# Attractors for Nonautonomous Parabolic Equations without Uniqueness

Authors: Cung The Anh; Nguyen Dinh Binh
Journal: International Journal of Differential Equations
 (2010)

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## 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.


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## Body

## 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 [7–9], 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, 13–15], 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 domainΩ⊂ℝN,N≥2:(1.1)∂u∂t-div(ρ(x)∇u)+f(x,u)=g(x,t),x∈Ω,t>τ,u|t=τ=uτ(x),x∈Ω,u|∂Ω=0,
where uτ∈L2(Ω) is given, and the coefficient ρ, the nonlinearity f, and the external force g satisfy the following conditions.(H1)
The functionρ:Ω→ℝ satisfies ρ∈L1(Ω) and, for some α∈(0,2), limx→zinf|x-z|-αρ(x)>0 for every z∈Ω¯.(H2)
f:Ω×ℝ→ℝ is a Caratheodory function, that is, the function f(·,u) is measurable and the function f(x,·) is continuous, and satisfies
(1.2)|f(x,u)|≤C1|u|p-1+h1(x),forsomep≥2,(1.3)uf(x,u)≥C2|u|p-h2(x),
where C1,C2 are positive constants; h1,h2 are nonnegative functions such that h1∈L∞(Ω) and h2∈L1(Ω) (see Remark 4.2 on a comment about conditions of h1).(H3)
g∈Lb2(ℝ+,L2(Ω)), where Lb2(ℝ+,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 caseg independent of time t, the existence and long-time behavior of solutions to problem (1.1) have been studied in [17–20]. 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 σ∈Σ=ℋw(g), the closure of the set {g(·+h)∣h∈ℝ+} in Lloc2(ℝ+,L2(Ω)) with the weak topology, we define the multivalued semiprocess Uσ:ℝd+×L2(Ω)↦2L2(Ω) as follows: (1.4)Uσ(t,τ,uτ)={u(t):u(·)∈𝒟τ,σ(uτ)}.
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}σ∈Σ. 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}σ∈Σ, which possesses a uniform global compact connected attractor 𝒜 in L2(Ω).It is worth noticing that under some additional conditions onf, for example, fu′(x,u)≥-C3 for all x∈Ω,u∈ℝ, or a weaker assumption (1.5)(f(x,u)-f(x,v))(u-v)⩾-C|u-v|2∀x∈Ω,u,v∈ℝ,
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 Section2, 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. LetN≥2, α∈(0,2), and(2.1)2α*={4α∈(2,∞)ifN=2,2NN-2+α∈(2,2NN-2)ifN≥3.
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 𝒟01(Ω,ρ), defined as the closure of C0∞(Ω) with respect to the norm (2.2)∥u∥𝒟01(Ω,ρ):=(∫Ωρ(x)|∇u|2dx)1/2.
The space 𝒟01(Ω,ρ) is a Hilbert space with respect to the scalar product (2.3)(u,v):=∫Ωρ(x)∇u∇vdx.
The following lemma comes from [21, Propositions 3.3–3.5].Lemma 2.1.
Assume thatΩ is a bounded domain in ℝN, N≥2, and ρ satisfies (H1). Then the following embeddings hold:(i)
𝒟01(Ω,ρ)↪L2α*(Ω) continuously,(ii)
𝒟01(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 that(2.4)(ej,ek)=δjk,-div(ρ(x)∇ej)=λjej,j,k=1,2,…,0<λ1≤λ2≤λ3≤…,λj→+∞,asj→∞.


### 2.2. The Translation-Bounded Functions

Definition 2.2.
Letℰ be a reflexive Banach space. A function φ∈Lloc2(ℝ+,ℰ) is said to be translation-bounded if
(2.5)∥φ∥Lb22=∥φ∥Lb2(ℝ+,ℰ)=supt∈ℝ+∫tt+1∥φ∥ℰ2ds<∞.We will denote byLb2(ℝ+,ℰ) the set of all translation-bounded functions in Lloc2(ℝ+,ℰ). Let g∈Lb2(ℝ+,ℰ) and ℋw(g) be the closure of the set {g(·+h)∣h∈ℝ+} in Lloc2(ℝ+,ℰ) with the weak topology. The following results are well-known.Lemma 2.3 (see [1, Chapter 5, Proposition 4.2]).
(1) For all σ∈ℋw(g),∥σ∥Lb22≤∥g∥Lb22.
(2) The translation group {T(h)} is weakly continuous on ℋw(g).
(3)T(h)ℋw(g)⊂ℋw(g) for h≥0.
(4)ℋw(g) is weakly compact.


### 2.3. Uniform Attractors of Multivalued Semiprocesses

Denoteℝd+={(t,τ):0≤τ≤t}. Let X be a complete metric space, let 𝒫(X) and ℬ(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 mapU:ℝd+×X→𝒫(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  x∈X,t,s,τ∈ℝ+,τ≤s≤t.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}σ∈Σ and define the map UΣ:ℝd+×X→𝒫(X) by UΣ(t,τ,x)=⋃σ∈ΣUσ(t,τ,x), which is also a multivalued semiprocess. For B⊂X, denote (2.6)γT,στ(B)=∪t≥TUσ(t,τ,B).Definition 2.5.
The family of MSP{Uσ}σ∈Σ is called uniformly asymptoticall upper semicompact if for any B∈ℬ(X) and τ∈ℝ+ such that, for some T=T(B,τ),γT,Στ(B)=⋃σ∈ΣγT,στ(B)∈ℬ(X), any sequence {ξn}, ξn∈Uσn(tn,τ,B),σn∈Σ,tn→+∞, is precompact in X.Definition 2.6.
The family of MSP{Uσ}σ∈Σ is called  pointwise dissipative if there exists B0∈ℬ(X) such that, for all x∈X,
(2.7)dist(UΣ(t,0,x),B0)→0,ast→∞.Definition 2.7.
LetX and Y be two metric spaces. The multivalued map F:X→Y is said to be w-upper semicontinuous (w-u.s.c.) at x0 if for any ϵ>0 there exists δ>0 such that
(2.8)F(x)⊂Oϵ(F(x0)),∀x∈Oδ(x0).
The map F is w-u.s.c. if it is w-u.s.c. at any x∈D(F)={y∈X:F(x)≠∅}.Definition 2.8.
The set𝒜 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,𝒜⊂UΣ(t,0,𝒜).(2)
It is uniformly attracting, that is, dist(UΣ(t,τ,B),𝒜)→0, as t→∞, for all B∈ℬ(X) and τ≥0.(3)
For any closed uniformly attracting setY, we have 𝒜⊂Y (minimality).The following result comes from [6, Theorem 2] and [10, Theorem 3.12].Theorem 2.9.
LetF(ℝ+,Z) be a space of functions with values in Z, where Z is a topological space, and let Σ⊂F(ℝ+,Z) be a compact metric space. Suppose that the family of multivalued semiprocesses {Uσ}σ∈Σ satisfies the following conditions.(1)
OnΣ is defined the continuous shift operator T(s)σ(t)=σ(t+s),s∈ℝ+ such that T(h)Σ⊂Σ, and for any (t,τ)∈ℝd+,σ∈Σ,s∈ℝ+,x∈X, one has
(2.9)Uσ(t+s,τ+s,x)=UT(s)σ(t,τ,x).
(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}σ∈Σ 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.


## 2.1. Function Space and Operator

We recall some basic results on the function space which we will use. LetN≥2, α∈(0,2), and(2.1)2α*={4α∈(2,∞)ifN=2,2NN-2+α∈(2,2NN-2)ifN≥3.
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 𝒟01(Ω,ρ), defined as the closure of C0∞(Ω) with respect to the norm (2.2)∥u∥𝒟01(Ω,ρ):=(∫Ωρ(x)|∇u|2dx)1/2.
The space 𝒟01(Ω,ρ) is a Hilbert space with respect to the scalar product (2.3)(u,v):=∫Ωρ(x)∇u∇vdx.
The following lemma comes from [21, Propositions 3.3–3.5].Lemma 2.1.
Assume thatΩ is a bounded domain in ℝN, N≥2, and ρ satisfies (H1). Then the following embeddings hold:(i)
𝒟01(Ω,ρ)↪L2α*(Ω) continuously,(ii)
𝒟01(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 that(2.4)(ej,ek)=δjk,-div(ρ(x)∇ej)=λjej,j,k=1,2,…,0<λ1≤λ2≤λ3≤…,λj→+∞,asj→∞.


## 2.2. The Translation-Bounded Functions

Definition 2.2.
Letℰ be a reflexive Banach space. A function φ∈Lloc2(ℝ+,ℰ) is said to be translation-bounded if
(2.5)∥φ∥Lb22=∥φ∥Lb2(ℝ+,ℰ)=supt∈ℝ+∫tt+1∥φ∥ℰ2ds<∞.We will denote byLb2(ℝ+,ℰ) the set of all translation-bounded functions in Lloc2(ℝ+,ℰ). Let g∈Lb2(ℝ+,ℰ) and ℋw(g) be the closure of the set {g(·+h)∣h∈ℝ+} in Lloc2(ℝ+,ℰ) with the weak topology. The following results are well-known.Lemma 2.3 (see [1, Chapter 5, Proposition 4.2]).
(1) For all σ∈ℋw(g),∥σ∥Lb22≤∥g∥Lb22.
(2) The translation group {T(h)} is weakly continuous on ℋw(g).
(3)T(h)ℋw(g)⊂ℋw(g) for h≥0.
(4)ℋw(g) is weakly compact.


## 2.3. Uniform Attractors of Multivalued Semiprocesses

Denoteℝd+={(t,τ):0≤τ≤t}. Let X be a complete metric space, let 𝒫(X) and ℬ(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 mapU:ℝd+×X→𝒫(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  x∈X,t,s,τ∈ℝ+,τ≤s≤t.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}σ∈Σ and define the map UΣ:ℝd+×X→𝒫(X) by UΣ(t,τ,x)=⋃σ∈ΣUσ(t,τ,x), which is also a multivalued semiprocess. For B⊂X, denote (2.6)γT,στ(B)=∪t≥TUσ(t,τ,B).Definition 2.5.
The family of MSP{Uσ}σ∈Σ is called uniformly asymptoticall upper semicompact if for any B∈ℬ(X) and τ∈ℝ+ such that, for some T=T(B,τ),γT,Στ(B)=⋃σ∈ΣγT,στ(B)∈ℬ(X), any sequence {ξn}, ξn∈Uσn(tn,τ,B),σn∈Σ,tn→+∞, is precompact in X.Definition 2.6.
The family of MSP{Uσ}σ∈Σ is called  pointwise dissipative if there exists B0∈ℬ(X) such that, for all x∈X,
(2.7)dist(UΣ(t,0,x),B0)→0,ast→∞.Definition 2.7.
LetX and Y be two metric spaces. The multivalued map F:X→Y is said to be w-upper semicontinuous (w-u.s.c.) at x0 if for any ϵ>0 there exists δ>0 such that
(2.8)F(x)⊂Oϵ(F(x0)),∀x∈Oδ(x0).
The map F is w-u.s.c. if it is w-u.s.c. at any x∈D(F)={y∈X:F(x)≠∅}.Definition 2.8.
The set𝒜 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,𝒜⊂UΣ(t,0,𝒜).(2)
It is uniformly attracting, that is, dist(UΣ(t,τ,B),𝒜)→0, as t→∞, for all B∈ℬ(X) and τ≥0.(3)
For any closed uniformly attracting setY, we have 𝒜⊂Y (minimality).The following result comes from [6, Theorem 2] and [10, Theorem 3.12].Theorem 2.9.
LetF(ℝ+,Z) be a space of functions with values in Z, where Z is a topological space, and let Σ⊂F(ℝ+,Z) be a compact metric space. Suppose that the family of multivalued semiprocesses {Uσ}σ∈Σ satisfies the following conditions.(1)
OnΣ is defined the continuous shift operator T(s)σ(t)=σ(t+s),s∈ℝ+ such that T(h)Σ⊂Σ, and for any (t,τ)∈ℝd+,σ∈Σ,s∈ℝ+,x∈X, one has
(2.9)Uσ(t+s,τ+s,x)=UT(s)σ(t,τ,x).
(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}σ∈Σ 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(3.1)A=-div(ρ(x)∇),V=L2(τ,T;𝒟01(Ω,ρ))∩Lp(τ,T;Lp(Ω)),V*=L2(τ,T;𝒟-1(Ω,ρ))+Lp′(τ,T;Lp′(Ω)),
where p' is the conjugate index of p. In what follows, we assume that uτ∈L2(Ω) is given.Definition 3.1.
A functionu(x,t) is called a weak solution of (1.1) on (τ,T) if and only if
(3.2)u∈V,∂u∂t∈V*,u|t=τ=uτ,a.e.inΩ,∫τT〈ut,φ〉+∫τT∫Ωρ∇u∇φ+∫τT〈f(x,u),φ〉=∫τT(g(t),φ),
for all test functions φ∈V.It follows from Theorem1.8 in [1, page 33] that if u∈V and du/dt∈V* then u∈C([τ,T];L2(Ω)). This makes the initial condition in problem (1.1) meaningful.Proposition 3.2.
For anyτ∈ℝ+,T>τ, and uτ∈L2(Ω) given, problem (1.1) has at least one weak solution on (τ,T).Proof.
The proof is classical, but we give somea priori estimates used later.
Consider the approximating solutionun(t) in the form(3.3)un(t)=∑k=1nunk(t)ek,
where {ej}j=1∞ are the eigenvectors of the operator A=-div(ρ(x)∇). We get un from solving the problem
(3.4)〈dundt,ek〉+〈Aun,ek〉+〈f(x,un),ek〉=(g,ek),(un(τ),ek)=(uτ,ek),k=1,…,n.
Using the Peano theorem, we get the local existence of un. We have
(3.5)12ddt∥un∥L2(Ω)2+∥un∥𝒟01(Ω,ρ)2+∫Ωf(x,un)un=∫Ωg(t)un.
Using hypothesis (1.3) and the Cauchy inequality, we get
(3.6)12ddt∥un∥L2(Ω)2+∥un∥𝒟01(Ω,ρ)2+C2∥un∥Lp(Ω)p-∥h2∥L1(Ω)≤12λ1∥g(t)∥L2(Ω)2+λ12∥un∥L2(Ω)2,
where λ1>0 is the first eigenvalue of the operator A in Ω with the homogeneous Dirichlet condition (noting that ∥u∥𝒟01(Ω,ρ)2≥λ1∥u∥L2(Ω)2). Hence
(3.7)ddt∥un∥L2(Ω)2+∥un∥𝒟01(Ω,ρ)2+2C2∥un∥Lp(Ω)p≤1λ1∥g(t)∥L2(Ω)2+2∥h2∥L1(Ω).
We show that the local solution un can be extended to the interval [τ,∞). Indeed, from (3.7) we have
(3.8)ddt∥un∥L2(Ω)2+λ1∥un∥L2(Ω)2≤2∥h2∥L1(Ω)+1λ1∥g(t)∥L2(Ω)2.
By the Gronwall inequality, we obtain
(3.9)∥un(t)∥L2(Ω)2⩽∥un(τ)∥L2(Ω)2e-λ1(t-τ)+2λ1(1-e-λ1(t-τ))∥h2∥L1(Ω)+1λ1∫τte-λ1(t-s)∥g(s)∥L2(Ω)2ds⩽∥un(τ)∥L2(Ω)2e-λ1(t-τ)+2λ1(1-e-λ1(t-τ))∥h2∥L1(Ω)+1λ1(1-e-λ1)∥g∥Lb22,
where we have used the fact that
(3.10)∫τte-λ1(t-s)∥g(s)∥L2(Ω)2ds⩽∫t-1te-λ1(t-s)∥g(s)∥L2(Ω)2ds+∫t-2t-1e-λ1(t-s)∥g(s)∥L2(Ω)2ds+⋯⩽∫t-1t∥g(s)∥L2(Ω)2ds+e-λ1∫t-2t-1∥g(s)∥L2(Ω)2ds+⋯⩽(1+e-λ1+e-2λ1+⋯)∥g∥Lb22=11-e-λ1∥g∥Lb22.
We now establish some a priori estimates for un. Integrating (3.7) on [τ,t],τ<t≤T, we have
(3.11)∥un(t)∥L2(Ω)2+∫τt∥un(s)∥𝒟01(Ω,ρ)2ds+2C2∫τt∥un(s)∥Lp(Ω)pds≤∥un(τ)∥L2(Ω)2+1λ1∫τt∥g(s)∥L2(Ω)2ds+2(t-τ)∥h2∥L1(Ω).
The last inequality implies that
(3.12){un}isboundedinL∞(τ,T;L2(Ω)),{un}isboundedinL2(τ,T;𝒟01(Ω,ρ)),{un}isboundedinLp(τ,T;Lp(Ω)).
Using hypothesis (1.2), one can prove that {f(x,un)} is bounded in Lp'(τ,T;Lp'(Ω)). By rewriting the equation as
(3.13)dundt=-Aun-f(x,un)+g,
we see that {dun/dt} is bounded in V* and, therefore, in Lp'(τ,T;𝒟-1(Ω,ρ)+Lp'(Ω)). Therefore, we have
(3.14)un⇀uinL2(τ,T;𝒟01(Ω,ρ)),dundt⇀dudtinV*,f(x,un)⇀ηinLp′(τ,T;Lp′(Ω)),Aun⇀AuinL2(τ,T;𝒟-1(Ω,ρ)),
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 by𝒟τ,σ(uτ) the set of all global weak solutions (defined for t≥τ) of problem (1.1) with the external force gσ instead of g and the initial datum u(τ)=uτ. We put Σ=ℋω(g), so it is clear that T(s)Σ⊂Σ, where T(s)σ=σ(·+s)=g(·+s),t≥0, and that this map is continuous. For each σ=gσ∈Σ, we define the map. (3.15)Uσ(t,τ,uτ)={u(t):u(·)∈𝒟τ,σ(uτ)}.Lemma 3.3.
Uσ(t,τ,uτ) is a strict multivalued semiprocess. Moreover,
(3.16)Uσ(t+s,τ+s,uτ)=UT(s)σ(t,τ,uτ)∀uτ∈L2(Ω),s≥0,t≥τ≥0.Proof.
Given thatz∈Uσ(t,τ,uτ), we have to prove that z∈Uσ(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 t≥s, 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 nowz∈Uσ(t,s,Uσ(s,τ,uτ)). Then there exist u∈𝒟τ,σ(uτ) and v∈𝒟s,σ(u(s)) such that z=v(t). Define the function(3.17)y(r)={u(r)ifr∈[τ,s],v(r-s)ifr∈[s,t+s].
It is easy to see that y∈𝒟τ,σ(uτ), so that z=y(t+s)∈Uσ(t+s,τ,uτ).
Letz∈Uσ(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 z∈Uτ,T(s)σ(uτ), then there is z∈𝒟τ,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 gσn→gσ in Σ. Then for any T>τ and tn→t0,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
(3.18)∥un(t)∥L2(Ω)2+∫st∥un(v)∥𝒟01(Ω,ρ)2dv+2C2∫st∥un(v)∥Lp(Ω)pdv≤∥un(s)∥L2(Ω)2+1λ1∫st∥gσn(v)∥L2(Ω)2dv+2(t-s)∥h2∥L1(Ω),
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
(3.19)un⇀uinL2(τ,T;𝒟01(Ω,ρ)),un→uinL2(τ,T;L2(Ω)),un(t,x)→u(t,x)fora.a.(t,x)∈(τ,T)×Ω,un(t)⇀u(t)inL2(Ω)uniformlyon[τ,T],
where u∈𝒟τ,σ(η).
Let nowtn→t0, 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 have(3.20)∥u(t0)∥L2(Ω)2≤lim infn→∞∥un(tn)∥L2(Ω)2.
Thus, if we can show that lim supn→∞∥un(tn)∥L2(Ω)2≤∥u(t0)∥L2(Ω)2, then the proof will be finished. It is easy to check that un and u satisfy the following inequalities:
(3.21)∥un(t)∥L2(Ω)2≤∥un(s)∥L2(Ω)2+2(t-s)∥h2∥L1(Ω)+∫st(gσn(v),un(v))dv,∥u(t)∥L2(Ω)2≤∥u(s)∥L2(Ω)2+2(t-s)∥h2∥L1(Ω)+∫st(gσ(v),u(v))ds,
for all t≥s,t,s∈[τ,T]. Therefore, the functions
(3.22)Jn(t)=∥un(t)∥L2(Ω)2-2t∥h2∥L1(Ω)-∫τt(gσn(s),un(s))ds,J(t)=∥u(t)∥L2(Ω)2-2t∥h2∥L1(Ω)-∫τt(gσ(s),u(s))ds
are continuous and nonincreasing on [τ,T]. Moreover, Jn(t)→J(t) for a.a. t∈(τ,T).
We now prove thatlim supJn(tn)≤J(t0), and this will imply that(3.23)lim supn→∞∥un(tn)∥L2(Ω)2≤∥u(t0)∥L2(Ω)2,
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
(3.24)Jn(tn)-J(t0)≤Jn(tm)-J(t0)≤|Jn(tm)-J(tm)|+|J(tm)-J(t0)|.
Hence for any ɛ>0, there exist tm and n0(tm) such that Jn(tn)-J(t0)≤ɛ for all n≥n0, and the result follows.Theorem 3.5.
Let conditions (H1)–(H3) hold. Then the family of multivalued semiprocesses {Uσ(t,τ)}σ∈Σ has a uniform global compact attractor 𝒜 in L2(Ω).Proof.
From (3.7), we obtain
(3.25)ddt∥u(t)∥L2(Ω)2+∥u(t)∥𝒟01(Ω,ρ)2+2C2∥u(t)∥Lp(Ω)p≤1λ1∥g(t)∥L2(Ω)2+2∥h2∥L1(Ω).
Hence, similar to (3.9), we get
(3.26)∥u(t)∥L2(Ω)2⩽∥u(0)∥L2(Ω)2e-λ1t+2λ1(1-e-λ1t)∥h2∥L1(Ω)+1λ1(1-e-λ1)∥g∥Lb22.
The last inequality implies that there is a positive constant R such that
(3.27)∥u(t)∥L2(Ω)2≤∥u(0)∥L2(Ω)2e-t+R2.
Hence the ball B0={u∈L2(Ω):∥u∥L2(Ω)≤R2+ϵ} is an absorbing set for the map (t,u)↦UΣ(t,0,u), that is, for any B∈ℬ(L2(Ω)) there exists T(B) such that UΣ(t,0,B)⊂B0, for all t≥T(B).
We define now the setK=UΣ(1,0,B0)¯. Lemma 3.4 implies that K is compact. Moreover, since B0 is absorbing, using Lemma 3.3 we have(3.28)Uσ(t,τ,B)=Uσ(t,t-1,Uσ(t-1,τ,B))=UT(t-1)σ(1,0,UT(τ)σ(t-1-τ,0,B))⊂UΣ(1,0,B0)⊂K,
for all σ∈Σ,B∈ℬ(L2(Ω)), and t≥T(B,τ). It follows that any sequence {ξn} such that {ξn}∈Uσn(tn,τ,B0),σn∈Σ,tn→+∞, B∈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 σ∈Σ.
Finally, let us prove that the map(σ,x)↦Uσ(t,τ,x) is upper semicontinuous for each fixed t≥τ≥0. Suppose that it is not true, that is, there exist u0∈L2(Ω),t≥τ≥0,σ0∈Σ,ϵ>0, δn→0,un∈Bδn(u0),σn→σ0, and ξn∈Uσn(t,τ,un) such that {ξn}∉Bϵ(Uσ0(t,τ,u0)). But Lemma 3.4 implies (up to a subsequence) that ξn→ξ∈Uσ0(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

Let𝒟τ,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 (4.1)Kt(uτ)={u(t):u(·)∈𝒟τ,T(uτ)}
is connected in L2(Ω).We define a sequence of smooth functionsψk:ℝ+→[0,1] satisfying(4.2)ψk(s)={1if0≤s≤k,0≤ψk(s)≤1ifk≤s≤k+1,0ifs≥k+1.
For every k≥1 we put fk(x,u)=ψk(|u|)f(x,u)+(1-ψk(|u|))|u|p-2u. Then fk(x,·)∈C(ℝ,ℝ) and sup|u|≤A|fk(x,u)-f(x,u)|→0 as k→∞, forallA>0.Letρϵ:ℝ→ℝ be a mollifier, that is, ρϵ∈C0∞(ℝ,ℝ) such that suppρϵ⊂Bϵ,∫ℝρϵ(s)ds=1,ρϵ(s)≥0, for all s∈ℝ, where Bϵ={u∈ℝ:|u|≤ϵ}.We define the functionsfkϵ(x,u)=∫ℝρϵ(s)fk(x,u-s)ds. Since, for any k≥1, 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 |u-s|<ϵk, we have(4.3)|fk(x,u)-fk(x,s)|≤1k.
We put fk(x,u)=fkϵk(x,u). Then fk(x,.)∈C∞(ℝ,ℝ), for all k≥1.We now prove the following lemma.Lemma 4.1.
For allk≥1, the following statements hold:
(4.4)sup|u|≤A|fk(x,u)-f(x,u)|→0,ask→∞,∀A>0,(4.5)|fk(x,u)|≤D1|u|p-1+D2(x),(4.6)fk(x,u)u≥D3|u|p-D4(x),(4.7)fuk(x,u)≥-D5(k),∀u∈ℝ,
where D2∈L∞(Ω) and D4∈L1(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
(4.8)|fk(x,u)-fk(x,u)|≤∫ℝρϵk(u-s)|fk(x,s)-fk(x,u)|ds≤1k,
we obtain that  for any A>0 and any u such that |u|≤A,
(4.9)|fk(x,u)-f(x,u)|≤|fk(x,u)-fk(x,u)|+|fk(x,u)-f(x,u)|,∀k≥A.
Hence (4.4) holds.
We prove thatfk satisfies conditions type of (H2). Indeed, we have(4.10)fk(x,u)u=ψk(|u|)f(x,u)u+(1-ψk(|u|))(|u|p-2u)u≥ψk(|u|)(C2|u|p-h2(x))+(1-ψk(|u|))|u|p≥C̃2|u|p-h̃2(x),
where C̃2=min{1,C2}, h̃2∈L1(Ω), and
(4.11)|fk(x,u)|≤D(|f(x,u)|+||u|p-2u|)≤D(C1|u|p-1+h1(x)+|u|p-1)≤D(C1+1)|u|p-1+Dh1(x),
for some constant D>0.
Now we check thatfk satisfies (4.5) and (4.6). Using the above estimates for fk, we have(4.12)|fk(x,u)|=|∫ℝρϵk(s)fk(x,u-s)ds|≤∫ℝρϵk(s)|fk(x,u-s)|ds≤∫ℝρϵk(s)(D(C1+1)|u-s|p-1+Dh1(x))ds≤∫ℝρϵk(s)(C̃|u|p-1+C̃|s|p-1+Dh1(x))ds≤∫ℝρϵk(s)(C̃|u|p-1+C̃|ϵk|p-1+Dh1(x))ds≤D1|u|p-1+D2(x),whereD2∈L∞(Ω).
On the other hand, using Young's inequality and the estimates for fk, we obtain
(4.13)fk(x,u)u=∫ℝρϵk(s)fk(x,u-s)(u-s)ds+∫ℝρϵk(s)fk(x,u-s)sds≥∫ℝρϵk(s)(C̃2|u-s|p-h̃2(x))ds-∫ℝρϵk(s)(C̃22D(C1+1)|fk(u-s)|p′+K0|s|p)ds≥∫ℝρϵk(s)(C̃2|u-s|p-h̃2(x))ds-∫ℝρϵk(s)(C̃22|u-s|p+D̃h1p′(x)+K0|ϵk|p)ds≥∫ℝρϵk(s)C̃22|u-s|pds-h̃2(x)-D̃h1p′(x)-K̃0≥D3|u|p-D4(x),withD4∈L1(Ω),
where in the last inequality we have used the fact that, for some M>0,
(4.14)|u|p=|u-s+s|p≤M(|u-s|p+|s|p)≤M(|u-s|p+ϵkp).
Let us show thatfuk(x,u)≥-C5(k),forallu∈ℝ. Indeed, if |u|>k+1, we have(4.15)fku(x,u)=∂fk(x,u)∂u=∂(|u|p-2u)∂u=(p-1)|u|p-2≥0.
Then for |u|>k+2, we get
(4.16)fuk(x,u)=∫ℝρϵk(s)fku(x,u)ds≥0.
Finally, if |u|≤k+2, we have
(4.17)|fuk(x,u)|≤∫ℝ|ρϵku(u-s)||fk(x,s)|ds≤D5(k),
where we have used the above estimate for fk with noting that h1∈L∞(Ω).Remark 4.2.
In fact, if we are concerned with the existence of the uniform global compact attractor inL2(Ω), we only need to assume that h1∈Lp′(Ω). The stronger assumption, namely, h1∈L∞(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 setKt(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 caset=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 A1∪A2=Kt*(uτ), A1∩A2≠∅. Let u1,u2∈𝒟τ,T(uτ) be such that u1(t*)∈U1, u2(t*)∈U2, where U1,U2 are disjoint open neighborhoods of A1,A2, respectively.
Letuik(t,γ),i=1,2, be equal to ui(t) if t∈[τ,γ] and be a solution of the problem(4.18)∂u∂t-div(σ(x)∇u)+fk(x,u)=g(t,x),(t,x)∈(γ,T)×Ω,u|∂Ω=0,u|t=γ=ui(γ,x)
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 uik(t,γ) are continuous on γ for each fixed k≥1 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 uik satisfy the estimate
(4.19)∥uik(t)∥L2(Ω)2+2∫γt∥uik(s)∥𝒟01(Ω,σ)2ds+2D3∫γt∥uik(s)∥Lp(Ω)p≤∥uik(γ)∥L2(Ω)2+K(∫γt∥g(s)∥L2(Ω)2ds+1).
Now we put(4.20)γ(λ)={τ-(T-τ)λifλ∈[-1,0],τ+(T-τ)λifλ∈[0,1]
and define the function
(4.21)φk(λ)(t)={u1k(t,γ(λ))ifλ∈[-1,0],u2k(t,γ(λ))ifλ∈[0,1].
We have φk(-1)(t)=u1k(t,T)=u1(t), φk(1)(t)=u2k(t,T)=u2(t). The map λ↦φk(λ)(t) is continuous for any fixed k≥1, t∈[τ,T] (note that u1k(t,τ)=u2k(t,τ)) and φk(-1)(t*)∈U1, φk(1)(t*)∈U2, so that there exists λk∈[-1,1] such that φk(λk)(t*)∉U1∪U2.
Denoteuk(t)=φk(λk)(t). Note that, for each k≥1, we have either uk(t)=u1k(t,γ(λk)) or uk(t)=u2k(t,γ(λk)). For some subsequence it is equal to one of them; say u1k(t,γ(λk)). Now we will consider the function u1k(t,γ(λk)),t∈[τ,T]. We have(4.22)uk(t)={u1(t)ift∈[τ,γ(λk)],u1k(t,γ(λk(t)))ift∈[γ(λk),T],
where γ(λk)→γ0∈[τ,T]. We define the function
(4.23)f̃k(x,v(t))={f(x,v(t))ift∈[τ,γ(λk)],fk(x,v(t))ift∈(γ(λk),T].
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, {f̃k(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 thatuk(t*)→u(t*) in L2(Ω) (see again [9]). From this we immediately obtain that u(t*)∉U1∪U2, which is a contradiction. This completes the proof.Corollary 4.4.
IfA⊂L2(Ω) is an arbitrary connected set, then
(4.24)Kt(A)={u(t):u(·)∈𝒟τ,T(A)}⊂L2(Ω)
is connected.Proof.
The proof is similar to the one of Corollary7 in [9], so we omit it.We now give the proof of the main theorem.Proof of Theorem1.1.
Under conditions (H1)–(H3), we have shown the existence of the uniform global compact attractor 𝒜 of the family of processes {Uσ}σ∈Σ 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).



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