τt‍‖u(r)‖T2dr+2∫τt‍(f(u(r)),u(r))Tdr=|uτ|T2+ɛ‖uτ‖T2+2∫τt‍(g(r),u(r))Tdr.
We will say that u satisfies the energy equality in (τ,T) if (3.12) is satisfied a.e. in (τ,T). Analogously, if u is a variational solution of (1.3), we will say that u satisfies the energy equality a.e. in (τ,+∞) if for each T>τ the restriction of u to Q~τ,T satisfies the energy equality (3.12) a.e. in (τ,T).For any functionv∈L2(τ,T;HT) and any t∈(τ,T], we put
(3.13)ηv,T(t):=limsup⁡h↓0h-1∫τt-h‍|v(r+h)-v(r)|T2dr.
Then ηv,T is a nondecreasing function. By Corollary 3.7, a variational solution u of (1.3) satisfies the energy equality a.e. in (τ,T) if and only if ηu,T(t)=0 for a.e. t∈(τ,T). In fact, using the continuity of the following mapping:
(3.14)t∈[τ,T]↦|uτ|T2+ɛ‖uτ‖T2+2∫τt‍[(g(r),u(r))T-‖u(r)‖T2-(f(u(r)),u(r))T]dr,
one can see that a variational solution u of (1.3) satisfies the energy equality a.e. in (τ,T) if and only if ηu,T(T)=0.The next lemma provides a sufficient condition foru to satisfy the energy equality a.e. in (τ,T).Lemma 3.8.
Letu be a variational solution of (2.1) and suppose that there exists a sequence {tn}⊂(τ,T) of Lebesgue points of |u|T2 such that tn→T and
(3.15)limsup⁡n⟶+∞⁡(|u(tn)|T2+ɛ‖u(tn)‖T2)≤|uτ|T2+ɛ‖uτ‖T2+2∫τT‍[(g(r),u(r))T-‖u(r)‖T2-(f(u(r)),u(r))T]dr.
Then, u satisfies the energy equality a.e. in (τ,T).Proof.
It is sufficient to prove thatηu,T(T)=0. Since tn→T and ηu,T is nondecreasing, by Corollary 3.7, we have
(3.16)ηu,T(T)≤limsup⁡n↑+∞ηu,T(tn)=limsup⁡n→+∞(2∫τtn‍|u(tn)|T2+ɛ‖u(tn)‖T2-|uτ|T2-ɛ‖uτ‖T255-2∫τtn‍[(g(r),u(r))T-‖u(r)‖T2-(f(u(r)),u(r))T]dr)≤limsup⁡n→+∞(|u(tn)|T2+ɛ‖u(tn)‖T2)-|uτ|T2-ɛ‖uτ‖T2-2∫τT‍[(g(r),u(r))T-‖u(r)‖T2-(f(u(r)),u(r))T]dr≤0.
This completes the proof.Proposition 3.9.
Letu,u¯ be two variational solutions of (2.1) corresponding to the initial data uτ,u¯τ∈Vτ∩Lp(Ωτ), respectively, which satisfy the energy equality a.e. in (τ,T). Then,
(3.17)|u(t)-u¯(t)|T2+ɛ‖u(t)-u¯(t)‖T2+2∫τt‍‖u(r)-u¯(r)‖T2dr≤e2l(t-τ)(|uτ-u¯τ|T2+ɛ‖uτ-u¯τ‖T2)a.e.t∈(τ,T).
Hence, it implies the uniqueness of variational solutions to (2.1) satisfying the energy equality in (τ,T).Proof.
Sinceu,u¯ satisfy the energy equation, ηu,T(t)=ηu¯,T(t)=0 for all t∈(τ,T) and
(3.18)|u(t)-u¯(t)|T2+ɛ‖u(t)-u¯(t)‖T2+2∫τt‍‖u(r)-u¯(r)‖T2dr+2∫τt‍(f(u(r))-f(u¯(r)),u(r)-u¯(r))Tdr≤|uτ-u¯τ|T2+ɛ‖uτ-u¯τ‖T2-2lim⁡h↓0h-1∫τt-h‍(u(r+h)-u(r),u¯(r+h)-u¯(r))Tdr.
On the other hand,
(3.19)|h-1∫τt-h‍(u(r+h)-u(r),u¯(r+h)-u¯(r))Tdr|2≤(h-1∫τt-h‍|u(r+h)-u(r)|2dr)(h-1∫τt-h‍|u¯(r+h)-u¯(r)|2dr),
so
(3.20)lim⁡h↓0h-1∫τt-h‍(u(r+h)-u(r),u¯(r+h)-u¯(r))Tdr=0.
Using this and (1.5) in (3.12), one can conclude
(3.21)|u(t)-u¯(t)|T2+ɛ‖u(t)-u¯(t)‖T2+2∫τt‍‖u(r)-u¯(r)‖T2dr≤|uτ-u¯τ|T2+ɛ‖uτ-u¯τ‖T2-2∫τt‍(f(u(r))-f(u¯(r)),u(r)-u¯(r))Tdr≤|uτ-u¯τ|T2+ɛ‖uτ-u¯τ‖T2+2l∫τt‍|u(r)-u¯(r)|T2dr.
By an application of Gronwall's inequality, we get (3.17).The method of penalization will now be used to prove the existence and uniqueness of a variational solution to problem (2.1) satisfying an energy equality a.e. in (τ,T) and, as a consequence, the existence and uniqueness of a variational solution to problem (1.3) satisfying the energy equality a.e. in (τ,∞).Theorem 3.10.
Letg∈L2(τ,T;HT),uτ∈Vτ∩Lp(Ωτ) be given. Then problem (2.1) has a unique variational solution satisfying the energy equality a.e. in (τ,T).Proof.
We divide the proof into two steps.
Step1. Existence of a weak solution to problem (2.12).
We will use the Galerkin method (see [20]). Take an orthonormal Hilbert basis {ej} of HT formed by elements of VT∩Lp(ΩT) such that the vector space generated by {ej} is dense in VT and Lp(ΩT). Then, one takes a sequence {uτm} converging to uτ in VT, with {uτm} in the vector space spanned by the m first {ej}. For each integer m≥1, one considers the approximation ukm(t)=∑j=1m‍γkm,j(t)ej, defined as a solution of
(3.22)(ukm′(t),ej)T+〈Ak(t)ukm(t),ej〉T+ɛ〈Ak(t)ukm′(t),ej〉T+(f(ukm(t)),ej)T=(g(t),ej)T,((ukm(τ),ej))T=((uτm,ej))T.
Multiplying (3.22) by γkm,j′(t) and summing from j=1 to m, we obtain
(3.23)(ukm′(t),ukm′(t))T+〈Ak(t)ukm(t),ukm′(t)〉+ɛ〈Ak(t)ukm′(t),ukm′(t)〉T+(f(ukm(t)),ukm′(t))T=(g(t),ukm′(t))T,
or
(3.24)|ukm′(t)|T2+12ddt‖ukm(t)‖T2+ɛ‖ukm′(t)‖T2+k((Pk(t)ukm(t),ukm′(t)))T+ɛk((Pk(t)ukm′(t),ukm′(t)))T+(f(ukm(t)),ukm′(t))T=(g(t),ukm′(t))T.
Thus,
(3.25)(g(t),ukm′(t))T≥|ukm′(t)|T2+ɛ‖ukm′(t)‖T2+12ddt[‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T]+ɛk((Pk(t)ukm′(t),ukm′(t)))T+(f(ukm(t)),ukm′(t))T=|ukm′(t)|T2+ɛ‖ukm′(t)‖T2+ɛk((Pk(t)ukm′(t),ukm′(t)))T+12ddt[‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx].
We have
(3.26)(g(t),ukm′(t))T≤12(|g(t)|T2+|ukm′(t)|T2),
so
(3.27)|g(t)|T2≥2ɛ‖ukm′(t)‖T2+ddt[‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx]+2ɛk((Pk(t)ukm′(t),ukm′(t)))T+|ukm′(t)|T2.
Integrating (3.27) on [τ,t],τ≤t≤T, we obtain
(3.28)2ɛ∫τt‍‖ukm′(r)‖T2dr+2ɛk∫τt‍((Pk(r)ukm′(r),ukm′(r)))Tdr+∫τt‍|ukm′(r)|T2dr+‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx≤∫τt‍|g(r)|T2dr+‖ukm(τ)‖T2+k((Pk(τ)ukm(τ),ukm(τ)))T+2∫ΩT‍F(ukm(x,τ))dx.
Since
(3.29)∫ΩT‍F(ukm(x,t))dx≥-β~|ΩT|+α~1‖ukm(t)‖Lp(ΩT)p,∫ΩT‍F(ukm(x,τ))dx≤β~|ΩT|+α~2‖uτm‖Lp(ΩT)p,
we have
(3.30)2ɛ∫τt‍‖ukm′(r)‖T2dr+2ɛk∫τt‍((Pk(r)ukm′(r),ukm′(r)))Tdr+∫τt‍|ukm′(r)|T2dr+‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T+2α~1‖ukm(t)‖Lp(ΩT)p≤∫τt|g(r)|T2dr+‖ukm(τ)‖T2+k((Pk(τ)ukm(τ),ukm(τ)))T+4β~|ΩT|+2α~2‖uτm‖Lp(ΩT)p.‍
From (3.30), we deduce that
{ukm} is bounded in L∞(τ,T;VT)∩Lp(Q~τ,T),
{ukm}⇀uk weakly in L∞(τ,T;VT)∩Lp(Q~τ,T),
{ukm′} is bounded in L2(τ,T;VT),
{ukm′}⇀uk′ weakly in L2(τ,T;VT).Since {ukm} is bounded in L∞(τ,T;VT)∩Lp(Q~τ,T), one can check that {f(ukm)} is bounded in Lq(τ,T;Lq(ΩT)) with q=p/(p-1), hence f(ukm)⇀η in Lq(τ,T;Lq(ΩT)). We now prove that η=f(uk).
Indeed, we have
VT⊂⊂HT⊂VT*,
{ukm} is bounded in L∞(τ,T;VT),
{ukm′} is bounded in L2(τ,T;VT*).By the Aubin-Lions lemma [20, Chapter 1], {ukm} is relatively compact in L2(τ,T;HT). Therefore, one can assume that ukm→uk strongly in L2(τ,T;HT), so ukm→uk a.e. in Q~τ,T. Since f is continuous, f(ukm)→f(uk) a.e. in Q~τ,T. Applying Lemma 1.3 in [20], we have
(3.31)f(ukm)⇀f(uk)weaklyinLq(τ,T;Lq(ΩT)).
This implies that uk is a weak solution of problem (2.12).
Step2. Existence of a variational solution to (2.1) satisfying the energy equality.From (3.30), we have
(3.32)k∫τT‍((Pk(r)ukm(r),ukm(r)))Tdr≤(T-τ)(∫τt|g(r)|T2dr+‖ukm(τ)‖T2+k((Pk(τ)ukm(τ),ukm(τ)))T‍+4β~|ΩT|+2α~2‖uτm‖Lp(ΩT)p∫τt|g(r)|T2dr+‖ukm(τ)‖T2+k((Pk(τ)ukm(τ),ukm(τ)))T‍).
Consider the function Φ:L2(τ,T;VT)→ℝ defined by
(3.33)Φ(v)=∫τT‍((Pk(t)v(t),v(t)))Tdt,v∈L2(τ,T;VT).
It is easy to see that Φ is a continuous and convex function. It follows that ∫τT‍((Pk(t)ukm(t),ukm(t)))Tdt is weakly lower semicontinuous in L2(τ,T;VT). Moreover, {ukm}⇀uk weakly in L2(τ,T;VT), hence
(3.34)k∫τT‍((Pk(t)uk(t),uk(t)))Tdt≤kliminf⁡m→∞∫τT‍((Pk(t)ukm(t),ukm(t)))Tdt≤(T-τ)(∫τt|g(r)|T2dr+‖uτ‖T2+k((Pk(τ)uτ,uτ))T+4β~|ΩT|+2α~2‖uτ‖Lp(ΩT)p‍).
Since {ukm′}⇀uk′ weakly in L2(τ,T;VT), then, reasoning as above,
(3.35)2ɛk∫τT‍((Pk(t)uk′(t),uk′(t)))Tdt≤2ϵkliminf⁡m→∞∫τT‍((Pk(t)ukm(t),ukm(t)))Tdt≤(∫τt‍|g(r)|T2dr+‖uτ‖T2+k((Pk(τ)uτ,uτ))T+4β~|ΩT|+2α~2‖uτ‖Lp(ΩT)p).
From the facts that ukm⇀uk weakly in L∞(τ,T;VT), ukm′⇀uk′ weakly in L2(τ,T;VT) and the weak lower semicontinuity of the norm, we deduce that
(3.36)2ɛ∫τt‍‖uk′(r)‖T2dr+2ɛk∫τt‍((Pk(r)uk′(r),uk′(r)))Tdr+∫τt‍|uk′(r)|T2dr+‖uk(t)‖T2+k∫τT‍((Pk(t)uk(t),uk(t)))T+2α~1‖uk(t)‖Lp(ΩT)p≤(5+T-τ)(∫τt‍|g(r)|T2dr+‖uτ‖T2+k((Pk(τ)uτ,uτ))T+4β~|ΩT|+2α~2‖uτ‖Lp(ΩT)p)=C.
Since uτ∈Vτ∩Lp(Ωτ), ((Pk(τ)uτ,uτ))T=0forallk≥1, (3.36) gives 
{uk} is bounded in L∞(τ,T;VT)∩Lp(τ,T;Lp(ΩT)),
{uk′} is bounded in L2(τ,T;VT),
{uk}⇀u weakly in L∞(τ,T;VT)∩Lp(τ,T;Lp(ΩT)),
{uk′}⇀u′ weakly in L2(τ,T;VT).From Lemma 2.1, we have
(3.37)∫τT‍‖P(t)u(t)‖T2dt≤liminf⁡k→∞∫τT‍((Pk(t)uk(t),uk(t)))Tdt≤liminf⁡k→∞Ck=0,thatis,P(t)u(t)=0a.e.in(τ,T)oru(t)∈Vta.e.in(τ,T),∫τT‖P(t)u′(t)‖T2dt≤liminf⁡k→∞∫τT((Pk(t)u′k(t),u′k(t)))Tdt≤liminf⁡k→∞Ck=0,thatis,P(t)u′(t)=0a.e.in(τ,T).
Moreover, (3.36) and the equality
(3.38)uk(t)-uk(s)=∫st‍uk′(r)dr,∀s,t∈[τ,T],∀k≥1,
give
(3.39)|uk(t)-uk(s)|T≤C1/2|t-s|1/2∀s,t∈[τ,T],∀k≥1.
It follows from (3.36) that ∥uk(t)∥T≤C for all t∈[τ,T] and each k≥1. Since the injection of VT into HT is compact, the set {v∈VT:∥v∥T2≤C} is compact in HT. By (3.39) and the Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by {uk}, such that
(3.40)uk→uinC([τ,T];HT)ask⟶+∞.
So, the condition (C4) is satisfied.On the other hand,{uk} is bounded in L∞(τ,T;VT) and {uk′}is bounded in L2(τ,T;VT), applying the Aubin-Lions lemma and Lemma 1.3 in [20, Chapter 1], one has
(3.41)f(uk)⇀f(u)weaklyinLq(τ,T,Lq(ΩT)).
Since uk is the weak solution of the problem
(3.42)(uk′(t),v)T+〈Ak(t)uk(t),v〉T+ɛ〈Ak(t)uk′(t),v〉T+(f(uk(t)),v)T=(g(t),v)T,∀v∈VT,((uk(τ),v))T=((uτ,v))T,
taking to the limit as k→+∞ and using the fact that P(t)u(t)=0,P(t)u′(t)=0 a.e. in (τ,T), we can conclude that u is the solution of (2.1).Now, we will show thatu satisfies the energy equality in (τ,T). Multiplying (3.22) by γkm,j and summing from j=1 to m, we obtain
(3.43)(ukm′(t),ukm(t))T+〈Ak(t)ukm(t),ukm(t)〉T+ɛ〈Ak(t)ukm′(t),ukm(t)〉T+(f(ukm(t)),ukm(t))T=(g(t),ukm(t))T.
Hence, we get
(3.44)12ddt|ukm(t)|T2+‖ukm(t)‖T2+k((Pk(t)ukm(t),ukm(t)))T+ϵ2ddt‖ukm(t)‖T2+ϵk((Pk(t)ukm′(t),ukm(t)))T+(f(ukm(t)),ukm(t))T=(g(t),ukm(t))T,
or
(3.45)|ukm(t)|T2+2∫τt‍‖ukm(r)‖T2dr+2k∫τT‍((Pk(r)ukm(r),ukm(r)))Tdr+ϵ‖ukm(t)‖T2+2kϵ∫τt‍((Pk(r)ukm′(r),ukm(r)))Tdr+2∫τt‍(f(ukm(r)),ukm(r))Tdr=2∫τt‍(g(r),ukm(r))Tdr+|uτm|T2+ɛ‖uτm‖T2.
Since
(3.46)((Pk(t)ukm′(t),ukm(t)))T≥12ddt((Pk(t)ukm(t),ukm(t)))T,((Pk(t)ukm(t),ukm(t)))T≥0,|ukm(t)|T2+2∫τt‍‖ukm(r)‖T2dr+ϵ‖ukm(t)‖T2+2∫τt‍(f(ukm(r)),ukm(r))Tdr≤2∫τt‍(g(r),ukm(r))Tdr+|uτm|T2+ɛ‖uτm‖T2+ɛk((Pk(t)uτm,uτm))T,
letting m→∞, we obtain
(3.47)|uk(T)|T2+2∫τT‍‖uk(r)‖T2dr+ϵ‖uk(T)‖T2+2∫τT‍(f(uk(r)),uk(r))Tdr≤2∫τT‍(g(r),uk(r))Tdr+|uτ|T2+ɛ‖uτ‖T2.
Now
(3.48)∫τT‍(f(uk(r)),uk(r))Tdr=∫τT‍(f(uk(r))-f(u(r)),uk(r)-u(r))Tdr+∫τT‍(f(uk(r)),u(r))Tdr+∫τT‍(f(u(r)),uk(r))Tdr-∫τT‍(f(u(r)),u(r))Tdr≥-l∫τT‍|uk(r)-u(r)|T2dr+∫τT‍(f(uk(r)),u(r))Tdr+∫τT‍(f(u(r)),uk(r))Tdr-∫τT‍(f(u(r)),u(r))Tdr.
This inequality and (3.47) give
(3.49)|uk(T)|T2+2∫τT‍‖uk(r)‖T2dr+ϵ‖uk(T)‖T2≤2∫τT‍(g(r),uk(r))Tdr+|uτ|T2+ɛ‖uτ‖T2+2l∫τT‍|uk(r)-u(r)|T2dr-∫τT‍(f(uk(r)),u(r))Tdr-∫τT‍(f(u(r)),uk(r))Tdr+∫τT‍(f(u(r)),u(r))Tdr.
Since uk⇀u weakly in L2(τ,T;VT), we get
(3.50)|u(T)|T2+ɛ‖u(T)‖T2≤|uτ|T2+ɛ‖uτ‖T2-2∫τT‍(f(u(r)),u(r))Tdr-2∫τT‍‖u(r)‖T2dr+2∫τT‍(g(r),u(r))Tdr.
Applying Lemma 3.8 with tn=T for all n, one concludes that u satisfies the energy equality on (τ,T), and the desired uniqueness follows from Proposition 3.9.


## 4. Existence of Pullback𝒟-Attractors

The aim of this section is to establish the existence of a pullback attractor for problem (1.3).Suppose thatg∈Lloc2(ℝN+1). Then, according to Theorem 3.10, for each τ∈ℝ and each uτ∈Vτ∩Lp(Ωτ) given, there exists a unique variational solution u(·;τ,uτ) of problem (1.3) satisfying the energy equality a.e. in (τ,T) for all T>τ.Define(4.1)U(t,τ)uτ:=u(t;τ,uτ),-∞<τ≤t<+∞,uτ∈Vτ∩Lp(Ωτ).
It is easy to check that the family of mappings {U(t,τ);-∞<τ≤t<+∞} is a process U(·,·).A uniform estimate inVT∩Lp(ΩT) will be obtained now for the variational solutions of (1.3) satisfying the energy equality, and since the compactness of the embedding VT∩Lp(ΩT)↪HT, this will immediately imply the existence of a pullback attractor for the process U(·,·). The proof requires the following lemma.Lemma 4.1 (see [21]).
LetX⊂Y be Banach spaces such that X is reflexive and the injection of X in Y is compact. Suppose that {vn} is a bounded sequence in L∞(t0,T;X) such that vn⇀v weakly in Lp(t0,T;X) for some p∈[1,+∞) and v∈C0([t0,T];Y). Then, v(t)∈X for all t∈[t0,T] and
(4.2)‖v(t)‖X≤liminf⁡n→+∞‖vn‖L∞(t0,T;X),∀t∈[t0,T].Proposition 4.2.
Suppose that the assumptions in Theorem3.10 hold and g∈Lloc2(ℝN+1) satisfies
(4.3)Cg,T=sup⁡t≤T∫t-1t‍|g(r)|T2dr<+∞.
Then, for any uτ∈Vτ∩Lp(Ωτ) given, the corresponding variational solution u of (1.3) satisfying the energy equality in (τ,T) also satisfies
(4.4)‖u(t)‖Lp(ΩT)p+‖u(t)‖T2≤C(e-σT(t-τ)(‖uτ‖τ2+‖uτ‖Lp(Ωτ)p)+1+11-e-σTCg,T),∀t∈[τ+1,T],
where σT=min{λ1,T/2,1/(ɛ+1),α1/α~2}, λ1,T>0 is the first eigenvalue of the operator -Δ in ΩT with the homogeneous Dirichlet condition, α1,α~2 are the constants in (H1), and the constant C is independent of t,τ,ɛ.Proof.
Assume thatukm are the Galerkin approximations of uk defined by (3.22). From (3.44) and (3.24), we have
(4.5)12ddt(|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2)+|ukm′(t)|T2+‖ukm(t)‖T2+ɛ‖ukm′(t)‖T2+(ɛ+1)k((Pk(t)ukm′(t),ukm(t)))T+k((Pk(t)ukm(t),ukm(t)))T+ɛk((Pk(t)ukm′(t),ukm′(t)))T+(f(ukm(t)),ukm′(t)+ukm(t))T=(g(t),ukm(t))T+(g(t),ukm′(t))T.
Moreover,
(4.6)((Pk(t)ukm′(t),ukm(t)))T≥12ddt((Pk(t)ukm(t),ukm(t)))T,(f(ukm(t)),ukm′(t))T=ddt∫ΩT‍F(ukm(x,t))dx,(g(t),ukm(t))T≤14η1|g(t)|T2+η1|ukm(t)|T2,∀η1>0,(g(t),ukm′(t))T≤14η2|g(t)|T2+η2|ukm′(t)|T2,∀η2>0,
so
(4.7)12ddt(2∫ΩT‍F(ukm(x,t))|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx)+|ukm′(t)|T2+‖ukm(t)‖T2+ɛ‖ukm′(t)‖T2+ɛk((Pk(t)ukm′(t),ukm′(t)))T+k((Pk(t)ukm(t),ukm(t)))T+(f(ukm(t)),ukm(t))T≤14η1|g(t)|T2+η1|ukm(t)|T2+14η2|g(t)|T2+η2|ukm′(t)|T2,∀η1,η2>0.
Since
(4.8)(f(ukm(t)),ukm(t))T=∫ΩT‍f(ukm(t))ukm(t)dx≥∫ΩT‍(-β+α1|ukm(t)|pdx)=-β|ΩT|+α1‖ukm(t)‖Lp(ΩT)p,
we have
(4.9)12ddt(2∫ΩT‍|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx{|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T})+(1-η2)|ukm′(t)|T2+12‖ukm(t)‖T2+(λ1,T2-η1)|ukm(t)|T2+ɛ‖ukm′(t)‖T2+ɛk((Pk(t)ukm′(t),ukm′(t)))T+k((Pk(t)ukm(t),ukm(t)))T+α1‖ukm(t)‖Lp(ΩT)p≤β|ΩT|+(14η1+14η2)|g(t)|T2.
Denote
(4.10)ykm(t):=|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T+2∫ΩT‍F(ukm(x,t))dx.
Choose η2<1 and η1 small enough such that σT<min{1/(ɛ+1),α1/α~2,λ1,T-2η1}, we have
(4.11)σTykm(t)=σT(2∫ΩT‍F(ukm(x,t))dx|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T22+2∫ΩT‍F(ukm(x,t))dx)≤σT(2∫ΩT‍|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T22+2∫ΩT‍(β~+α~2|ukm(t)|p)dx+2β~|ΩT|)≤σT(|ukm(t)|T2+(ɛ+1)‖ukm(t)‖T2+(ɛ+1)k((Pk(t)ukm(t),ukm(t)))T555+2α~2‖ukm(t)‖Lp(ΩT)pdx((Pk(t)ukm(t),ukm(t)))T)≤2(1-η2)|ukm′(t)|T2+212‖ukm(t)‖T2+2(λ1,T2-η1)|ukm(t)|T2+2ɛ‖ukm′(t)‖T2555+2ɛk((Pk(t)ukm′(t),ukm′(t)))T+2k((Pk(t)ukm(t),ukm(t)))T+2α1‖ukm(t)‖Lp(ΩT)p.
Hence, we have
(4.12)ddtykm(t)+σTykm(t)≤C(1+|g(t)|T2).
By Gronwall’s lemma, we get
(4.13)ykm(t)≤e-σT(t-τ)ykm(τ)+C(1+e-σTt∫τt‍eσTs|g(s)|T2ds).
Now, observe that
(4.14)ykm(τ)=|uτm|T2+(ɛ+1)‖uτm‖T2+(ɛ+1)k((Pk(τ)uτm,uτm))T+2∫ΩT‍F(uτm)dx+2β~|ΩT|≤(1λ1,T+ɛ+1)‖uτm‖T2+(ɛ+1)k((Pk(τ)uτm,uτm))T+2∫ΩT‍(β~+α~2|uτm|p)dx+2β~|ΩT|=CT‖uτm‖T2+(ɛ+1)k((Pk(τ)uτm,uτm))T+2α~2‖uτm‖Lp(ΩT)p+4β~|ΩT|≤CT(1+‖uτm‖T2+‖uτm‖Lp(ΩT)p)+(ɛ+1)k((Pk(τ)uτm,uτm))T.
Since
(4.15)2∫ΩT‍F(ukm(x,t))dx+2β~|ΩT|≥2α~1‖ukm‖Lp(ΩT)p,
so combining with (4.13) and (4.14), we have
(4.16)2α~1‖ukm‖Lp(ΩT)p+‖ukm(t)‖T2≤C(e-σT(t-τ)(1+‖uτm‖T2+‖uτm‖Lp(ΩT)p)+1+e-σTt∫τt‍eσTs|g(s)|T2ds)+e-σT(t-τ)(ɛ+1)k((Pk(τ)uτm,uτm))T,
where C is independent of t,τ,ɛ, and k.
Now, it is known thatukm⇀uk*-weakly in L∞(τ,T;VT) as m→+∞. Hence, by (4.16) and Lemma 4.1, we can conclude that
(4.17)2α~1‖uk‖Lp(ΩT)p+‖uk(t)‖T2≤C(e-σT(t-τ)(1+‖uτ‖τ2+‖uτ‖Lp(Ωτ)p)+1+e-σTt∫τt‍eσTs|g(s)|T2ds)+e-σT(t-τ)(ɛ+1)k((Pk(τ)uτ,uτ))τ=C(e-σT(t-τ)(1+‖uτ‖τ2+‖uτ‖Lp(Ωτ)p)+1+e-σTt∫τt‍eσTs|g(s)|T2ds).
Finally, since uk⇀u in L2(τ,T;VT) as k→+∞, we get
(4.18)2α~1‖u(t)‖Lp(ΩT)p+‖u(t)‖T2≤C(e-σT(t-τ)(1+‖uτ‖τ2+‖uτ‖Lp(Ωτ)p)+1+e-σTt∫τt‍eσTs|g(s)|T2ds)≤C(e-σT(t-τ)(‖uτ‖τ2+‖uτ‖Lp(Ωτ)p)+1+11-eσTCg,T),
where we have used the fact that
(4.19)∫τte-σT(t-s)|g(s)|T2ds≤∫t-1te-σT(t-s)|g(s)|T2ds+∫t-2t-1e-σT(t-s)|g(s)|T2ds+⋯≤(1+e-σT+e-2σT+⋯)Cg,T=11-e-σTCg,T.Letℛ be the set of all r(t) such that
(4.20)lim⁡t→-∞etσt(‖r(t)‖t2+‖r(t)‖Lp(Ωt)p)=0.
Denote by 𝒟 the class of all families 𝒟^={D(t);D(t)∈Vt∩Lp(Ωt)),D(t)≠∅,t∈ℝ} such that D(t)⊂B(r(t)) for some r(t)∈ℛ.For eacht∈ℝ define
(4.21)r02(t)=2C(1+11-e-σtCg,t),
and consider the family of closed balls ℬ^={B(t);t∈ℝ}, where
(4.22)B(t)={v∈Vt:‖v‖t≤r0(t)},t∈R.
Then using (4.4), it is not difficult to check that ℬ^ is pullback 𝒟-absorbing for the process U(·,·). Moreover, by the compactness of the injection of Vt into Ht, it is clear that B(t) is a compact set of Ht for any t∈ℝ. Then, it follows from Theorem 2.6 that the process U(·,·) has a pullback 𝒟-attractor 𝒜^ɛ={Aɛ(t):t∈ℝ} in a family of spaces {Ht}.


## 5. The Upper Semicontinuity of Pullback𝒟-Attractors at ɛ=0

It is proved in [4], when ɛ=0, the existence of a pullback 𝒟-attractor 𝒜^0={A0(t):t∈ℝ} in a family of spaces {Ht} for problem (P0). The aim of this section is to prove the upper semicontinuity of pullback attractors 𝒜^ɛ at ɛ=0 in {Ht}, that is,
(5.1)lim⁡ɛ→0sup⁡t∈Idistt(Aɛ(t),A0(t))=0,
where I is an arbitrary bounded interval in ℝ.The following lemma is the key of this section.Lemma 5.1.
For eacht∈ℝ, each T>0, and each compact subset K of Vt-T, we have
(5.2)|Uɛ(t,τ)uτ-U0(t,τ)uτ|t2≤Cɛ,∀τ∈[t-T,t],∀uτ∈K,
where the constant C is independent of τ and uτ.Proof.
DenoteUɛ(t,τ)uτ by u(t), and U0(t,τ)uτ by v(t). Let w(t)=u(t)-v(t), we have
(5.3)wt-ɛΔut-Δw+f(u)-f(v)=0.
Multiplying this equation by w and integrating over Ωt, we get
(5.4)12ddt|w|t2-ɛ(Δut,w)t+‖w‖t2+(f(u)-f(v),w)t=0.
We have
(5.5)(f(u)-f(v),w)t=∫Ωt‍(f(u)-f(v))(u-v)≥-l|u-v|t2,ɛ(Δut,w)t=-ɛ(∇ut,∇w)t≤ɛ‖ut‖t⋅‖w‖t.
Applying (5.5) in (5.4), we have
(5.6)ddt|w|t2≤2l|w|t2+2ɛ‖ut‖t⋅‖w‖t.
Hence
(5.7)|w(t)|t2≤2ɛ∫τt‍(∫τt‍‖ut(s)‖t2)1/2e2l(t-s)‖ut(s)‖t⋅‖w(s)‖t≤2ɛe2lT(∫τt‍(∫τt‍‖w(s)‖t2)1/2‖ut(s)‖t2)1/2(∫τt‍(∫τt‍‖w(s)‖t2)1/2‖w(s)‖t2)1/2.
Now, we estimate the term on the right-hand side of (5.7). Multiplying the first equation in (1.3) by ut and integrating over Ωt, we obtain
(5.8)|ut|t2+ɛ‖ut‖t2+12ddt‖u‖t2+ddt∫Ωt‍F(u)≤∫Ωt‍g(t)ut.
Using Cauchy’s inequality, we conclude that
(5.9)ddt(‖u(t)‖t2+2∫Ωt‍F(u(t)))+2ɛ‖ut(t)‖t2≤12|g(t)|t2.
Integrating (5.9) from τ to t, τ∈[t-T,t], we find that
(5.10)‖u‖t2+2∫Ωt‍F(u(t))+2ɛ∫τt‍‖ut(s)‖t2≤‖uτ‖t2+2∫Ωt‍F(uτ)+12∫τt‍|g(s)|t2.
Since ∫ΩtF(u(t))≥-β~|Ωt|+α~1∥u(t)∥Lp(Ωt)p, we have
(5.11)2ɛ∫τt‍‖ut(s)‖t2≤‖uτ‖t2+2∫Ωt‍F(uτ)+12∫τt‍|g(s)|t2+C.
From (3.29), we obtain
(5.12)‖uτ‖t2+2∫Ωt‍F(uτ)≤C‖uτ‖t2+C‖uτ‖Lp(Ωt)p+C.
Combining (5.10) and (5.12), we see that
(5.13)∫τt‍‖ut(s)‖t2≤Cɛ(1+‖uτ‖t2+‖uτ‖Lp(Ωt)p+∫τt‍|g(s)|t2)≤Cɛ(1+‖uτ‖t2+‖uτ‖Lp(Ωt)p+∫t-Tt‍|g(s)|t2)≤C(K,T,g,t)ɛ,
because of uτ∈K and g∈Lloc2(ℝn+1). Now, using (5.13) in (5.7), we get
(5.14)|w(t)|t2≤C(K,T,g,t)ɛ(∫τt‍‖w(s)‖t2)1/2.
Using (4.4) and noting that τ∈[t-T,t], we have
(5.15)‖w(t)‖t2≤‖u(t)‖t2+‖v(t)‖t2≤2C(e-σt(t-τ)(‖uτ‖t2+‖uτ‖Lp(Ωt)p+1)+1+e-σtt∫τt‍eσts|g(s)|t2)≤2C(‖uτ‖t2+‖uτ‖Lp(Ωt)p+2+e-σtt∫τt‍eσts|g(s)|t2)≤C(K)(1+∫τt‍|g(s)|t2)≤C(K)(1+∫t-Tt‍|g(s)|t2).
Thus,
(5.16)∫τt‖w(t)‖t2≤C(K)(t-τ+∫τt‍∫t-Tt‍|g(s)|t2)≤C(K)(T+∬t-Tt‍|g(s)|t2)≤C(K,T,g,t).
Combining (5.14) and (5.16) we get
(5.17)|w(t)|t2≤C(K,T,g,t)ɛ.
The proof is complete.Theorem 5.2.
Ifg∈Lloc2(ℝN+1) satisfies (4.3), then for any bounded interval I∈ℝ, the family of pullback 𝒟-attractors {Aɛ(·):ɛ∈[0,1]} is upper semicontinuous in L2(Ωt) at 0 for any t∈I, that is,
(5.18)lim⁡ɛ→0sup⁡t∈Idist⁡L2(Ωt)(Aɛ(t),A0(t))=0.Proof.
We will verify the conditions (i)–(iii) in Theorem2.8. First, condition (i) follows directly from Lemma 5.1.
LetB(·)=B¯(r0(·)) be the corresponding family of pullback 𝒟-absorbing sets of (1.3), which is uniform with respect to ɛ. By the definition of pullback 𝒟-absorbing sets, for any t∈ℝ, there exists τ0=τ0(t)≤t such that
(5.19)⋃τ≤τ0‍Uɛ(t,τ)B(τ)⊂B(t)=B¯(r0(t)).
By Theorem 2.6, we see that
(5.20)Aɛ(t)=⋂s≤t‍⋃τ≤s‍Uɛ(t,τ)B(τ)¯.
From (5.19) and (5.20), we get
(5.21)Aɛ(t)⊂B¯(r0(t)).
Now, for given t0∈ℝ, we can write
(5.22)⋃ɛ∈[0,1]‍⋃t≤t0‍Aɛ(t)⊂⋃t≤t0‍B¯(r0(t)).
Because limt→-∞supr0(t)<+∞ due to (4.4), from (5.22) we have
(5.23)⋃ɛ∈[0,1]‍⋃t≤t0‍Aɛ(t)isboundedinL2(Ωt0)forgivent0,
that is, condition (ii) is satisfied. From (5.21), we can find that, for each t∈ℝ,
(5.24)⋃‍0<ɛ≤1Aɛ(t)⊂B¯(r0(t)),
thus ⋃0<ɛ≤1Aɛ(t) is bounded in H01(Ωt), hence
(5.25)⋃0<ɛ≤1‍Aɛ(t)¯iscompactinL2(Ωt),
since H01(Ωt)⊂L2(Ωt) compactly. Then condition (iii) holds.


## 6. Conclusion

In this paper we have proved the existence and uniqueness of variational solutions satisfying the energy equality to a class of nonautonomous nonclassical diffusion equations in noncylindrical domains. We have also proved the existence of pullback attractors𝒜^ɛ of the process generated by this class of solutions and the upper semicontinuity of {𝒜^ɛ:ɛ∈[0,1]} at ɛ=0, which means that the pullback attractors 𝒜^ɛ of the nonclassical diffusion equations converge to the pullback attractor 𝒜^0 of the reaction-diffusion equation in the sense of the Hausdorff semidistance. As far as we know, this is the first result on the existence and long-time behavior of solutions to the nonclassical diffusion equations in noncylindrical domains. The result is obtained under the assumption (1.1) of spatial domains which are expanding in time. This assumption may be replaced by the assumption that the spatial domains Ωt in ℝN are obtained from a bounded base domain Ω by a C2-diffeomorphism, which is continuously differentiable in the time variable and are contained, in the past, in a common bounded domain (see [5] for the related results in the case ɛ=0).It is noticed that the obtained results seem to be not very satisfying because although the processU(·,·) associated to problem (1.3) is constructed in the family of spaces H01(Ωt)∩Lp(Ωt), we are only able to prove the existence and upper semicontinuity of the pullback attractor in L2(Ωt). It would be very interesting if one can show the existence and upper semicontinuity of the pullback attractor in the space H01(Ωt)∩Lp(Ωt). For nonclassical diffusion equations in cylindrical domains, this question has been solved very recently in [15, 22] by using the so-called asymptotic a priori estimate method. However, in the case of non-cylindrical domains, this question seems to be more difficult and is still completely open.



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*Source: 875913-2012-08-15.xml*