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

Yongqin Xie,Kaixuan Zhu,Jun Li

doi:10.1186/s13662-020-03146-2

Abstract

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p-1 (p geq 2). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

Highlights

In this paper, we consider the following perturbed nonclassical diffusion equation: ut – ν ut – u + f (u) = g in × [0, ∞). (1.1)The problem is supplemented with the boundary condition u(x, t)|∂ = 0 for all t ≥ 0 (1.2)and the initial condition u(x, t)|t=0 = u0(x), (1.3)where is a bounded smooth domain in Rn (n ≥ 3), ν ∈ [0, +∞) is a perturbed parameter, and g ∈ L2( ) is a given external force term.Xie et al Advances in Difference EquationsThe nonlinearity f ∈ C1 fulfills f (0) = 0 and satisfies the following arbitrary-order polynomial growth condition: γ1|s|p – β1 ≤ f (s)s ≤ γ2|s|p + β2, p ≥ 2, (1.4)and the dissipative condition f (s) ≥ –l
In order to overcome the difficulty mentioned above, we introduce the asymptotic contractive function method to verify asymptotic compactness of a family of semigroups for autonomous dynamical systems by referring to the methods and ideas in [2, 3]
We present the following theorem to verify upper semicontinuity of global attractors in autonomous dynamical systems

Summary

Introduction

Our main purpose is to consider upper semicontinuity of attractors for Eq (1.1) with the nonlinearity satisfying arbitrary-order polynomial growth condition, which makes the Sobolev compact embedding no longer valid and brings more difficulties for verifying the corresponding asymptotic compactness of the family of solutions semigroup {Sν(t)}t≥0, ν ∈ [0, +∞). Theorem 1.1 (Global attractors) Let ⊂ Rn be a bounded domain with smooth boundary, ν ∈ I ⊂ [0, +∞), f satisfy (1.4)–(1.5), and g ∈ L2( ).

Results

Conclusion