Existence of pullback attractors for pullback asymptotically compact processes

• Published in: Nonlinear analysis Vol. 72; no. 3; pp. 1967 - 1976
• Main Authors: Caraballo, Tomás, Carvalho, Alexandre N., Langa, José A., Rivero, Felipe
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.02.2010; Elsevier
• Subjects: Evolution processes; Exact sciences and technology; Global analysis, analysis on manifolds; Group theory; Group theory and generalizations; Mathematical analysis; Mathematics; Non-autonomous dynamical systems; Ordinary differential equations; Pullback asymptotic compactness; Pullback attractors; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Evolution processes; Non-autonomous dynamical systems; Pullback asymptotic compactness; Pullback attractors; Damping; Autonomous system; Nonlinear analysis; Asymptotic behavior; Semigroup; Dynamical system; Mathematical model; Global attractor
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2009.09.037

This paper is concerned with the existence of pullback attractors for evolution processes. Our aim is to provide results that extend the following results for autonomous evolution processes (semigroups) (i) An autonomous evolution process which is bounded, dissipative and asymptotically compact has a global attractor. (ii) An autonomous evolution process which is bounded, point dissipative and asymptotically compact has a global attractor. The extension of such results requires the introduction of new concepts and brings up some important differences between the asymptotic properties of autonomous and non-autonomous evolution processes. An application to damped wave problem with non-autonomous damping is considered.