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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012Β (2012), Article IDΒ 875913, 30 pages
http://dx.doi.org/10.1155/2012/875913
Research Article

Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains

1Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam

Received 26 March 2012; Accepted 20 May 2012

Academic Editor: Ram U.Β Verma

Copyright Β© 2012 Cung The Anh and Nguyen Duong Toan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation π‘’π‘‘βˆ’πœ€Ξ”π‘’π‘‘βˆ’Ξ”π‘’+𝑓(𝑒)=𝑔(π‘₯,𝑑),πœ€βˆˆ(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 ξπ’œπœ€, which is upper semicontinuous at πœ€=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 {Ω𝑑}π‘‘βˆˆβ„ be a family of nonempty bounded open subsets of ℝ𝑁 such that 𝑠<π‘‘βŸΉΞ©π‘ βŠ‚Ξ©π‘‘.(1.1) From now on, we will frequently use the following notations: π‘„πœ,π‘‡ξšβˆΆ=π‘‘βˆˆ(𝜏,𝑇)Ω𝑑𝑄×{𝑑},𝜏∢=π‘‘βˆˆ(𝜏,∞)Ω𝑑×{𝑑},βˆ€πœβˆˆβ„,𝜏,π‘‡ξšβˆΆ=π‘‘βˆˆ(𝜏,𝑇)πœ•Ξ©π‘‘ξ“Γ—{𝑑},𝜏∢=π‘‘βˆˆ(𝜏,∞)πœ•Ξ©π‘‘Γ—{𝑑}.(1.2) In this paper we consider the following nonautonomous equation: ξ€·π‘ƒπœ–ξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©π‘’π‘‘βˆ’πœ€Ξ”π‘’π‘‘βˆ’Ξ”π‘’+𝑓(𝑒)=𝑔(π‘₯,𝑑)inπ‘„πœ,βˆ‘π‘’=0on𝜏,𝑒||𝑑=𝜏=π‘’πœ(π‘₯),π‘₯∈Ω𝜏,(1.3) where πœ€βˆˆ[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 πœ€>0, and when πœ€=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 𝑑 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 𝑑. It is noticed that this question for problem (1.3) in the case πœ€=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 π‘“βˆˆπΆ1(ℝ,ℝ) satisfies that βˆ’π›½+𝛼1|𝑠|𝑝≀𝑓(𝑠)𝑠≀𝛽+𝛼2|𝑠|𝑝,𝑓(1.4)ξ…ž(𝑠)β‰₯βˆ’β„“,(1.5) for some 𝑝β‰₯2, where 𝛼1,𝛼2,𝛽,β„“ are nonnegative constants. By (1.4), there exist nonnegative constants 𝛼1,𝛼2,̃𝛽 such that βˆ’Μƒπ›½+𝛼1|𝑠|𝑝̃≀𝐹(𝑠)≀𝛽+𝛼2|𝑠|𝑝,(1.6) where ∫𝐹(𝑒)=𝑒0𝑓(𝑠)𝑑𝑠 is the primitive of 𝑓. (H2) The external force π‘”βˆˆπΏ2loc(ℝ𝑁+1).(H3) The initial datum π‘’πœβˆˆπ»10(Ω𝜏)βˆ©πΏπ‘(Ω𝜏) is given.

Since the open set Ω𝑑 changes with time 𝑑, 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 πœ€=0, that is, for the classical reaction-diffusion equation, has been derived recently in [4]. In the case πœ€>0, since (1.3) contains the term βˆ’πœ€Ξ”π‘’π‘‘, this is essentially different from the classical reaction-diffusion equation. For example, the reaction-diffusion equation has some kind of β€œregularity”; 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 πœ€>0, because of βˆ’Ξ”π‘’π‘‘, if the initial datum π‘’πœ belongs to 𝐻10(Ω𝜏)βˆ©πΏπ‘(Ω𝜏), the solution 𝑒(𝑑) with the initial condition 𝑒(𝜏)=π‘’πœ is always in 𝐻10(Ω𝑑)βˆ©πΏπ‘(Ω𝑑) 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 ξπ’œπœ€ for the process associated to problem (1.3). Finally, we study the continuous dependence on πœ€ 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 πœ€β†’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 ξπ’œπœ€ in 𝐿2(Ω𝑑) at πœ€=0. The last result means that the pullback attractors ξπ’œπœ€ of the nonclassical diffusion equations converge to the pullback attractor ξπ’œ0 of the classical reaction-diffusion equations as πœ€β†’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 ξπ’œπœ€ in an appropriate framework. The upper semicontinuity of pullback attractors ξπ’œπœ€ at πœ€=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 π»π‘ŸβˆΆ=𝐿2(Ξ©π‘Ÿ) and π‘‰π‘ŸβˆΆ=𝐻10(Ξ©π‘Ÿ) for each π‘Ÿβˆˆβ„, and denote by (β‹…,β‹…)π‘Ÿ and |β‹…|π‘Ÿ the usual inner product and associated norm in π»π‘Ÿ and by ((β‹…,β‹…)) and β€–β‹…β€–π‘Ÿ the usual gradient inner product and associated norm in π‘‰π‘Ÿ. For each 𝑠<𝑑, consider 𝑉𝑠 as a closed subspace of 𝑉𝑑 with the functions belonging to 𝑉𝑠 being trivially extended by zero. It follows from (1.1) that {𝑉𝑑}π‘‘βˆˆ[𝜏,𝑇] can be considered as a family of closed subspaces of 𝑉𝑇 for each 𝑇>𝜏 with 𝑠<π‘‘βŸΉπ‘‰π‘ βŠ‚π‘‰π‘‘.(1.7) In addition, π»π‘Ÿ will be identified with its topological dual π»βˆ—π‘Ÿ by means of the Riesz theorem and π‘‰π‘Ÿ will be considered as a subspace of π»βˆ—π‘Ÿ with π‘£βˆˆπ‘‰π‘Ÿ identified with the element π‘“π‘£βˆˆπ»βˆ—π‘Ÿ defined by 𝑓𝑣(β„Ž)=(𝑣,β„Ž)π‘Ÿ,β„Žβˆˆπ»π‘Ÿ.(1.8) The duality product between π‘‰βˆ—π‘Ÿ and π‘‰π‘Ÿ will be denoted by βŸ¨β‹…,β‹…βŸ©.

2. Preliminaries

2.1. Penalty Method

To study problem (1.3), for each 𝑇>𝜏, we consider the following auxiliary problem: π‘’π‘‘βˆ’πœ€Ξ”π‘’π‘‘βˆ’Ξ”π‘’+𝑓(𝑒)=𝑔(π‘₯,𝑑)inπ‘„πœ,𝑇,𝑒=0on𝜏,𝑇,𝑒||𝑑=𝜏=π‘’πœ(π‘₯),π‘₯∈Ω𝜏,(2.1) where πœβˆˆβ„, π‘’πœβˆΆΞ©πœβ†’β„ and π‘”βˆΆπ‘„πœβ†’β„ 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 π‘‘βˆˆ[𝜏,𝑇] denote by π‘‰βŸ‚π‘‘ξ€½βˆΆ=π‘£βˆˆπ‘‰π‘‡βˆΆ((𝑣,πœ”))𝑇=0,βˆ€πœ”βˆˆπ‘‰π‘‘ξ€Ύ(2.2) the orthogonal subspace of 𝑉𝑑 with respect the inner product in 𝑉𝑇 and by 𝑃(𝑑)βˆˆβ„’(𝑉𝑇) the orthogonal projection operator from 𝑉𝑇 onto π‘‰βŸ‚π‘‡, which is defined as 𝑃(𝑑)π‘£βˆˆπ‘‰βŸ‚π‘‘,π‘£βˆ’π‘ƒ(𝑑)π‘£βˆˆπ‘‰π‘‘,(2.3) for each π‘£βˆˆπ‘‰π‘‡. Finally, define 𝑃(𝑑)=𝑃(𝑇) for all 𝑑>𝑇 and observe that 𝑃(𝑇) is the zero of β„’(𝑉𝑇).

We will now approximate 𝑃(𝑑) by operators which are more regular in time. Consider the family 𝑝(𝑑;β‹…,β‹…) of symmetric bilinear forms on 𝑉𝑇 defined by 𝑝(𝑑;𝑣,πœ”)∢=((𝑃(𝑑)𝑣,πœ”))𝑇,βˆ€π‘£,πœ”βˆˆπ‘‰π‘‡,βˆ€π‘‘β‰₯𝜏.(2.4) It can be proved that the mapping [𝜏,∞)βˆ‹π‘‘β†¦π‘(𝑑;𝑣,πœ”)βˆˆβ„ is measurable for all 𝑣,πœ”βˆˆπ‘‰π‘‡. Moreover, |𝑝(𝑑;𝑣,πœ”)|β‰€β€–π‘£β€–π‘‡β€–πœ”β€–π‘‡. For each integer π‘˜β‰₯1 and each 𝑑β‰₯𝜏, we define π‘π‘˜ξ€œ(𝑑;𝑣,πœ”)∢=π‘˜01/π‘˜π‘(𝑑+π‘Ÿ;𝑣,πœ”)π‘‘π‘Ÿ,βˆ€π‘£,πœ”βˆˆπ‘‰π‘‡,βˆ€π‘‘β‰₯𝜏,(2.5) and denote by π‘ƒπ‘˜(𝑑)βˆˆβ„’(𝑉𝑇) the associated operator defined by π‘ƒξ€·ξ€·π‘˜(𝑑)𝑣,πœ”ξ€Έξ€Έπ‘‡βˆΆ=π‘π‘˜(𝑑;𝑣,πœ”),βˆ€π‘£,πœ”βˆˆπ‘‰π‘‡,βˆ€π‘‘β‰₯𝜏.(2.6)

Lemma 2.1 (see [2, 4]). For any integer 1β‰€β„Žβ‰€π‘˜, any 𝑑β‰₯𝜏 and every 𝑣,πœ”βˆˆπ‘‰π‘‡, π‘π‘˜(𝑑;𝑣,πœ”)=π‘π‘˜(𝑑;πœ”,𝑣),0β‰€π‘β„Ž(𝑑;𝑣,𝑣)β‰€π‘π‘˜(𝑑;𝑣,𝑣)≀𝑝(𝑑;𝑣,𝑣)=‖𝑃(𝑑)𝑣‖2𝑇≀‖𝑣‖2𝑇,π‘ξ…žπ‘˜π‘‘(𝑑;𝑣,𝑣)∢=π‘π‘‘π‘‘π‘˜ξ‚€π‘ξ‚€1(𝑑;𝑣,𝑣)=π‘˜π‘‘+π‘˜ξ‚ξ‚π‘ƒ;𝑣,π‘£βˆ’π‘(𝑑;𝑣,𝑣)≀0,ξ€·ξ€·π‘˜(𝑑)𝑣,𝑧𝑇=0,βˆ€π‘§βˆˆπ‘‰π‘‘.(2.7) Moreover,   for every sequence {π‘£π‘˜}βŠ‚πΏ2(𝜏,𝑇;𝑉𝑇) weak convergent to 𝑣 in 𝐿2(𝜏,𝑇;𝑉𝑇), liminfπ‘˜β†’+βˆžξ€œπ‘‡πœπ‘π‘˜ξ€·π‘‘;π‘£π‘˜(𝑑),π‘£π‘˜ξ€Έξ€œ(𝑑)𝑑𝑑β‰₯π‘‡πœπ‘(𝑑;𝑣(𝑑),𝑣(𝑑))𝑑𝑑.(2.8)

Let π½βˆΆπ‘‰π‘‡β†’π‘‰βˆ—π‘‡ be the Riesz isomorphism defined by βŸ¨π½π‘£,πœ”βŸ©π‘‡βˆΆ=((𝑣,πœ”))𝑇,βˆ€π‘£,πœ”βˆˆπ‘‰π‘‡,(2.9) and for each integer π‘˜β‰₯1 and each π‘‘βˆˆ[𝜏,𝑇], we denote π΄π‘˜(𝑑)∢=βˆ’Ξ”+π‘˜π½π‘ƒπ‘˜(𝑑).(2.10) Obviously, π΄π‘˜(𝑑)βˆˆβ„’(𝑉𝑇,π‘‰βˆ—π‘‡),π‘‘βˆˆ[𝜏,𝑇], is a family of symmetric linear operators such that the mapping π‘‘βˆˆ[𝜏,𝑇]β†¦π΄π‘˜(𝑑)βˆˆβ„’(𝑉𝑇,π‘‰βˆ—π‘‡) is measurable, bounded, and satisfies βŸ¨π΄π‘˜(𝑑)𝑣,π‘£βŸ©π‘‡β‰₯‖𝑣‖2𝑇,βˆ€π‘£βˆˆπ‘‰π‘‡[].,βˆ€π‘‘βˆˆπœ,𝑇(2.11) Let π‘’πœβˆˆπ‘‰π‘‡ be given and for each π‘˜β‰₯1 consider the following problem: ξ€·π‘’ξ…žπ‘˜ξ€Έ(𝑑),𝑣𝑇+βŸ¨π΄π‘˜(𝑑)π‘’π‘˜(𝑑),π‘£βŸ©π‘‡ξ«π΄+πœ€π‘˜(𝑑)π‘’ξ…žπ‘˜ξ¬(𝑑),𝑣𝑇+ξ€·π‘“ξ€·π‘’π‘˜ξ€Έξ€Έ(𝑑),𝑣𝑇=(𝑔(𝑑),𝑣)𝑇,βˆ€π‘£βˆˆπ‘‰π‘‡,π‘’ξ€·ξ€·π‘˜(𝜏),𝑣𝑇=π‘’ξ€·ξ€·πœ,𝑣𝑇.(2.12)

The idea of the penalty method is as follows: for each π‘˜β‰₯1 we first prove the existence of a solution π‘’π‘˜ to problem (2.12) (a problem in a cylindrical domain) using standard methods such as the Galerkin method, and then show that π‘’π‘˜ 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 Ω𝑑 changes with time 𝑑, 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 π’Ÿ-attractors which is a modification of the theory in [16].

Consider a process π‘ˆ(β‹…,β‹…) on a family of metric spaces {(𝑋𝑑,𝑑𝑑);π‘‘βˆˆβ„}, that is, a family {π‘ˆ(𝑑,𝜏);βˆ’βˆž<πœβ‰€π‘‘<+∞} of mappings π‘ˆ(𝑑,𝜏)βˆΆπ‘‹πœβ†’π‘‹π‘‘ such that π‘ˆ(𝜏,𝜏)π‘₯=π‘₯ for all π‘₯βˆˆπ‘‹πœ and π‘ˆ(𝑑,𝜏)=π‘ˆ(𝑑,π‘Ÿ)π‘ˆ(π‘Ÿ,𝜏)βˆ€πœβ‰€π‘Ÿβ‰€π‘‘.(2.13) In addition, suppose π’Ÿ is a nonempty class of parameterized sets of the form ξπ’Ÿ={𝐷(𝑑);𝐷(𝑑)βŠ‚π‘‹π‘‘,𝐷(𝑑)β‰ βˆ…,π‘‘βˆˆβ„}.

Definition 2.2 (see [4]). The process π‘ˆ(β‹…,β‹…) is said to be pullback π’Ÿ-asymptotically compact if the sequence {π‘ˆ(𝑑,πœπ‘›)π‘₯𝑛} is relatively compact in 𝑋𝑑 for any π‘‘βˆˆβ„, any ξπ’Ÿβˆˆπ’Ÿ, and any sequences {πœπ‘›} and {π‘₯𝑛} with πœπ‘›β†’βˆ’βˆž and π‘₯π‘›βˆˆπ·(πœπ‘›).

Definition 2.3 (see [4]). A family ξβ„¬βˆˆπ’Ÿ is said to be pullback π’Ÿ-absorbing for the process π‘ˆ(β‹…,β‹…) if for any π‘‘βˆˆβ„ and any ξπ’Ÿβˆˆπ’Ÿ, there exists 𝜏0(𝑑,π’Ÿ)≀𝑑 such that π‘ˆ(𝑑,𝜏)π’Ÿ(𝜏)βŠ‚β„¬(𝑑),(2.14) for all πœβ‰€πœ0(𝑑,π’Ÿ).

Remark 2.4. Note that if ξβ„¬βˆˆπ’Ÿ is pullback π’Ÿ-absorbing for the process π‘ˆ(β‹…,β‹…) and 𝐡(𝑑) is a compact subset of 𝑋𝑑 for any π‘‘βˆˆβ„, then the process π‘ˆ(β‹…,β‹…) is pullback π’Ÿ-asymptotically compact.

For each π‘‘βˆˆβ„, let dist𝑑(𝐷1,𝐷2) be the Hausdorff semi-distance between nonempty subsets 𝐷1 and 𝐷2 of 𝑋𝑑, which is defined as dist𝑑𝐷1,𝐷2ξ€Έ=supπ‘₯∈𝐷1infπ‘¦βˆˆπ·2𝑑𝑋𝑑(π‘₯,𝑦)for𝐷1,𝐷2βŠ‚π‘‹π‘‘.(2.15)

Definition 2.5 (see [4]). The family ξπ’œ={𝐴(𝑑);𝐴(𝑑)βŠ‚π‘‹π‘‘,𝐴(𝑑)β‰ βˆ…,π‘‘βˆˆβ„} is said to be a pullback π’Ÿ-attractor for π‘ˆ(β‹…,β‹…) if (1)π‘Ž(𝑑) is a compact set of 𝑋𝑑 for all π‘‘βˆˆβ„, (2)ξπ’œ is pullback π’Ÿ-attracting, that is, limπœβ†’βˆ’βˆždist𝑑(π‘ˆ(𝑑,𝜏)𝐷(𝜏),𝐴(𝑑))=0βˆ€π’Ÿβˆˆπ’Ÿ,βˆ€π‘‘βˆˆβ„,(2.16)(3)ξπ’œ is invariant, that is, π‘ˆ(𝑑,𝜏)𝐴(𝜏)=𝐴(𝑑)forβˆ’βˆž<πœβ‰€π‘‘<+∞.(2.17)

Theorem 2.6 (see [4]). Suppose that the process π‘ˆ(β‹…,β‹…) is pullback π’Ÿ-asymptotically compact and that ξπ΅βˆˆπ’Ÿ is a family of pullback π’Ÿ-absorbing sets for π‘ˆ(β‹…,β‹…). Then, the family ξπ’œ={𝐴(𝑑);π‘‘βˆˆβ„} defined by 𝐴(𝑑)∢=Ξ›(𝐡,𝑑),π‘‘βˆˆβ„, where for each ξπ·βˆˆπ’Ÿ and π‘‘βˆˆβ„, Λ𝐷,π‘‘βˆΆ=π‘ β‰€π‘‘ξšπœβ‰€π‘ π‘ˆ(𝑑,𝜏)𝐷(𝜏)𝑋𝑑closurein𝑋𝑑(2.18) is a pullback π’Ÿ-attractor for π‘ˆ(β‹…,β‹…), which in addition satisfies 𝐴(𝑑)=ξšξπ·βˆˆπ’ŸΞ›ξ‚€ξξ‚π·,𝑑𝑋𝑑.(2.19) Furthermore, ξπ’œ is minimal in the sense that if 𝐢={𝐢(𝑑);π‘‘βˆˆβ„} is a family of nonempty sets such that 𝐢(𝑑) is a closed subset of 𝑋𝑑 and limπœβ†’βˆ’βˆždist𝑑(π‘ˆ(𝑑,𝜏)𝐡(𝜏),𝐢(𝑑))=0foranyπ‘‘βˆˆβ„,then𝐴(𝑑)βŠ‚πΆ(𝑑)βˆ€π‘‘βˆˆβ„.(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 {π‘ˆπœ€(β‹…,β‹…)βˆΆπœ€βˆˆ[0,1]} be a family of evolution processes in a family of Banach spaces {𝑋𝑑} with corresponding pullback π’Ÿ-attractors {π΄πœ€(𝑑)βˆΆπœ€βˆˆ[0,1],π‘‘βˆˆβ„}. For any bounded interval πΌβˆˆβ„, we say that {π΄πœ€(β‹…)} is upper semicontinuous at πœ€=0 for π‘‘βˆˆπΌ if limsupπœ€β†’0π‘‘βˆˆπΌdistπ‘‘ξ€·π΄πœ€(𝑑),𝐴0ξ€Έ(𝑑)=0.(2.21)

Theorem 2.8. Let {π‘ˆπœ€(β‹…,β‹…)βˆΆπœ€βˆˆ[0,1]} be a family of processes with corresponding pullback π’Ÿ-attractors {π΄πœ€(β‹…)βˆΆπœ€βˆˆ[0,1]}. Then, for any bounded interval πΌβŠ‚β„, {π΄πœ€(β‹…)βˆΆπœ€βˆˆ[0,1]} is upper semicontinuous at 0 for π‘‘βˆˆπΌ if for each π‘‘βˆˆβ„, for each 𝑇>0, and for each compact subset 𝐾 of π‘‹π‘‘βˆ’πœ, the following conditions hold: (i)sup𝜏∈[0,𝑇]supπ‘₯∈𝐾dist𝑑(π‘ˆπœ€(𝑑,π‘‘βˆ’πœ)π‘₯,π‘ˆ0(𝑑,π‘‘βˆ’πœ)π‘₯)β†’0asπœ€β†’0,(ii)β‹ƒπœ€βˆˆ[0,1]⋃𝑑≀𝑑0π΄πœ€(𝑑)isboundedforgiven𝑑0,(iii)⋃0<πœ€β‰€1π΄πœ€(𝑑)iscompactforeachπ‘‘βˆˆβ„.

3. Existence and Uniqueness of Variational Solutions

For each 𝑇>𝜏, denote ξ‚π‘„πœ,π‘‡βˆΆ=Ξ©π‘‡π‘ˆΓ—(𝜏,𝑇),𝜏,π‘‡ξ‚†βˆΆ=Ξ¦βˆˆπΏβˆžξ€·πœ,𝑇;π‘‰π‘‡ξ€Έβˆ©πΏπ‘ξ‚€ξ‚π‘„πœ,𝑇,Ξ¦β€²βˆˆπΏ2ξ€·πœ,𝑇;𝑉𝑇,Ξ¦(𝜏)=Ξ¦(𝑇)=0,Ξ¦(𝑑)βˆˆπ‘‰π‘‘ξ‚‡.a.e.in(𝜏,𝑇)(3.1)

Definition 3.1. A variational solution of (2.1) is a function 𝑒 such that (C1)π‘’βˆˆπΏβˆž(𝜏,𝑇,𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇), π‘’ξ…žβˆˆπΏ2(𝜏,𝑇;𝑉𝑇),(C2)for all Ξ¦βˆˆπ‘ˆπœ,𝑇, ξ€œπ‘‡πœξ€Ίβˆ’ξ€·π‘’(𝑑),Ξ¦ξ…žξ€Έ(𝑑)𝑇+((𝑒(𝑑),Ξ¦(𝑑)))𝑇𝑒+πœ€ξ€·ξ€·ξ…ž(𝑑),Ξ¦(𝑑)𝑇+(𝑓(𝑒),Ξ¦(𝑑))𝑇=ξ€œπ‘‘π‘‘π‘‡πœ(𝑔(𝑑),Ξ¦(𝑑))𝑇𝑑𝑑,(3.2)(C3)𝑒(𝑑)βˆˆπ‘‰π‘‘ a.e. in (𝜏,𝑇), (C4)limπ‘‘β†“πœ(π‘‘βˆ’πœ)βˆ’1βˆ«π‘‘πœ|𝑒(π‘Ÿ)βˆ’π‘’πœ|2π‘‡π‘‘π‘Ÿ=0.

Remark 3.2. If 𝑇2>𝑇1>𝜏 and 𝑒 is a variational solution of (2.1) with 𝑇=𝑇2, then the restriction of 𝑒 to ξ‚π‘„πœ,𝑇1 is a variational solution of (2.1) with 𝑇=𝑇1.

Denote ξ‚π‘„πœβˆΆ=U𝑇>πœξ‚π‘„(𝜏,𝑇).

Definition 3.3. A variational solution of (1.3) is a function ξ‚π‘„π‘’βˆΆπœβ†’β„ such that for each 𝑇>𝜏, its restriction to ξ‚π‘„πœ,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 π‘£βˆˆπΏ2(𝜏,𝑇,𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇) and there exist πœ‰βˆˆπΏ2(𝜏,𝑇,π‘‰βˆ—π‘‡) and πœ‚βˆˆπΏπ‘/π‘βˆ’1(ξ‚π‘„πœ,𝑇) such that ξ€œπ‘‡πœξ€·π‘£(𝑑),Ξ¦ξ…žξ€Έ(𝑑)π‘‡ξ€œπ‘‘π‘‘=βˆ’π‘‡πœβŸ¨πœ‰(𝑑),Ξ¦(𝑑)βŸ©π‘‡ξ€œπ‘‘π‘‘βˆ’π‘‡πœ(πœ‚(𝑑),Ξ¦(𝑑))𝑇𝑑𝑑,(3.3) for every function Ξ¦βˆˆπ‘ˆπœ,𝑇.
For each 0<β„Ž<π‘‡βˆ’πœ, define π‘£β„Ž by π‘£β„Žξ‚»β„ŽβˆΆ=βˆ’1(𝑣(𝑑+β„Ž)βˆ’π‘£(𝑑))a.e.in(𝜏,π‘‡βˆ’β„Ž),0a.e.in(π‘‡βˆ’β„Ž,𝑇).(3.4) Then limβ„Žβ†“0ξ€œπ‘‡πœξ€·π‘£β„Žξ€Έ(𝑑),πœ”(𝑑)π‘‡ξ€œπ‘‘π‘‘=π‘‡πœβŸ¨πœ‰(𝑑),πœ”(𝑑)βŸ©π‘‡ξ€œπ‘‘π‘‘+π‘‡πœ(πœ‚(𝑑),πœ”(𝑑))𝑇𝑑𝑑,(3.5) for every function πœ”βˆˆπΏ2(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇) such that πœ”(𝑑)βˆˆπ‘‰π‘‘ a.e. in (𝜏,𝑇).

Remark 3.5. If 𝜏<𝑇′<𝑇 and Φ∈𝐿2(𝜏,𝑇′;𝑉𝑇)βˆ©πΏπ‘(Ω𝑇×(𝜏,𝑇′)), with Ξ¦β€²βˆˆπΏ2(𝜏,𝑇′;𝐻𝑇) satisfies Ξ¦(𝜏)=Ξ¦(𝑇′)=0 and Ξ¦(𝑑)βˆˆπ‘‰π‘‘ a.e. in (𝜏,𝑇′), then the trivial extension Φ of Ξ¦ satisfies ξ‚Ξ¦βˆˆπ‘ˆπœ,𝑇, with (Φ)ξ…ž=ξ‚Ξ¦ξ…ž. Using the open sets ξΞ©π‘‘βˆΆ=Ω𝑑+π‘‡βˆ’π‘‡ξ…ž,πœβ‰€π‘‘β‰€π‘‡β€², it is easy to see that under the conditions of (3.5), one also has limβ„Žβ†“0ξ€œπ‘‡β€²πœβˆ’β„Žξ€·π‘£β„Žξ€Έ(𝑑),πœ”(𝑑)π‘‡ξ€œπ‘‘π‘‘=π‘‡β€²πœβŸ¨πœ‰(𝑑),πœ”(𝑑)βŸ©π‘‡ξ€œπ‘‘π‘‘+π‘‡β€²πœ(πœ‚(𝑑),πœ”(𝑑))𝑇𝑑𝑑,(3.6) for every πœβ‰€π‘‡β€²β‰€π‘‡ and every function πœ”βˆˆπΏ2(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇) such that πœ”(𝑑)βˆˆπ‘‰π‘‘ a.e. in (𝜏,𝑇).

Lemma 3.6 (see [4]). Let π‘£π‘–βˆˆπΏ2(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇),𝑖=1,2, be two functions such that 𝑣𝑖(𝑑)βˆˆπ‘‰π‘‘ a.e. in (𝜏,𝑇) for 𝑖=1,2. Assume that there exist πœ‰π‘–βˆˆπΏ2(𝜏,𝑇;π‘‰βˆ—π‘‡),πœ‚π‘–βˆˆπΏπ‘/π‘βˆ’1(ξ‚π‘„πœ,𝑇),𝑖=1,2 such that ξ€œπ‘‡πœξ€·π‘£π‘–(𝑑),Ξ¦ξ…žξ€Έ(𝑑)π‘‡ξ€œπ‘‘π‘‘=βˆ’π‘‡πœβŸ¨πœ‰π‘–(𝑑),Ξ¦(𝑑)βŸ©π‘‡ξ€œπ‘‘π‘‘βˆ’π‘‡πœξ€·πœ‚π‘–ξ€Έ(𝑑),Ξ¦(𝑑)𝑇𝑑𝑑,𝑖=1,2,(3.7) for every function Ξ¦βˆˆπ‘ˆπœ,𝑇. Then,   for every pair πœβ‰€π‘ <𝑑≀𝑇 of Lebesgue points of the inner product function (𝑣1,𝑣2)𝑇 it holds 𝑣1(𝑑),𝑣2(𝑑)π‘‡βˆ’ξ€·π‘£1(𝑠),𝑣2(𝑠)𝑇=ξ€œπ‘‘π‘ βŸ¨πœ‰1(π‘Ÿ),𝑣2(π‘Ÿ)βŸ©π‘‡ξ€œπ‘‘π‘Ÿ+π‘‘π‘ βŸ¨πœ‰2(π‘Ÿ),𝑣1(π‘Ÿ)βŸ©π‘‡+ξ€œπ‘‘π‘Ÿπ‘‘π‘ ξ€·πœ‚1(π‘Ÿ),𝑣2(ξ€Έπ‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿ+π‘‘π‘ ξ€·πœ‚2(π‘Ÿ),𝑣1(ξ€Έπ‘Ÿ)π‘‡π‘‘π‘Ÿ+limβ„Žβ†“0β„Žβˆ’1ξ€œπ‘ π‘‘βˆ’β„Žξ€·π‘£1(π‘Ÿ+β„Ž)βˆ’π‘£1(π‘Ÿ),𝑣2(π‘Ÿ+β„Ž)βˆ’π‘£2ξ€Έ(π‘Ÿ)π‘‡π‘‘π‘Ÿ.(3.8)

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

Corollary 3.7. If 𝑒 is a variational solution of (2.1), then for every Lebesgue point π‘‘βˆˆ(𝜏,𝑇) of |𝑒|2𝑇 it holds ||||𝑒(𝑑)2𝑇+πœ€β€–π‘’(𝑑)β€–2π‘‡ξ€œ+2π‘‘πœ(β€–π‘’π‘Ÿ)β€–2π‘‡ξ€œπ‘‘r+2π‘‘πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))𝑇=||π‘’π‘‘π‘Ÿπœ||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œ+2π‘‘πœ(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ+limβ„Žβ†“0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Ž||||𝑒(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ)2π‘‡π‘‘π‘Ÿ.(3.9)

Proof. If 𝑒 is a variational solution of (2.1), then we have ξ€œπ‘‡πœξ‚ƒβˆ’ξ€·π‘’(𝑑),Ξ¦ξ…žξ€Έ(𝑑)𝑇+πœ€ξ‚€ξ‚€πœ•π‘’πœ•π‘‘,Ξ¦(𝑑)𝑇+((𝑒(𝑑),Ξ¦(𝑑)))𝑇+(𝑓(𝑒(𝑑)),Ξ¦(𝑑))𝑇=ξ€œπ‘‘π‘‘π‘‡πœ(𝑔(𝑑),Ξ¦(𝑑))𝑇𝑑𝑑.(3.10) Applying Lemma 3.6 with 𝑣1=𝑣2=𝑒, we get ||||𝑒(𝑑)2π‘‡βˆ’||π‘’πœ||2π‘‡ξ€œ=βˆ’π‘‘πœπœ€ξ‚€ξ‚€πœ•π‘’πœ•π‘‘+𝑒(π‘Ÿ),𝑒(π‘Ÿ)ξ‚ξ‚π‘‡ξ€œπ‘‘π‘Ÿβˆ’π‘‘πœπœ€ξ‚€ξ‚€πœ•π‘’πœ•π‘‘+𝑒(π‘Ÿ),𝑒(π‘Ÿ)ξ‚ξ‚π‘‡βˆ’ξ€œπ‘‘π‘Ÿπ‘‘πœ(𝑓(𝑒(π‘Ÿ))βˆ’π‘”(π‘Ÿ),𝑒(π‘Ÿ))π‘‡ξ€œπ‘‘π‘Ÿβˆ’π‘‘πœ(𝑓(𝑒(π‘Ÿ))βˆ’π‘”(π‘Ÿ),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ+limβ„Žβ†’0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Ž||||𝑒(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ=βˆ’2π‘‘πœβ€–π‘’(π‘Ÿ)β€–2π‘‡ξ€œπ‘‘π‘Ÿβˆ’2π‘‘πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€œπ‘‘π‘Ÿ+2π‘‘πœ(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿβˆ’πœ€β€–π‘’(𝑑)β€–2𝑇‖‖𝑒+πœ€πœβ€–β€–2𝑇+limβ„Žβ†’0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Ž||||𝑒(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ)2π‘‡π‘‘π‘Ÿ.(3.11) Hence, it implies the desired result.

The aim of this section is to obtain a variational solution of (2.1) such that ||||𝑒(𝑑)2𝑇+πœ€β€–π‘’(𝑑)β€–2π‘‡ξ€œ+2π‘‘πœ(β€–π‘’π‘Ÿ)β€–2π‘‡ξ€œπ‘‘π‘Ÿ+2π‘‘πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))𝑇=||π‘’π‘‘π‘Ÿπœ||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œ+2π‘‘πœ(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ.(3.12) We will say that 𝑒 satisfies the energy equality in (𝜏,𝑇) if (3.12) is satisfied a.e. in (𝜏,𝑇). Analogously, if 𝑒 is a variational solution of (1.3), we will say that 𝑒 satisfies the energy equality a.e. in (𝜏,+∞) if for each 𝑇>𝜏 the restriction of 𝑒 to ξ‚π‘„πœ,𝑇 satisfies the energy equality (3.12) a.e. in (𝜏,𝑇).

For any function π‘£βˆˆπΏ2(𝜏,𝑇;𝐻𝑇) and any π‘‘βˆˆ(𝜏,𝑇], we put πœ‚π‘£,𝑇(𝑑)∢=limsupβ„Žβ†“0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Ž||𝑣||(π‘Ÿ+β„Ž)βˆ’π‘£(π‘Ÿ)2π‘‡π‘‘π‘Ÿ.(3.13) Then πœ‚π‘£,𝑇 is a nondecreasing function. By Corollary 3.7, a variational solution 𝑒 of (1.3) satisfies the energy equality a.e. in (𝜏,𝑇) if and only if πœ‚π‘’,𝑇(𝑑)=0 for a.e. π‘‘βˆˆ(𝜏,𝑇). In fact, using the continuity of the following mapping: []↦||π‘’π‘‘βˆˆπœ,π‘‡πœ||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œ+2π‘‘πœξ€Ί(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡βˆ’β€–π‘’(π‘Ÿ)β€–2π‘‡βˆ’(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€»π‘‘π‘Ÿ,(3.14) one can see that a variational solution 𝑒 of (1.3) satisfies the energy equality a.e. in (𝜏,𝑇) if and only if πœ‚π‘’,𝑇(𝑇)=0.

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

Lemma 3.8. Let 𝑒 be a variational solution of (2.1) and suppose that there exists a sequence {𝑑𝑛}βŠ‚(𝜏,𝑇) of Lebesgue points of |𝑒|2𝑇 such that 𝑑𝑛→𝑇 and limsupn⟢+βˆžξ‚€||𝑒𝑑𝑛||2𝑇‖‖𝑒𝑑+πœ€π‘›ξ€Έβ€–β€–2𝑇≀||π‘’πœ||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œ+2π‘‡πœξ€Ί(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡βˆ’β€–π‘’(π‘Ÿ)β€–2π‘‡βˆ’(𝑓(𝑒(π‘Ÿ)),𝑒(r))π‘‡ξ€»π‘‘π‘Ÿ.(3.15) Then, 𝑒 satisfies the energy equality a.e. in (𝜏,𝑇).

Proof. It is sufficient to prove that πœ‚π‘’,𝑇(𝑇)=0. Since 𝑑𝑛→𝑇 and πœ‚π‘’,𝑇 is nondecreasing, by Corollary 3.7, we have πœ‚π‘’,𝑇(𝑇)≀limsup𝑛↑+βˆžπœ‚π‘’,𝑇𝑑𝑛=limsup𝑛→+βˆžξ‚΅||𝑒𝑑𝑛||2𝑇‖‖𝑒𝑑+πœ€π‘›ξ€Έβ€–β€–2π‘‡βˆ’||π‘’πœ||2π‘‡β€–β€–π‘’βˆ’πœ€πœβ€–β€–2π‘‡ξ€œβˆ’2π‘‘π‘›πœξ€Ί(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡βˆ’β€–π‘’(π‘Ÿ)β€–2π‘‡βˆ’(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€»ξ‚Άπ‘‘π‘Ÿβ‰€limsup𝑛→+βˆžξ‚€||𝑒𝑑𝑛||2𝑇‖‖𝑒𝑑+πœ€π‘›ξ€Έβ€–β€–2π‘‡ξ‚βˆ’||π‘’πœ||2π‘‡β€–β€–π‘’βˆ’πœ€πœβ€–β€–2π‘‡ξ€œβˆ’2π‘‡πœξ€Ί(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡βˆ’β€–π‘’(π‘Ÿ)β€–2π‘‡βˆ’(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€»π‘‘π‘Ÿβ‰€0.(3.16) This completes the proof.

Proposition 3.9. Let 𝑒,𝑒 be two variational solutions of (2.1) corresponding to the initial data π‘’πœ,π‘’πœβˆˆπ‘‰πœβˆ©πΏπ‘(Ω𝜏), respectively, which satisfy the energy equality a.e. in (𝜏,𝑇). Then, ||𝑒(𝑑)βˆ’||𝑒(𝑑)2𝑇‖‖+πœ€π‘’(𝑑)βˆ’β€–β€–π‘’(𝑑)2π‘‡ξ€œ+2π‘‘πœβ€–β€–π‘’(π‘Ÿ)βˆ’β€–β€–π‘’(π‘Ÿ)2π‘‡π‘‘π‘Ÿβ‰€π‘’2β„“(π‘‘βˆ’πœ)ξ‚€||π‘’πœβˆ’π‘’πœ||2𝑇‖‖𝑒+πœ€πœβˆ’π‘’πœβ€–β€–2𝑇a.e.π‘‘βˆˆ(𝜏,𝑇).(3.17) Hence, it implies the uniqueness of variational solutions to (2.1) satisfying the energy equality in (𝜏,𝑇).

Proof. Since 𝑒,𝑒 satisfy the energy equation, πœ‚π‘’,𝑇(𝑑)=πœ‚π‘’,𝑇(𝑑)=0 for all π‘‘βˆˆ(𝜏,𝑇) and ||𝑒(𝑑)βˆ’||𝑒(𝑑)2𝑇‖‖+πœ€π‘’(𝑑)βˆ’β€–β€–π‘’(𝑑)2π‘‡ξ€œ+2π‘‘πœβ€–β€–π‘’(π‘Ÿ)βˆ’β€–β€–π‘’(π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2π‘‘πœξ€·ξ€·π‘“(𝑒(π‘Ÿ))βˆ’π‘“ξ€Έπ‘’(π‘Ÿ),𝑒(π‘Ÿ)βˆ’ξ€Έπ‘’(π‘Ÿ)𝑇≀||π‘’π‘‘π‘Ÿπœβˆ’π‘’πœ||2𝑇‖‖𝑒+πœ€πœβˆ’π‘’πœβ€–β€–2π‘‡βˆ’2limβ„Žβ†“0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Žξ€·π‘’(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ),𝑒(π‘Ÿ+β„Ž)βˆ’ξ€Έπ‘’(π‘Ÿ)π‘‡π‘‘π‘Ÿ.(3.18) On the other hand, ||||β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Žξ€·π‘’(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ),𝑒(π‘Ÿ+β„Ž)βˆ’ξ€Έπ‘’(π‘Ÿ)𝑇||||π‘‘π‘Ÿ2β‰€ξ‚΅β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Ž||𝑒||(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ)2β„Žπ‘‘π‘Ÿξ‚Άξ‚΅βˆ’1ξ€œπœπ‘‘βˆ’β„Ž||𝑒(π‘Ÿ+β„Ž)βˆ’π‘’||(π‘Ÿ)2ξ‚Ά,π‘‘π‘Ÿ(3.19) so limβ„Žβ†“0β„Žβˆ’1ξ€œπœπ‘‘βˆ’β„Žξ€·π‘’(π‘Ÿ+β„Ž)βˆ’π‘’(π‘Ÿ),𝑒(π‘Ÿ+β„Ž)βˆ’π‘’ξ€Έ(π‘Ÿ)π‘‡π‘‘π‘Ÿ=0.(3.20) Using this and (1.5) in (3.12), one can conclude ||𝑒(𝑑)βˆ’||𝑒(𝑑)2𝑇‖‖+πœ€π‘’(𝑑)βˆ’β€–β€–π‘’(𝑑)2π‘‡ξ€œ+2π‘‘πœβ€–β€–π‘’(π‘Ÿ)βˆ’β€–β€–π‘’(π‘Ÿ)2𝑇≀||π‘’π‘‘π‘Ÿπœβˆ’π‘’πœ||2𝑇‖‖𝑒+πœ€πœβˆ’π‘’πœβ€–β€–2π‘‡ξ€œβˆ’2π‘‘πœξ€·ξ€·π‘“(𝑒(π‘Ÿ))βˆ’π‘“ξ€Έπ‘’(π‘Ÿ),𝑒(π‘Ÿ)βˆ’ξ€Έπ‘’(π‘Ÿ)𝑇≀||π‘’π‘‘π‘Ÿπœβˆ’π‘’πœ||2𝑇‖‖𝑒+πœ€πœβˆ’π‘’πœβ€–β€–2π‘‡ξ€œ+2β„“π‘‘πœ||𝑒(π‘Ÿ)βˆ’π‘’||(π‘Ÿ)2π‘‡π‘‘π‘Ÿ.(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 π‘”βˆˆπΏ2(𝜏,𝑇;𝐻𝑇),π‘’πœβˆˆπ‘‰πœβˆ©πΏπ‘(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.
Step 1. 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 π‘‰π‘‡βˆ©πΏπ‘(Ω𝑇) such that the vector space generated by {𝑒𝑗} is dense in 𝑉𝑇 and 𝐿𝑝(Ω𝑇). Then, one takes a sequence {π‘’πœπ‘š} converging to π‘’πœ in 𝑉𝑇, with {π‘’πœπ‘š} in the vector space spanned by the π‘š first {𝑒𝑗}. For each integer π‘šβ‰₯1, one considers the approximation π‘’π‘˜π‘šβˆ‘(𝑑)=π‘šπ‘—=1π›Ύπ‘˜π‘š,𝑗(𝑑)𝑒𝑗, defined as a solution of ξ‚€π‘’ξ…žπ‘˜π‘š(𝑑),𝑒𝑗𝑇+ξ«π΄π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),𝑒𝑗𝑇𝐴+πœ€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),𝑒𝑗𝑇+ξ€·π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),𝑒𝑗𝑇=𝑔(𝑑),𝑒𝑗𝑇,π‘’ξ€·ξ€·π‘˜π‘š(𝜏),𝑒𝑗𝑇=π‘’ξ€·ξ€·πœπ‘š,𝑒𝑗𝑇.(3.22) Multiplying (3.22) by π›Ύξ…žπ‘˜π‘š,𝑗(𝑑) and summing from 𝑗=1 to π‘š, we obtain ξ‚€π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇+ξ‚¬π΄π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘šξ‚­ξ‚¬π΄(𝑑)+πœ€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘šξ‚­(𝑑)𝑇+ξ‚€π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇=𝑔(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇,(3.23) or |||π‘’ξ…žπ‘˜π‘š|||(𝑑)2𝑇+12π‘‘β€–β€–π‘’π‘‘π‘‘π‘˜π‘šβ€–β€–(𝑑)2𝑇‖‖𝑒+πœ€ξ…žπ‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘š(𝑑)𝑇𝑃+πœ€π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘š(𝑑)𝑇+ξ‚€π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇=𝑔(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇.(3.24) Thus, 𝑔(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇β‰₯|||π‘’ξ…žπ‘˜π‘š|||(𝑑)2𝑇‖‖𝑒+πœ€ξ…žπ‘˜π‘šβ€–β€–(𝑑)2𝑇+12π‘‘ξ‚ƒβ€–β€–π‘’π‘‘π‘‘π‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇𝑃+πœ€π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘š(𝑑)𝑇+ξ‚€π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇=|||π‘’ξ…žπ‘˜π‘š(|||𝑑)2𝑇‖‖𝑒+πœ€ξ…žπ‘˜π‘š(‖‖𝑑)2𝑇𝑃+πœ€π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘š(𝑑)𝑇+12π‘‘ξ‚Έβ€–β€–π‘’π‘‘π‘‘π‘˜π‘š(‖‖𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)ξ€Έξ€Έπ‘‡ξ€œ+2Ξ©π‘‡πΉξ€·π‘’π‘˜π‘š(ξ€Έξ‚Ή.π‘₯,𝑑)𝑑π‘₯(3.25) We have 𝑔(𝑑),π‘’ξ…žπ‘˜π‘šξ‚(𝑑)𝑇≀12ξ‚΅||||𝑔(𝑑)2𝑇+|||π‘’ξ…žπ‘˜π‘š|||(𝑑)2𝑇,(3.26) so ||||𝑔(𝑑)2𝑇‖‖𝑒β‰₯2πœ€ξ…žπ‘˜π‘šβ€–β€–(𝑑)2𝑇+π‘‘ξ‚Έβ€–β€–π‘’π‘‘π‘‘π‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)ξ€Έξ€Έπ‘‡ξ€œ+2Ξ©π‘‡πΉξ€·π‘’π‘˜π‘šξ€Έξ‚Ήπ‘ƒ(π‘₯,𝑑)𝑑π‘₯+2πœ€π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’ξ…žπ‘˜π‘š(𝑑)𝑇+|||π‘’ξ…žπ‘˜π‘š|||(𝑑)2𝑇.(3.27) Integrating (3.27) on [𝜏,𝑑],πœβ‰€π‘‘β‰€π‘‡, we obtain ξ€œ2πœ€π‘‘πœβ€–β€–π‘’ξ…žπ‘˜π‘š(β€–β€–π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2πœ€π‘˜π‘‘πœπ‘ƒξ‚€ξ‚€π‘˜(π‘Ÿ)π‘’ξ…žπ‘˜π‘š(π‘Ÿ),π‘’ξ…žπ‘˜π‘š(π‘Ÿ)ξ‚ξ‚π‘‡ξ€œπ‘‘π‘Ÿ+π‘‘πœ|||π‘’ξ…žπ‘˜π‘š(|||π‘Ÿ)2𝑇+β€–β€–π‘’π‘‘π‘Ÿπ‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)ξ€Έξ€Έπ‘‡ξ€œ+2Ξ©π‘‡πΉξ€·π‘’π‘˜π‘šξ€Έβ‰€ξ€œ(π‘₯,𝑑)𝑑π‘₯π‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+π‘˜π‘š(β€–β€–πœ)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’π‘˜π‘š(𝜏),π‘’π‘˜π‘š(𝜏)ξ€Έξ€Έπ‘‡ξ€œ+2Ξ©π‘‡πΉξ€·π‘’π‘˜π‘š(ξ€Έπ‘₯,𝜏)𝑑π‘₯.(3.28) Since ξ€œΞ©π‘‡πΉξ€·π‘’π‘˜π‘š(̃𝛽||Ξ©π‘₯,𝑑)𝑑π‘₯β‰₯βˆ’π‘‡||+𝛼1β€–β€–π‘’π‘˜π‘š(‖‖𝑑)𝑝𝐿𝑝Ω𝑇,ξ€œΞ©π‘‡πΉξ€·π‘’π‘˜π‘šξ€ΈΜƒπ›½||Ξ©(π‘₯,𝜏)𝑑π‘₯≀𝑇||+𝛼2β€–β€–π‘’πœπ‘šβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έ,(3.29) we have ξ€œ2πœ€π‘‘πœβ€–β€–π‘’ξ…žπ‘˜π‘š(β€–β€–π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2πœ€π‘˜π‘‘πœπ‘ƒξ‚€ξ‚€π‘˜(π‘Ÿ)π‘’ξ…žπ‘˜π‘š(π‘Ÿ),π‘’ξ…žπ‘˜π‘š(π‘Ÿ)ξ‚ξ‚π‘‡ξ€œπ‘‘π‘Ÿ+π‘‘πœ|||π‘’ξ…žπ‘˜π‘š(|||π‘Ÿ)2𝑇+β€–β€–π‘’π‘‘π‘Ÿπ‘˜π‘š(‖‖𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇+2𝛼1β€–β€–π‘’π‘˜π‘š(‖‖𝑑)π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έβ‰€ξ€œπ‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+π‘˜π‘šβ€–β€–(𝜏)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’π‘˜π‘š(𝜏),π‘’π‘˜π‘š(𝜏)𝑇̃𝛽||Ξ©+4𝑇||+2𝛼2β€–β€–π‘’πœπ‘šβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έ.(3.30) From (3.30), we deduce thatο‚·{π‘’π‘˜π‘š} is bounded in 𝐿∞(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇),ο‚·{π‘’π‘˜π‘š}β‡€π‘’π‘˜ weakly in 𝐿∞(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇),ο‚·{π‘’ξ…žπ‘˜π‘š} is bounded in 𝐿2(𝜏,𝑇;𝑉𝑇),ο‚·{π‘’ξ…žπ‘˜π‘š}β‡€π‘’ξ…žπ‘˜ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇).Since {π‘’π‘˜π‘š} is bounded in 𝐿∞(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(ξ‚π‘„πœ,𝑇), one can check that {𝑓(π‘’π‘˜π‘š)} is bounded in πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ω𝑇)) with π‘ž=𝑝/(π‘βˆ’1), hence 𝑓(π‘’π‘˜π‘š)β‡€πœ‚ in πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ω𝑇)). We now prove that πœ‚=𝑓(π‘’π‘˜).
Indeed, we haveο‚·π‘‰π‘‡βŠ‚βŠ‚π»π‘‡βŠ‚π‘‰βˆ—π‘‡, ο‚·{π‘’π‘˜π‘š} is bounded in 𝐿∞(𝜏,𝑇;𝑉𝑇),ο‚·{π‘’ξ…žπ‘˜π‘š} is bounded in 𝐿2(𝜏,𝑇;π‘‰βˆ—π‘‡).By the Aubin-Lions lemma [20, Chapter 1], {π‘’π‘˜π‘š} is relatively compact in 𝐿2(𝜏,𝑇;𝐻𝑇). Therefore, one can assume that π‘’π‘˜π‘šβ†’π‘’π‘˜ strongly in 𝐿2(𝜏,𝑇;𝐻𝑇), so π‘’π‘˜π‘šβ†’π‘’π‘˜ a.e. in ξ‚π‘„πœ,𝑇. Since 𝑓 is continuous, 𝑓(π‘’π‘˜π‘š)→𝑓(π‘’π‘˜) a.e. in ξ‚π‘„πœ,𝑇. Applying Lemma 1.3 in [20], we have π‘“ξ€·π‘’π‘˜π‘šξ€Έξ€·π‘’β‡€π‘“π‘˜ξ€ΈweaklyinπΏπ‘žξ€·πœ,𝑇;πΏπ‘žξ€·Ξ©π‘‡.ξ€Έξ€Έ(3.31) This implies that π‘’π‘˜ 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 π‘˜ξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(π‘Ÿ)π‘’π‘˜π‘š(π‘Ÿ),π‘’π‘˜π‘š(π‘Ÿ)ξ€Έξ€Έπ‘‡ξ‚΅ξ€œπ‘‘π‘Ÿβ‰€(π‘‡βˆ’πœ)π‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+π‘˜π‘š(β€–β€–πœ)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’π‘˜π‘š(𝜏),π‘’π‘˜π‘š(𝜏)𝑇̃𝛽||Ξ©+4𝑇||+2𝛼2β€–β€–π‘’πœπ‘šβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έξ‚Ά.(3.32) Consider the function Φ∢𝐿2(𝜏,𝑇;𝑉𝑇)→ℝ defined by ξ€œΞ¦(𝑣)=π‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)𝑣(𝑑),𝑣(𝑑)𝑇𝑑𝑑,π‘£βˆˆπΏ2ξ€·πœ,𝑇;𝑉𝑇.(3.33) It is easy to see that Ξ¦ is a continuous and convex function. It follows that βˆ«π‘‡πœ((π‘ƒπ‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)))𝑇𝑑𝑑 is weakly lower semicontinuous in 𝐿2(𝜏,𝑇;𝑉𝑇). Moreover, {π‘’π‘˜π‘š}β‡€π‘’π‘˜ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇), hence π‘˜ξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜(𝑑),π‘’π‘˜(𝑑)ξ€Έξ€Έπ‘‡π‘‘π‘‘β‰€π‘˜liminfπ‘šβ†’βˆžξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)ξ€Έξ€Έπ‘‡ξ‚΅ξ€œπ‘‘π‘‘β‰€(π‘‡βˆ’πœ)π‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœβ€–β€–2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’πœ,π‘’πœξ€Έξ€Έπ‘‡Μƒπ›½||Ξ©+4𝑇||+2𝛼2β€–β€–π‘’πœβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έξ‚Ά.(3.34) Since {π‘’ξ…žπ‘˜π‘š}β‡€π‘’ξ…žπ‘˜ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇), then, reasoning as above, ξ€œ2πœ€π‘˜π‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’ξ…žπ‘˜(𝑑),π‘’ξ…žπ‘˜(𝑑)𝑇𝑑𝑑≀2πœ–π‘˜liminfπ‘šβ†’βˆžξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)ξ€Έξ€Έπ‘‡β‰€ξ‚΅ξ€œπ‘‘π‘‘π‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœβ€–β€–2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’πœ,π‘’πœξ€Έξ€Έπ‘‡Μƒπ›½||Ξ©+4𝑇||+2𝛼2β€–β€–π‘’πœβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έξ‚Ά.(3.35) From the facts that π‘’π‘˜π‘šβ‡€π‘’π‘˜ weakly in 𝐿∞(𝜏,𝑇;𝑉𝑇), π‘’ξ…žπ‘˜π‘šβ‡€π‘’ξ…žπ‘˜ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇) and the weak lower semicontinuity of the norm, we deduce that ξ€œ2πœ€π‘‘πœβ€–β€–π‘’ξ…žπ‘˜(β€–β€–π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2πœ€π‘˜π‘‘πœπ‘ƒξ€·ξ€·π‘˜(π‘Ÿ)π‘’ξ…žπ‘˜(π‘Ÿ),π‘’ξ…žπ‘˜(π‘Ÿ)ξ€Έξ€Έπ‘‡ξ€œπ‘‘π‘Ÿ+π‘‘πœ||π‘’ξ…žπ‘˜(||π‘Ÿ)2𝑇+β€–β€–π‘’π‘‘π‘Ÿπ‘˜(‖‖𝑑)2π‘‡ξ€œ+π‘˜π‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜(𝑑),π‘’π‘˜(𝑑)𝑇+2𝛼1β€–β€–π‘’π‘˜(‖‖𝑑)π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έξ‚΅ξ€œβ‰€(5+π‘‡βˆ’πœ)π‘‘πœ||||𝑔(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœβ€–β€–2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝜏)π‘’πœ,π‘’πœξ€Έξ€Έπ‘‡Μƒπ›½||Ξ©+4𝑇||+2𝛼2β€–β€–π‘’πœβ€–β€–π‘πΏπ‘ξ€·Ξ©π‘‡ξ€Έξ‚Ά=𝐢.(3.36) Since π‘’πœβˆˆπ‘‰πœβˆ©πΏπ‘(Ω𝜏), ((π‘ƒπ‘˜(𝜏)π‘’πœ,π‘’πœ))𝑇=0forallπ‘˜β‰₯1, (3.36) gives ο‚·{π‘’π‘˜} is bounded in 𝐿∞(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(𝜏,𝑇;𝐿𝑝(Ω𝑇)),ο‚·{π‘’ξ…žπ‘˜} is bounded in 𝐿2(𝜏,𝑇;𝑉𝑇),ο‚·{π‘’π‘˜}⇀𝑒 weakly in 𝐿∞(𝜏,𝑇;𝑉𝑇)βˆ©πΏπ‘(𝜏,𝑇;𝐿𝑝(Ω𝑇)),ο‚·{π‘’ξ…žπ‘˜}⇀𝑒′ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇).From Lemma 2.1, we have ξ€œπ‘‡πœβ€–π‘ƒ(𝑑)𝑒(𝑑)β€–2𝑇𝑑𝑑≀liminfπ‘˜β†’βˆžξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜(𝑑),π‘’π‘˜(𝑑)𝑇𝑑𝑑≀liminfπ‘˜β†’βˆžπΆπ‘˜=0,thatis,𝑃(𝑑)𝑒(𝑑)=0a.e.in(𝜏,𝑇)or𝑒(𝑑)βˆˆπ‘‰π‘‘ξ€œa.e.in(𝜏,𝑇),π‘‡πœβ€–β€–π‘ƒ(𝑑)π‘’ξ…žβ€–β€–(𝑑)2𝑇𝑑𝑑≀liminfπ‘˜β†’βˆžξ€œπ‘‡πœπ‘ƒξ€·ξ€·π‘˜(𝑑)π‘’ξ…žπ‘˜(𝑑),π‘’ξ…žπ‘˜(𝑑)𝑇𝑑𝑑≀liminfπ‘˜β†’βˆžπΆπ‘˜=0,thatis,𝑃(𝑑)π‘’ξ…ž(𝑑)=0a.e.in(𝜏,𝑇).(3.37) Moreover, (3.36) and the equality π‘’π‘˜(𝑑)βˆ’π‘’π‘˜(ξ€œπ‘ )=π‘‘π‘ π‘’ξ…žπ‘˜([]π‘Ÿ)π‘‘π‘Ÿ,βˆ€π‘ ,π‘‘βˆˆπœ,𝑇,βˆ€π‘˜β‰₯1,(3.38) give ||π‘’π‘˜(𝑑)βˆ’π‘’π‘˜||(𝑠)𝑇≀𝐢1/2|π‘‘βˆ’π‘ |1/2[]βˆ€π‘ ,π‘‘βˆˆπœ,𝑇,βˆ€π‘˜β‰₯1.(3.39) It follows from (3.36) that β€–π‘’π‘˜(𝑑)‖𝑇≀𝐢 for all π‘‘βˆˆ[𝜏,𝑇] and each π‘˜β‰₯1. Since the injection of 𝑉𝑇 into 𝐻𝑇 is compact, the set {π‘£βˆˆπ‘‰π‘‡βˆΆβ€–π‘£β€–2𝑇≀𝐢} is compact in 𝐻𝑇. By (3.39) and the Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by {π‘’π‘˜}, such that π‘’π‘˜ξ€·[]→𝑒in𝐢𝜏,𝑇;𝐻𝑇asπ‘˜βŸΆ+∞.(3.40) So, the condition (C4) is satisfied.

On the other hand, {π‘’π‘˜} is bounded in 𝐿∞(𝜏,𝑇;𝑉𝑇) and {π‘’ξ…žπ‘˜}is bounded in 𝐿2(𝜏,𝑇;𝑉𝑇), applying the Aubin-Lions lemma and Lemma 1.3 in [20, Chapter 1], one has π‘“ξ€·π‘’π‘˜ξ€Έβ‡€π‘“(𝑒)weaklyinπΏπ‘žξ€·πœ,𝑇,πΏπ‘žξ€·Ξ©π‘‡ξ€Έξ€Έ.(3.41) Since π‘’π‘˜ is the weak solution of the problem ξ€·π‘’ξ…žπ‘˜ξ€Έ(𝑑),𝑣𝑇+βŸ¨π΄π‘˜(𝑑)π‘’π‘˜(𝑑),π‘£βŸ©π‘‡ξ«π΄+πœ€π‘˜(𝑑)π‘’ξ…žπ‘˜ξ¬(𝑑),𝑣𝑇+ξ€·π‘“ξ€·π‘’π‘˜ξ€Έξ€Έ(𝑑),𝑣𝑇=(𝑔(𝑑),𝑣)𝑇,βˆ€π‘£βˆˆπ‘‰π‘‡,π‘’ξ€·ξ€·π‘˜(𝜏),𝑣𝑇=π‘’ξ€·ξ€·πœ,𝑣𝑇,(3.42) taking to the limit as π‘˜β†’+∞ and using the fact that 𝑃(𝑑)𝑒(𝑑)=0,𝑃(𝑑)𝑒′(𝑑)=0 a.e. in (𝜏,𝑇), we can conclude that 𝑒 is the solution of (2.1).

Now, we will show that 𝑒 satisfies the energy equality in (𝜏,𝑇). Multiplying (3.22) by π›Ύπ‘˜π‘š,𝑗 and summing from 𝑗=1 to π‘š, we obtain ξ‚€π‘’ξ…žπ‘˜π‘š(𝑑),π‘’π‘˜π‘šξ‚(𝑑)𝑇+ξ«π΄π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘šξ¬(𝑑)𝑇𝐴+πœ€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’π‘˜π‘šξ‚­(𝑑)𝑇+ξ€·π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),π‘’π‘˜π‘šξ€Έ(𝑑)𝑇=𝑔(𝑑),π‘’π‘˜π‘šξ€Έ(𝑑)𝑇.(3.43) Hence, we get 12𝑑||π‘’π‘‘π‘‘π‘˜π‘š||(𝑑)2𝑇+β€–β€–π‘’π‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇+πœ–2π‘‘β€–β€–π‘’π‘‘π‘‘π‘˜π‘šβ€–β€–(𝑑)2𝑇𝑃+πœ–π‘˜ξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇+ξ€·π‘“ξ€·π‘’π‘˜π‘šξ€Έ(𝑑),π‘’π‘˜π‘šξ€Έ(𝑑)𝑇=𝑔(𝑑),π‘’π‘˜π‘šξ€Έ(𝑑)𝑇,(3.44) or ||π‘’π‘˜π‘š||(𝑑)2π‘‡ξ€œ+2π‘‘πœβ€–β€–π‘’π‘˜π‘šβ€–β€–(π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2π‘˜π‘‡πœπ‘ƒξ€·ξ€·π‘˜(π‘Ÿ)π‘’π‘˜π‘š(π‘Ÿ),π‘’π‘˜π‘š(π‘Ÿ)ξ€Έξ€Έπ‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœ–π‘˜π‘š(‖‖𝑑)2π‘‡ξ€œ+2π‘˜πœ–π‘‘πœπ‘ƒξ‚€ξ‚€π‘˜(π‘Ÿ)π‘’ξ…žπ‘˜π‘š(π‘Ÿ),π‘’π‘˜π‘š(π‘Ÿ)ξ‚ξ‚π‘‡ξ€œπ‘‘π‘Ÿ+2π‘‘πœξ€·π‘“ξ€·π‘’π‘˜π‘š(ξ€Έπ‘Ÿ),π‘’π‘˜π‘š(ξ€Έπ‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿ=2π‘‘πœξ€·π‘”(π‘Ÿ),π‘’π‘˜π‘šξ€Έ(π‘Ÿ)𝑇||π‘’π‘‘π‘Ÿ+πœπ‘š||2𝑇‖‖𝑒+πœ€πœπ‘šβ€–β€–2𝑇.(3.45) Since π‘ƒξ‚€ξ‚€π‘˜(𝑑)π‘’ξ…žπ‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇β‰₯12π‘‘π‘ƒπ‘‘π‘‘ξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇,π‘ƒξ€·ξ€·π‘˜(𝑑)π‘’π‘˜π‘š(𝑑),π‘’π‘˜π‘š(𝑑)𝑇||𝑒β‰₯0,π‘˜π‘š||(𝑑)2π‘‡ξ€œ+2π‘‘πœβ€–β€–π‘’π‘˜π‘šβ€–β€–(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœ–π‘˜π‘šβ€–β€–(𝑑)2π‘‡ξ€œ+2π‘‘πœξ€·π‘“ξ€·π‘’π‘˜π‘šξ€Έ(π‘Ÿ),π‘’π‘˜π‘šξ€Έ(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿβ‰€2π‘‘πœξ€·π‘”(π‘Ÿ),π‘’π‘˜π‘šξ€Έ(π‘Ÿ)𝑇||π‘’π‘‘π‘Ÿ+πœπ‘š||2𝑇‖‖𝑒+πœ€πœπ‘šβ€–β€–2𝑇𝑃+πœ€π‘˜ξ€·ξ€·π‘˜(𝑑)π‘’πœπ‘š,π‘’πœπ‘šξ€Έξ€Έπ‘‡,(3.46) letting π‘šβ†’βˆž, we obtain ||π‘’π‘˜||(𝑇)2π‘‡ξ€œ+2π‘‡πœβ€–β€–π‘’π‘˜β€–β€–(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœ–π‘˜β€–β€–(𝑇)2π‘‡ξ€œ+2π‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έ(π‘Ÿ),π‘’π‘˜ξ€Έ(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿβ‰€2π‘‡πœξ€·π‘”(π‘Ÿ),π‘’π‘˜ξ€Έ(π‘Ÿ)𝑇||π‘’π‘‘π‘Ÿ+𝜏||2𝑇‖‖𝑒+πœ€πœβ€–β€–2𝑇.(3.47) Now ξ€œπ‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έ(π‘Ÿ),π‘’π‘˜ξ€Έ(π‘Ÿ)𝑇=ξ€œπ‘‘π‘Ÿπ‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έ(π‘Ÿ)βˆ’π‘“(𝑒(π‘Ÿ)),π‘’π‘˜ξ€Έ(π‘Ÿ)βˆ’π‘’(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿ+π‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έξ€Έ(π‘Ÿ),𝑒(π‘Ÿ)𝑇+ξ€œπ‘‘π‘Ÿπ‘‡πœξ€·π‘“(𝑒(π‘Ÿ)),π‘’π‘˜ξ€Έ(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿβˆ’π‘‡πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€œπ‘‘π‘Ÿβ‰₯βˆ’β„“π‘‡πœ||π‘’π‘˜||(π‘Ÿ)βˆ’π‘’(π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+π‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έξ€Έ(π‘Ÿ),𝑒(π‘Ÿ)𝑇+ξ€œπ‘‘π‘Ÿπ‘‡πœξ€·π‘“(𝑒(π‘Ÿ)),π‘’π‘˜ξ€Έ(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿβˆ’π‘‡πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ.(3.48) This inequality and (3.47) give ||π‘’π‘˜||(𝑇)2π‘‡ξ€œ+2π‘‡πœβ€–β€–π‘’π‘˜β€–β€–(π‘Ÿ)2π‘‡β€–β€–π‘’π‘‘π‘Ÿ+πœ–π‘˜β€–β€–(𝑇)2π‘‡ξ€œβ‰€2π‘‡πœξ€·π‘”(π‘Ÿ),π‘’π‘˜ξ€Έ(π‘Ÿ)𝑇||π‘’π‘‘π‘Ÿ+𝜏||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œ+2π‘™π‘‡πœ||π‘’π‘˜||(π‘Ÿ)βˆ’π‘’(π‘Ÿ)2π‘‡βˆ’ξ€œπ‘‘π‘Ÿπ‘‡πœξ€·π‘“ξ€·π‘’π‘˜ξ€Έξ€Έ(π‘Ÿ),𝑒(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿβˆ’π‘‡πœξ€·π‘“(𝑒(π‘Ÿ)),π‘’π‘˜ξ€Έ(π‘Ÿ)π‘‡ξ€œπ‘‘π‘Ÿ+π‘‡πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ.(3.49) Since π‘’π‘˜β‡€π‘’ weakly in 𝐿2(𝜏,𝑇;𝑉𝑇), we get ||||𝑒(𝑇)2𝑇+πœ€β€–π‘’(𝑇)β€–2𝑇≀||π‘’πœ||2𝑇‖‖𝑒+πœ€πœβ€–β€–2π‘‡ξ€œβˆ’2π‘‡πœ(𝑓(𝑒(π‘Ÿ)),𝑒(π‘Ÿ))π‘‡ξ€œπ‘‘π‘Ÿβˆ’2π‘‡πœβ€–β€–π‘’(π‘Ÿ)2π‘‡ξ€œπ‘‘π‘Ÿ+2π‘‡πœ(𝑔(π‘Ÿ),𝑒(π‘Ÿ))π‘‡π‘‘π‘Ÿ.(3.50) Applying Lemma 3.8 with 𝑑𝑛=𝑇 for all 𝑛, one concludes that 𝑒 satisfies the energy equality on (𝜏,𝑇), and the desired uniqueness follows from Proposition 3.9.

4. Existence of Pullback π’Ÿ-Attractors

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

Suppose that