Pathwise upper semi-continuity of random pullback attractors along the time axis

• DOI: 10.1016/j.physd.2018.03.002
• Publication Date: 2018-03-20T19:36:10Z
• Authors: Hongyong Cui; Peter E. Kloeden; Fuke Wu
• Keywords: Pullback; Random compact set; Multiplicative noise; Uniform limit theorem

Abstract The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form { A t ( ⋅ ) } t ∈ R with each A t ( ⋅ ) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t ↦ A t ( ω ) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪ s ∈ I A s ( ω ) , where I ⊂ R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.