The Existence Of Attractors For Two Classes Of Non-classical Diffusion Equations In Unbounded Domains

In this paper, we devote the long-time behavior of two classes of non-classical diffusion equations with critical nonlinearity on the whole space R3, and prove the existence of global attractor for the corresponding equation.This paper is divided into three sections.In Section1, we introduce research background of the problems and recall some preliminaries used in the main part of the paper.In Section2, we study the non-classical diffusion equations with critical non-linearity on the whole space R3, we decompose the domain R3into bounded and unbounded parts. For the unbounded parts, we appeal to the idea of uniform esti-mates to show that the norm of solutions are less than ε when t is large enough. For the bounded parts, we again apply the decompose techniques to prove the property of asymptotic smoothness of solutions. Ultimately, we prove the existence of global attractors for the problem in H1(R3).In Section3, we consider the non-classical diffusion equations with fading mem-ory and with critical nonlinearity on R3. we use some priori estimates and the compactness transitivity lemma to overcome the obstacle of the validation of com-pactness generated by the fading memory, and apply the idea in Section2to prove the existence of global attractors for the problem in the weak topological space H1(R3)×Lμ2(R+,H1(R3)).