Pullback attractors for a class of non-autonomous nonclassical diffusion equations
Introduction
In this paper, we consider the following non-autonomous equation: where is a bounded domain in with smooth boundary , the nonlinearity and the external force satisfy some specified conditions later, and . This equation is called the nonclassical diffusion equation when , and when , it turns out to be the classical reaction–diffusion equation.
Nonclassical parabolic equations arise as models to describe physical phenomena, such as non-Newtonian flows, soil mechanics, and heat conduction (see, e.g., [1], [2], [3], [4]). The long-time behavior of solutions to problem (1.1) in the autonomous case, that is, the case when is independent of time , has been studied in recent years [5], [6], [7], [8]. In this paper, we will study the long-time behavior of solutions to problem (1.1) by allowing the external force to depend on time .
Non-autonomous equations appear in many applications in the natural sciences, so they are of great importance and interest. The long-time behavior of solutions of such equations have been studied extensively in recent years. The first attempt was to extend the notion of a global attractor to the non-autonomous case, leading to the concept of the so-called uniform attractor (see [9]). It is remarkable that the conditions ensuring the existence of the uniform attractor parallel those for the autonomous case. However, one disadvantage of the uniform attractor is that it need not to be “invariant”, unlike the global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many non-autonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems. In order to overcome this drawback, a new concept, called a pulback attractor, has been introduced for the non-autonomous case. The theory of pullback attractors has been developed for both non-autonomous and random dynamical systems and it has been shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems (see [10] and references therein).
To study problem (1.1), we assume the following conditions:
- (H1)
The nonlinearity satisfies where if and if , , is the first eigenvalue of in with the homogeneous Dirichlet condition, and is the primitive of .
- (H2)
The external force satisfies where when and when .
- (H3)
The initial datum is given.
In this paper, we try to overcome this difficulty by using the asymptotic a priori estimate method, which was initiated in [15] for autonomous equations and developed in [16] for non-autonomous equations, to prove the asymptotic compactness of the corresponding process. We first use this method to prove the existence of a pullback -attractor in . Then by verifying Condition (PDC) introduced in [12], we obtain the existence of a pullback -attractor in . Next, we study the continuous dependence of solutions to problem (1.1) on as . Hence, using an abstract result derived recently by Carvalho et al. [17], we prove the upper semicontinuity of pullback -attractors in at .
Because of Condition (1.4) in (H1), we are in the subcritical case, so some of the results obtained are weaker than the existing ones in the autonomous case [6]. On the other hand, although pullback attractors are shown to exist in , we are only able to prove the upper semicontinuity of the pullback attractor in . It is interesting if one can show the existence of a pullback attractor in the critical case and prove the upper semicontinuity of the pullback attractor in the space . However, to the best of our knowledge, this is still an open problem.
The rest of the paper is organized as follows. In the next section, for the convenience of the reader, we recall some results on pullback -attractors. In Section 3, we prove the existence of pullback -attractors for problem (1.1) when . The upper semicontinuity of pullback -attractors at is investigated in Section 4.
Section snippets
Preliminaries
Let be a metric space with metric . Denote by the set of all bounded subsets of . For , the Hausdorff semi-distance between and is defined by Let be a process in . The process is said to be norm-to-weak continuous if , as in , for all . The following result is useful for proving the norm-to-weak continuity of a process.
Proposition 2.1 Let be two Banach spaces, and be respectively their dual spaces. [18]
Existence of global solutions
Denote by the norm and scalar product of and , respectively.
Definition 3.1 A function is called a weak solution of (1.1) on if and only if and for all test functions .
We first prove the following lemma.
Lemma 3.1 If is bounded in , then is bounded in , where .
Proof We have and . Define ; then
The upper semicontinuity of pullback -attractors at
First, we consider problem (1.1) when . Under Conditions (H1)–(H3), when , one can prove the existence of a weak solution by using the standard Galerkin method (see e.g. [9], [21]). Then, using arguments as in [12], one can prove the existence of a pullback -attractor in the space for problem . The aim of this section is to prove the upper semicontinuity of pullback attractors at in .
The following lemma is the key of this section.
Lemma 4.1 For each , each
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