Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains

Abstract

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation 𝑢_𝑡−𝜀Δ𝑢_𝑡−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 𝒜_𝜀, which is upper semicontinuous at 𝜀=0.

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 {Ω_𝑡}_𝑡∈ℝ be a family of nonempty bounded open subsets of ℝ^𝑁 such that 𝑠<𝑡⟹Ω_𝑠⊂Ω_𝑡.(1.1) From now on, we will frequently use the following notations: 𝑄𝑄_𝜏,𝑇∶=_{𝑡∈(𝜏,𝑇)}Ω_𝑡×{𝑡},_𝜏∶=_{𝑡∈(𝜏,∞)}Ω_𝑡×{𝑡},∀𝜏∈ℝ,_𝜏,𝑇∶=_{𝑡∈(𝜏,𝑇)}𝜕Ω_𝑡×{𝑡},_𝜏∶=_{𝑡∈(𝜏,∞)}𝜕Ω_𝑡×{𝑡}.(1.2) In this paper we consider the following nonautonomous equation: 𝑃_𝜖⎧⎪⎨⎪⎩𝑢^𝑡𝑢=−0𝜀Δ𝑢_𝑡−Δ𝑢+𝑓(𝑢)=𝑔(𝑥,𝑡)ionn𝑄∑_𝜏𝜏,𝑢||_𝑡=𝜏,=𝑢_𝜏(𝑥),𝑥∈Ω_𝜏,(1.3) where 𝜀∈[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 𝜀>0, and when 𝜀=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 𝜀=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 function 𝑓∈𝐶¹(ℝ,ℝ) satisfies that −𝛽+𝛼₁|𝑠|𝑓^𝑝≤𝑓(𝑠)𝑠≤𝛽+𝛼₂|𝑠|^𝑝,(1.4)_(𝑠)≥−ℓ,(1.5) for some 𝑝≥2, where 𝛼₁,𝛼₂,𝛽,ℓ are nonnegative constants. By (1.4), there exist nonnegative constants 𝛼₁,𝛼₂,𝛽̃ such that −𝛽̃+𝛼₁|𝑠|^𝑝≤𝐹(𝑠)≤𝛽̃+𝛼₂|𝑠|^𝑝,(1.6) where 𝐹(𝑢)=∫^𝑢0𝑓(𝑠)𝑑𝑠 is the primitive of 𝑓. (H2) The external force 𝑔∈𝐿^2loc(ℝ^𝑁+1).(H3) The initial datum 𝑢_𝜏∈𝐻¹⁰(Ω_𝜏)∩𝐿^𝑝(Ω_𝜏) is given.

Since the open set Ω_𝑡 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 𝜀=0, that is, for the classical reaction-diffusion equation, has been derived recently in [4]. In the case 𝜀>0, since (1.3) contains the term −𝜀Δ𝑢_𝑡, 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 𝜀>0, because of −Δ𝑢_𝑡, if the initial datum 𝑢_𝜏 belongs to 𝐻¹⁰(Ω_𝜏)∩𝐿^𝑝(Ω_𝜏), the solution 𝑢(𝑡) with the initial condition 𝑢(𝜏)=𝑢_𝜏 is always in 𝐻¹⁰(Ω_𝑡)∩𝐿^𝑝(Ω_𝑡) 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_𝜀 for the process associated to problem (1.3). Finally, we study the continuous dependence on 𝜀 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 𝜀→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_𝜀 in 𝐿²(Ω_𝑡) at 𝜀=0. The last result means that the pullback attractors 𝒜_𝜀 of the nonclassical diffusion equations converge to the pullback attractor 𝒜₀ of the classical reaction-diffusion equations as 𝜀→0, in the sense of the Hausdorff semidistance.

The paper is organized as follows. In Section 2, 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_𝜀 in an appropriate framework. The upper semicontinuity of pullback attractors 𝒜_𝜀 at 𝜀=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 𝐻_𝑟∶=𝐿²(Ω_𝑟) and 𝑉_𝑟∶=𝐻¹⁰(Ω_𝑟) for each 𝑟∈ℝ, and denote by (⋅,⋅)_𝑟 and |⋅|_𝑟 the usual inner product and associated norm in 𝐻_𝑟 and by ((⋅,⋅)) and ‖⋅‖_𝑟 the usual gradient inner product and associated norm in 𝑉_𝑟. For each 𝑠<𝑡, consider 𝑉_𝑠 as a closed subspace of 𝑉_𝑡 with the functions belonging to 𝑉_𝑠 being trivially extended by zero. It follows from (1.1) that {𝑉_𝑡}_{𝑡∈[𝜏,𝑇]} can be considered as a family of closed subspaces of 𝑉_𝑇 for each 𝑇>𝜏 with 𝑠<𝑡⟹𝑉_𝑠⊂𝑉_𝑡.(1.7) In addition, 𝐻_𝑟 will be identified with its topological dual 𝐻^∗𝑟 by means of the Riesz theorem and 𝑉_𝑟 will be considered as a subspace of 𝐻^∗𝑟 with 𝑣∈𝑉_𝑟 identified with the element 𝑓_𝑣∈𝐻^∗𝑟 defined by 𝑓_𝑣(ℎ)=(𝑣,ℎ)_𝑟,ℎ∈𝐻_𝑟.(1.8) The duality product between 𝑉^∗𝑟 and 𝑉_𝑟 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: 𝑢_𝑡−𝜀Δ𝑢_𝑡−Δ𝑢𝑢=+0𝑓(𝑢o)n=𝑔_𝜏,𝑇(𝑥,𝑡)in𝑄_𝜏,𝑇,𝑢||_𝑡=𝜏,=𝑢_𝜏(𝑥),𝑥∈Ω_𝜏,(2.1) where 𝜏∈ℝ, 𝑢_𝜏∶Ω_𝜏→ℝ and 𝑔∶𝑄_𝜏→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 𝑉^⟂𝑡∶=𝑣∈𝑉_𝑇∶((𝑣,𝜔))_𝑇=0,∀𝜔∈𝑉_𝑡(2.2) the orthogonal subspace of 𝑉_𝑡 with respect the inner product in 𝑉_𝑇 and by 𝑃(𝑡)∈ℒ(𝑉_𝑇) the orthogonal projection operator from 𝑉_𝑇 onto 𝑉^⟂𝑇, which is defined as 𝑃(𝑡)𝑣∈𝑉^⟂𝑡,𝑣−𝑃(𝑡)𝑣∈𝑉_𝑡,(2.3) for each 𝑣∈𝑉_𝑇. Finally, define 𝑃(𝑡)=𝑃(𝑇) for all 𝑡>𝑇 and observe that 𝑃(𝑇) is the zero of ℒ(𝑉_𝑇).

We will now approximate 𝑃(𝑡) by operators which are more regular in time. Consider the family 𝑝(𝑡;⋅,⋅) of symmetric bilinear forms on 𝑉_𝑇 defined by 𝑝(𝑡;𝑣,𝜔)∶=((𝑃(𝑡)𝑣,𝜔))_𝑇,∀𝑣,𝜔∈𝑉_𝑇,∀𝑡≥𝜏.(2.4) It can be proved that the mapping [𝜏,∞)∋𝑡↦𝑝(𝑡;𝑣,𝜔)∈ℝ is measurable for all 𝑣,𝜔∈𝑉_𝑇. Moreover, |𝑝(𝑡;𝑣,𝜔)|≤‖𝑣‖_𝑇‖𝜔‖_𝑇. For each integer 𝑘≥1 and each 𝑡≥𝜏, we define 𝑝_𝑘(𝑡;𝑣,𝜔)∶=𝑘^01/𝑘𝑝(𝑡+𝑟;𝑣,𝜔)𝑑𝑟,∀𝑣,𝜔∈𝑉_𝑇,∀𝑡≥𝜏,(2.5) and denote by 𝑃_𝑘(𝑡)∈ℒ(𝑉_𝑇) the associated operator defined by 𝑃_𝑘(𝑡)𝑣,𝜔_𝑇∶=𝑝_𝑘(𝑡;𝑣,𝜔),∀𝑣,𝜔∈𝑉_𝑇,∀𝑡≥𝜏.(2.6)

Lemma 2.1 (see [2, 4]). For any integer 1≤ℎ≤𝑘, any 𝑡≥𝜏 and every 𝑣,𝜔∈𝑉_𝑇, 0≤𝑝_ℎ𝑝_𝑘(𝑡;𝑣,𝜔)=𝑝_𝑘(𝑡;𝜔,𝑣),𝑝_𝑘(𝑡;𝑣,𝑣)≤𝑝_𝑘(𝑡;𝑣,𝑣)≤𝑝(𝑡;𝑣,𝑣)=‖𝑃(𝑡)𝑣‖^2𝑇≤‖𝑣‖^2𝑇,(𝑡;𝑣,𝑣)∶=𝑑𝑑𝑡𝑝^𝑘(𝑡;𝑣,𝑣)=𝑘𝑝𝑡+1𝑃_𝑘𝑘;𝑣,𝑣−𝑝(𝑡;𝑣,𝑣)≤0,(𝑡)𝑣,𝑧_𝑇=0,∀𝑧∈𝑉_𝑡.(2.7) Moreover,   for every sequence {𝑣_𝑘}⊂𝐿²(𝜏,𝑇;𝑉_𝑇) weak convergent to 𝑣 in 𝐿²(𝜏,𝑇;𝑉_𝑇), liminf_𝑘→+∞^𝑇𝜏𝑝_𝑘𝑡;𝑣_𝑘(𝑡),𝑣_𝑘(𝑡)𝑑𝑡≥^𝑇𝜏𝑝(𝑡;𝑣(𝑡),𝑣(𝑡))𝑑𝑡.(2.8)

Let 𝐽∶𝑉_𝑇→𝑉^∗𝑇 be the Riesz isomorphism defined by ⟨𝐽𝑣,𝜔⟩_𝑇∶=((𝑣,𝜔))_𝑇,∀𝑣,𝜔∈𝑉_𝑇,(2.9) and for each integer 𝑘≥1 and each 𝑡∈[𝜏,𝑇], we denote 𝐴_𝑘(𝑡)∶=−Δ+𝑘𝐽𝑃_𝑘(𝑡).(2.10) Obviously, 𝐴_𝑘(𝑡)∈ℒ(𝑉_𝑇,𝑉^∗𝑇),𝑡∈[𝜏,𝑇], is a family of symmetric linear operators such that the mapping 𝑡∈[𝜏,𝑇]↦𝐴_𝑘(𝑡)∈ℒ(𝑉_𝑇,𝑉^∗𝑇) is measurable, bounded, and satisfies ⟨𝐴_𝑘(𝑡)𝑣,𝑣⟩_𝑇≥‖𝑣‖^2𝑇,∀𝑣∈𝑉_𝑇,∀𝑡∈[𝜏,𝑇].(2.11) Let 𝑢_𝜏∈𝑉_𝑇 be given and for each 𝑘≥1 consider the following problem: 𝑢^𝑘(𝑡),𝑣_𝑇+⟨𝐴_𝑘(𝑡)𝑢_𝑘(𝑡),𝑣⟩_𝑇𝑢_𝑘+𝜀𝐴_𝑘(𝑡)𝑢^𝑘(𝑡),𝑣_𝑇+𝑓𝑢_𝑘(𝑡),𝑣_𝑇=(𝑔(𝑡),𝑣)_𝑇,∀𝑣∈𝑉_𝑇,(𝜏),𝑣_𝑇=𝑢_𝜏,𝑣_𝑇.(2.12)

The idea of the penalty method is as follows: for each 𝑘≥1 we first prove the existence of a solution 𝑢_𝑘 to problem (2.12) (a problem in a cylindrical domain) using standard methods such as the Galerkin method, and then show that 𝑢_𝑘 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 set Ω_𝑡 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 process 𝑈(⋅,⋅) on a family of metric spaces {(𝑋_𝑡,𝑑_𝑡);𝑡∈ℝ}, that is, a family {𝑈(𝑡,𝜏);−∞<𝜏≤𝑡<+∞} of mappings 𝑈(𝑡,𝜏)∶𝑋_𝜏→𝑋_𝑡 such that 𝑈(𝜏,𝜏)𝑥=𝑥 for all 𝑥∈𝑋_𝜏 and 𝑈(𝑡,𝜏)=𝑈(𝑡,𝑟)𝑈(𝑟,𝜏)∀𝜏≤𝑟≤𝑡.(2.13) In addition, suppose 𝒟 is a nonempty class of parameterized sets of the form 𝒟={𝐷(𝑡);𝐷(𝑡)⊂𝑋_𝑡,𝐷(𝑡)≠∅,𝑡∈ℝ}.

Definition 2.2 (see [4]). The process 𝑈(⋅,⋅) is said to be pullback 𝒟-asymptotically compact if the sequence {𝑈(𝑡,𝜏_𝑛)𝑥_𝑛} is relatively compact in 𝑋_𝑡 for any 𝑡∈ℝ, any 𝒟∈𝒟, and any sequences {𝜏_𝑛} and {𝑥_𝑛} with 𝜏_𝑛→−∞ and 𝑥_𝑛∈𝐷(𝜏_𝑛).

Definition 2.3 (see [4]). A family ℬ∈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 𝑈(𝑡,𝜏)𝒟(𝜏)⊂ℬ(𝑡),(2.14) for all 𝜏≤𝜏₀(𝑡,𝒟).

Remark 2.4. Note that if ℬ∈𝒟 is pullback 𝒟-absorbing for the process 𝑈(⋅,⋅) and 𝐵(𝑡) is a compact subset of 𝑋_𝑡 for any 𝑡∈ℝ, then the process 𝑈(⋅,⋅) is pullback 𝒟-asymptotically compact.

For each 𝑡∈ℝ, let dist_𝑡(𝐷₁,𝐷₂) be the Hausdorff semi-distance between nonempty subsets 𝐷₁ and 𝐷₂ of 𝑋_𝑡, which is defined as dist_𝑡𝐷₁,𝐷₂=sup_𝑥∈𝐷1inf_𝑦∈𝐷2𝑑_𝑋𝑡(𝑥,𝑦)for𝐷₁,𝐷₂⊂𝑋_𝑡.(2.15)

Definition 2.5 (see [4]). The family 𝒜={𝐴(𝑡);𝐴(𝑡)⊂𝑋_𝑡,𝐴(𝑡)≠∅,𝑡∈ℝ} is said to be a pullback 𝒟-attractor for 𝑈(⋅,⋅) if (1)𝑎(𝑡) is a compact set of 𝑋_𝑡 for all 𝑡∈ℝ, (2)𝒜 is pullback 𝒟-attracting, that is, lim_{𝜏→−∞}dist_𝑡(𝑈(𝑡,𝜏)𝐷(𝜏),𝐴(𝑡))=0∀𝒟∈𝒟,∀𝑡∈ℝ,(2.16)(3)𝒜 is invariant, that is, 𝑈(𝑡,𝜏)𝐴(𝜏)=𝐴(𝑡)for−∞<𝜏≤𝑡<+∞.(2.17)

Theorem 2.6 (see [4]). Suppose that the process 𝑈(⋅,⋅) 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, Λ𝐷,𝑡∶=_𝑠≤𝑡_𝜏≤𝑠𝑈(𝑡,𝜏)𝐷(𝜏)^𝑋𝑡closurein𝑋_𝑡(2.18) is a pullback 𝒟-attractor for 𝑈(⋅,⋅), which in addition satisfies 𝐴(𝑡)=𝐷_∈𝒟Λ𝐷,𝑡^𝑋𝑡.(2.19) 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 𝑋_𝑡 and lim_{𝜏→−∞}dist_𝑡(𝑈(𝑡,𝜏)𝐵(𝜏),𝐶(𝑡))=0forany𝑡∈ℝ,then𝐴(𝑡)⊂𝐶(𝑡)∀𝑡∈ℝ.(2.20)

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 {𝑈_𝜀(⋅,⋅)∶𝜀∈[0,1]} be a family of evolution processes in a family of Banach spaces {𝑋_𝑡} with corresponding pullback 𝒟-attractors {𝐴_𝜀(𝑡)∶𝜀∈[0,1],𝑡∈ℝ}. For any bounded interval 𝐼∈ℝ, we say that {𝐴_𝜀(⋅)} is upper semicontinuous at 𝜀=0 for 𝑡∈𝐼 if limsup_{𝜀→0𝑡∈𝐼}dist_𝑡𝐴_𝜀(𝑡),𝐴₀(𝑡)=0.(2.21)

Theorem 2.8. Let {𝑈_𝜀(⋅,⋅)∶𝜀∈[0,1]} be a family of processes with corresponding pullback 𝒟-attractors {𝐴_𝜀(⋅)∶𝜀∈[0,1]}. Then, for any bounded interval 𝐼⊂ℝ, {𝐴_𝜀(⋅)∶𝜀∈[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 𝑋_𝑡−𝜏, the following conditions hold: (i)sup_{𝜏∈[0,𝑇]}sup_𝑥∈𝐾dist_𝑡(𝑈_𝜀(𝑡,𝑡−𝜏)𝑥,𝑈₀(𝑡,𝑡−𝜏)𝑥)→0as𝜀→0,(ii)⋃_𝜀∈[0,1]⋃_𝑡≤𝑡0𝐴_𝜀(𝑡)isboundedforgiven𝑡₀,(iii)⋃_0<𝜀≤1𝐴_𝜀(𝑡)iscompactforeach𝑡∈ℝ.

3. Existence and Uniqueness of Variational Solutions

For each 𝑇>𝜏, denote 𝑈𝑄_𝜏,𝑇∶=Ω_𝑇×(𝜏,𝑇),_𝜏,𝑇Φ(𝜏)∶==ΦΦ(𝑇∈)=𝐿^∞0,Φ𝜏,(𝑇𝑡);𝑉∈_𝑇𝑉_𝑡∩𝐿^𝑝𝑄^𝜏,𝑇,Φ′∈𝐿²𝜏,𝑇;𝑉_𝑇,a.e.in(𝜏,𝑇).(3.1)

Definition 3.1. A variational solution of (2.1) is a function 𝑢 such that (C1)𝑢∈𝐿^∞(𝜏,𝑇,𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇), 𝑢^∈𝐿²(𝜏,𝑇;𝑉_𝑇),(C2)for all Φ∈𝑈_𝜏,𝑇, ^𝑇𝜏−𝑢(𝑡),Φ^=(𝑡)_𝑇+((𝑢(𝑡),Φ(𝑡)))_𝑇+𝜀𝑢^(𝑡),Φ(𝑡)_𝑇+(𝑓(𝑢),Φ(𝑡))_𝑇𝑑𝑡_𝑇𝜏(𝑔(𝑡),Φ(𝑡))_𝑇𝑑𝑡,(3.2)(C3)𝑢(𝑡)∈𝑉_𝑡 a.e. in (𝜏,𝑇), (C4)lim_𝑡↓𝜏(𝑡−𝜏)⁻¹∫^𝑡𝜏|𝑢(𝑟)−𝑢_𝜏|^2𝑇𝑑𝑟=0.

Remark 3.2. If 𝑇₂>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.

Denote 𝑄_𝜏∶=U_𝑇>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 that 𝑣∈𝐿²(𝜏,𝑇,𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇) and there exist 𝜉∈𝐿²(𝜏,𝑇,𝑉^∗𝑇) and 𝜂∈𝐿^{𝑝/𝑝−1}(𝑄_𝜏,𝑇) such that ^𝑇𝜏𝑣(𝑡),Φ^(𝑡)_𝑇𝑑𝑡=−^𝑇𝜏⟨𝜉(𝑡),Φ(𝑡)⟩_𝑇𝑑𝑡−^𝑇𝜏(𝜂(𝑡),Φ(𝑡))_𝑇𝑑𝑡,(3.3) for every function Φ∈𝑈_𝜏,𝑇.
For each 0<ℎ<𝑇−𝜏, define 𝑣_ℎ by 𝑣_ℎ∶=ℎ⁻¹0(𝑣(𝑡+ℎ)−𝑣(𝑡))aa..ee..iinn((𝜏𝑇,−𝑇ℎ−,ℎ𝑇)),.(3.4) Then lim_ℎ↓0^𝑇𝜏𝑣_ℎ(𝑡),𝜔(𝑡)_𝑇𝑑𝑡=^𝑇𝜏⟨𝜉(𝑡),𝜔(𝑡)⟩_𝑇𝑑𝑡+^𝑇𝜏(𝜂(𝑡),𝜔(𝑡))_𝑇𝑑𝑡,(3.5) for every function 𝜔∈𝐿²(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇) such that 𝜔(𝑡)∈𝑉_𝑡 a.e. in (𝜏,𝑇).

Remark 3.5. If 𝜏<𝑇′<𝑇 and Φ∈𝐿²(𝜏,𝑇′;𝑉_𝑇)∩𝐿^𝑝(Ω_𝑇×(𝜏,𝑇′)), with Φ′∈𝐿²(𝜏,𝑇′;𝐻_𝑇) satisfies Φ(𝜏)=Φ(𝑇′)=0 and Φ(𝑡)∈𝑉_𝑡 a.e. in (𝜏,𝑇′), then the trivial extension Φ of Φ satisfies Φ∈𝑈_𝜏,𝑇, with (Φ)^=Φ^. Using the open sets Ω_𝑡∶=Ω_{𝑡+𝑇−𝑇},𝜏≤𝑡≤𝑇′, it is easy to see that under the conditions of (3.5), one also has lim_ℎ↓0^{𝑇′𝜏−ℎ}𝑣_ℎ(𝑡),𝜔(𝑡)_𝑇𝑑𝑡=^𝑇′_𝜏⟨𝜉(𝑡),𝜔(𝑡)⟩_𝑇𝑑𝑡+^𝑇′_𝜏(𝜂(𝑡),𝜔(𝑡))_𝑇𝑑𝑡,(3.6) for every 𝜏≤𝑇′≤𝑇 and every function 𝜔∈𝐿²(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇) such that 𝜔(𝑡)∈𝑉_𝑡 a.e. in (𝜏,𝑇).

Lemma 3.6 (see [4]). Let 𝑣_𝑖∈𝐿²(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇),𝑖=1,2, be two functions such that 𝑣_𝑖(𝑡)∈𝑉_𝑡 a.e. in (𝜏,𝑇) for 𝑖=1,2. Assume that there exist 𝜉_𝑖∈𝐿²(𝜏,𝑇;𝑉^∗𝑇),𝜂_𝑖∈𝐿^{𝑝/𝑝−1}(𝑄_𝜏,𝑇),𝑖=1,2 such that ^𝑇𝜏𝑣_𝑖(𝑡),Φ^(𝑡)_𝑇𝑑𝑡=−^𝑇𝜏⟨𝜉_𝑖(𝑡),Φ(𝑡)⟩_𝑇𝑑𝑡−^𝑇𝜏𝜂_𝑖(𝑡),Φ(𝑡)_𝑇𝑑𝑡,𝑖=1,2,(3.7) for every function Φ∈𝑈_𝜏,𝑇. Then,   for every pair 𝜏≤𝑠<𝑡≤𝑇 of Lebesgue points of the inner product function (𝑣₁,𝑣₂)_𝑇 it holds 𝑣₁(𝑡),𝑣₂(𝑡)_𝑇−𝑣₁(𝑠),𝑣₂(𝑠)_𝑇=+^𝑡𝑠⟨𝜉₁(𝑟),𝑣₂(𝑟)⟩_𝑇𝑑𝑟+^𝑡𝑠⟨𝜉₂(𝑟),𝑣₁(𝑟)⟩_𝑇𝑑𝑟_𝑡𝑠+lim_ℎ↓0𝜂₁(𝑟),𝑣₂(𝑟)_𝑇𝑑𝑟+^𝑡𝑠𝜂₂(𝑟),𝑣₁(𝑟)_𝑇𝑑𝑟ℎ⁻¹^{𝑠𝑡−ℎ}𝑣₁(𝑟+ℎ)−𝑣₁(𝑟),𝑣₂(𝑟+ℎ)−𝑣₂(𝑟)_𝑇𝑑𝑟.(3.8)

If 𝑢 is a variational solution of problem (2.1), then 𝜏 is the Lebesgue point of |𝑢|^2𝑇 since the condition (C4) is satisfied. The next corollary gives an obvious consequence of (3.8).

Corollary 3.7. If 𝑢 is a variational solution of (2.1), then for every Lebesgue point 𝑡∈(𝜏,𝑇) of |𝑢|^2𝑇 it holds ||𝑢(𝑡)||=^2𝑇+||𝑢𝜀‖𝑢(𝑡)‖^2𝑇+2^𝑡𝜏‖𝑢(𝑟)‖^2𝑇𝑑r+2^𝑡𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇+2^𝑡𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟+lim_ℎ↓0ℎ⁻¹^{𝜏𝑡−ℎ}||𝑢(𝑟+ℎ)−𝑢(𝑟)||^2𝑇𝑑𝑟.(3.9)

Proof. If 𝑢 is a variational solution of (2.1), then we have ^𝑇𝜏−𝑢(𝑡),Φ^(𝑡)_𝑇+𝜀𝜕𝑢𝜕𝑡,Φ(𝑡)_𝑇=+((𝑢(𝑡),Φ(𝑡)))_𝑇+(𝑓(𝑢(𝑡)),Φ(𝑡))_𝑇𝑑𝑡_𝑇𝜏(𝑔(𝑡),Φ(𝑡))_𝑇𝑑𝑡.(3.10) Applying Lemma 3.6 with 𝑣₁=𝑣₂=𝑢, we get ||𝑢(𝑡)||^2𝑇−||𝑢_𝜏||^2𝑇=−^𝑡𝜏−𝜀𝜕𝜕𝑢𝑡+𝑢(𝑟),𝑢(𝑟)_𝑇𝑑𝑟−^𝑡𝜏𝜀𝜕𝜕𝑢𝑡+𝑢(𝑟),𝑢(𝑟)_𝑇𝑑𝑟_𝑡𝜏+lim(_ℎ→0𝑓(𝑢(𝑟))−𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟−^𝑡𝜏(𝑓(𝑢(𝑟))−𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟ℎ⁻¹^{𝜏𝑡−ℎ}=−2_𝑡𝜏||𝑢(𝑟+ℎ)−𝑢(𝑟)||^2𝑇𝑑𝑟−𝜀‖𝑢(‖𝑡𝑢)‖(_2𝑇𝑟)‖^2𝑇𝑑𝑟−2^𝑡𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟+2^𝑡𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟+𝜀‖‖𝑢_𝜏‖‖^2𝑇+lim_ℎ→0ℎ⁻¹^{𝜏𝑡−ℎ}||𝑢(𝑟+ℎ)−𝑢(𝑟)||^2𝑇𝑑𝑟.(3.11) Hence, it implies the desired result.

The aim of this section is to obtain a variational solution of (2.1) such that ||𝑢(𝑡)||=^2𝑇+||𝑢𝜀‖𝑢(𝑡)‖^2𝑇+2^𝑡𝜏‖𝑢(𝑟)‖^2𝑇𝑑𝑟+2^𝑡𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇+2^𝑡𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟.(3.12) 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 function 𝑣∈𝐿²(𝜏,𝑇;𝐻_𝑇) and any 𝑡∈(𝜏,𝑇], we put 𝜂_𝑣,𝑇(𝑡)∶=limsup_ℎ↓0ℎ⁻¹^{𝜏𝑡−ℎ}||𝑣(𝑟+ℎ)−𝑣(𝑟)||^2𝑇𝑑𝑟.(3.13) Then 𝜂_𝑣,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 𝜂_𝑢,𝑇(𝑡)=0 for a.e. 𝑡∈(𝜏,𝑇). In fact, using the continuity of the following mapping: 𝑡∈[𝜏,𝑇]↦||𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇+2^𝑡𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇−‖𝑢(𝑟)‖^2𝑇−(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟,(3.14) one can see that a variational solution 𝑢 of (1.3) satisfies the energy equality a.e. in (𝜏,𝑇) if and only if 𝜂_𝑢,𝑇(𝑇)=0.

The next lemma provides a sufficient condition for 𝑢 to satisfy the energy equality a.e. in (𝜏,𝑇).

Lemma 3.8. Let 𝑢 be a variational solution of (2.1) and suppose that there exists a sequence {𝑡_𝑛}⊂(𝜏,𝑇) of Lebesgue points of |𝑢|^2𝑇 such that 𝑡_𝑛→𝑇 and lims≤up_n⟶+∞||𝑢_𝜏||𝑢𝑡^𝑛||^2𝑇+𝜀‖‖𝑢𝑡_𝑛‖‖^2𝑇||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇+2^𝑇𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇−‖𝑢(𝑟)‖^2𝑇−(𝑓(𝑢(𝑟)),𝑢(r))_𝑇𝑑𝑟.(3.15) Then, 𝑢 satisfies the energy equality a.e. in (𝜏,𝑇).

Proof. It is sufficient to prove that 𝜂_𝑢,𝑇(𝑇)=0. Since 𝑡_𝑛→𝑇 and 𝜂_𝑢,𝑇 is nondecreasing, by Corollary 3.7, we have 𝜂_𝑢,𝑇(𝑇)≤limsup_𝑛↑+∞𝜂_𝑢,𝑇𝑡_𝑛=limsup_𝑛→+∞||𝑢𝑡^𝑛||^2𝑇+𝜀‖‖𝑢𝑡_𝑛‖‖^2𝑇−||𝑢_𝜏||^2𝑇−𝜀‖‖𝑢_𝜏‖‖^2𝑇≤limsup_𝑛→+∞−2^𝑡𝑛_𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇−‖𝑢(𝑟)‖^2𝑇−(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟||𝑢𝑡^𝑛||^2𝑇+𝜀‖‖𝑢𝑡_𝑛‖‖^2𝑇−||𝑢^𝜏||^2𝑇−𝜀‖‖𝑢_𝜏‖‖^2𝑇≤0−.2^𝑇𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇−‖𝑢(𝑟)‖^2𝑇−(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟(3.16) This completes the proof.

Proposition 3.9. Let 𝑢,𝑢 be two variational solutions of (2.1) corresponding to the initial data 𝑢_𝜏,𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝(Ω_𝜏), respectively, which satisfy the energy equality a.e. in (𝜏,𝑇). Then, ||𝑢(𝑡)≤−𝑒𝑢(𝑡)||^2𝑇_{2ℓ(𝑡−𝜏)}+𝜀‖‖𝑢(𝑡)−𝑢(𝑡)‖‖^2𝑇+2^𝑡𝜏‖‖𝑢(𝑟)−𝑢(𝑟)‖‖^2𝑇𝑑𝑟||𝑢^𝜏−𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏−𝑢_𝜏‖‖^2𝑇a.e.𝑡∈(𝜏,𝑇).(3.17) Hence, it implies the uniqueness of variational solutions to (2.1) satisfying the energy equality in (𝜏,𝑇).

Proof. Since 𝑢,𝑢 satisfy the energy equation, 𝜂_𝑢,𝑇(𝑡)=𝜂_𝑢,𝑇(𝑡)=0 for all 𝑡∈(𝜏,𝑇) and ||𝑢(𝑡)+−2𝑢(_𝑡𝜏𝑡)||^2𝑇+𝜀‖‖𝑢(𝑡)−𝑢(𝑡)‖‖^2𝑇+2^𝑡𝜏‖‖𝑢(𝑟)−𝑢(𝑟)‖‖^2𝑇𝑑𝑟≤||𝑢𝑓(𝑢(𝑟))−𝑓𝑢(𝑟),𝑢(𝑟)−𝑢(𝑟)_𝑇𝑑𝑟_𝜏−𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏−𝑢_𝜏‖‖^2𝑇−2lim_ℎ↓0ℎ⁻¹^{𝜏𝑡−ℎ}𝑢(𝑟+ℎ)−𝑢(𝑟),𝑢(𝑟+ℎ)−𝑢(𝑟)_𝑇𝑑𝑟.(3.18) On the other hand, ||||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,(3.19) so lim_ℎ↓0ℎ⁻¹^{𝜏𝑡−ℎ}𝑢(𝑟+ℎ)−𝑢(𝑟),𝑢(𝑟+ℎ)−𝑢(𝑟)_𝑇𝑑𝑟=0.(3.20) Using this and (1.5) in (3.12), one can conclude ||𝑢(𝑡)≤−||𝑢𝑢(𝑡)||^2𝑇+𝜀‖‖𝑢(𝑡)−𝑢(𝑡)‖‖^2𝑇+2^𝑡𝜏‖‖𝑢(𝑟)−𝑢(𝑟)‖‖^2𝑇𝑑𝑟_𝜏≤||𝑢−𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏−𝑢_𝜏‖‖^2𝑇−2^𝑡𝜏𝑓(𝑢(𝑟))−𝑓𝑢(𝑟),𝑢(𝑟)−𝑢(𝑟)_𝑇𝑑𝑟_𝜏−𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏−𝑢_𝜏‖‖^2𝑇+2ℓ^𝑡𝜏||𝑢(𝑟)−𝑢(𝑟)||^2𝑇𝑑𝑟.(3.21) 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. Let 𝑔∈𝐿²(𝜏,𝑇;𝐻_𝑇),𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝(Ω_𝜏) 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.
Step 1. Existence of a weak solution to problem (2.12).
We will use the Galerkin method (see [20]). Take an orthonormal Hilbert basis {𝑒_𝑗} of 𝐻_𝑇 formed by elements of 𝑉_𝑇∩𝐿^𝑝(Ω_𝑇) such that the vector space generated by {𝑒_𝑗} is dense in 𝑉_𝑇 and 𝐿^𝑝(Ω_𝑇). Then, one takes a sequence {𝑢_𝜏𝑚} converging to 𝑢_𝜏 in 𝑉_𝑇, with {𝑢_𝜏𝑚} in the vector space spanned by the 𝑚 first {𝑒_𝑗}. For each integer 𝑚≥1, one considers the approximation 𝑢_𝑘𝑚(𝑡)=∑^𝑚𝑗=1𝛾_{𝑘𝑚,𝑗}(𝑡)𝑒_𝑗, defined as a solution of 𝑢^𝑘_𝑚(𝑡),𝑒_𝑗_𝑇+𝐴_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑒_𝑗𝑢_𝑇𝑘𝑚+𝜀𝐴_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑒_𝑗_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑒_𝑗_𝑇=𝑔(𝑡),𝑒_𝑗_𝑇,(𝜏),𝑒_𝑗_𝑇=𝑢_𝜏𝑚,𝑒_𝑗_𝑇.(3.22) Multiplying (3.22) by 𝛾^𝑘_𝑚,𝑗(𝑡) and summing from 𝑗=1 to 𝑚, we obtain 𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝐴_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)+𝜀𝐴_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇=𝑔(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇,(3.23) or |||𝑢_𝑘𝑚(𝑡)|||^2𝑇+12𝑑𝑑𝑡‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇=𝑔(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇.(3.24) Thus, 𝑔(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇≥|||𝑢+^𝑘_𝑚𝜀𝑘(𝑡)|||^2𝑇𝑃+_𝑘𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+12𝑑𝑑𝑡‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇=|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇+𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+12𝑑𝑑𝑡‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥.(3.25) We have 𝑔(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇≤12||𝑔(𝑡)||^2𝑇+|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇,(3.26) so ||𝑔(𝑡)||^2𝑇≥2𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+𝑑𝑑𝑡‖‖𝑢^𝑘𝑚+2𝜀𝑘𝑃_𝑘(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇.(3.27) Integrating (3.27) on [𝜏,𝑡],𝜏≤𝑡≤𝑇, we obtain 2𝜀^𝑡𝜏‖‖𝑢^𝑘_𝑚(𝑟)‖‖^2𝑇𝑑𝑟+2𝜀𝑘^𝑡𝜏𝑃^𝑘(𝑟)𝑢^𝑘_𝑚(𝑟),𝑢^𝑘_𝑚(𝑟)_𝑇𝑑𝑟+^𝑡𝜏|||𝑢_𝑘𝑚+‖‖𝑢(𝑟)|||^2𝑇𝑑𝑟_𝑘𝑚≤(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥_𝑡𝜏||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝑘𝑚(𝜏)‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝑘𝑚(𝜏),𝑢_𝑘𝑚(𝜏)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝜏)𝑑𝑥.(3.28) Since _Ω𝑇𝐹_Ω𝑇𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥≥−𝛽̃||Ω_𝑇||+𝛼₁‖‖𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇},𝐹𝑢_𝑘𝑚(𝑥,𝜏)𝑑𝑥≤𝛽̃||Ω_𝑇||+𝛼₂‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇},(3.29) we have 2𝜀^𝑡𝜏‖‖𝑢^𝑘_𝑚(𝑟)‖‖^2𝑇𝑑𝑟+2𝜀𝑘^𝑡𝜏𝑃^𝑘(𝑟)𝑢^𝑘_𝑚(𝑟),𝑢^𝑘_𝑚(𝑟)_𝑇𝑑𝑟+^𝑡𝜏|||𝑢_𝑘𝑚+‖‖𝑢(𝑟)|||^2𝑇𝑑𝑟_𝑘𝑚(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2𝛼₁‖‖𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇}≤^𝑡𝜏||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝑘𝑚(𝜏)‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝑘𝑚(𝜏),𝑢_𝑘𝑚(𝜏)_𝑇+4𝛽̃||Ω_𝑇||+2𝛼₂‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇}.(3.30) From (3.30), we deduce that{𝑢_𝑘𝑚} is bounded in 𝐿^∞(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇),{𝑢_𝑘𝑚}⇀𝑢_𝑘 weakly in 𝐿^∞(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇),{𝑢^𝑘_𝑚} is bounded in 𝐿²(𝜏,𝑇;𝑉_𝑇),{𝑢^𝑘_𝑚}⇀𝑢^𝑘 weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇).Since {𝑢_𝑘𝑚} is bounded in 𝐿^∞(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝑄_𝜏,𝑇), one can check that {𝑓(𝑢_𝑘𝑚)} is bounded in 𝐿^𝑞(𝜏,𝑇;𝐿^𝑞(Ω_𝑇)) with 𝑞=𝑝/(𝑝−1), hence 𝑓(𝑢_𝑘𝑚)⇀𝜂 in 𝐿^𝑞(𝜏,𝑇;𝐿^𝑞(Ω_𝑇)). We now prove that 𝜂=𝑓(𝑢_𝑘).
Indeed, we have𝑉_𝑇⊂⊂𝐻_𝑇⊂𝑉^∗𝑇, {𝑢_𝑘𝑚} is bounded in 𝐿^∞(𝜏,𝑇;𝑉_𝑇),{𝑢^𝑘_𝑚} is bounded in 𝐿²(𝜏,𝑇;𝑉^∗𝑇).By the Aubin-Lions lemma [20, Chapter 1], {𝑢_𝑘𝑚} is relatively compact in 𝐿²(𝜏,𝑇;𝐻_𝑇). Therefore, one can assume that 𝑢_𝑘𝑚→𝑢_𝑘 strongly in 𝐿²(𝜏,𝑇;𝐻_𝑇), so 𝑢_𝑘𝑚→𝑢_𝑘 a.e. in 𝑄_𝜏,𝑇. Since 𝑓 is continuous, 𝑓(𝑢_𝑘𝑚)→𝑓(𝑢_𝑘) a.e. in 𝑄_𝜏,𝑇. Applying Lemma 1.3 in [20], we have 𝑓𝑢_𝑘𝑚⇀𝑓𝑢_𝑘weaklyin𝐿^𝑞𝜏,𝑇;𝐿^𝑞Ω_𝑇.(3.31) This implies that 𝑢_𝑘 is a weak solution of problem (2.12).
Step 2. Existence of a variational solution to (2.1) satisfying the energy equality.

From (3.30), we have 𝑘^𝑇𝜏≤𝑃(𝑇_𝑘(−𝑟)𝜏𝑢)_𝑘𝑚(𝑟_𝑡𝜏),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝑘𝑚(𝜏)‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝑘𝑚(𝜏),𝑢_𝑘𝑚(𝜏)_𝑇+4𝛽̃||Ω_𝑇||+2𝛼₂‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇}.(3.32) Consider the function Φ∶𝐿²(𝜏,𝑇;𝑉_𝑇)→ℝ defined by Φ(𝑣)=^𝑇𝜏𝑃_𝑘(𝑡)𝑣(𝑡),𝑣(𝑡)_𝑇𝑑𝑡,𝑣∈𝐿²𝜏,𝑇;𝑉_𝑇.(3.33) It is easy to see that Φ is a continuous and convex function. It follows that ∫^𝑇𝜏((𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)))_𝑇𝑑𝑡 is weakly lower semicontinuous in 𝐿²(𝜏,𝑇;𝑉_𝑇). Moreover, {𝑢_𝑘𝑚}⇀𝑢_𝑘 weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇), hence 𝑘^𝑇𝜏≤𝑘𝑃_𝑘li(m𝑡)i𝑢n_𝑘f_𝑚→∞(𝑡),𝑢_𝑘(𝑡)_𝑇𝑑𝑡^𝑇𝜏𝑃_𝑘≤(𝑇−𝜏)_𝑡𝜏(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇𝑑𝑡||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝜏‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝜏,𝑢_𝜏_𝑇+4𝛽̃||Ω_𝑇||+2𝛼₂‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑇}.(3.34) Since {𝑢^𝑘_𝑚}⇀𝑢^𝑘 weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇), then, reasoning as above, 2𝜀𝑘≤^𝑇𝜏2𝜖𝑘𝑃_𝑘lim(𝑡)i𝑢n^𝑘f_𝑚→∞(𝑡),𝑢^𝑘(𝑡)_𝑇𝑑𝑡^𝑇𝜏𝑃_𝑘≤(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇𝑑𝑡_𝑡𝜏||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝜏‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝜏,𝑢_𝜏_𝑇+4𝛽̃||Ω_𝑇||+2𝛼₂‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑇}.(3.35) From the facts that 𝑢_𝑘𝑚⇀𝑢_𝑘 weakly in 𝐿^∞(𝜏,𝑇;𝑉_𝑇), 𝑢^𝑘_𝑚⇀𝑢^𝑘 weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇) and the weak lower semicontinuity of the norm, we deduce that 2𝜀^𝑡𝜏+‖‖𝑢‖‖^𝑘𝑢(𝑟)‖‖^2𝑇𝑑𝑟+2𝜀𝑘^𝑡𝜏𝑃_𝑘(𝑟)𝑢^𝑘(𝑟),𝑢^𝑘(𝑟)_𝑇𝑑𝑟+^𝑡𝜏||𝑢^𝑘(𝑟)||^2𝑇𝑑𝑟_𝑘(𝑡)‖‖^2𝑇+𝑘^𝑇𝜏𝑃_𝑘(𝑡)𝑢_𝑘(𝑡),𝑢_𝑘(𝑡)_𝑇+2𝛼₁‖‖𝑢_𝑘(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇}≤(5+𝑇−𝜏)^𝑡𝜏||𝑔(𝑟)||^2𝑇𝑑𝑟+‖‖𝑢_𝜏‖‖^2𝑇+𝑘𝑃_𝑘(𝜏)𝑢_𝜏,𝑢_𝜏_𝑇+4𝛽̃||Ω_𝑇||+2𝛼₂‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑇}=𝐶.(3.36) Since 𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝(Ω_𝜏), ((𝑃_𝑘(𝜏)𝑢_𝜏,𝑢_𝜏))_𝑇=0forall𝑘≥1, (3.36) gives {𝑢_𝑘} is bounded in 𝐿^∞(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝜏,𝑇;𝐿^𝑝(Ω_𝑇)),{𝑢^𝑘} is bounded in 𝐿²(𝜏,𝑇;𝑉_𝑇),{𝑢_𝑘}⇀𝑢 weakly in 𝐿^∞(𝜏,𝑇;𝑉_𝑇)∩𝐿^𝑝(𝜏,𝑇;𝐿^𝑝(Ω_𝑇)),{𝑢^𝑘}⇀𝑢′ weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇).From Lemma 2.1, we have ^𝑇𝜏‖𝑃(𝑡)𝑢(𝑡)‖^2𝑇𝑑𝑡≤liminf_𝑘→∞^𝑇𝜏𝑃_𝑘(𝑡)𝑢_𝑘(𝑡),𝑢_𝑘(𝑡)_𝑇𝑑𝑡≤liminf_𝑘→∞𝐶thatis,𝑃(𝑡)𝑢(𝑡)=0a.e.in(𝜏,𝑇)or𝑢(𝑡)∈𝑉_𝑡𝑘=0,a.e.in(𝜏,𝑇),_𝑇𝜏‖‖𝑃(𝑡)𝑢^(𝑡)‖‖^2𝑇𝑑𝑡≤liminf_𝑘→∞^𝑇𝜏𝑃_𝑘(𝑡)𝑢^𝑘(𝑡),𝑢^𝑘(𝑡)_𝑇𝑑𝑡≤liminf_𝑘→∞𝐶thatis,𝑃(𝑡)𝑢_𝑘=0,(𝑡)=0a.e.in(𝜏,𝑇).(3.37) Moreover, (3.36) and the equality 𝑢_𝑘(𝑡)−𝑢_𝑘(𝑠)=^𝑡𝑠𝑢^𝑘(𝑟)𝑑𝑟,∀𝑠,𝑡∈[𝜏,𝑇],∀𝑘≥1,(3.38) give ||𝑢_𝑘(𝑡)−𝑢_𝑘(𝑠)||_𝑇≤𝐶^1/2|𝑡−𝑠|^1/2∀𝑠,𝑡∈[𝜏,𝑇],∀𝑘≥1.(3.39) It follows from (3.36) that ‖𝑢_𝑘(𝑡)‖_𝑇≤𝐶 for all 𝑡∈[𝜏,𝑇] and each 𝑘≥1. Since the injection of 𝑉_𝑇 into 𝐻_𝑇 is compact, the set {𝑣∈𝑉_𝑇∶‖𝑣‖^2𝑇≤𝐶} is compact in 𝐻_𝑇. By (3.39) and the Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by {𝑢_𝑘}, such that 𝑢_𝑘→𝑢in𝐶[𝜏,𝑇];𝐻_𝑇as𝑘⟶+∞.(3.40) So, the condition (C4) is satisfied.

On the other hand, {𝑢_𝑘} is bounded in 𝐿^∞(𝜏,𝑇;𝑉_𝑇) and {𝑢^𝑘}is bounded in 𝐿²(𝜏,𝑇;𝑉_𝑇), applying the Aubin-Lions lemma and Lemma 1.3 in [20, Chapter 1], one has 𝑓𝑢_𝑘⇀𝑓(𝑢)weaklyin𝐿^𝑞𝜏,𝑇,𝐿^𝑞Ω_𝑇.(3.41) Since 𝑢_𝑘 is the weak solution of the problem 𝑢^𝑘(𝑡),𝑣_𝑇+⟨𝐴_𝑘(𝑡)𝑢_𝑘(𝑡),𝑣⟩_𝑇+𝜀𝐴_𝑘(𝑡)𝑢^𝑘(𝑡),𝑣𝑢_𝑇𝑘+𝑓𝑢_𝑘(𝑡),𝑣_𝑇=(𝑔(𝑡),𝑣)_𝑇,∀𝑣∈𝑉_𝑇,(𝜏),𝑣_𝑇=𝑢_𝜏,𝑣_𝑇,(3.42) 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 that 𝑢 satisfies the energy equality in (𝜏,𝑇). Multiplying (3.22) by 𝛾_{𝑘𝑚,𝑗} and summing from 𝑗=1 to 𝑚, we obtain 𝑢^𝑘_𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝐴_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝜀𝐴_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇=𝑔(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇.(3.43) Hence, we get 12𝑑𝑑𝑡||𝑢^𝑘𝑚(𝑡)||^2𝑇+‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝜖2𝑑𝑑𝑡‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+𝜖𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇=𝑔(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇,(3.44) or ||𝑢_𝑘𝑚(+𝑡)||𝜖^2𝑇‖‖+𝑢2_𝑘𝑚^𝑡𝜏‖‖𝑢_𝑘𝑚(𝑟)‖‖^2𝑇𝑑𝑟+2𝑘^𝑇𝜏𝑃_𝑘(𝑟)𝑢_𝑘𝑚(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟=2_𝑡𝜏(𝑡)‖‖^2𝑇+2𝑘𝜖^𝑡𝜏𝑃^𝑘(𝑟)𝑢^𝑘_𝑚(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟+2^𝑡𝜏𝑓𝑢_𝑘𝑚(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟𝑔(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟+||𝑢_𝜏𝑚||^2𝑇+𝜀‖‖𝑢_𝜏𝑚‖‖^2𝑇.(3.45) Since 𝑃^𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢𝑃_𝑘𝑚𝑘(𝑡)_𝑇≥12𝑑𝑑𝑡𝑃^𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇,||𝑢(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≥0,_𝑘𝑚(≤𝑡)||2^2𝑇+_𝑡𝜏2^𝑡𝜏‖‖𝑢_𝑘𝑚(𝑟)‖‖^2𝑇𝑑𝑟+𝜖‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+2^𝑡𝜏𝑓𝑢_𝑘𝑚(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟𝑔(𝑟),𝑢_𝑘𝑚(𝑟)_𝑇𝑑𝑟+||𝑢_𝜏𝑚||^2𝑇+𝜀‖‖𝑢_𝜏𝑚‖‖^2𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇,(3.46) letting 𝑚→∞, we obtain ||𝑢_𝑘(𝑇≤)||2^2𝑇+_𝑇𝜏2^𝑇𝜏‖‖𝑢_𝑘(𝑟)‖‖^2𝑇𝑑𝑟+𝜖‖‖𝑢_𝑘(𝑇)‖‖^2𝑇+2^𝑇𝜏𝑓𝑢_𝑘(𝑟),𝑢_𝑘(𝑟)_𝑇𝑑𝑟𝑔(𝑟),𝑢_𝑘(𝑟)_𝑇𝑑𝑟+||𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇.(3.47) Now ^𝑇𝜏𝑓𝑢_𝑘=(𝑟),𝑢_𝑘(𝑟)_𝑇𝑑𝑟_𝑇𝜏+𝑓𝑢_𝑘(𝑟)−𝑓(𝑢(𝑟)),𝑢_𝑘(𝑟)−𝑢(𝑟)_𝑇𝑑𝑟+^𝑇𝜏𝑓𝑢_𝑘(𝑟),𝑢(𝑟)_𝑇𝑑𝑟_𝑇𝜏≥−ℓ_𝑇𝜏𝑓(𝑢(𝑟)),𝑢_𝑘(𝑟)_𝑇𝑑𝑟−^𝑇𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟+||𝑢_𝑘(𝑟)−𝑢(𝑟)||^2𝑇𝑑𝑟+^𝑇𝜏𝑓𝑢_𝑘(𝑟),𝑢(𝑟)_𝑇𝑑𝑟_𝑇𝜏𝑓(𝑢(𝑟)),𝑢_𝑘(𝑟)_𝑇𝑑𝑟−^𝑇𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟.(3.48) This inequality and (3.47) give ||𝑢_𝑘(𝑇)||^2𝑇+2^𝑇𝜏‖‖𝑢_𝑘(𝑟)‖‖^2𝑇𝑑𝑟+𝜖‖‖𝑢_𝑘(𝑇)‖‖^2𝑇≤−2^𝑇𝜏𝑔(𝑟),𝑢_𝑘(𝑟)_𝑇𝑑𝑟+||𝑢_𝜏||^2𝑇+𝜀‖‖𝑢_𝜏‖‖^2𝑇+2𝑙^𝑇𝜏||𝑢_𝑘(𝑟)−𝑢(𝑟)||^2𝑇𝑑𝑟_𝑇𝜏𝑓𝑢_𝑘(𝑟),𝑢(𝑟)_𝑇𝑑𝑟−^𝑇𝜏𝑓(𝑢(𝑟)),𝑢_𝑘(𝑟)_𝑇𝑑𝑟+^𝑇𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟.(3.49) Since 𝑢_𝑘⇀𝑢 weakly in 𝐿²(𝜏,𝑇;𝑉_𝑇), we get ||𝑢(𝑇)||^2𝑇+𝜀‖𝑢(𝑇)‖^2𝑇≤−||𝑢_𝜏2||^2𝑇_𝑇𝜏+𝜀‖‖𝑢_𝜏‖‖^2𝑇−2^𝑇𝜏(𝑓(𝑢(𝑟)),𝑢(𝑟))_𝑇𝑑𝑟‖𝑢(𝑟)‖^2𝑇𝑑𝑟+2^𝑇𝜏(𝑔(𝑟),𝑢(𝑟))_𝑇𝑑𝑟.(3.50) Applying Lemma 3.8 with 𝑡_𝑛=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 Pullback 𝒟-Attractors

The aim of this section is to establish the existence of a pullback attractor for problem (1.3).

Suppose that 𝑔∈𝐿^2loc(ℝ^𝑁+1). Then, according to Theorem 3.10, for each 𝜏∈ℝ and each 𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝(Ω_𝜏) given, there exists a unique variational solution 𝑢(⋅;𝜏,𝑢_𝜏) of problem (1.3) satisfying the energy equality a.e. in (𝜏,𝑇) for all 𝑇>𝜏.

Define 𝑈(𝑡,𝜏)𝑢_𝜏∶=𝑢𝑡;𝜏,𝑢_𝜏,−∞<𝜏≤𝑡<+∞,𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝Ω_𝜏.(4.1) It is easy to check that the family of mappings {𝑈(𝑡,𝜏);−∞<𝜏≤𝑡<+∞} is a process 𝑈(⋅,⋅).

A uniform estimate in 𝑉_𝑇∩𝐿^𝑝(Ω_𝑇) will be obtained now for the variational solutions of (1.3) satisfying the energy equality, and since the compactness of the embedding 𝑉_𝑇∩𝐿^𝑝(Ω_𝑇)↪𝐻_𝑇, 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]). Let 𝑋⊂𝑌 be Banach spaces such that 𝑋 is reflexive and the injection of 𝑋 in 𝑌 is compact. Suppose that {𝑣_𝑛} is a bounded sequence in 𝐿^∞(𝑡₀,𝑇;𝑋) such that 𝑣_𝑛⇀𝑣 weakly in 𝐿^𝑝(𝑡₀,𝑇;𝑋) for some 𝑝∈[1,+∞) and 𝑣∈𝐶⁰([𝑡₀,𝑇];𝑌). Then, 𝑣(𝑡)∈𝑋 for all 𝑡∈[𝑡₀,𝑇] and ‖𝑣(𝑡)‖_𝑋≤liminf_𝑛→+∞‖‖𝑣_𝑛‖‖_{𝐿∞(𝑡0,𝑇;𝑋)},∀𝑡∈𝑡₀,𝑇.(4.2)

Proposition 4.2. Suppose that the assumptions in Theorem 3.10 hold and 𝑔∈𝐿^2loc(ℝ^𝑁+1) satisfies 𝐶_𝑔,𝑇=sup_𝑡≤𝑇^𝑡𝑡−1||𝑔(𝑟)||^2𝑇𝑑𝑟<+∞.(4.3) Then, for any 𝑢_𝜏∈𝑉_𝜏∩𝐿^𝑝(Ω_𝜏) given, the corresponding variational solution 𝑢 of (1.3) satisfying the energy equality in (𝜏,𝑇) also satisfies ‖𝑢(𝑡)‖^𝑝𝐿_{𝑝Ω𝑇}+‖𝑢(𝑡)‖^2𝑇≤𝐶𝑒^{−𝜎𝑇(𝑡−𝜏)}‖‖𝑢^𝜏‖‖^2𝜏+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝜏}+1+11−𝑒^−𝜎𝑇𝐶_𝑔,𝑇,∀𝑡∈[𝜏+1,𝑇],(4.4) where 𝜎_𝑇=min{𝜆_1,𝑇/2,1/(𝜀+1),𝛼₁/𝛼₂}, 𝜆_1,𝑇>0 is the first eigenvalue of the operator −Δ in Ω_𝑇 with the homogeneous Dirichlet condition, 𝛼₁,𝛼₂ are the constants in (𝐻1), and the constant 𝐶 is independent of 𝑡,𝜏,𝜀.

Proof. Assume that 𝑢_𝑘𝑚 are the Galerkin approximations of 𝑢_𝑘 defined by (3.22). From (3.44) and (3.24), we have 12𝑑𝑑𝑡||𝑢^𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇+‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)+𝑢_𝑘𝑚(𝑡)_𝑇=𝑔(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝑔(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇.(4.5) Moreover, 𝑃^𝑘(𝑡𝑓)𝑢𝑢^𝑘_𝑚𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≥12𝑑𝑑𝑡𝑃^𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇,(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇=𝑑𝑑𝑡_Ω𝑇𝑔(𝑡),𝑢_𝑘𝑚𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥,(𝑡)_𝑇≤14𝜂₁𝑔(𝑡),𝑢_𝑘𝑚||𝑔(𝑡)||^2𝑇+𝜂₁||𝑢_𝑘𝑚(𝑡)||^2𝑇,∀𝜂₁>0,(𝑡)_𝑇≤14𝜂₂||𝑔(𝑡)||^2𝑇+𝜂₂|||𝑢_𝑘𝑚(𝑡)|||^2𝑇,∀𝜂₂>0,(4.6) so 12𝑑𝑑𝑡||𝑢^𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥+|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇+‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝑓𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≤14𝜂₁||𝑔(𝑡)||^2𝑇+𝜂₁||𝑢_𝑘𝑚(𝑡)||^2𝑇+14𝜂₂||𝑔(𝑡)||^2𝑇+𝜂₂|||𝑢_𝑘𝑚(𝑡)|||^2𝑇,∀𝜂₁,𝜂₂>0.(4.7) Since 𝑓𝑢≥_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇=_Ω𝑇𝑓𝑢_𝑘𝑚(𝑡)𝑢_𝑘𝑚(𝑡)𝑑𝑥_Ω𝑇−𝛽+𝛼₁||𝑢_𝑘𝑚(𝑡)||^𝑝𝑑𝑥=−𝛽||Ω_𝑇||+𝛼₁‖‖𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇},(4.8) we have 12𝑑𝑑𝑡||𝑢^𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2∫_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥+1−𝜂₂|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇+12‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+𝜆^1,𝑇2−𝜂¹||𝑢^𝑘𝑚(𝑡)||^2𝑇+𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+𝛼₁‖‖𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇}≤𝛽||Ω_𝑇||+14𝜂₁+14𝜂₂||𝑔(𝑡)||^2𝑇.(4.9) Denote 𝑦_𝑘𝑚(𝑡)∶=||𝑢_𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥.(4.10) Choose 𝜂₂<1 and 𝜂₁ small enough such that 𝜎_𝑇<min{1/(𝜀+1),𝛼₁/𝛼₂,𝜆_1,𝑇−2𝜂₁}, we have 𝜎_𝑇𝑦_𝑘𝑚(𝑡)=𝜎_𝑇||𝑢^𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≤𝜎_𝑇+2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥||𝑢^𝑘𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≤𝜎_𝑇+2_Ω𝑇𝛽̃+𝛼₂||𝑢_𝑘𝑚(𝑡)||^𝑝𝑑𝑥+2𝛽̃||Ω_𝑇||||𝑢^𝑘_𝑚(𝑡)||^2𝑇+(𝜀+1)‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇≤2+12𝛼−₂‖‖𝜂₂𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇}𝑑𝑥|||𝑢^𝑘_𝑚(𝑡)|||^2𝑇+221‖‖𝑢^𝑘𝑚(𝑡)‖‖^2𝑇+2𝜆2^1,𝑇−𝜂¹||𝑢^𝑘𝑚(𝑡)||^2𝑇+2𝜀‖‖𝑢^𝑘_𝑚(𝑡)‖‖^2𝑇+2𝜀𝑘𝑃_𝑘(𝑡)𝑢^𝑘_𝑚(𝑡),𝑢^𝑘_𝑚(𝑡)_𝑇+2𝑘𝑃_𝑘(𝑡)𝑢_𝑘𝑚(𝑡),𝑢_𝑘𝑚(𝑡)_𝑇+2𝛼₁‖‖𝑢_𝑘𝑚(𝑡)‖‖^𝑝𝐿_{𝑝Ω𝑇}.(4.11) Hence, we have 𝑑𝑑𝑡𝑦^𝑘𝑚(𝑡)+𝜎_𝑇𝑦_𝑘𝑚(𝑡)≤𝐶1+||𝑔(𝑡)||^2𝑇.(4.12) By Gronwall’s lemma, we get 𝑦_𝑘𝑚(𝑡)≤𝑒^{−𝜎𝑇(𝑡−𝜏)}𝑦_𝑘𝑚(𝜏)+𝐶1+𝑒^{−𝜎𝑇𝑡}^𝑡𝜏𝑒^𝜎𝑇𝑠||𝑔(𝑠)||^2𝑇𝑑𝑠.(4.13) Now, observe that 𝑦_𝑘𝑚(𝜏)=||𝑢_𝜏𝑚||^2𝑇+(𝜀+1)‖‖𝑢_𝜏𝑚‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝜏)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇≤+21_Ω𝑇𝐹𝑢_𝜏𝑚𝑑𝑥+2𝛽̃||Ω_𝑇||𝜆_1,𝑇+𝜀+1‖‖𝑢_𝜏𝑚‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝜏)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇=𝐶+_𝑇2_Ω𝑇𝛽̃+𝛼₂||𝑢_𝜏𝑚||^𝑝𝑑𝑥+2𝛽̃||Ω_𝑇||≤𝐶_𝑇‖‖𝑢_𝜏𝑚‖‖^2𝑇+(𝜀+1)𝑘𝑃_𝑘(𝜏)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇+2𝛼₂‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇}+4𝛽̃||Ω_𝑇||1+‖‖𝑢^𝜏_𝑚‖‖^2𝑇+‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇}+(𝜀+1)𝑘𝑃^𝑘(𝜏)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇.(4.14) Since 2_Ω𝑇𝐹𝑢_𝑘𝑚(𝑥,𝑡)𝑑𝑥+2𝛽̃||Ω_𝑇||≥2𝛼₁‖‖𝑢_𝑘𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇},(4.15) so combining with (4.13) and (4.14), we have 2𝛼₁‖‖𝑢_𝑘𝑚‖‖^𝑝𝐿_𝑝(Ω𝑇)+‖‖𝑢_𝑘𝑚(𝑡)‖‖^2𝑇≤𝐶𝑒^{−𝜎𝑇(𝑡−𝜏)}1+‖‖𝑢^𝜏_𝑚‖‖^2𝑇+‖‖𝑢_𝜏𝑚‖‖^𝑝𝐿_{𝑝Ω𝑇}+1+𝑒^{−𝜎𝑇𝑡}^𝑡𝜏+𝑒_{−𝜎𝑇(𝑡−𝜏)}𝑒^𝜎𝑇𝑠||𝑔(𝑠)||^2𝑇𝑑𝑠(𝜀+1)𝑘𝑃_𝑘(𝜏)𝑢_𝜏𝑚,𝑢_𝜏𝑚_𝑇,(4.16) where 𝐶 is independent of 𝑡,𝜏,𝜀, and 𝑘.
Now, it is known that 𝑢_𝑘𝑚⇀𝑢_𝑘∗-weakly in 𝐿^∞(𝜏,𝑇;𝑉_𝑇) as 𝑚→+∞. Hence, by (4.16) and Lemma 4.1, we can conclude that 2𝛼₁‖‖𝑢_𝑘‖‖^𝑝𝐿_{𝑝Ω𝑇}+‖‖𝑢_𝑘(𝑡)‖‖^2𝑇≤𝐶𝑒^{−𝜎𝑇(𝑡−𝜏)}1+‖‖𝑢^𝜏‖‖^2𝜏+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝜏}+1+𝑒^{−𝜎𝑇𝑡}^𝑡𝜏+𝑒_{−𝜎𝑇(𝑡−𝜏)}𝑒^𝜎𝑇𝑠||𝑔(𝑠)||^2𝑇𝑑𝑠(𝜀+1)𝑘𝑃_𝑘(𝜏)𝑢_𝜏,𝑢_𝜏_𝜏=𝐶𝑒^{−𝜎𝑇(𝑡−𝜏)}1+‖‖𝑢^𝜏‖‖^2𝜏+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝜏}+1+𝑒^{−𝜎𝑇𝑡}^𝑡𝜏𝑒^𝜎𝑇𝑠||𝑔(𝑠)||^2𝑇𝑑𝑠.(4.17) Finally, since 𝑢_𝑘⇀𝑢 in 𝐿²(𝜏,𝑇;𝑉_𝑇) as 𝑘→+∞, we get 2𝛼₁‖𝑢(𝑡)‖^𝑝𝐿_{𝑝Ω𝑇}+‖𝑢(𝑡)‖^2𝑇≤𝐶𝑒^{−𝜎𝑇(𝑡−𝜏)}1+‖‖𝑢^𝜏‖‖^2𝜏+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝜏}+1+𝑒^{−𝜎𝑇𝑡}^𝑡𝜏≤𝐶𝑒_{−𝜎𝑇(𝑡−𝜏)}𝑒^𝜎𝑇𝑠||𝑔(𝑠)||^2𝑇𝑑𝑠‖‖𝑢^𝜏‖‖^2𝜏+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝜏}+1+1−1𝑒^𝜎𝑇𝐶_𝑔,𝑇,(4.18) where we have used the fact that ^𝑡𝜏𝑒^{−𝜎𝑇(𝑡−𝑠)}||𝑔(𝑠)||^2𝑇𝑑𝑠≤≤1^𝑡𝑡−1+𝑒𝑒^{−𝜎𝑇(𝑡−𝑠)}_−𝜎𝑇||𝑔(𝑠)||^2𝑇𝑑𝑠+^{𝑡−1𝑡−2}𝑒^{−𝜎𝑇(𝑡−𝑠)}||𝑔(𝑠)||^2𝑇𝑑𝑠+⋯+𝑒^−2𝜎𝑇+⋯𝐶_𝑔,𝑇=11−𝑒^−𝜎𝑇𝐶_𝑔,𝑇.(4.19)

Let ℛ be the set of all 𝑟(𝑡) such that lim_{𝑡→−∞}𝑒^𝑡𝜎𝑡‖𝑟(𝑡)‖^2𝑡+‖𝑟(𝑡)‖^𝑝𝐿_{𝑝Ω𝑡}=0.(4.20) Denote by 𝒟 the class of all families 𝒟={𝐷(𝑡);𝐷(𝑡)∈𝑉_𝑡∩𝐿^𝑝(Ω_𝑡)),𝐷(𝑡)≠∅,𝑡∈ℝ} such that 𝐷(𝑡)⊂𝐵(𝑟(𝑡)) for some 𝑟(𝑡)∈ℛ.

For each 𝑡∈ℝ define 𝑟²⁰(𝑡)=2𝐶1+11−𝑒^−𝜎𝑡𝐶_𝑔,𝑡,(4.21) and consider the family of closed balls ℬ={𝐵(𝑡);𝑡∈ℝ}, where 𝐵(𝑡)=𝑣∈𝑉_𝑡∶‖𝑣‖_𝑡≤𝑟₀(𝑡),𝑡∈ℝ.(4.22) 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 𝑉_𝑡 into 𝐻_𝑡, it is clear that 𝐵(𝑡) is a compact set of 𝐻_𝑡 for any 𝑡∈ℝ. Then, it follows from Theorem 2.6 that the process 𝑈(⋅,⋅) has a pullback 𝒟-attractor 𝒜_𝜀={𝐴_𝜀(𝑡)∶𝑡∈ℝ} in a family of spaces {𝐻_𝑡}.

5. The Upper Semicontinuity of Pullback 𝒟-Attractors at 𝜀=0

It is proved in [4], when 𝜀=0, the existence of a pullback 𝒟-attractor 𝒜₀={𝐴₀(𝑡)∶𝑡∈ℝ} in a family of spaces {𝐻_𝑡} for problem (𝑃₀). The aim of this section is to prove the upper semicontinuity of pullback attractors 𝒜_𝜀 at 𝜀=0 in {𝐻_𝑡}, that is, lim_𝜀→0sup_𝑡∈𝐼dist_𝑡𝐴_𝜀(𝑡),𝐴₀(𝑡)=0,(5.1) where 𝐼 is an arbitrary bounded interval in ℝ.

The following lemma is the key of this section.

Lemma 5.1. For each 𝑡∈ℝ, each 𝑇>0, and each compact subset K of 𝑉_𝑡−𝑇, we have ||𝑈_𝜀(𝑡,𝜏)𝑢_𝜏−𝑈₀(𝑡,𝜏)𝑢_𝜏||^2𝑡≤𝐶√𝜀,∀𝜏∈[𝑡−𝑇,𝑡],∀𝑢_𝜏∈𝐾,(5.2) where the constant 𝐶 is independent of 𝜏 and 𝑢_𝜏.

Proof. Denote 𝑈_𝜀(𝑡,𝜏)𝑢_𝜏 by 𝑢(𝑡), and 𝑈₀(𝑡,𝜏)𝑢_𝜏 by 𝑣(𝑡). Let 𝑤(𝑡)=𝑢(𝑡)−𝑣(𝑡), we have 𝑤_𝑡−𝜀Δ𝑢_𝑡−Δ𝑤+𝑓(𝑢)−𝑓(𝑣)=0.(5.3) Multiplying this equation by 𝑤 and integrating over Ω_𝑡, we get 12𝑑𝑑𝑡|𝑤|^2𝑡−𝜀Δ𝑢_𝑡,𝑤_𝑡+‖𝑤‖^2𝑡+(𝑓(𝑢)−𝑓(𝑣),𝑤)_𝑡=0.(5.4) We have (𝑓(𝑢)−𝜀𝑓(Δ𝑣𝑢)_𝑡,𝑤)_𝑡=_Ω𝑡(𝑓(𝑢)−𝑓(𝑣))(𝑢−𝑣)≥−ℓ|𝑢−𝑣|^2𝑡,,𝑤_𝑡=−𝜀∇𝑢_𝑡,∇𝑤_𝑡≤𝜀‖‖𝑢_𝑡‖‖_𝑡⋅‖𝑤‖_𝑡.(5.5) Applying (5.5) in (5.4), we have 𝑑𝑑𝑡|𝑤|^2𝑡≤2ℓ|𝑤|^2𝑡+2𝜀‖‖𝑢_𝑡‖‖_𝑡⋅‖𝑤‖_𝑡.(5.6) Hence ||𝑤(𝑡)||^2𝑡≤2𝜀^𝑡𝜏𝑒^{2ℓ(𝑡−𝑠)}‖‖𝑢_𝑡(𝑠)‖‖_𝑡⋅‖𝑤(𝑠)‖_𝑡≤2𝜀𝑒^2ℓ𝑇^𝑡𝜏‖‖𝑢_𝑡(𝑠)‖‖^2𝑡^1/2^𝑡𝜏‖𝑤(𝑠)‖^2𝑡^1/2.(5.7) Now, we estimate the term on the right-hand side of (5.7). Multiplying the first equation in (1.3) by 𝑢_𝑡 and integrating over Ω_𝑡, we obtain ||𝑢_𝑡||^2𝑡+𝜀‖‖𝑢_𝑡‖‖^2𝑡+12𝑑𝑑𝑡‖𝑢‖^2𝑡+𝑑𝑑𝑡_Ω𝑡𝐹(𝑢)≤_Ω𝑡𝑔(𝑡)𝑢_𝑡.(5.8) Using Cauchy’s inequality, we conclude that 𝑑𝑑𝑡‖𝑢(𝑡)‖^2𝑡+2_Ω𝑡𝐹(𝑢(𝑡))+2𝜀‖‖𝑢_𝑡(𝑡)‖‖^2𝑡≤12||𝑔(𝑡)||^2𝑡.(5.9) Integrating (5.9) from 𝜏 to 𝑡, 𝜏∈[𝑡−𝑇,𝑡], we find that ‖𝑢‖^2𝑡+2_Ω𝑡𝐹(𝑢(𝑡))+2𝜀^𝑡𝜏‖‖𝑢_𝑡(𝑠)‖‖^2𝑡≤‖‖𝑢_𝜏‖‖^2𝑡+2_Ω𝑡𝐹𝑢_𝜏+12^𝑡𝜏||𝑔(𝑠)||^2𝑡.(5.10) Since ∫_Ω𝑡𝐹(𝑢(𝑡))≥−𝛽̃|Ω_𝑡|+𝛼₁‖𝑢(𝑡)‖^𝑝𝐿_𝑝(Ω𝑡), we have 2𝜀^𝑡𝜏‖‖𝑢_𝑡(𝑠)‖‖^2𝑡≤‖‖𝑢_𝜏‖‖^2𝑡+2_Ω𝑡𝐹𝑢_𝜏+12^𝑡𝜏||𝑔(𝑠)||^2𝑡+𝐶.(5.11) From (3.29), we obtain ‖‖𝑢_𝜏‖‖^2𝑡+2_Ω𝑡𝐹𝑢_𝜏≤𝐶‖‖𝑢_𝜏‖‖^2𝑡+𝐶‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑡}+𝐶.(5.12) Combining (5.10) and (5.12), we see that ^𝑡𝜏‖‖𝑢_𝑡(𝑠)‖‖^2𝑡≤𝐶𝜀1+‖‖𝑢^𝜏‖‖^2𝑡+‖‖𝑢_𝜏‖‖^𝑝𝐿_𝑝(Ω𝑡)+^𝑡𝜏||𝑔(𝑠)||^2𝑡≤𝐶𝜀1+‖‖𝑢^𝜏‖‖^2𝑡+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑡}+^{𝑡𝑡−𝑇}||𝑔(𝑠)||^2𝑡≤𝐶(𝐾,𝑇,𝑔,𝑡)𝜀,(5.13) because of 𝑢_𝜏∈𝐾 and 𝑔∈𝐿^2loc(ℝ^𝑛+1). Now, using (5.13) in (5.7), we get ||𝑤(𝑡)||^2𝑡≤𝐶(𝐾,𝑇,𝑔,𝑡)√𝜀^𝑡𝜏‖𝑤(𝑠)‖^2𝑡^1/2.(5.14) Using (4.4) and noting that 𝜏∈[𝑡−𝑇,𝑡], we have ‖𝑤(𝑡)‖^2𝑡≤‖𝑢(𝑡)‖^2𝑡+‖𝑣(𝑡)‖^2𝑡≤2𝐶𝑒^{−𝜎𝑡(𝑡−𝜏)}‖‖𝑢^𝜏‖‖^2𝑡+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑡}+1+1+𝑒^{−𝜎𝑡𝑡}^𝑡𝜏𝑒^𝜎𝑡𝑠||𝑔(𝑠)||^2𝑡≤2𝐶‖‖𝑢_𝜏‖‖^2𝑡+‖‖𝑢_𝜏‖‖^𝑝𝐿_{𝑝Ω𝑡}+2+𝑒^{−𝜎𝑡𝑡}^𝑡𝜏𝑒^𝜎𝑡𝑠||𝑔(𝑠)||^2𝑡≤𝐶(𝐾)1+_𝑡𝜏||𝑔(𝑠)||^2𝑡≤𝐶(𝐾)1+_{𝑡𝑡−𝑇}||𝑔(𝑠)||^2𝑡.(5.15) Thus, ^𝑡𝜏‖𝑤(𝑡)‖^2𝑡≤𝐶(𝐾)𝑡−𝜏+^𝑡𝜏^{𝑡𝑡−𝑇}||𝑔(𝑠)||^2𝑡≤𝐶(𝐾)𝑇+_{𝑡𝑡−𝑇}||𝑔(𝑠)||^2𝑡≤𝐶(𝐾,𝑇,𝑔,𝑡).(5.16) Combining (5.14) and (5.16) we get ||𝑤(𝑡)||^2𝑡≤𝐶(𝐾,𝑇,𝑔,𝑡)√𝜀.(5.17) The proof is complete.

Theorem 5.2. If 𝑔∈𝐿^2loc(ℝ^𝑁+1) satisfies (4.3), then for any bounded interval 𝐼∈ℝ, the family of pullback 𝒟-attractors {𝐴_𝜀(⋅)∶𝜀∈[0,1]} is upper semicontinuous in 𝐿²(Ω_𝑡) at 0 for any 𝑡∈𝐼, that is, lim_𝜀→0sup_𝑡∈𝐼dist_𝐿²_(Ω𝑡)𝐴_𝜀(𝑡),𝐴₀(𝑡)=0.(5.18)

Proof. We will verify the conditions (i)–(iii) in Theorem 2.8. First, condition (i) follows directly from Lemma 5.1.
Let 𝐵(⋅)=𝐵(𝑟₀(⋅)) be the corresponding family of pullback 𝒟-absorbing sets of (1.3), which is uniform with respect to 𝜀. By the definition of pullback 𝒟-absorbing sets, for any 𝑡∈ℝ, there exists 𝜏₀=𝜏₀(𝑡)≤𝑡 such that _𝜏_≤𝜏0𝑈_𝜀(𝑡,𝜏)𝐵(𝜏)⊂𝐵(𝑡)=𝐵𝑟₀(𝑡).(5.19) By Theorem 2.6, we see that 𝐴_𝜀(𝑡)=_𝑠≤𝑡_𝜏≤𝑠𝑈_𝜀(𝑡,𝜏)𝐵(𝜏).(5.20) From (5.19) and (5.20), we get 𝐴_𝜀(𝑡)⊂𝐵𝑟₀(𝑡).(5.21) Now, for given 𝑡₀∈ℝ, we can write _[]_𝜀∈0,1_𝑡≤𝑡0𝐴_𝜀(𝑡)⊂_𝑡≤𝑡0𝐵𝑟₀(𝑡).(5.22) Because lim_{𝑡→−∞}sup𝑟₀(𝑡)<+∞ due to (4.4), from (5.22) we have _[]_𝜀∈0,1_𝑡≤𝑡0𝐴_𝜀(𝑡)isboundedin𝐿²Ω_𝑡0forgiven𝑡₀,(5.23) that is, condition (ii) is satisfied. From (5.21), we can find that, for each 𝑡∈ℝ, _0<_𝜀≤1𝐴_𝜀(𝑡)⊂𝐵𝑟₀(𝑡),(5.24) thus ⋃_0<𝜀≤1𝐴_𝜀(𝑡) is bounded in 𝐻¹⁰(Ω_𝑡), hence ₀_<𝜀≤1𝐴_𝜀(𝑡)iscompactin𝐿²Ω_𝑡,(5.25) since 𝐻¹⁰(Ω_𝑡)⊂𝐿²(Ω_𝑡) 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 attractors 𝒜_𝜀 of the process generated by this class of solutions and the upper semicontinuity of {𝒜_𝜀∶𝜀∈[0,1]} at 𝜀=0, which means that the pullback attractors 𝒜_𝜀 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 Ω_𝑡 in ℝ^𝑁 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 𝜀=0).

It is noticed that the obtained results seem to be not very satisfying because although the process 𝑈(⋅,⋅) associated to problem (1.3) is constructed in the family of spaces 𝐻¹⁰(Ω_𝑡)∩𝐿^𝑝(Ω_𝑡), we are only able to prove the existence and upper semicontinuity of the pullback attractor in 𝐿²(Ω_𝑡). It would be very interesting if one can show the existence and upper semicontinuity of the pullback attractor in the space 𝐻¹⁰(Ω_𝑡)∩𝐿^𝑝(Ω_𝑡). 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.

Acknowledgment

The authors would like to thank the anonymous referee for the helpful comments and suggestions which improved the presentation of the paper.