Research Article | Open Access
Cung The Anh, Nguyen Duong Toan, "Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 875913, 30 pages, 2012. https://doi.org/10.1155/2012/875913
Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains
The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation , in a noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor , which is upper semicontinuous at .
In recent years, the evolution equations on noncylindrical domains, that is, spatial domains which vary in time so their Cartesian products with the time variable are noncylindrical sets, have been investigated extensively (see, e.g., [1–3]).Much of the progress has been made for nested spatial domains which expand in time.However, the results focus mainly on formulation of the problems and existence and uniqueness theory, while the existence of attractors of such systems has been less considered, except some recent works for the reaction-diffusion equation (or the heat equation) [4, 5]. This is not really surprising since such systems are intrinsically nonautonomous even if the equations themselves contain no time-dependent terms and require the concept of a nonautonomous attractor, which has only been introduced in recent years.
In this paper, we consider a class of nonautonomous nonclassical diffusion equations on bounded spatial domains which are expanding in time. First, we show how the first initial boundary value problem for these equations can be formulated as a variational problem with appropriate function spaces, and then we establish the existence and uniqueness over a finite time interval of variational solutions. Next, we show that the process of two parameter generated by such solutions has a nonautonomous pullback attractor. Finally, we study the upper semicontinuity of the obtained pullback attractor.
Let be a family of nonempty bounded open subsets of such that From now on, we will frequently use the following notations: In this paper we consider the following nonautonomous equation: where , the nonlinear term and the external force satisfy some conditions specified later on. This equation is called the nonclassical diffusion equation when , and when , it turns to be the classical reaction-diffusion equation.
Nonclassical diffusion equation arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanics, and heat conduction (see, e.g., [6–9]). In the last few years, the existence and long-time behavior of solutions to nonclassical diffusion equations has attracted the attention of many mathematicians. However, to the best of our knowledge, all existing results are devoted to the study of the equation in cylindrical domains. For example, under a Sobolev growth rate of the nonlinearity , problem (1.3) in cylindrical domains has been studied [10–13] for the autonomous case, that is the case not depending on time and in [14, 15] for the nonautonomous case. In this paper, we will study the existence and long-time behavior of solutions to problem (1.3) in the case of noncylindrical domains, the nonlinearity of polynomial type satisfying some dissipativity condition, and the external force depending on time . It is noticed that this question for problem (1.3) in the case , that is, for the reaction-diffusion equation, has only been studied recently in [4, 5].
In order to study problem (1.3), we make the following assumptions. (H1) The function satisfies that for some , where are nonnegative constants. By (1.4), there exist nonnegative constants such that where is the primitive of . (H2) The external force .(H3) The initial datum is given.
Since the open set changes with time , problem (1.3) is nonautonomous even when the external force is independent of time. Thus, in order to study the long-time behavior of solutions to (1.3), we use the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of these dynamical systems (see  and references therein). The existence of a pullback attractor for problem (1.3) in the case , that is, for the classical reaction-diffusion equation, has been derived recently in . In the case , since (1.3) contains the term , this is essentially different from the classical reaction-diffusion equation. For example, the reaction-diffusion equation has some kind of “regularity”; for example, although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity, and hence we can use the compact Sobolev embeddings to obtain the existence of attractors easily. However, for problem (1.3) when , because of , if the initial datum belongs to , the solution with the initial condition is always in and has no higher regularity, which is similar to hyperbolic equations. This brings some difficulty in establishing the existence of attractors for the nonclassical diffusion equations. Other difficulty arises since the considered domain is not cylindrical, so the standard techniques used for studying evolution equations in cylindrical domains cannot be used directly. Therefore, up to now, although there are many results on attractors for evolution equations in cylindrical domains (see, e.g., [17, 18]), little seems to be known for the equations in noncylindrical domains.
In this paper, we first exploit the penalty method to prove the existence and uniqueness of a variational solution satisfying the energy equality to problem (1.3). Next, we prove the existence of a pullback attractor for the process associated to problem (1.3). Finally, we study the continuous dependence on of the solutions to problem (1.3), in particular we show that the solutions of the nonclassical diffusion equations converge to the solution of the classical reaction-diffusion equation as . Hence using an abstract result derived recently by Carvalho et al.  and techniques similar to the ones used in , we prove the upper semicontinuity of pullback attractors in at . The last result means that the pullback attractors of the nonclassical diffusion equations converge to the pullback attractor of the classical reaction-diffusion equations as , in the sense of the Hausdorff semidistance.
The paper is organized as follows. In Section 2, for the convenience of readers, we recall some results on the penalty method and the theory of pullback attractors. After some preliminary results in Section 2, we proceed by a penalty method to solve approximated problem, and then we also prove the existence and uniqueness of the solution to problem (1.3) in Section 3. In Section 4, a uniform estimate for the solutions is then obtained under an additional assumption of the external force , and this will lead to the proof of existence of a pullback attractor in an appropriate framework. The upper semicontinuity of pullback attractors at is investigated in Section 5. In the last section, we give some discussions and related open problems.
Notations. In what follows, we will introduce some notations which are frequently used in the paper. Denote and for each , and denote by and the usual inner product and associated norm in and by and the usual gradient inner product and associated norm in . For each , consider as a closed subspace of with the functions belonging to being trivially extended by zero. It follows from (1.1) that can be considered as a family of closed subspaces of for each with In addition, will be identified with its topological dual by means of the Riesz theorem and will be considered as a subspace of with identified with the element defined by The duality product between and will be denoted by .
2.1. Penalty Method
To study problem (1.3), for each , we consider the following auxiliary problem: where , and are given functions.
The method of penalization due to Lions (see ) will be used to prove the existence and uniqueness of a solution to problem (2.1) satisfying an energy equality a.e. in and, as a consequence, the existence and uniqueness of a solution to problem (1.3) satisfying the energy equality a.e. in . To begin, fix and for each denote by the orthogonal subspace of with respect the inner product in and by the orthogonal projection operator from onto , which is defined as for each . Finally, define for all and observe that is the zero of .
We will now approximate by operators which are more regular in time. Consider the family of symmetric bilinear forms on defined by It can be proved that the mapping is measurable for all . Moreover, . For each integer and each , we define and denote by the associated operator defined by
Let be the Riesz isomorphism defined by and for each integer and each , we denote Obviously, , is a family of symmetric linear operators such that the mapping is measurable, bounded, and satisfies Let be given and for each consider the following problem:
The idea of the penalty method is as follows: for each we first prove the existence of a solution to problem (2.12) (a problem in a cylindrical domain) using standard methods such as the Galerkin method, and then show that converges to a solution to problem (2.1) (a problem in a noncylindrical domain) in some suitable sense, and as a consequence, the existence of a solution to problem (1.3) (see Section 3 for details).
2.2. Pullback Attractors
Since the open set changes with time , problem (1.3) is nonautonomous even when the external force is independent of time. Thus, in order to study the long-time behavior of solutions to (1.3), we use the theory of pullback -attractors which is a modification of the theory in .
Consider a process on a family of metric spaces , that is, a family of mappings such that for all and In addition, suppose is a nonempty class of parameterized sets of the form .
Definition 2.2 (see ). The process is said to be pullback -asymptotically compact if the sequence is relatively compact in for any , any , and any sequences and with and .
Definition 2.3 (see ). A family is said to be pullback -absorbing for the process if for any and any , there exists such that for all .
Remark 2.4. Note that if is pullback -absorbing for the process and is a compact subset of for any , then the process is pullback -asymptotically compact.
For each , let be the Hausdorff semi-distance between nonempty subsets and of , which is defined as
Definition 2.5 (see ). The family is said to be a pullback -attractor for if (1) is a compact set of for all , (2) is pullback -attracting, that is, (3) is invariant, that is,
Theorem 2.6 (see ). Suppose that the process is pullback -asymptotically compact and that is a family of pullback -absorbing sets for . Then, the family defined by , where for each and , is a pullback -attractor for , which in addition satisfies Furthermore, is minimal in the sense that if is a family of nonempty sets such that is a closed subset of and
2.3. The Upper Semicontinuity of Pullback Attractors
Definition 2.7. Let be a family of evolution processes in a family of Banach spaces with corresponding pullback -attractors . For any bounded interval , we say that is upper semicontinuous at for if
Theorem 2.8. Let be a family of processes with corresponding pullback -attractors . Then, for any bounded interval , is upper semicontinuous at for if for each , for each , and for each compact subset of , the following conditions hold: (i),(ii),(iii).
3. Existence and Uniqueness of Variational Solutions
For each , denote
Definition 3.1. A variational solution of (2.1) is a function such that (C1), ,(C2)for all , (C3) a.e. in , (C4).
To prove the uniqueness of variational solutions to problem (2.1), we need the following lemmas.
Lemma 3.4 (see ). Assume that and there exist and such that
for every function .
For each , define by Then for every function such that a.e. in .
Remark 3.5. If and , with satisfies and a.e. in , then the trivial extension of satisfies , with . Using the open sets , it is easy to see that under the conditions of (3.5), one also has for every and every function such that a.e. in .
Lemma 3.6 (see ). Let , be two functions such that a.e. in for . Assume that there exist such that for every function . Then, for every pair of Lebesgue points of the inner product function it holds
Corollary 3.7. If is a variational solution of (2.1), then for every Lebesgue point of it holds
The aim of this section is to obtain a variational solution of (2.1) such that We will say that satisfies the energy equality in if (3.12) is satisfied a.e. in . Analogously, if is a variational solution of (1.3), we will say that satisfies the energy equality a.e. in if for each the restriction of to satisfies the energy equality (3.12) a.e. in .
For any function and any , we put Then is a nondecreasing function. By Corollary 3.7, a variational solution of (1.3) satisfies the energy equality a.e. in if and only if for a.e. . In fact, using the continuity of the following mapping: one can see that a variational solution of (1.3) satisfies the energy equality a.e. in if and only if .
The next lemma provides a sufficient condition for to satisfy the energy equality a.e. in .
Lemma 3.8. Let be a variational solution of (2.1) and suppose that there exists a sequence of Lebesgue points of such that and Then, satisfies the energy equality a.e. in .
Proof. It is sufficient to prove that . Since and is nondecreasing, by Corollary 3.7, we have This completes the proof.
Proposition 3.9. Let be two variational solutions of (2.1) corresponding to the initial data , respectively, which satisfy the energy equality a.e. in . Then, Hence, it implies the uniqueness of variational solutions to (2.1) satisfying the energy equality in .
The method of penalization will now be used to prove the existence and uniqueness of a variational solution to problem (2.1) satisfying an energy equality a.e. in and, as a consequence, the existence and uniqueness of a variational solution to problem (1.3) satisfying the energy equality a.e. in .
Theorem 3.10. Let be given. Then problem (2.1) has a unique variational solution satisfying the energy equality a.e. in .
Proof. We divide the proof into two steps.
Step 1. Existence of a weak solution to problem (2.12).
We will use the Galerkin method (see ). Take an orthonormal Hilbert basis of formed by elements of such that the vector space generated by is dense in and . Then, one takes a sequence converging to in , with in the vector space spanned by the first . For each integer , one considers the approximation , defined as a solution of Multiplying (3.22) by and summing from to , we obtain or Thus, We have so Integrating (3.27) on , we obtain Since we have From (3.30), we deduce that is bounded in , weakly in , is bounded in , weakly in .Since is bounded in , one can check that is bounded in with , hence in . We now prove that .
Indeed, we have, is bounded in , is bounded in .By the Aubin-Lions lemma [20, Chapter 1], is relatively compact in . Therefore, one can assume that strongly in , so a.e. in . Since is continuous, a.e. in . Applying Lemma 1.3 in , we have This implies that is a weak solution of problem (2.12).
Step 2. Existence of a variational solution to (2.1) satisfying the energy equality.
From (3.30), we have Consider the function defined by It is easy to see that is a continuous and convex function. It follows that is weakly lower semicontinuous in . Moreover, weakly in , hence Since weakly in , then, reasoning as above, From the facts that weakly in , weakly in and the weak lower semicontinuity of the norm, we deduce that Since , , (3.36) gives is bounded in , is bounded in , weakly in , weakly in .From Lemma 2.1, we have Moreover, (3.36) and the equality give It follows from (3.36) that for all and each . Since the injection of into is compact, the set is compact in . By (3.39) and the Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by , such that So, the condition (C4) is satisfied.
On the other hand, is bounded in and is bounded in , applying the Aubin-Lions lemma and Lemma 1.3 in [20, Chapter 1], one has Since is the weak solution of the problem taking to the limit as and using the fact that a.e. in , we can conclude that is the solution of (2.1).
Now, we will show that satisfies the energy equality in . Multiplying (3.22) by and summing from to