Asymptotic Behavior for a Class of Nonclassical Parabolic Equations

Yanjun Zhang,Qiaozhen Ma

doi:10.1155/2013/204270

Abstract

This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations ut-εΔut-ωΔu+f(u)=g(x) with critical nonlinearity, where ε∈[0,1] and ω>0 are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for g(x)∈H-1(Ω), which are independent of the parameter ε. Secondly, some uniformly (with respect to ε∈[0,1]) asymptotic regularity about the solutions has been established for g(x)∈L2(Ω), which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ε∈[0,1]). Finally, as an application of this regularity result, a family {ℰε}ε∈[0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation (ε=0), the upper semicontinuity, at ε=0, of the global attractors has been proved.

Highlights

We study the long-time behavior of the following class of nonclassical parabolic equations: ut − εΔut − ωΔu + f (u) = g (x), in Ω × R+, u (x, 0) = u0 (x), (Eε) u|∂Ω = 0, where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, ε ∈ [0, 1] and ω > 0 are two parameters, the external force g is time independent, and the nonlinearity f satisfies some specified conditions later
In [10] the author proved the existence of a class of attractors in H2 ∩ H01 with initial data u0 ∈ H2 ∩ H01 and the upper semicontinuity of attractors in H01 under subcritical assumptions and g(x) = 0 in the case of N ≤ 3
In this paper, inspired by the ideas in [17, 18] and motivated by the dynamical results in [19,20,21,22], we study the uniform qualitative analysis for the solutions of the nonclassical parabolic equations (Eε) and give some information about the relation between the solutions of (E0) and those of (Eε)

Summary

Introduction

We study the long-time behavior of the following class of nonclassical parabolic equations: ut − εΔut − ωΔu + f (u) = g (x) , in Ω × R+, u (x, 0) = u0 (x) ,. In this paper, inspired by the ideas in [17, 18] and motivated by the dynamical results in [19,20,21,22], we study the uniform (with respect to the parameter ε ∈ [0, 1]) qualitative analysis (a priori estimates) for the solutions of the nonclassical parabolic equations (Eε) and give some information about the relation between the solutions of (E0) and those of (Eε).

Main Results

Preliminaries

Uniformly Decaying Estimates in H

Proof of the Main Results

Applications of Theorem 1