Asymptotic behavior for the semilinear reaction–diffusion equations with memory

Abstract

AbstractIn 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 $L^{2}(\varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0}^{1}( \varOmega ))$L2(Ω)×Lμ2(R;H01(Ω)) 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 $L^{p}( \varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0}^{1}(\varOmega ))$Lp(Ω)×Lμ2(R;H01(Ω)) with initial data in $L^{2}(\varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0} ^{1}(\varOmega ))$L2(Ω)×Lμ2(R;H01(Ω)).