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Uniform global attractors for non-autonomous dissipative dynamical systems
July 2017, 22(5): 2011-2051. doi: 10.3934/dcdsb.2017119
Multivalued non-autonomous random dynamical systems for wave equations without uniqueness
Bixiang Wang
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA
Received July 2015 Revised April 2016 Published March 2017
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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 $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.
Keywords: Random attractor, multivalued system, measurability, wave equation, unbounded domain.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L3.
Citation: Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
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