Uniform Attractors for Nonclassical Diffusion Equations with Memory

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)).

Asymptotic behavior for the semilinear reaction–diffusion equations with memory

“…Definition 2.7 ( [18,24]) Let X be a Banach space and B be a bounded subset of X. We call a function φ(•, •), defined on X × X, a contractive function if, for any sequence…”

“…To overcome the difficulty of the noncompact embedding, in [18], using the idea of the contractive function method, the authors consider the asymptotic behavior of nonautonomous wave equations, they prove that the family of processes is uniformly asymptotically compact under the nonlinearity f satisfying critical growth (see, e.g., [18]). In [23,24], the authors consider the asymptotic behavior of nonclassical diffusion equations with the nonlinearity satisfying arbitrary polynomial growth by the (uniform) asymptotic contractive function method, but the initial data z 0 and the solution z of Eq. (1.1) belong to the same space z 0 , z(t) ∈ H 1 0 (Ω)) × L 2 μ (R; H 1 0 (Ω)).…”

“…In this paper, we take the extension of the method in [18,23,24,27,28] to the normto-weak continuous semigroup on a product space. The abstract result, which proves the existence of bi-space global attractors, is obtained (see Theorem 2.9).…”

Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

“…Asymptotic behavior similar to Eq. (1.1) has been investigated in many documents during the last years (see, e.g., [2,3,[11][12][13][14][15] and the references therein).When ν = 0, Eq. (1.1) can be simplified to a usual reaction-diffusion equation, so the dynamical behavior of this equations has been investigated in many documents (see, e.g., [16][17][18][19] and the references therein).…”

Asymptotic Regularity and Uniform Attractor for Non-autonomous Viscoelastic Equations with Memory

“…Maybe, we could establish the existence of uniform attractors of (1) using the method in [16,17], but the regularity and structure cannot obtain directly. In this paper, we will apply the techniques introduced in Sun [14] to overcome the difficulty due to the critical nonlinearity, and establish the asymptotic regularity of the solutions.…”