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)(H,V)-pullback attractor A={A(τ)}τ∈RA={A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of HH in the topology of VV, and moreover⋃s⩽τA(s)is precompact in  V,∀τ∈R.A non-autonomous Fitz-Hugh–Nagumo equation is studied as a specific example of the reaction–diffusion system.