2015/07/08 by Bixiang Wang, Wang, Bixiang
Computer Science · Engineering · Mathematics · #37L30 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 35B40 #Secondary 35B41 #Stability and Controllability of Differential Equations #math.AP #math.DS #msc:35B40 #msc:35B41 #msc:37L30
paper · pdf · doi:10.48550/arxiv.1507.02013
arxiv created 2015/07/08 · openalex publication_date 2015/07/08 · arxiv updated 2015/07/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on Rn is overcome by the uniform estimates on the tails of solutions.