Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains

This paper is concerned with the long term behavior of the solutions of the non-autonomous Benjamin-Bona-Mahony equation driven by nonlinear colored noise with continuous coefficients defined on three-dimensional unbounded channels. The solutions of the equation are not unique and hence generate a multivalued non-autonomous random dynamical system. We first prove the measurability of the multivalued random system by the idea of weak upper semicontinuity of solutions, and then establish the existence and uniqueness of tempered pullback random attractors based on the pullback asymptotic compactness of solutions. The difficulty of the non-compactness of Sobolev embeddings on unbounded domains is overcome by the methods of the spectral decomposition inside bounded domains as well as the uniform tail-estimates of solutions outside bounded domains.