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International Journal of Differential Equations
Volume 2010 (2010), Article ID 103510, 17 pages
http://dx.doi.org/10.1155/2010/103510
Research Article

Attractors for Nonautonomous Parabolic Equations without Uniqueness

1Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam
2Department of Applied Mathematics and Informatics, Hanoi University of Technology, 1 Dai Co Viet, Hai Ba Trung, 10408 Hanoi, Vietnam

Received 9 September 2009; Revised 12 February 2010; Accepted 8 April 2010

Academic Editor: Igor D. Chueshov

Copyright © 2010 Cung The Anh and Nguyen Dinh Binh. 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

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

1. Introduction

The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to attack the problem for a dissipative dynamical system is to consider its global attractor. The existence of the global attractor has been derived for a large class of PDEs (see [1, 2] and references therein), for both autonomous and nonautonomous equations. However, these researches may not be applied to a wide class of problems, in which solutions may not be unique. Good examples of such systems are differential inclusions, variational inequalities, control infinite-dimensional systems, and also some partial differential equations for which solutions may not be known to be unique as, for example, some certain semilinear wave equations with high-power nonlinearities, the incompressible Navier-Stokes equation in three-space dimension, the Ginzburg-Landau equation, and so forth. For the qualitative analysis of the abovementioned systems from the point of view of the theory of dynamical systems, it is necessary to develop a corresponding theory for multivalued semigroups.

In the last years, there have been some theories for which one can treat multivalued semiflows and their asymptotic behavior, including generalized semiflows theory of Ball [3], theory of trajectory attractors of Chepyzhov and Vishik [4], and theories of multivalued semiflows and semiprocesses of Melnik and Valero [5, 6]. Thanks to these theories, several results concerning attractors in the case of equations without uniqueness have been obtained recently for differential inclusion [5, 6], parabolic equations [79], the phase-field equation [10], the wave equation [11], the three-dimensional Navier-Stokes equation [3, 12], and so forth. On the other hand, when a problem does not possess the property of uniqueness, we have a set of solutions corresponding to each initial datum. We can speak then about a set of values attained by the solutions for every fixed moment of time. It is interesting to study the topological properties of such set and, in particular, its connectedness. This property is known as the Kneser property in the literature. The Kneser property has been studied for some parabolic equations [9, 1315], semilinear wave equations [11], and so forth. By results in [5, 6], the Kneser property implies the connectedness of the global attractor. Although the existence of global attractor and the Kneser property have been derived for some classes of partial differential equations without uniqueness, to the best of our knowledge, little seems to be known for nonautonomous degenerate equations.

In this paper we study the following nonautonomous semilinear degenerate parabolic equation with variable, nonnegative coefficients, defined on a bounded domain Ω𝑁,𝑁2:𝜕𝑢𝜕𝑡div(𝜌(𝑥)𝑢)+𝑓(𝑥,𝑢)=𝑔(𝑥,𝑡),𝑥Ω,𝑡>𝜏,𝑢|𝑡=𝜏=𝑢𝜏𝑢|(𝑥),𝑥Ω,𝜕Ω=0,(1.1) where 𝑢𝜏𝐿2(Ω) is given, and the coefficient 𝜌, the nonlinearity 𝑓, and the external force 𝑔 satisfy the following conditions.(𝐻1)The function 𝜌Ω satisfies 𝜌𝐿1(Ω) and, for some 𝛼(0,2), lim𝑥𝑧inf|𝑥𝑧|𝛼𝜌(𝑥)>0 for every 𝑧Ω.(𝐻2)𝑓Ω× is a Caratheodory function, that is, the function 𝑓(,𝑢) is measurable and the function 𝑓(𝑥,) is continuous, and satisfies ||||𝑓(𝑥,𝑢)𝐶1|𝑢|𝑝1+1(𝑥),forsome𝑝2,(1.2)𝑢𝑓(𝑥,𝑢)𝐶2|𝑢|𝑝2(𝑥),(1.3) where 𝐶1,𝐶2 are positive constants; 1,2 are nonnegative functions such that 1𝐿(Ω) and 2𝐿1(Ω) (see Remark 4.2 on a comment about conditions of 1).(𝐻3)𝑔𝐿2𝑏(+,𝐿2(Ω)), where 𝐿2𝑏(+,𝐿2(Ω)) is the set of all translation-bounded functions (see Section 2.2 for its definition).

The degeneracy of problem (1.1) is considered in the sense that the measurable, nonnegative diffusion coefficient 𝜌(𝑥) is allowed to have at most a finite number of (essential) zeroes at some points. The physical motivation of the assumption (𝐻1) is related to the modelling of reaction diffusion processes in composite materials, occupying a bounded domain Ω, in which at some points they behave as perfect insulator. Following [16, page 79], when at some points the medium is perfectly insulating, it is natural to assume that 𝜌(𝑥) vanishes at these points. Note that, in various diffusion processes, the equation involves diffusion of the type 𝜌(𝑥)|𝑥|𝛼,𝛼(0,2).

In the autonomous case, which is the case 𝑔 independent of time 𝑡, the existence and long-time behavior of solutions to problem (1.1) have been studied in [1720]. In this paper we continue studying the long-time behavior of solutions to problem (1.1) by allowing the external force 𝑔 to be dependent on time 𝑡. Moreover, the conditions imposed on the nonlinearity 𝑓 provide global existence of a weak solution to problem (1.1), but not uniqueness. Let 𝒟𝜏,𝜎(𝑢𝜏) be the set of all global weak solutions of problem (1.1) with the external force 𝑔𝜎 instead of 𝑔 and initial datum 𝑢(𝜏)=𝑢𝜏. For each 𝜎Σ=𝑤(𝑔), the closure of the set {𝑔(+)+} in 𝐿2loc(+,𝐿2(Ω)) with the weak topology, we define the multivalued semiprocess 𝑈𝜎+𝑑×𝐿2(Ω)2𝐿2(Ω) as follows: 𝑈𝜎𝑡,𝜏,𝑢𝜏=𝑢(𝑡)𝑢()𝒟𝜏,𝜎𝑢𝜏.(1.4) We prove that 𝑈𝜎 is a strict multivalued semiprocess and then use the theory of multivalued semiprocesses of Melnik and Valero [6] to prove the existence of a uniform global compact attractor for the family of multivalued semiprocesses {𝑈𝜎}𝜎Σ. Finally, following the general lines of the approach in [9, 11, 14, 15], we prove that the Kneser property holds for the set of all weak solutions, that is, the set of values attained by the solutions at every moment of time is connected. Thanks to the Kneser property, the uniform global attractor derived above is connected in 𝐿2(Ω). We summarize our main results in the following theorem.

Theorem 1.1. Under conditions (𝐻1)–(𝐻3), problem (1.1) defines a family of strict multivalued semiprocesses {𝑈𝜎}𝜎Σ, which possesses a uniform global compact connected attractor 𝒜 in 𝐿2(Ω).

It is worth noticing that under some additional conditions on 𝑓, for example, 𝑓𝑢(𝑥,𝑢)𝐶3 for all 𝑥Ω,𝑢, or a weaker assumption (𝑓(𝑥,𝑢)𝑓(𝑥,𝑣))(𝑢𝑣)𝐶|𝑢𝑣|2𝑥Ω,𝑢,𝑣,(1.5) one can prove that the weak solution of problem (1.1) is unique. Then the multivalued semiprocess 𝑈𝜎 turns to be a single-valued one and the uniform compact global attractor 𝒜 derived in Theorem 1.1 is exactly the usual uniform attractor for the family of single-valued semiprocesses [1].

The rest of the paper is organized as follows. In Section 2, for convenience of readers, we recall some results on function spaces and uniform global attractors for multivalued semiprocesses. Section 3 is devoted to prove the global existence of a weak solution and the existence of a uniform global attractor of the family of multivalued semiprocesses associated to problem (1.1). In the last section, we prove the Kneser property for the solutions. As a result, we obtain the connectedness of the uniform global attractor.

2. Preliminaries

2.1. Function Space and Operator

We recall some basic results on the function space which we will use. Let 𝑁2, 𝛼(0,2), and2𝛼=4𝛼(2,)if𝑁=2,2𝑁𝑁2+𝛼2,2𝑁𝑁2if𝑁3.(2.1) The exponent 2𝛼 has the role of the critical exponent in the classical Sobolev embedding.

The natural energy space for problem (1.1) involves the space 𝒟10(Ω,𝜌), defined as the closure of 𝐶0(Ω) with respect to the norm 𝑢𝒟10(Ω,𝜌)=Ω||||𝜌(𝑥)𝑢2𝑑𝑥1/2.(2.2) The space 𝒟10(Ω,𝜌) is a Hilbert space with respect to the scalar product (𝑢,𝑣)=Ω𝜌(𝑥)𝑢𝑣𝑑𝑥.(2.3) The following lemma comes from [21, Propositions 3.3–3.5].

Lemma 2.1. Assume that Ω is a bounded domain in 𝑁, 𝑁2, and 𝜌 satisfies (𝐻1). Then the following embeddings hold: (i)𝒟10(Ω,𝜌)𝐿2𝛼(Ω) continuously,(ii)𝒟10(Ω,𝜌)𝐿𝑝(Ω) compactly if 𝑝[1,2𝛼).

It is known (see [19]) that there exists a complete orthonormal system of eigenvectors (𝑒𝑗,𝜆𝑗) of the operator 𝐴=div(𝜌(𝑥)) such that𝑒𝑗,𝑒𝑘=𝛿𝑗𝑘𝜌,div(𝑥)𝑒𝑗=𝜆𝑗𝑒𝑗,𝑗,𝑘=1,2,,0<𝜆1𝜆2𝜆3,𝜆𝑗+,as𝑗.(2.4)

2.2. The Translation-Bounded Functions

Definition 2.2. Let be a reflexive Banach space. A function 𝜑𝐿2loc(+,) is said to be translation-bounded if 𝜑2𝐿2𝑏=𝜑𝐿2𝑏(+,)=sup𝑡+𝑡𝑡+1𝜑2𝑑𝑠<.(2.5)

We will denote by 𝐿2𝑏(+,) the set of all translation-bounded functions in 𝐿2loc(+,). Let 𝑔𝐿2𝑏(+,) and 𝑤(𝑔) be the closure of the set {𝑔(+)+} in 𝐿2loc(+,) with the weak topology. The following results are well-known.

Lemma 2.3 (see [1, Chapter 5, Proposition 4.2]). (1) For all 𝜎𝑤(𝑔),𝜎2𝐿2𝑏𝑔2𝐿2𝑏.
(2) The translation group {𝑇()} is weakly continuous on 𝑤(𝑔).
(3)𝑇()𝑤(g)𝑤(𝑔) for 0.
(4)𝑤(𝑔) is weakly compact.

2.3. Uniform Attractors of Multivalued Semiprocesses

Denote +𝑑={(𝑡,𝜏)0𝜏𝑡}. Let 𝑋 be a complete metric space, let 𝒫(𝑋) and (𝑋) be the set of all nonempty subsets and the set of all nonempty bounded subsets of the space 𝑋, respectively, and let Σ be a compact metric space.

Definition 2.4. The map 𝑈+𝑑×𝑋𝒫(𝑋) is called a multivalued semiprocess (MSP) if (1)𝑈(𝜏,𝜏,)=Id (the identity map) (2)𝑈(𝑡,𝜏,𝑥)𝑈(𝑡,𝑠,𝑈(𝑠,𝜏,𝑥)), for all 𝑥𝑋,𝑡,𝑠,𝜏+,𝜏𝑠𝑡.It is called a strict multivalued semiprocess if 𝑈(𝑡,𝜏,𝑥)=𝑈(𝑡,𝑠,𝑈(𝑠,𝜏,𝑥)).

We consider the family of MSP {𝑈𝜎}𝜎Σ and define the map 𝑈Σ+𝑑×𝑋𝒫(𝑋) by 𝑈Σ(𝑡,𝜏,𝑥)=𝜎Σ𝑈𝜎(𝑡,𝜏,𝑥), which is also a multivalued semiprocess. For 𝐵𝑋, denote 𝛾𝜏𝑇,𝜎(𝐵)=𝑡𝑇𝑈𝜎(𝑡,𝜏,𝐵).(2.6)

Definition 2.5. The family of MSP {𝑈𝜎}𝜎Σ is called uniformly asymptoticall upper semicompact if for any 𝐵(𝑋) and 𝜏+ such that, for some 𝑇=𝑇(𝐵,𝜏),𝛾𝜏𝑇,Σ(𝐵)=𝜎Σ𝛾𝜏𝑇,𝜎(𝐵)(𝑋), any sequence {𝜉𝑛}, 𝜉𝑛𝑈𝜎𝑛(𝑡𝑛,𝜏,𝐵),𝜎𝑛Σ,𝑡𝑛+, is precompact in 𝑋.

Definition 2.6. The family of MSP {𝑈𝜎}𝜎Σ is called pointwise dissipative if there exists 𝐵0(𝑋) such that, for all 𝑥𝑋, 𝑈distΣ(𝑡,0,𝑥),𝐵00,as𝑡.(2.7)

Definition 2.7. Let 𝑋 and 𝑌 be two metric spaces. The multivalued map 𝐹𝑋𝑌 is said to be w-upper semicontinuous (w-u.s.c.) at 𝑥0 if for any 𝜖>0 there exists 𝛿>0 such that 𝐹(𝑥)𝑂𝜖𝐹𝑥0,𝑥𝑂𝛿𝑥0.(2.8) The map 𝐹 is w-u.s.c. if it is w-u.s.c. at any 𝑥𝐷(𝐹)={𝑦𝑋𝐹(𝑥)}.

Definition 2.8. The set 𝒜 is called a uniform global attractor for the family of multivalued semiprocesses 𝑈Σ if the following are satisfied.(1)It is negatively semiinvariant, that is, 𝒜𝑈Σ(𝑡,0,𝒜).(2)It is uniformly attracting, that is, dist(𝑈Σ(𝑡,𝜏,𝐵),𝒜)0, as 𝑡, for all 𝐵(𝑋) and 𝜏0.(3)For any closed uniformly attracting set 𝑌, we have 𝒜𝑌 (minimality).

The following result comes from [6, Theorem 2] and [10, Theorem 3.12].

Theorem 2.9. Let 𝐹(+,𝑍) be a space of functions with values in 𝑍, where 𝑍 is a topological space, and let Σ𝐹(+,𝑍) be a compact metric space. Suppose that the family of multivalued semiprocesses {𝑈𝜎}𝜎Σ satisfies the following conditions. (1)On Σ is defined the continuous shift operator 𝑇(𝑠)𝜎(𝑡)=𝜎(𝑡+𝑠),𝑠+ such that 𝑇()ΣΣ, and for any (𝑡,𝜏)+𝑑,𝜎Σ,𝑠+,𝑥𝑋, one has 𝑈𝜎(𝑡+𝑠,𝜏+𝑠,𝑥)=𝑈𝑇(𝑠)𝜎(𝑡,𝜏,𝑥).(2.9)(2)𝑈𝜎 is uniformly asymtopically upper semicompact.(3)𝑈𝜎 is pointwise dissipative.(4)The map (𝑥,𝜎)𝑈𝜎(𝑡,0,𝑥) has closed values and is 𝑤-upper semicontinuous.
Then the family of multivalued semiprocesses {𝑈𝜎}𝜎Σ has a uniform global compact attractor 𝒜. Moreover, if Σ is a connected space, the map (𝑥,𝜎)𝑈𝜎(𝑡,0,𝑥) is upper semicontinuous with connected values and the global attractor 𝒜 is contained in a connected bounded subset of 𝑋, then 𝒜 is a connected set.

3. Existence of Uniform Global Attractors

We denote 𝐴=div(𝜌(𝑥)),𝑉=𝐿2𝜏,𝑇;𝒟10(Ω,𝜌)𝐿𝑝(𝜏,𝑇;𝐿𝑝𝑉(Ω)),=𝐿2𝜏,𝑇;𝒟1(Ω,𝜌)+𝐿𝑝𝜏,𝑇;𝐿𝑝,(Ω)(3.1) where 𝑝 is the conjugate index of 𝑝. In what follows, we assume that 𝑢𝜏𝐿2(Ω) is given.

Definition 3.1. A function 𝑢(𝑥,𝑡) is called a weak solution of (1.1) on (𝜏,𝑇) if and only if 𝑢𝑉,𝜕𝑢𝜕𝑡𝑉,𝑢|𝑡=𝜏=𝑢𝜏,a.e.inΩ,𝑇𝜏𝑢𝑡,𝜑+𝑇𝜏Ω𝜌𝑢𝜑+𝑇𝜏𝑓(𝑥,𝑢),𝜑=𝑇𝜏(𝑔(𝑡),𝜑),(3.2) for all test functions 𝜑𝑉.

It follows from Theorem 1.8 in [1, page 33] that if 𝑢𝑉 and 𝑑𝑢/𝑑𝑡𝑉 then 𝑢𝐶([𝜏,𝑇];𝐿2(Ω)). This makes the initial condition in problem (1.1) meaningful.

Proposition 3.2. For any 𝜏+,𝑇>𝜏, and 𝑢𝜏𝐿2(Ω) given, problem (1.1) has at least one weak solution on (𝜏,𝑇).

Proof. The proof is classical, but we give some a priori estimates used later.
Consider the approximating solution 𝑢𝑛(𝑡) in the form𝑢𝑛(𝑡)=𝑛𝑘=1𝑢𝑛𝑘(𝑡)𝑒𝑘,(3.3) where {𝑒𝑗}𝑗=1 are the eigenvectors of the operator 𝐴=div(𝜌(𝑥)). We get 𝑢𝑛 from solving the problem 𝑑𝑢𝑛𝑑𝑡,𝑒𝑘+𝐴𝑢𝑛,𝑒𝑘𝑓+𝑥,𝑢𝑛,𝑒𝑘=𝑔,𝑒𝑘,𝑢𝑛(𝜏),𝑒𝑘=𝑢𝜏,𝑒𝑘,𝑘=1,,𝑛.(3.4) Using the Peano theorem, we get the local existence of 𝑢𝑛. We have 12𝑑𝑢𝑑𝑡𝑛2𝐿2(Ω)+𝑢𝑛2𝒟10(Ω,𝜌)+Ω𝑓𝑥,𝑢𝑛𝑢𝑛=Ω𝑔(𝑡)𝑢𝑛.(3.5) Using hypothesis (1.3) and the Cauchy inequality, we get 12𝑑𝑢𝑑𝑡𝑛2𝐿2(Ω)+𝑢𝑛2𝒟10(Ω,𝜌)+𝐶2𝑢𝑛𝑝𝐿𝑝(Ω)2𝐿1(Ω)12𝜆1𝑔(𝑡)2𝐿2(Ω)+𝜆12𝑢𝑛2𝐿2(Ω),(3.6) where 𝜆1>0 is the first eigenvalue of the operator 𝐴 in Ω with the homogeneous Dirichlet condition (noting that 𝑢2𝒟10(Ω,𝜌)𝜆1𝑢2𝐿2(Ω)). Hence 𝑑𝑢𝑑𝑡𝑛2𝐿2(Ω)+𝑢𝑛2𝒟10(Ω,𝜌)+2𝐶2𝑢𝑛𝑝𝐿𝑝(Ω)1𝜆1𝑔(𝑡)2𝐿2(Ω)+22𝐿1(Ω).(3.7) We show that the local solution 𝑢𝑛 can be extended to the interval [𝜏,). Indeed, from (3.7) we have 𝑑𝑢𝑑𝑡𝑛2𝐿2(Ω)+𝜆1𝑢𝑛2𝐿2(Ω)22𝐿1(Ω)+1𝜆1𝑔(𝑡)2𝐿2(Ω).(3.8) By the Gronwall inequality, we obtain 𝑢𝑛(𝑡)2𝐿2(Ω)𝑢𝑛(𝜏)2𝐿2(Ω)𝑒𝜆1(𝑡𝜏)+2𝜆11𝑒𝜆1(𝑡𝜏)2𝐿1(Ω)+1𝜆1𝑡𝜏𝑒𝜆1(𝑡𝑠)𝑔(𝑠)2𝐿2(Ω)𝑢𝑑𝑠𝑛(𝜏)2𝐿2(Ω)𝑒𝜆1(𝑡𝜏)+2𝜆11𝑒𝜆1(𝑡𝜏)2𝐿1(Ω)+1𝜆11𝑒𝜆1𝑔2𝐿2𝑏,(3.9) where we have used the fact that 𝑡𝜏𝑒𝜆1(𝑡𝑠)𝑔(𝑠)2𝐿2(Ω)𝑑𝑠𝑡𝑡1𝑒𝜆1(𝑡𝑠)𝑔(𝑠)2𝐿2(Ω)𝑑𝑠+𝑡1𝑡2𝑒𝜆1(𝑡𝑠)𝑔(𝑠)2𝐿2(Ω)𝑑𝑠+𝑡𝑡1𝑔(𝑠)2𝐿2(Ω)𝑑𝑠+𝑒𝜆1𝑡1𝑡2𝑔(𝑠)2𝐿2(Ω)𝑑𝑠+1+𝑒𝜆1+𝑒2𝜆1+𝑔2𝐿2𝑏=11𝑒𝜆1𝑔2𝐿2𝑏.(3.10) We now establish some a priori estimates for 𝑢𝑛. Integrating (3.7) on [𝜏,𝑡],𝜏<𝑡𝑇, we have 𝑢𝑛(𝑡)2𝐿2(Ω)+𝑡𝜏𝑢𝑛(𝑠)2𝒟10(Ω,𝜌)𝑑𝑠+2𝐶2𝑡𝜏𝑢𝑛(𝑠)𝑝𝐿𝑝(Ω)𝑢𝑑𝑠𝑛(𝜏)2𝐿2(Ω)+1𝜆1𝑡𝜏𝑔(𝑠)2𝐿2(Ω)𝑑𝑠+2(𝑡𝜏)2𝐿1(Ω).(3.11) The last inequality implies that 𝑢𝑛isboundedin𝐿𝜏,𝑇;𝐿2,𝑢(Ω)𝑛isboundedin𝐿2𝜏,𝑇;𝒟10,𝑢(Ω,𝜌)𝑛isboundedin𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω)).(3.12) Using hypothesis (1.2), one can prove that {𝑓(𝑥,𝑢𝑛)} is bounded in 𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω)). By rewriting the equation as 𝑑𝑢𝑛𝑑𝑡=𝐴𝑢𝑛𝑓𝑥,𝑢𝑛+𝑔,(3.13) we see that {𝑑𝑢𝑛/𝑑𝑡} is bounded in 𝑉 and, therefore, in 𝐿𝑝(𝜏,𝑇;𝒟1(Ω,𝜌)+𝐿𝑝(Ω)). Therefore, we have 𝑢𝑛𝑢in𝐿2𝜏,𝑇;𝒟10,(Ω,𝜌)𝑑𝑢𝑛𝑑𝑡𝑑𝑢𝑑𝑡in𝑉,𝑓𝑥,𝑢𝑛𝜂in𝐿𝑝𝜏,𝑇;𝐿𝑝,(Ω)𝐴𝑢𝑛𝐴𝑢in𝐿2𝜏,𝑇;𝒟1,(Ω,𝜌)(3.14) up to a subsequence. Hence by standard arguments [22, Chapter 1], one can show that 𝑢 is a weak solution of problem (1.1).

Denote by 𝒟𝜏,𝜎(𝑢𝜏) the set of all global weak solutions (defined for 𝑡𝜏) of problem (1.1) with the external force 𝑔𝜎 instead of 𝑔 and the initial datum 𝑢(𝜏)=𝑢𝜏. We put Σ=𝜔(𝑔), so it is clear that 𝑇(𝑠)ΣΣ, where 𝑇(𝑠)𝜎=𝜎(+𝑠)=𝑔(+𝑠),𝑡0, and that this map is continuous. For each 𝜎=𝑔𝜎Σ, we define the map. 𝑈𝜎𝑡,𝜏,𝑢𝜏=𝑢(𝑡)𝑢()𝒟𝜏,𝜎𝑢𝜏.(3.15)

Lemma 3.3. 𝑈𝜎(𝑡,𝜏,𝑢𝜏) is a strict multivalued semiprocess. Moreover, 𝑈𝜎𝑡+𝑠,𝜏+𝑠,𝑢𝜏=𝑈𝑇(𝑠)𝜎𝑡,𝜏,𝑢𝜏𝑢𝜏𝐿2(Ω),𝑠0,𝑡𝜏0.(3.16)

Proof. Given that 𝑧𝑈𝜎(𝑡,𝜏,𝑢𝜏), we have to prove that 𝑧𝑈𝜎(𝑡,𝑠,𝑈𝜎(𝑠,𝜏,𝑢𝜏)). Take 𝑦()𝒟𝜏,𝜎(𝑢𝜏) such that 𝑦(𝜏)=𝑢𝜏 and 𝑦(𝑡)=𝑧. Clearly, 𝑦(𝑠)𝑈𝜎(𝑠,𝜏,𝑢𝜏). Then if we define 𝑧(𝑡)=𝑦(𝑡) for 𝑡𝑠, then we have 𝑧(𝑠)=𝑦(𝑠) and obviously 𝑧()𝒟𝑠,𝜎(𝑦(𝑠)). Consequently, 𝑧(𝑡)𝑈𝜎(𝑡,𝑠,𝑈𝜎(𝑠,𝜏,𝑢𝜏)).
Let now 𝑧𝑈𝜎(𝑡,𝑠,𝑈𝜎(𝑠,𝜏,𝑢𝜏)). Then there exist 𝑢𝒟𝜏,𝜎(𝑢𝜏) and 𝑣𝒟𝑠,𝜎(𝑢(𝑠)) such that 𝑧=𝑣(𝑡). Define the function[],[].𝑦(𝑟)=𝑢(𝑟)if𝑟𝜏,𝑠𝑣(𝑟𝑠)if𝑟𝑠,𝑡+𝑠(3.17) It is easy to see that 𝑦𝒟𝜏,𝜎(𝑢𝜏), so that 𝑧=𝑦(𝑡+𝑠)𝑈𝜎(𝑡+𝑠,𝜏,𝑢𝜏).
Let 𝑧𝑈𝜎(𝑡+𝑠,𝜏+𝑠,𝑢𝜏). Then there exists 𝑢()𝒟𝜏+𝑠,𝜎(𝑢𝜏) such that 𝑧=𝑢(𝑡+𝑠) and 𝑣()=𝑢(+𝑠)𝒟𝜏,𝑇(𝑠)𝜎(𝑢𝜏), so that 𝑧=𝑣(𝑡)𝑈𝑇(𝑠)𝜎(𝑡,𝜏,𝑢𝜏). Conversely, if 𝑧𝑈𝜏,𝑇(𝑠)𝜎(𝑢𝜏), then there is 𝑧𝒟𝜏,𝑇(𝑠)𝜎(𝑢𝜏) such that 𝑧=𝑢(𝑡) and 𝑣()=𝑢(𝑠+)𝒟𝜏+𝑠,𝜎(𝑢𝜏) so that 𝑧=𝑣(𝑡+𝑠)𝑈𝜎(𝑡+𝑠,𝜏+𝑠,𝑢𝜏).

Lemma 3.4. Let conditions (𝐻1)–(𝐻3) hold and let {𝑢𝑛}𝒟𝜏,𝜎𝑛(𝑢𝑛(𝜏)) be an arbitrary sequence of solutions of (1.1) with initial data 𝑢𝑛(𝜏)𝜂 weakly in 𝐿2(Ω) and external forces 𝑔𝜎𝑛𝑔𝜎 in Σ. Then for any 𝑇>𝜏 and 𝑡𝑛𝑡0,𝑡𝑛,𝑡0(𝜏,𝑇], there exists a subsequence such that 𝑢𝑛(𝑡𝑛)𝑢(𝑡0) strongly in 𝐿2(Ω), where 𝑢()𝒟𝜏,𝜎(𝜂) is a weak solution of (1.1) with initial datum 𝑢(𝜏)=𝜂.

Proof. Repeating the proof of inequality (3.11), we see that the solution 𝑢𝑛 satisfies 𝑢𝑛(𝑡)2𝐿2(Ω)+𝑡𝑠𝑢𝑛(𝑣)2𝒟10(Ω,𝜌)𝑑𝑣+2𝐶2𝑡𝑠𝑢𝑛(𝑣)𝑝𝐿𝑝(Ω)𝑢𝑑𝑣𝑛(𝑠)2𝐿2(Ω)+1𝜆1𝑡𝑠𝑔𝜎𝑛(𝑣)2𝐿2(Ω)𝑑𝑣+2(𝑡𝑠)2𝐿1(Ω),(3.18) and a similar inequality holds for the solution 𝑢. Hence, by the arguments as in the proof of Proposition 3.2 and the Aubin-Lions lemma [22], we infer up to a subsequence that 𝑢𝑛𝑢in𝐿2𝜏,𝑇;𝒟10,𝑢(Ω,𝜌)𝑛𝑢in𝐿2𝜏,𝑇;𝐿2,𝑢(Ω)𝑛𝑢(𝑡,𝑥)𝑢(𝑡,𝑥)fora.a.(𝑡,𝑥)(𝜏,𝑇)×Ω,𝑛(𝑡)𝑢(𝑡)in𝐿2[],(Ω)uniformlyon𝜏,𝑇(3.19) where 𝑢𝒟𝜏,𝜎(𝜂).
Let now 𝑡𝑛𝑡0, with 𝑡𝑛,𝑡0(𝜏,𝑇]. We will prove that 𝑢𝑛(𝑡𝑛)𝑢(𝑡0) strongly in 𝐿2(Ω). Since 𝑢𝑛(𝑡𝑛)𝑢(𝑡0) weakly in 𝐿2(Ω), we have𝑢(𝑡0)2𝐿2(Ω)liminf𝑛𝑢𝑛(𝑡𝑛)2𝐿2(Ω).(3.20) Thus, if we can show that limsup𝑛𝑢𝑛(𝑡𝑛)2𝐿2(Ω)𝑢(𝑡0)2𝐿2(Ω), then the proof will be finished. It is easy to check that 𝑢𝑛 and 𝑢 satisfy the following inequalities: 𝑢𝑛(𝑡)2𝐿2(Ω)𝑢𝑛(𝑠)2𝐿2(Ω)+2(𝑡𝑠)2𝐿1(Ω)+𝑡𝑠𝑔𝜎𝑛(𝑣),𝑢𝑛(𝑣)𝑑𝑣,𝑢(𝑡)2𝐿2(Ω)𝑢(𝑠)2𝐿2(Ω)+2(𝑡𝑠)2𝐿1(Ω)+𝑡𝑠𝑔𝜎(𝑣),𝑢(𝑣)𝑑𝑠,(3.21) for all 𝑡𝑠,𝑡,𝑠[𝜏,𝑇]. Therefore, the functions 𝐽𝑛(𝑢𝑡)=𝑛(𝑡)2𝐿2(Ω)2𝑡2𝐿1(Ω)𝑡𝜏𝑔𝜎𝑛(𝑠),𝑢𝑛(𝑠)𝑑𝑠,𝐽(𝑡)=𝑢(𝑡)2𝐿2(Ω)2𝑡2𝐿1(Ω)𝑡𝜏𝑔𝜎(𝑠),𝑢(𝑠)𝑑𝑠(3.22) are continuous and nonincreasing on [𝜏,𝑇]. Moreover, 𝐽𝑛(𝑡)𝐽(𝑡) for a.a. 𝑡(𝜏,𝑇).
We now prove that limsup𝐽𝑛(𝑡𝑛)𝐽(𝑡0), and this will imply thatlimsup𝑛𝑢𝑛(𝑡𝑛)2𝐿2(Ω)𝑢(𝑡0)2𝐿2(Ω),(3.23) as desired. Indeed, suppose that {𝑡𝑚} is an increasing sequence in (𝜏,𝑡0) such that 𝐽𝑛(𝑡𝑚)𝐽(𝑡𝑚) as 𝑛. We can assume that 𝑡𝑚<𝑡𝑛, so that 𝐽𝑛𝑡𝑛𝑡𝐽0𝐽𝑛𝑡𝑚𝑡𝐽0||𝐽𝑛𝑡𝑚𝑡𝐽𝑚||+||𝐽𝑡𝑚𝑡𝐽0||.(3.24) Hence for any 𝜀>0, there exist 𝑡𝑚 and 𝑛0(𝑡𝑚) such that 𝐽𝑛(𝑡𝑛)𝐽(𝑡0)𝜀 for all 𝑛𝑛0, and the result follows.

Theorem 3.5. Let conditions (𝐻1)–(𝐻3) hold. Then the family of multivalued semiprocesses {𝑈𝜎(𝑡,𝜏)}𝜎Σ has a uniform global compact attractor 𝒜 in 𝐿2(Ω).

Proof. From (3.7), we obtain 𝑑𝑑𝑡𝑢(𝑡)2𝐿2(Ω)+𝑢(𝑡)2𝒟10(Ω,𝜌)+2𝐶2𝑢(𝑡)𝑝𝐿𝑝(Ω)1𝜆1𝑔(𝑡)2𝐿2(Ω)+22𝐿1(Ω).(3.25) Hence, similar to (3.9), we get 𝑢(𝑡)2𝐿2(Ω)𝑢(0)2𝐿2(Ω)𝑒𝜆1𝑡+2𝜆11𝑒𝜆1𝑡2𝐿1(Ω)+1𝜆11𝑒𝜆1𝑔2𝐿2𝑏.(3.26) The last inequality implies that there is a positive constant 𝑅 such that 𝑢(𝑡)2𝐿2(Ω)𝑢(0)2𝐿2(Ω)𝑒𝑡+𝑅2.(3.27) Hence the ball 𝐵0={𝑢𝐿2(Ω)𝑢𝐿2(Ω)𝑅2+𝜖} is an absorbing set for the map (𝑡,𝑢)𝑈Σ(𝑡,0,𝑢), that is, for any 𝐵(𝐿2(Ω)) there exists 𝑇(𝐵) such that 𝑈Σ(𝑡,0,𝐵)𝐵0, for all 𝑡𝑇(𝐵).
We define now the set 𝐾=𝑈Σ(1,0,𝐵0). Lemma 3.4 implies that 𝐾 is compact. Moreover, since 𝐵0 is absorbing, using Lemma 3.3 we have𝑈𝜎(𝑡,𝜏,𝐵)=𝑈𝜎𝑡,𝑡1,𝑈𝜎(𝑡1,𝜏,𝐵)=𝑈𝑇(𝑡1)𝜎1,0,𝑈𝑇(𝜏)𝜎(𝑡1𝜏,0,𝐵)𝑈Σ1,0,𝐵0𝐾,(3.28) for all 𝜎Σ,𝐵(𝐿2(Ω)), and 𝑡𝑇(𝐵,𝜏). It follows that any sequence {𝜉𝑛} such that {𝜉𝑛}𝑈𝜎𝑛(𝑡𝑛,𝜏,𝐵0),𝜎𝑛Σ,𝑡𝑛+, 𝐵(𝐿2(Ω)), is precompact in 𝐿2(Ω). It is a consequence of Lemma 3.4 that the map 𝑈𝜎 has compact values for any 𝜎Σ.
Finally, let us prove that the map (𝜎,𝑥)𝑈𝜎(𝑡,𝜏,𝑥) is upper semicontinuous for each fixed 𝑡𝜏0. Suppose that it is not true, that is, there exist 𝑢0𝐿2(Ω),𝑡𝜏0,𝜎0Σ,𝜖>0, 𝛿𝑛0,𝑢𝑛𝐵𝛿𝑛(𝑢0),𝜎𝑛𝜎0, and 𝜉𝑛𝑈𝜎𝑛(𝑡,𝜏,𝑢𝑛) such that {𝜉𝑛}𝐵𝜖(𝑈𝜎0(𝑡,𝜏,𝑢0)). But Lemma 3.4 implies (up to a subsequence) that 𝜉𝑛𝜉𝑈𝜎0(𝑡,𝜏,𝑢0), which is a contradiction. The existence of the uniform global compact attractor follows then from Theorem 2.9.

4. The Kneser Property and Connectedness of the Attractors

Let 𝒟𝜏,𝑇(𝑢𝜏) be the set of all weak solutions of the problem (1.1) on (𝜏,𝑇) with the initial datum 𝑢(𝜏)=𝑢𝜏. In this section we will check that the set 𝐾𝑡𝑢𝜏=𝑢(𝑡)𝑢()𝒟𝜏,𝑇𝑢𝜏(4.1) is connected in 𝐿2(Ω).

We define a sequence of smooth functions 𝜓𝑘+[0,1] satisfying𝜓𝑘(𝑠)=1if0𝑠𝑘,0𝜓𝑘(𝑠)1if𝑘𝑠𝑘+1,0if𝑠𝑘+1.(4.2) For every 𝑘1 we put 𝑓𝑘(𝑥,𝑢)=𝜓𝑘(|𝑢|)𝑓(𝑥,𝑢)+(1𝜓𝑘(|𝑢|))|𝑢|𝑝2𝑢. Then 𝑓𝑘(𝑥,)𝐶(,) and sup|𝑢|𝐴|𝑓𝑘(𝑥,𝑢)𝑓(𝑥,𝑢)|0 as 𝑘, forall𝐴>0.

Let 𝜌𝜖 be a mollifier, that is, 𝜌𝜖𝐶0(,) such that supp𝜌𝜖𝐵𝜖,𝜌𝜖(𝑠)𝑑𝑠=1,𝜌𝜖(𝑠)0, for all 𝑠, where 𝐵𝜖={𝑢|𝑢|𝜖}.

We define the functions 𝑓𝜖𝑘(𝑥,𝑢)=𝜌𝜖(𝑠)𝑓𝑘(𝑥,𝑢𝑠)𝑑𝑠. Since, for any 𝑘1, 𝑓𝑘(𝑥,.) is uniformly continuous on 𝐵𝑘+1, there exists 𝜖𝑘(0,1) such that, for any 𝑢 satisfying |𝑢|𝑘 and for all 𝑠 for which |𝑢𝑠|<𝜖𝑘, we have||𝑓𝑘(𝑥,𝑢)𝑓𝑘||1(𝑥,𝑠)𝑘.(4.3) We put 𝑓𝑘(𝑥,𝑢)=𝑓𝜖𝑘𝑘(𝑥,𝑢). Then 𝑓𝑘(𝑥,.)𝐶(,), for all 𝑘1.

We now prove the following lemma.

Lemma 4.1. For all 𝑘1, the following statements hold: sup|𝑢|𝐴||𝑓𝑘||||𝑓(𝑥,𝑢)𝑓(𝑥,𝑢)0,as𝑘,𝐴>0,(4.4)𝑘||(𝑥,𝑢)𝐷1|𝑢|𝑝1+𝐷2𝑓(𝑥),(4.5)𝑘(𝑥,𝑢)𝑢𝐷3|𝑢|𝑝𝐷4𝑓(𝑥),(4.6)𝑘𝑢(𝑥,𝑢)𝐷5(𝑘),𝑢,(4.7) where 𝐷2𝐿(Ω) and 𝐷4𝐿1(Ω) are nonnegative functions, 𝐷5(𝑘) is a nonnegative number, and the positive constants 𝐷1,𝐷3 do not depend on 𝑘.

Proof. Because of (4.3), for any 𝑢 such that |𝑢|<𝑘, we have ||𝑓𝑘(𝑥,𝑢)𝑓𝑘(||𝑥,𝑢)𝜌𝜖𝑘(||𝑓𝑢𝑠)𝑘(𝑥,𝑠)𝑓𝑘(||1𝑥,𝑢)𝑑𝑠𝑘,(4.8) we obtain that for any 𝐴>0 and any 𝑢 such that |𝑢|𝐴, ||𝑓𝑘||||𝑓(𝑥,𝑢)𝑓(𝑥,𝑢)𝑘(𝑥,𝑢)𝑓𝑘||+||𝑓(𝑥,𝑢)𝑘||(𝑥,𝑢)𝑓(𝑥,𝑢),𝑘𝐴.(4.9) Hence (4.4) holds.
We prove that 𝑓𝑘 satisfies conditions type of (𝐻2). Indeed, we have𝑓𝑘(𝑥,𝑢)𝑢=𝜓𝑘(|𝑢|)𝑓(𝑥,𝑢)𝑢+1𝜓𝑘(|𝑢|)|𝑢|𝑝2𝑢𝑢𝜓𝑘𝐶(|𝑢|)2|𝑢|𝑝2+(𝑥)1𝜓𝑘(|𝑢|)|𝑢|𝑝𝐶2|𝑢|𝑝2(𝑥),(4.10) where 𝐶2=min{1,𝐶2}, 2𝐿1(Ω), and ||𝑓𝑘||||||+||(𝑥,𝑢)𝐷𝑓(𝑥,𝑢)|𝑢|𝑝2𝑢||𝐶𝐷1|𝑢|𝑝1+1(𝑥)+|𝑢|𝑝1𝐶𝐷1+1|𝑢|𝑝1+𝐷1(𝑥),(4.11) for some constant 𝐷>0.
Now we check that 𝑓𝑘 satisfies (4.5) and (4.6). Using the above estimates for 𝑓𝑘, we have||𝑓𝑘||=||||(𝑥,𝑢)𝜌𝜖k(𝑠)𝑓𝑘||||(𝑥,𝑢𝑠)𝑑𝑠𝜌𝜖𝑘||𝑓(𝑠)𝑘||(𝑥,𝑢𝑠)𝑑𝑠𝜌𝜖𝑘𝐷𝐶(𝑠)1+1|𝑢𝑠|𝑝1+𝐷1(𝑥)𝑑𝑠𝜌𝜖𝑘(𝑠)𝐶|𝑢|𝑝1+𝐶|𝑠|𝑝1+𝐷1(𝑥)𝑑𝑠𝜌𝜖𝑘𝐶(𝑠)|𝑢|𝑝1+𝐶||𝜖𝑘||𝑝1+𝐷1(𝑥)𝑑𝑠𝐷1|𝑢|𝑝1+𝐷2(𝑥),where𝐷2𝐿(Ω).(4.12) On the other hand, using Young's inequality and the estimates for 𝑓𝑘, we obtain 𝑓𝑘(𝑥,𝑢)𝑢=𝜌𝜖𝑘(𝑠)𝑓𝑘(𝑥,𝑢𝑠)(𝑢𝑠)𝑑𝑠+𝜌𝜖𝑘(𝑠)𝑓𝑘(𝑥,𝑢𝑠)𝑠𝑑𝑠𝜌𝜖𝑘𝐶(𝑠)2|𝑢𝑠|𝑝2(𝑥)𝑑𝑠𝜌𝜖𝑘𝐶(𝑠)2𝐶2𝐷1||𝑓+1𝑘||(𝑢𝑠)𝑝+𝐾0|𝑠|𝑝𝑑𝑠𝜌𝜖𝑘𝐶(𝑠)2|𝑢𝑠|𝑝2(𝑥)𝑑𝑠𝜌𝜖𝑘𝐶(𝑠)22|𝑢𝑠|𝑝+𝐷𝑝1(𝑥)+𝐾0||𝜖𝑘||𝑝𝑑𝑠𝜌𝜖𝑘(𝐶𝑠)22|𝑢𝑠|𝑝𝑑𝑠2(𝑥)𝐷𝑝1(𝐾𝑥)0𝐷3|𝑢|𝑝𝐷4(𝑥),with𝐷4𝐿1(Ω),(4.13) where in the last inequality we have used the fact that, for some 𝑀>0, |𝑢|𝑝=|𝑢𝑠+𝑠|𝑝𝑀(|𝑢𝑠|𝑝+|𝑠|𝑝)𝑀|𝑢𝑠|𝑝+𝜖𝑝𝑘.(4.14)
Let us show that 𝑓𝑘𝑢(𝑥,𝑢)𝐶5(𝑘),forall𝑢. Indeed, if |𝑢|>𝑘+1, we have𝑓𝑘𝑢(𝑥,𝑢)=𝜕𝑓𝑘(𝑥,𝑢)=𝜕𝜕𝑢|𝑢|𝑝2𝑢𝜕𝑢=(𝑝1)|𝑢|𝑝20.(4.15) Then for |𝑢|>𝑘+2, we get 𝑓𝑘𝑢(𝑥,𝑢)=𝜌𝜖𝑘(𝑠)𝑓𝑘𝑢(𝑥,𝑢)𝑑𝑠0.(4.16) Finally, if |𝑢|𝑘+2, we have ||𝑓𝑘𝑢(||𝑥,𝑢)||𝜌𝜖𝑘𝑢(||||𝑓𝑢𝑠)𝑘(||𝑥,𝑠)𝑑𝑠𝐷5(𝑘),(4.17) where we have used the above estimate for 𝑓𝑘 with noting that 1𝐿(Ω).

Remark 4.2. In fact, if we are concerned with the existence of the uniform global compact attractor in 𝐿2(Ω), we only need to assume that 1𝐿𝑝(Ω). The stronger assumption, namely, 1𝐿(Ω), is used to prove (4.7), which is nessceary for the proof of the Kneser property and the connectedness of the uniform global attractor.

We are now in a position to prove the following theorem.

Theorem 4.3. The set 𝐾𝑡(𝑢𝜏) is connected in 𝐿2(Ω) for any 𝑡[𝜏,𝑇].

Proof. The proof is quite standard (see, e.g., [9, 15]), so we only give its sketch.
The case 𝑡=𝜏 is obvious. Suppose then, that for some 𝑡(𝜏,𝑇], the set 𝐾𝑡(𝑢𝜏) is not connected. Then there exist two compact sets 𝐴1,𝐴2 in 𝐿2(Ω) such that 𝐴1𝐴2=𝐾𝑡(𝑢𝜏), 𝐴1𝐴2. Let 𝑢1,𝑢2𝒟𝜏,𝑇(𝑢𝜏) be such that 𝑢1(𝑡)𝑈1, 𝑢2(𝑡)𝑈2, where 𝑈1,𝑈2 are disjoint open neighborhoods of 𝐴1,𝐴2, respectively.
Let 𝑢𝑘𝑖(𝑡,𝛾),𝑖=1,2, be equal to 𝑢𝑖(𝑡) if 𝑡[𝜏,𝛾] and be a solution of the problem𝜕𝑢𝜕𝑡div(𝜎(𝑥)𝑢)+𝑓𝑘(𝑥,𝑢)=𝑔(𝑡,𝑥),(𝑡,𝑥)(𝛾,𝑇)×Ω,𝑢|𝜕Ω=0,𝑢|𝑡=𝛾=𝑢𝑖(𝛾,𝑥)(4.18) if 𝑡[𝛾,𝑇]. Since Lemma 4.1 and Proposition 3.2, problem (4.18) has at least one weak solution. It follows from (4.7) that this solution is unique. Also, the maps 𝑢𝑘𝑖(𝑡,𝛾) are continuous on 𝛾 for each fixed 𝑘1 and 𝑡[𝜏,𝑇]. For details of the proof of these facts, see, for example, [9]. Using (4.6), one can prove that the functions 𝑢𝑘𝑖 satisfy the estimate 𝑢𝑘𝑖(𝑡)2𝐿2(Ω)+2𝑡𝛾𝑢𝑘𝑖(𝑠)2𝒟10(Ω,𝜎)𝑑𝑠+2𝐷3𝑡𝛾𝑢𝑘𝑖(𝑠)𝑝𝐿𝑝(Ω)𝑢𝑘𝑖(𝛾)2𝐿2(Ω)+𝐾𝑡𝛾𝑔(𝑠)2𝐿2(Ω).𝑑𝑠+1(4.19)
Now we put[],[]𝛾(𝜆)=𝜏(𝑇𝜏)𝜆if𝜆1,0𝜏+(𝑇𝜏)𝜆if𝜆0,1(4.20) and define the function 𝜑𝑘𝑢(𝜆)(𝑡)=𝑘1[],𝑢(𝑡,𝛾(𝜆))if𝜆1,0𝑘2[].(𝑡,𝛾(𝜆))if𝜆0,1(4.21) We have 𝜑𝑘(1)(𝑡)=𝑢𝑘1(𝑡,𝑇)=𝑢1(𝑡), 𝜑𝑘(1)(𝑡)=𝑢𝑘2(𝑡,𝑇)=𝑢2(𝑡). The map 𝜆𝜑𝑘(𝜆)(𝑡) is continuous for any fixed 𝑘1, 𝑡[𝜏,𝑇] (note that 𝑢𝑘1(𝑡,𝜏)=𝑢𝑘2(𝑡,𝜏)) and 𝜑𝑘(1)(𝑡)𝑈1, 𝜑𝑘(1)(𝑡)𝑈2, so that there exists 𝜆𝑘[1,1] such that 𝜑𝑘(𝜆𝑘)(𝑡)𝑈1𝑈2.
Denote 𝑢𝑘(𝑡)=𝜑𝑘(𝜆𝑘)(𝑡). Note that, for each 𝑘1, we have either 𝑢𝑘(𝑡)=𝑢𝑘1(𝑡,𝛾(𝜆𝑘)) or 𝑢𝑘(𝑡)=𝑢𝑘2(𝑡,𝛾(𝜆𝑘)). For some subsequence it is equal to one of them; say 𝑢𝑘1(𝑡,𝛾(𝜆𝑘)). Now we will consider the function 𝑢𝑘1(𝑡,𝛾(𝜆𝑘)),𝑡[𝜏,𝑇]. We have𝑢𝑘𝑢(𝑡)=1𝜆(𝑡)if𝑡𝜏,𝛾𝑘,𝑢𝑘1𝜆𝑡,𝛾𝑘𝛾𝜆(𝑡)if𝑡𝑘,,𝑇(4.22) where 𝛾(𝜆𝑘)𝛾0[𝜏,𝑇]. We define the function 𝑓𝑘𝜆(𝑥,𝑣(𝑡))=𝑓(𝑥,𝑣(𝑡))if𝑡𝜏,𝛾𝑘,𝑓𝑘𝛾𝜆(𝑥,𝑣(𝑡))if𝑡𝑘.,𝑇(4.23) By the continuity, 𝑢1(𝛾(𝜆𝑘))𝑢1(𝛾0) as 𝑘. Moreover, from (4.18), (4.5), and (4.6), one can show that {𝑢𝑘} is bounded in 𝑉, {𝑓𝑘(𝑥,𝑢𝑘)} is bounded in 𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ω)), and, therefore, {𝑑𝑢𝑘/𝑑𝑡} is bounded in 𝐿𝑝(𝜏,𝑇;𝑉). By the Aubin-Lions lemma [22], we can prove that 𝑢𝑘 converges weakly to a weak solution 𝑢 of (1.1) satisfying 𝑢(𝜏)=𝑢𝜏.
Finally, we can prove also that 𝑢𝑘(𝑡)𝑢(𝑡) in 𝐿2(Ω) (see again [9]). From this we immediately obtain that 𝑢(𝑡)𝑈1𝑈2, which is a contradiction. This completes the proof.

Corollary 4.4. If 𝐴𝐿2(Ω) is an arbitrary connected set, then 𝐾𝑡(𝐴)=𝑢(𝑡)𝑢()𝒟𝜏,𝑇(𝐴)𝐿2(Ω)(4.24) is connected.

Proof. The proof is similar to the one of Corollary 7 in [9], so we omit it.

We now give the proof of the main theorem.

Proof of Theorem 1.1. Under conditions (𝐻1)–(𝐻3), we have shown the existence of the uniform global compact attractor 𝒜 of the family of processes {𝑈𝜎}𝜎Σ in Theorem 3.5. In addition, 𝑈𝜎 is upper semicontinuous and has the compact values. Moreover, 𝑈𝜎 has the connected values since Theorem 4.3. Since both 𝐿2(Ω) and Σ are connected, 𝒜 is connected in 𝐿2(Ω).

Acknowledgments

The authors would like to thank the reviewers for their helpful comments and suggestions. This work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).

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