In this paper, we prove the existence of pullback attractors in $C_{H_0^1\left(\mathrm{\Omega }\right)}$ for a nonclassical diffusion equation with delay term g(t, u_t) which contains some hereditary characteristics. We consider two types of nonlinearity f: one is the case of critical growth and the other one is the polynomial growth of arbitrary order p − 1(p ≥ 2).

In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in $\mathcal{H}_{t}$Ht for a class of nonclassical reaction–diffusion equations with the forcing term $g(x)\in H^{-1}(\varOmega )$g(x)∈H−1(Ω) and the nonlinearity f satisfying the polynomial growth of arbitrary $p-1$p−1 ( $p\geq 2$p≥2 ) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).

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.

In this paper, we prove the existence of the pullback attractors in $L^2\left({\mathbb{R}}^N\right)$ , $L^p\left({\mathbb{R}}^N\right)$ , and $H^1\left({\mathbb{R}}^N\right)$ for a nonlinear reaction-diffusion equation in unbounded domains, in which the nonlinearity f satisfies the polynomial growth of arbitrary order p − 1(p > 2) and the external forcing $g\in L_{loc}^2\left(\mathbb{R};L^2\left({\mathbb{R}}^N\right)\right)$ only subjects to the weaker integrability condition ${\int }_{-\infty }^t{e}^{\alpha s}{\Vert g\left(s\right)\Vert }_2^{2}ds$ .