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In this paper, we mainly study the long-time dynamical behaviors of 2Dnonlocal stochastic Swift-Hohenberg equations with multiplicative noise fromtwo perspectives. Firstly, by adopting the analytic semigroup theory, we provethe 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 extendthe classical "stochastic Gronwall's lemma", making it more convenient inapplications. Based on this improvement, we are allowed to use the analyticsemigroup theory to establish the existence of ergodic invariant measures.

journal of mathematical physics

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