摘要：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 $L^,(\varOmega )\times L_{\mu }^,(\mathbb{R}; H_(^)( \varOmega ))$ 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 $L^{p}( \varOmega )\times L_{\mu }^,(\mathbb{R}; H_(^)(\varOmega ))$ with initial data in $L^,(\varOmega )\times L_{\mu }^,(\mathbb{R}; H_( ^)(\varOmega ))$.
关键词：Reaction–diffusion equation ; Memory ; The bi-spaces asymptotic compact ; Global attractor ; Arbitrary polynomial growth ;
