Uniform Attractors of Nonclassical Diffusion Equations Lacking Instantaneous Damping on RN with Memory

In this paper, we consider the non-autonomous nonclassical diffusion equations on R-N with hereditary memory u(t) - Delta u(t) - integral(infinity)(0) kappa(s)Delta u(t - s)ds + f (x, u) = g(x, t). The main characteristics of the model is that the equation does not contain a term of the form -Delta u, which contributes to an instantaneous damping. We first investigate the existence and uniqueness of weak solutions to the initial-boundary-value problem for above-mentioned equation. Next, we study the long-time dynamical behavior of the solutions in the weak topological space H-1(R-N) x L-mu(2) (R+, H-1(R-N)), where the nonlinearity is critical and the time-dependent forcing term is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in (Conti et al. in Commun. Pure Appl. Anal. 19:2035-2050, 2020) in the non-autonomous and unbouded domain cases which have not been studied before.