Regularity and structure of pullback attractors for reaction–diffusion type systems without uniqueness. (July 2016)

• Record Type: Journal Article
• Authors: Cui, Hongyong; Langa, José A.; Li, Yangrong
• Is Part Of: Nonlinear analysis. Volume 140(2016)
• Page Start: 208
• Page End: 235
• Publication Date: 2016-07
• Subjects: Pullback attractors -- Multi-valued non-autonomous dynamical systems -- Structure of pullback attractors; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.03.012
• Languages: English
• ISSNs: 0362-546X

Abstract

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.