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Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

Arrieta Algarra, José María and Carvalho, Alexandre N. and Langa, José A. and Rodríguez Bernal, Aníbal (2012) Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations. Journal of Dynamics and Differential Equations, 24 (3). pp. 427-481. ISSN 1040-7294

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Abstract

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Item Type:Article
Uncontrolled Keywords:Nonautonomous dynamical systems; Hyperbolic global bounded solutions; Unstable manifolds; Dichotomy; Singular perturbations; Attractors; Lower semicontinuity
Subjects:Sciences > Mathematics > Differential equations
ID Code:16761
Deposited On:18 Oct 2012 11:02
Last Modified:07 Feb 2014 09:35

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