Robustness of nonautonomous attractors for a family of nonlocal reaction-diffusion equations without uniqueness

In this paper, we consider a nonautonomous nonlocal reaction-diffusion equation with a small perturbation in the nonlocal diffusion term and the nonautonomous force. Under the assumptions imposed on the viscosity function, the uniqueness of weak solutions cannot be guaranteed. In this multi-valued framework, the existence of weak solutions and minimal pullback attractors in the -norm is analysed. In addition, some relationships between the attractors of the universe of fixed bounded sets and those associated to a universe given by a tempered condition are established. Finally, the upper semicontinuity property of pullback attractors w.r.t. the parameter is proved. Indeed, under suitable assumptions, we prove that the family of pullback attractors converges to the corresponding global compact attractor associated with the autonomous nonlocal limit problem when the parameter goes to zero.