Abstract
In this paper, we consider the dynamics of a reaction–diffusion equation with fading memory and nonlinearity satisfying arbitrary polynomial growth condition. Firstly, we prove a criterion in a general setting as an alternative method (or technique) to the existence of the bi-spaces attractors for the nonlinear evolutionary equations (see Theorem 2.14). Secondly, we prove the asymptotic compactness of the semigroup on L^{2}(varOmega )times L_{mu }^{2}(mathbb{R}; H_{0}^{1}( varOmega )) by using the contractive function, and the global attractor is confirmed. Finally, the bi-spaces global attractor is obtained by verifying the asymptotic compactness of the semigroup on L^{p}( varOmega )times L_{mu }^{2}(mathbb{R}; H_{0}^{1}(varOmega )) with initial data in L^{2}(varOmega )times L_{mu }^{2}(mathbb{R}; H_{0} ^{1}(varOmega )).
Highlights
1 Introduction The aim of this paper is to analyze the long-time behavior of solutions of the following semilinear reaction–diffusion equations with fading memory:
With Dirichlet boundary condition, where g ∈ L2(Ω), Ω ⊂ Rn (n ≥ 3) is a bounded domain with smooth boundary, nonlinearity f, and the memory kernel function k(s) satisfying posterior adaptation hypothesis which will be given below. This kind of integro-differential reaction–diffusion equation is well known and can be interpreted, for instance, as a model of heat diffusion with memory which accounts for a reaction process depending on the temperature itself
This equation appears as the model of polymers and high viscosity liquids, the diffusion process is influenced by the past historical growth, which is mainly represented in the convolution term of fading memory characterizing diffusion species to a suitable memory core [12]
Summary
The aim of this paper is to analyze the long-time behavior of solutions of the following semilinear reaction–diffusion equations with fading memory:. Lemma 2.8 ([13, 18, 24]) Let X be a Banach space and B be a bounded subset of X, {S(t)}t≥0 is a semigroup with a bounded absorbing set B0 on X. From Lemma 2.2, {S(t)}t≥0 is a norm-to-weak continuous semigroup on Z and possesses a bounded absorbing set B0. According to Theorem 2.9, the key to proving the existence of (L2, Lp)-global attractors is to verify that semigroup {S(t)}t≥0 is (L2, Lp)-asymptotically compact. Proof By Lemma 2.9, we only need to verify that {S(t)}t≥0 is (L2, Lp)-asymptotically compact Combining this embedding Lp(Ω) → L2(Ω) is continuous with the definitions of L2 and Lp, we have that the semigroup {S(t)}t≥0 has an (L2, L2)-bounded absorbing set B2.
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