Uniform Attractors for Nonclassical Diffusion Equations with Memory

Yongqin Xie,Ye Zeng,Yanan Li

doi:10.1155/2016/5340489

Abstract

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearityfsatisfies the polynomial growth of arbitrary order and the time-dependent forcing termgis only translation-bounded inLloc2(R;L2(Ω)).

Highlights

The existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearity f satisfies the polynomial growth of arbitrary order and the time-dependent forcing term g is only translation-bounded in L2loc(R; L2(Ω))
We prove some weak continuity for the family of processes and obtain the structure of the compact uniform attractors
The main results of this paper are given expression to in the following two theorems, which will be proved in Sections 2 and 3, respectively

Summary

Introduction

The existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearity f satisfies the polynomial growth of arbitrary order and the time-dependent forcing term g is only translation-bounded in L2loc(R; L2(Ω)). The past history uτ(τ − s) of the variable u satisfies the condition as follows: there exist positive constants R and 󰜚 ≤ δ (from (5)), such that s)󵄩󵄩󵄩󵄩20 ds

Results

Conclusion