Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise

TL;DR

This paper investigates the long-term behavior of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise, proving upper semi-continuity of attractors and establishing invariant measures using advanced analytical techniques.

Contribution

It introduces new results on the upper semi-continuity of random attractors and the existence of invariant measures for the stochastic Swift-Hohenberg equation with multiplicative noise.

Findings

Proved upper semi-continuity of random attractors as noise coefficient approaches zero.

Extended stochastic Gronwall's lemma for better applicability.

Established existence of ergodic invariant measures.

Abstract

In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space $H_0^2(U)$, as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures.