%0 Journal Article
%T Asymptotic Behavior for a Class of Nonclassical Parabolic Equations
%A Yanjun Zhang
%A Qiaozhen Ma
%J ISRN Applied Mathematics
%D 2013
%R 10.1155/2013/204270
%X This paper is devoted to the qualitative analysis of a class of nonclassical parabolic equations with critical nonlinearity, where and are two parameters. Firstly, we establish some uniform decay estimates for the solutions of the problem for , which are independent of the parameter . Secondly, some uniformly (with respect to ) asymptotic regularity about the solutions has been established for , which shows that the solutions are exponentially approaching a more regular, fixed subset uniformly (with respect to ). Finally, as an application of this regularity result, a family of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with the reaction diffusion equation ( ), the upper semicontinuity, at , of the global attractors has been proved. 1. Introduction We study the long-time behavior of the following class of nonclassical parabolic equations: where is a bounded domain with smooth boundary , and are two parameters, the external force is time independent, and the nonlinearity satisfies some specified conditions later. When for the fixed constant , equation is a usual reaction-diffusion equation, and its asymptotic behavior has been studied extensively in terms of attractors by many authors; see [1¨C5]. For each fixed , equation is a nonclassical reaction-diffusion equation, which arises as models to describe physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction; see [6¨C8] and references therein. Aifantis in [6] provided a quite, general approach for obtaining these equations. The asymptotic behavior of the solutions for this equation has been studied by many authors; see [9¨C16]. For the fixed constant , any , and the long-time behavior of the solutions of has been considered by some researchers; see [10, 13]. In [10] the author proved the existence of a class of attractors in with initial data and the upper semicontinuity of attractors in under subcritical assumptions and in the case of . In [13] similar results have been shown when and . In this paper, inspired by the ideas in [17, 18] and motivated by the dynamical results in [19¨C22], we study the uniform (with respect to the parameter ) qualitative analysis (a priori estimates) for the solutions of the nonclassical parabolic equations and then give some information about the relation between the solutions of and those of . Our main difficulty comes from the critical nonlinearity and the uniformness with respect to . This paper is organized as follows. In Section 2, we introduce basic notations and state our main results. In
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2013/204270/