Regularity and structure of pullback attractors for reaction–diffusion type systems without uniqueness. (July 2016) - Cui, Hongyong; Langa, José A.; Li, Yangrong
Abstract: 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 ( τ ) τ ∈ 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 ⋃ s ⩽ τ A ( 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.