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Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

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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.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    José M. Arrieta & Aníbal Rodriguez-Bernal

  2. Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, São Carlos, SP, 13560-970, Brazil

    Alexandre N. Carvalho

  3. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, Sevilla, 41012, Spain

    José A. Langa

Authors
  1. José M. Arrieta
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  2. Alexandre N. Carvalho
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  3. José A. Langa
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  4. Aníbal Rodriguez-Bernal
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Correspondence to José M. Arrieta.

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Dedicated to Jack K. Hale, In Memoriam.

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Arrieta, J.M., Carvalho, A.N., Langa, J.A. et al. Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations. J Dyn Diff Equat 24, 427–481 (2012). https://doi.org/10.1007/s10884-012-9269-y

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  • DOI: https://doi.org/10.1007/s10884-012-9269-y

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