Multivalued non-autonomous random dynamical systems for wave equations without uniqueness

Bixiang Wang

doi:10.3934/dcdsb.2017119

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2017, Volume 22, Issue 5: 2011-2051. Doi: 10.3934/dcdsb.2017119

This issue Previous Article Averaging of fuzzy integral equations Next Article Uniform global attractors for non-autonomous dissipative dynamical systems

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: June 30, 2015

Revised: March 31, 2016

Published: August 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:35B40;Secondary:35B41, 37L30.

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