International Journal of Differential Equations

Volume 2010, Article ID 103510, 17 pages

http://dx.doi.org/10.1155/2010/103510

## Attractors for Nonautonomous Parabolic Equations without Uniqueness

^{1}Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, 10307 Hanoi, Vietnam^{2}Department 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 [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 is given, and the coefficient , the nonlinearity , and the external force satisfy the following conditions.()The function satisfies and, for some , for every () is a Caratheodory function, that is, the function is measurable and the function is continuous, and satisfies where are positive constants; are nonnegative functions such that and (see Remark 4.2 on a comment about conditions of ).(), where 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 is allowed to have at most a finite number of (essential) zeroes at some points. The physical motivation of the assumption is related to the modelling of reaction diffusion processes in composite materials, occupying a bounded domain , 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 vanishes at these points. Note that, in various diffusion processes, the equation involves diffusion of the type .

In the autonomous case, which is the case independent of time , 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 to be dependent on time . Moreover, the conditions imposed on the nonlinearity provide global existence of a weak solution to problem (1.1), but not uniqueness. Let be the set of all global weak solutions of problem (1.1) with the external force instead of and initial datum For each , the closure of the set in with the weak topology, we define the multivalued semiprocess as follows: We prove that 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 . 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 . We summarize our main results in the following theorem.

Theorem 1.1. *Under conditions ()–(), problem (1.1) defines a family of strict multivalued semiprocesses , which possesses a uniform global compact connected attractor in .*

It is worth noticing that under some additional conditions on , for example, for all , or a weaker assumption one can prove that the weak solution of problem (1.1) is unique. Then the multivalued semiprocess turns to be a single-valued one and the uniform compact global attractor 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 , , and The exponent has the role of the critical exponent in the classical Sobolev embedding.

The natural energy space for problem (1.1) involves the space , defined as the closure of with respect to the norm The space is a Hilbert space with respect to the scalar product The following lemma comes from [21, Propositions 3–3.].

Lemma 2.1. *Assume that is a bounded domain in , , and satisfies . Then the following embeddings hold: *(i)* continuously,*(ii)* compactly if .*

It is known (see [19]) that there exists a complete orthonormal system of eigenvectors of the operator such that

##### 2.2. The Translation-Bounded Functions

*Definition 2.2. *Let be a reflexive Banach space. A function is said to be translation-bounded if

We will denote by the set of all translation-bounded functions in . Let and be the closure of the set in with the weak topology. The following results are well-known.

Lemma 2.3 (see [1, Chapter 5, Proposition ]). * For all . ** The translation group is weakly continuous on . ** for . ** is weakly compact.*

##### 2.3. Uniform Attractors of Multivalued Semiprocesses

Denote . Let be a complete metric space, let and be the set of all nonempty subsets and the set of all nonempty bounded subsets of the space , respectively, and let be a compact metric space.

*Definition 2.4. *The map is called a multivalued semiprocess (MSP) if (1) (the identity map) (2) for all It is called a strict multivalued semiprocess if .

We consider the family of MSP and define the map by , which is also a multivalued semiprocess. For , denote

*Definition 2.5. *The family of MSP is called uniformly asymptoticall upper semicompact if for any and such that, for some , any sequence , , is precompact in .

*Definition 2.6. *The family of MSP is called pointwise dissipative if there exists such that, for all ,

*Definition 2.7. *Let and be two metric spaces. The multivalued map is said to be w-upper semicontinuous (w-u.s.c.) at if for any there exists such that
The map is w-u.s.c. if it is w-u.s.c. at any .

*Definition 2.8. *The set is called a uniform global attractor for the family of multivalued semiprocesses if the following are satisfied.()It is negatively semiinvariant, that is, .()It is uniformly attracting, that is, dist, as for all and .()For any closed uniformly attracting set , we have (minimality).

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

Theorem 2.9. *Let be a space of functions with values in , where is a topological space, and let be a compact metric space. Suppose that the family of multivalued semiprocesses satisfies the following conditions. *()*On is defined the continuous shift operator such that , and for any , one has
() is uniformly asymtopically upper semicompact.() is pointwise dissipative.()The map has closed values and is -upper semicontinuous.*

*Then the family of multivalued semiprocesses has a uniform global compact attractor . Moreover, if is a connected space, the map is upper semicontinuous with connected values and the global attractor is contained in a connected bounded subset of , then is a connected set.*

#### 3. Existence of Uniform Global Attractors

We denote where is the conjugate index of . In what follows, we assume that is given.

*Definition 3.1. *A function is called a weak solution of (1.1) on if and only if
for all test functions .

It follows from Theorem in [1, page 33] that if and then . This makes the initial condition in problem (1.1) meaningful.

Proposition 3.2. *For any and given, problem (1.1) has at least one weak solution on .*

*Proof. *The proof is classical, but we give some * a priori* estimates used later.

Consider the approximating solution in the form
where are the eigenvectors of the operator . We get from solving the problem
Using the Peano theorem, we get the local existence of . We have
Using hypothesis (1.3) and the Cauchy inequality, we get
where is the first eigenvalue of the operator in with the homogeneous Dirichlet condition (noting that ). Hence
We show that the local solution can be extended to the interval . Indeed, from (3.7) we have
By the Gronwall inequality, we obtain
where we have used the fact that
We now establish some * a priori* estimates for . Integrating (3.7) on we have
The last inequality implies that
Using hypothesis (1.2), one can prove that is bounded in . By rewriting the equation as
we see that is bounded in and, therefore, in . Therefore, we have
up to a subsequence. Hence by standard arguments [22, Chapter 1], one can show that is a weak solution of problem (1.1).

Denote by the set of all global weak solutions (defined for ) of problem (1.1) with the external force instead of and the initial datum . We put , so it is clear that , where , and that this map is continuous. For each , we define the map.

Lemma 3.3. * is a strict multivalued semiprocess. Moreover,
*

*Proof. *Given that , we have to prove that . Take such that and . Clearly, . Then if we define for , then we have and obviously . Consequently, .

Let now . Then there exist and such that . Define the function
It is easy to see that , so that .

Let . Then there exists such that and , so that . Conversely, if , then there is such that and so that .

Lemma 3.4. *Let conditions ()–() hold and let be an arbitrary sequence of solutions of (1.1) with initial data weakly in and external forces in . Then for any and , there exists a subsequence such that strongly in , where is a weak solution of (1.1) with initial datum .*

*Proof. *Repeating the proof of inequality (3.11), we see that the solution satisfies
and a similar inequality holds for the solution . 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
where .

Let now with . We will prove that strongly in . Since weakly in , we have
Thus, if we can show that , then the proof will be finished. It is easy to check that and satisfy the following inequalities:
for all . Therefore, the functions
are continuous and nonincreasing on . Moreover, for a.a. .

We now prove that , and this will imply that
as desired. Indeed, suppose that is an increasing sequence in such that as . We can assume that , so that
Hence for any , there exist and such that for all , and the result follows.

Theorem 3.5. *Let conditions ()–() hold. Then the family of multivalued semiprocesses has a uniform global compact attractor in .*

*Proof. *From (3.7), we obtain
Hence, similar to (3.9), we get
The last inequality implies that there is a positive constant such that
Hence the ball is an absorbing set for the map , that is, for any there exists such that , for all .

We define now the set . Lemma 3.4 implies that is compact. Moreover, since is absorbing, using Lemma 3.3 we have
for all , and . It follows that any sequence such that , , is precompact in . It is a consequence of Lemma 3.4 that the map has compact values for any .

Finally, let us prove that the map is upper semicontinuous for each fixed . Suppose that it is not true, that is, there exist , , and such that . But Lemma 3.4 implies (up to a subsequence) that , 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 be the set of all weak solutions of the problem (1.1) on with the initial datum In this section we will check that the set is connected in .

We define a sequence of smooth functions satisfying For every we put . Then and as , .

Let be a mollifier, that is, such that , for all , where .

We define the functions . Since, for any , is uniformly continuous on , there exists such that, for any satisfying and for all for which , we have We put . Then , for all .

We now prove the following lemma.

Lemma 4.1. *For all , the following statements hold:
**
where and are nonnegative functions, is a nonnegative number, and the positive constants do not depend on .*

*Proof. *Because of (4.3), for any such that , we have
we obtain that for any and any such that ,
Hence (4.4) holds.

We prove that satisfies conditions type of . Indeed, we have
where , , and
for some constant .

Now we check that satisfies (4.5) and (4.6). Using the above estimates for , we have
On the other hand, using Young's inequality and the estimates for , we obtain
where in the last inequality we have used the fact that, for some ,

Let us show that . Indeed, if , we have
Then for , we get
Finally, if , we have
where we have used the above estimate for with noting that .

*Remark 4.2. *In fact, if we are concerned with the existence of the uniform global compact attractor in , we only need to assume that . The stronger assumption, namely, , 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 is connected in for any .*

*Proof. *The proof is quite standard (see, e.g., [9, 15]), so we only give its sketch.

The case is obvious. Suppose then, that for some , the set is not connected. Then there exist two compact sets in such that , . Let be such that , , where are disjoint open neighborhoods of , respectively.

Let , be equal to if and be a solution of the problem
if . 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 are continuous on for each fixed and . For details of the proof of these facts, see, for example, [9]. Using (4.6), one can prove that the functions satisfy the estimate

Now we put
and define the function
We have , . The map is continuous for any fixed , (note that ) and , , so that there exists such that .

Denote . Note that, for each , we have either or . For some subsequence it is equal to one of them; say . Now we will consider the function . We have
where . We define the function
By the continuity, as . Moreover, from (4.18), (4.5), and (4.6), one can show that is bounded in , is bounded in , and, therefore, is bounded in . By the Aubin-Lions lemma [22], we can prove that converges weakly to a weak solution of (1.1) satisfying .

Finally, we can prove also that in (see again [9]). From this we immediately obtain that , which is a contradiction. This completes the proof.

Corollary 4.4. * If is an arbitrary connected set, then
**
is connected.*

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

We now give the proof of the main theorem.

*Proof of Theorem 1.1. *Under conditions ()–(), we have shown the existence of the uniform global compact attractor of the family of processes in Theorem 3.5. In addition, is upper semicontinuous and has the compact values. Moreover, has the connected values since Theorem 4.3. Since both and are connected, is connected in .

#### 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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