Regularity of pullback attractors and equilibria for a stochastic non-autonomous reaction-diffusion equations perturbed by a multiplicative noise

TL;DR

This paper investigates the existence, properties, and stability of pullback attractors and equilibria in stochastic non-autonomous reaction-diffusion equations with multiplicative noise, advancing understanding of long-term behavior in such systems.

Contribution

It establishes the existence and upper semi-continuity of pullback attractors in non-initial spaces for stochastic PDEs and proves conditions for unique equilibria.

Findings

Existence of pullback attractors in non-initial spaces.

Upper semi-continuity of attractors under perturbations.

Conditions for unique equilibrium in stochastic systems.

Abstract

In this paper, a standard about the existence and upper semi-continuity of pullback attractors in the non-initial space is established for some classes of non-autonomous SPDE. This pullback attractor, which is the omega-limit set of the absorbing set constructed in the initial space, is completely determined by the asymptotic compactness of solutions in both the initial and non-initial spaces. As applications, the existences and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion equation driven by a multiplicative noise. Finally we show that under some additional conditions the cocycle admits a unique equilibrium.