Pullback attractors for a class of non-autonomous nonclassical diffusion equations

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Abstract

We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation utεΔutΔu+f(u)=g(t), ε[0,1], in a bounded domain in RN. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors Aεˆ in the space H01(Ω) and the upper semicontinuity of Aεˆ at ε=0.

Introduction

In this paper, we consider the following non-autonomous equation: (Pϵ){utεΔutΔu+f(u)=g(t),xΩ,t>τ,u|Ω=0,u|t=τ=uτ(x),xΩ, where Ω is a bounded domain in RN(N3) with smooth boundary Ω, the nonlinearity f and the external force g satisfy some specified conditions later, and ε[0,1]. This equation is called the nonclassical diffusion equation when ε>0, and when ε=0, 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 g is independent of time t, 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 g to depend on time t.

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 fC1(R,R) satisfies f(u)uμu2C,f(u),f(0)=0,|f(u)|C(1+|u|ρ),lim inf|u|uf(u)κF(u)u20,for some κ>0,lim inf|u|F(u)u20, where 0<ρ<min{N+2N2,2+4N} if ε>0 and 0<ρNN2 if ε=0, μ<λ1, λ1>0 is the first eigenvalue of Δ in Ω with the homogeneous Dirichlet condition, and F(u)=0uf(s)ds is the primitive of f.

  • (H2)

    The external force gLloc2(R,L2(Ω)) satisfies g(t)L2(Ω)2Meγ|t|,tR, where γ<σ<min{2λ12λ1+1,2κ} when ε>0 and γ<2λ12μ when ε=0.

  • (H3)

    The initial datum uτH01(Ω) is given.

The main aim of this paper is to prove the existence of pullback attractors Aεˆ={Aε(t):tR}, ε[0,1], for problem (1.1) and to prove the upper semicontinuity of Aεˆ at ε=0. The existence of pullback D-attractors for problem (1.1) in the case ε=0 has been studied in [11], [12], [13], [14]. In the case ε>0, since Eq. (1.1) contains the term εΔut, this is essentially different from the classical reaction–diffusion equation. For example, the reaction–diffusion equation has some kind of “regularity”; e.g., 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 embedding to obtain the existence of attractors easily. However, for problem (1.1) when ε>0, because of Δut, if the initial datum uτ belongs to H01(Ω), the solution u(t) with the initial condition u(τ)=uτ is always in H01(Ω) and has no higher regularity, which is similar to hyperbolic equations. This brings some difficulty in establishing the existence of pullback attractors for nonclassical diffusion equations.

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 D-attractor in L2NN2(Ω). Then by verifying Condition (PDC) introduced in [12], we obtain the existence of a pullback D-attractor in H01(Ω). Next, we study the continuous dependence of solutions to problem (1.1) on ε as ε0. Hence, using an abstract result derived recently by Carvalho et al. [17], we prove the upper semicontinuity of pullback D-attractors Aˆε in L2(Ω) at ε=0.

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 H01(Ω), we are only able to prove the upper semicontinuity of the pullback attractor in L2(Ω). 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 H01(Ω). 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 D-attractors. In Section 3, we prove the existence of pullback D-attractors Aεˆ for problem (1.1) when ε>0. The upper semicontinuity of pullback D-attractors Aˆε at ε=0 is investigated in Section 4.

Section snippets

Preliminaries

Let X be a metric space with metric d. Denote by B(X) the set of all bounded subsets of X. For A,BX, the Hausdorff semi-distance between A and B is defined by dist(A,B)=supxAinfyBd(x,y). Let {U(t,τ):tτ,τR} be a process in X. The process {U(t,τ)} is said to be norm-to-weak continuous if U(t,τ)xnU(t,τ)x, as xnx in X, for all tτ,τR. The following result is useful for proving the norm-to-weak continuity of a process.

Proposition 2.1

[18]

Let X,Y be two Banach spaces, and X,Y be respectively their dual spaces.

Existence of global solutions

Denote by |.|2,(.,.),.,((.,.)) the norm and scalar product of L2(Ω) and H01(Ω), respectively.

Definition 3.1

A function u(x,t) is called a weak solution of (1.1) on (τ,T) if and only if uC([τ,T];H01(Ω)),utL2(τ,T;H01(Ω)),u|t=τ=uτa.e. in Ω, and τTΩ(utφ+εutφ+uφ+f(u)φ)=τTΩgφ, for all test functions φC([τ,T];H01(Ω)).

We first prove the following lemma.

Lemma 3.1

If {un} is bounded in L(τ,T;H01(Ω)), then {f(un)} is bounded in Lq(τ,T;Lq(Ω)), where q=2(N+2)Nρ.

Proof

We have 1<q and 2<ρq<2NN2. Define a=N2N; then a(0,1

The upper semicontinuity of pullback D-attractors at ε=0

First, we consider problem (1.1) when ε=0. Under Conditions (H1)–(H3), when ε=0, 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 D-attractor A0ˆ={A0(t):tR} in the space H01(Ω) for problem (P0). The aim of this section is to prove the upper semicontinuity of pullback attractors Aϵˆ at ε=0 in L2(Ω).

The following lemma is the key of this section.

Lemma 4.1

For each tR, each

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