Multivalued Random Dynamics of Fractional Reaction–Diffusion–AdvectionEquations Driven by Superlinear Colored Noise on Unbounded Domains

This article investigates the multivalued random dynamics generated by solution operators of fractional random reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains. The equations under consideration exhibit non-Lipschitz continuous nonlinear drift and diffusion terms, which lead to the non-uniqueness of solutions and hence generate multivalued random dynamical systems. Our goals are to demonstrate: (i) the existence of solutions for fractional reaction–diffusion–advection equations driven by superlinear colored noise on unbounded domains; and (ii) the existence and uniqueness of pullback random attractors for these systems. The measurability of the random attractors is demonstrated using a method that relies on the weak upper semicontinuity of the solutions. The asymptotic compactness of solution operators is obtained by employing Ball’s approach to energy equations in order to address the non-compactness of Sobolev embeddings on the entire space.