Attractors for Nonautonomous Parabolic Equations without Uniqueness

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.


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.

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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 domain where u τ ∈ L 2 Ω is given, and the coefficient ρ, the nonlinearity f, and the external force g satisfy the following conditions.
• is continuous, and satisfies 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 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 D τ,σ u τ be the set of all global weak solutions of problem 1.1 with the external force g σ instead of g and initial datum u τ u τ .For each σ ∈ Σ H w g , the closure of the set {g with the weak topology, we define the multivalued semiprocess U σ : R d × L 2 Ω → 2 L 2 Ω as follows: We prove that U σ 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 {U σ } 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 L 2 Ω .We summarize our main results in the following theorem.It is worth noticing that under some additional conditions on f, for example, f u x, u ≥ −C 3 for all x ∈ Ω, u ∈ R, or a weaker assumption one can prove that the weak solution of problem 1.1 is unique.Then the multivalued semiprocess U σ 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.

Function Space and Operator
We recall some basic results on the function space which we will use.Let N ≥ 2, α ∈ 0, 2 , and

2.1
The exponent 2 * α has the role of the critical exponent in the classical Sobolev embedding.The natural energy space for problem 1.1 involves the space D 1 0 Ω, ρ , defined as the closure of C ∞ 0 Ω with respect to the norm

2.2
The space D 1 0 Ω, ρ is a Hilbert space with respect to the scalar product The following lemma comes from 21, Propositions 3.3-3.5 .
Lemma 2.1.Assume that Ω is a bounded domain in R N , N ≥ 2, and ρ satisfies H1 .Then the following embeddings hold: It is known see 19 that there exists a complete orthonormal system of eigenvectors e j , λ j of the operator A div ρ x ∇ such that e j , e k δ jk , − div ρ x ∇e j λ j e j , j,k 1, 2, . . ., 2.4

The Translation-Bounded Functions
2.5 We will denote by with the weak topology.The following results are well-known.
4 H w g is weakly compact.

Uniform Attractors of Multivalued Semiprocesses
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.
We consider the family of MSP {U σ } σ∈Σ and define the map U Σ : R d × X → P X by U Σ t, τ, x σ∈Σ U σ t, τ, x , which is also a multivalued semiprocess.For B ⊂ X, denote

2.6
Definition 2.5.The family of MSP {U σ } σ∈Σ is called uniformly asymptoticall upper semicompact if for any B ∈ B X and τ ∈ R such that, for some Definition 2.7.Let X and Y be two metric spaces.The multivalued map F : X → Y is said to be w-upper semicontinuous w-u.s.c. at x 0 if for any > 0 there exists δ > 0 such that The map F is w-u.s.c.if it is w-u.s.c. at any x ∈ D F {y ∈ X : F x / ∅}.

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Definition 2.8.The set A is called a uniform global attractor for the family of multivalued semiprocesses U Σ if the following are satisfied.
2 It is uniformly attracting, that is, dist U Σ t, τ, B , A → 0, as t → ∞, for all B ∈ B X and τ ≥ 0.
3 For any closed uniformly attracting set Y , we have A ⊂ Y minimality .
The following result comes from 6, Theorem 2 and 10, Theorem 3.12 .
Theorem 2.9.Let F 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 {U σ } σ∈Σ satisfies the following conditions.
1 On Σ 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 ∈ R d , σ ∈ Σ, s ∈ R , x ∈ X, one has 2 U σ is uniformly asymtopically upper semicompact.
3 U σ is pointwise dissipative.4 The map x, σ → U σ t, 0, x has closed values and is w-upper semicontinuous.
Then the family of multivalued semiprocesses {U σ } σ∈Σ has a uniform global compact attractor A. Moreover, if Σ is a connected space, the map x, σ → U σ 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.

Existence of Uniform Global Attractors
We denote where p is the conjugate index of p.In what follows, we assume that u τ ∈ L 2 Ω is given.

3.4
Using the Peano theorem, we get the local existence of u n .We have 1 2

3.5
Using hypothesis 1.3 and the Cauchy inequality, we get 1 2 where λ 1 > 0 is the first eigenvalue of the operator A in Ω with the homogeneous Dirichlet condition noting that u 2 We show that the local solution u n can be extended to the interval τ, ∞ .Indeed, from 3.7 we have

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By the Gronwall inequality, we obtain 3.9 where we have used the fact that

3.10
We now establish some a priori estimates for u n .Integrating 3.7 on τ, t , τ < t ≤ T, we have

3.11
The last inequality implies that

3.12
Using hypothesis 1.2 , one can prove that {f x, u n } is bounded in L p τ, T; L p Ω .By rewriting the equation as we see that {du n /dt} is bounded in V * and, therefore, in L p τ, T; D −1 Ω, ρ L p Ω .Therefore, we have Denote by D τ,σ u τ the set of all global weak solutions defined for t ≥ τ of problem 1.1 with the external force g σ instead of g and the initial datum u τ u τ .We put Σ H ω 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 g σ ∈ Σ, we define the map.

3.17
It is easy to see that y ∈ D τ,σ u τ , so that z y t s ∈ U σ t s, τ, u τ .Let z ∈ U σ t s, τ s, u τ .Then there exists u • ∈ D τ s,σ u τ such that z u t s and v Lemma 3.4.Let conditions (H1)-(H3) hold and let {u n } ⊂ D τ,σ n u n τ be an arbitrary sequence of solutions of 1.1 with initial data u n τ → η weakly in L 2 Ω and external forces g σ n → g σ in Σ.Then for any T > τ and t n → t 0 , t n , t 0 ∈ τ, T , there exists a subsequence such that u n t n → u t 0 strongly in L 2 Ω , where u • ∈ D τ,σ η is a weak solution of 1.1 with initial datum u τ η.

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Proof.Repeating the proof of inequality 3.11 , we see that the solution u n satisfies 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
where u ∈ D τ,σ η .Let now t n → t 0 , with t n , t 0 ∈ τ, T .We will prove that u n t n → u t 0 strongly in L 2 Ω .Since u n t n → u t 0 weakly in L 2 Ω , we have 3.20 Thus, if we can show that lim sup n → ∞ u n t n 2 , then the proof will be finished.It is easy to check that u n and u satisfy the following inequalities: for all t ≥ s, t, s ∈ τ, T .Therefore, the functions are continuous and nonincreasing on τ, T .Moreover, J n t → J t for a.a.t ∈ τ, T .We now prove that lim sup J n t n ≤ J t 0 , and this will imply that lim sup as desired.Indeed, suppose that {t m } is an increasing sequence in τ, t 0 such that J n t m → J t m as n → ∞.We can assume that t m < t n , so that

3.24
Hence for any ε > 0, there exist t m and n 0 t m such that J n t n − J t 0 ≤ ε for all n ≥ n 0 , and the result follows.Proof.From 3.7 , we obtain Hence, similar to 3.9 , we get

3.26
The last inequality implies that there is a positive constant R such that

3.27
Hence the ball } is an absorbing set for the map t, u → U Σ t, 0, u , that is, for any B ∈ B L 2 Ω there exists T B such that U Σ t, 0, B ⊂ B 0 , for all t ≥ T B .
We define now the set K U Σ 1, 0, B 0 .Lemma 3.4 implies that K is compact.Moreover, since B 0 is absorbing, using Lemma 3.3 we have It is a consequence of Lemma 3.4 that the map U σ has compact values for any σ ∈ Σ.

The Kneser Property and Connectedness of the Attractors
Let D τ,T u τ be the set of all weak solutions of the problem 1.1 on τ, T with the initial datum u τ u τ .In this section we will check that the set We define a sequence of smooth functions ψ k : R → 0, 1 satisfying

4.2
For every k ≥ 1 we put Let ρ : R → R be a mollifier, that is, ρ ∈ C ∞ 0 R, R such that supp ρ ⊂ B , R ρ s ds 1, ρ s ≥ 0, for all s ∈ R, where B {u ∈ R : |u| ≤ }.We define the functions f k x, u R ρ s f k x, u − s ds.Since, for any k ≥ 1, f k x, . is uniformly continuous on B k 1 , there exists k ∈ 0, 1 such that, for any u satisfying |u| ≤ k and for all s for which |u − s| < k , we have We put f k x, u f k k x, u .Then f k x, .∈ C ∞ R, R , for all k ≥ 1.We now prove the following lemma.Lemma 4.1.For all k ≥ 1, the following statements hold: Proof.Because of 4.3 , for any u such that |u| < k, we have we obtain that for any A > 0 and any u such that |u| ≤ A, Hence 4.4 holds.We prove that f k satisfies conditions type of H2 .Indeed, we have where C 2 min{1, C 2 }, h 2 ∈ L 1 Ω , and for some constant D > 0. Now we check that f k satisfies 4.5 and 4.6 .Using the above estimates for f k , we have

International Journal of Differential Equations
On the other hand, using Young's inequality and the estimates for f k , we obtain where in the last inequality we have used the fact that, for some M > 0, Then for |u| > k 2, we get Finally, if |u| ≤ k 2, we have where we have used the above estimate for f k with noting that h 1 ∈ L ∞ Ω .
Remark 4.2.In fact, if we are concerned with the existence of the uniform global compact attractor in L 2 Ω , we only need to assume that h 1 ∈ L p Ω .The stronger assumption, namely, h 1 ∈ 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.

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Theorem 4.3.The set K t u τ is connected in L 2 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 K t * u τ is not connected.Then there exist two compact sets

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 .

. 7 where D 2 ∈
L ∞ Ω and D 4 ∈ L 1 Ω are nonnegative functions, D 5 k is a nonnegative number, and the positive constants D 1 , D 3 do not depend on k.
Definition 3.1.A function u x, t is called a weak solution of 1.1 on τ, T if and only ifIt follows from Theorem 1.8 in 1, page 33 that if u ∈ V and du/dt ∈ V * then u ∈ C τ, T ; L 2 Ω .This makes the initial condition in problem 1.1 meaningful.For any τ ∈ R , T > τ, and u τ ∈ L 2 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 u n t in the form τ , e k , k 1, . . ., n.
and t ≥ T B, τ .It follows that any sequence {ξ n } such that {ξ n u 1 , u 2 ∈ D τ,T u τ be such that u 1 t * ∈ U 1 , u 2 t * ∈ U 2 , where U 1 , U 2 are disjoint open neighborhoods of A 1 , A 2 , respectively.Let u k i t, γ , i 1, 2, be equal to u i t if t ∈ τ, γ and be a solution of the problem ∈ 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 u k i 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 u k i satisfy the estimate .The map λ → ϕ k λ t is continuous for anyfixed k ≥ 1, t ∈ τ, T note that u k 1 t, τ u k 2 t, τ and ϕ k −1 t * ∈ U 1 , ϕ k 1 t * ∈ U 2 , so that there exists λ k ∈ −1, 1 such that ϕ k λ k t * / ∈ U 1 ∪ U 2 .Denote u k t ϕ k λ k t .Note that, for each k ≥ 1, we have eitheru k t u k 1 t, γ λ k or u k t u k 2 t, γ λ k .For some subsequence it is equal to one of them; say u k 1 t, γ λ k .Now we will consider the function u k 1 t, γ λ k , t ∈ τ, T .We have k t if t ∈ γ λ k , T , 4.22