Upper semicontinuity of pullback D$\mathcal {D}$‐attractors for nonlinear parabolic equation with nonstandard growth condition

Abstract

This paper is devoted to the well‐posed problem and the existence of pullback ‐attractors for a class of nonlinear parabolic equation with nonstandard growth condition. First, by making use of Galerkin's method and monotone operator method, the existence of solutions is proved in Orlicz–Sobolev space with variable exponents depending on time and space, then the uniqueness and continuity of solutions are also obtained. Finally, by verifying the pullback ‐asymptotic compactness, the existence of pullback ‐attractors is proved. In particular, the upper semicontinuity of pullback ‐attractors of the corresponding equation with respect to the disturbance parameter λ is also proved.

Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter

“…When ε(t) is zero (or a positive constant) and memory term is non-degenerate, it is easy to show that the equation (1.1) becomes the usual reaction-diffusion equation (or nonclassical diffusion equation) with memory, under these circumstances, the asymptotic behavior of solutions has been researched by many scholars in recent years (see [9,17,18,19,31,35,36,37]). Especially to deserve to be mentioned, more recently, the authors considered the existence, regularity and upper semicontinuity of global and uniform attractors for autonomous and non-autonomous nonclassical diffusion equation lacking instantaneous damping −∆u in bounded and unbounded domain when the nonlinearity satisfies critical exponential growth and polynomial growth of arbitrary order respectively, see [10,29,32,33,36,38,39].…”