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)).

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 $L^{2}(\varOmega )\times L_{\mu }^{2}(\mathbb{R}; H_{0}^{1}( \varOmega ))$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)) .

In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [7,14] as the perturbed parameters \begin{document}$ \nu = 0 $\end{document}, but the nonlinearity \begin{document}$ f $\end{document} satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.

This paper considers the asymptotic behavior of nonclassical diffusion equation with memory and lacking instantaneous damping on time-dependent space. The existence and regularity of time-dependent pullback global attractors are proved by using the contractive process method and a new analytical technique. It is remarkable that the nonlinearity $ f $ satisfies the polynomial growth of arbitrary order.

We study the long-term behavior of the non-autonomous wave equations with a new-type nonlocal weak damping and critical nonlinearity. After proving the global well-posedness, we first study the asymptotic regularity of solutions, which shows that the solutions are exponentially approaching a more regular bounded subset. Based on this regularity result, we finally obtain the existence and smoothness of uniform exponential attractors.