Multivalued Non-Autonomous Random Dynamical Systems for Wave Equations without Uniqueness

TL;DR

This paper studies multivalued non-autonomous random dynamical systems generated by stochastic wave equations with non-Lipschitz nonlinearities, establishing measurability and asymptotic properties without requiring uniqueness of solutions.

Contribution

It introduces weak upper semicontinuity for multivalued functions and proves measurability of pullback attractors for wave equations on unbounded domains.

Findings

Proved measurability of pullback attractors independent of probability space completeness.

Established asymptotic compactness of solutions using energy estimates.

Overcame non-compactness issues via tail estimates of solutions.

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 $R^n$ is overcome by the uniform estimates on the tails of solutions.