Set-Valued and Variational Analysis

, Volume 21, Issue 1, pp 127–149 | Cite as

On the Theory of Global Attractors and Lyapunov Functionals

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Abstract

From the point of view of longterm dynamics, we study multivalued and single-valued semigroups of operators acting on complete metric spaces. We provide necessary and sufficient conditions for the existence of the global attractor under minimal requirements in terms of continuity of the semigroup. In the case of single-valued semigroups possessing a Lyapunov functional, we exhibit a simple proof of the existence and the characterization of the attractor in terms of the unstable set of stationary points. As an application, we consider the multivalued semigroup generated by the equation ruling the evolution of the specific humidity in a system of moist air, and we prove the existence of a regular global attractor.

Keywords

Multivalued dynamical systems Global attractors Lyapunov function Unstable set Stationary points 

Mathematics Subject Classifications (2010)

37L05 37L45 47H04 47H20 
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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentIndiana UniversityBloomingtonUSA

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