This paper investigates the nonlocal reaction-diffusion equations with time-dependent advection term driven by nonlinear colored noise. The nonlinear terms are assumed continuous but not necessarily Lipschitz continuous. Together with the nonlinear advection term α⃗ϵ(t)⋅∇uγ and the nonlocal diffusion coefficient a(l(u)), this introduces technical difficulties in the analysis, which leads to the non-uniqueness of solutions and the equations generates a multivalued nonautonomous random dynamical system. For such systems, we establish the existence and uniqueness of the pullback random attractors. Specifically, the measurability of the random attractors is obtained via weak upper semicontinuity for multivalued functions, and asymptotic compactness follows from Ball’s energy equation method.
Ding et al. (Mon,) studied this question.