Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness

In this paper, we study the pullback attractor for a general reaction-diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction-diffusion system is proved to have an invariant (H, V)-pullback attractor A = {A(tau)}(tau is an element of R). This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover boolean OR(s <=tau) A(s) is precompact in V, for all tau is an element of R. A non-autonomous Fitz-Hugh-Nagumo equation is studied as a specific example of the reaction-diffusion system. (C) 2016 Elsevier Ltd. All rights reserved.