Nonlinear evolution inclusions driven by nonconvex sweeping processes: existence, uniqueness and global attractor

This paper is concerned with the study of a broad class of nonsmooth dynamical systems in infinite-dimensional Hilbert spaces, each composed of a semilinear differential inclusion coupled with a nonconvex sweeping process involving a uniformly prox-regular moving set and a nonlinear perturbation. Under the appropriate conditions, we first prove the existence of mild solutions to the nonsmooth dynamical system under consideration by applying the well-known Kakutani–Ky Fan fixed point theorem for multivalued maps, the theory of measures of noncompactness and the method of variational analysis. Then, when the infinitesimal generator A is dissipative or the multi-valued perturbation F is locally Lipschitz in the Hausdorff sense, the uniqueness results are obtained. Moreover, a framework for studying the long-time dynamical behavior of the solution set to the nonsmooth systems is established, in which three theorems for determining the existence of global attractors are delivered. Finally, four concrete examples are explored and numerical experiments are carried out to demonstrate consistency with the theoretical results.