Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations |
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Authors: | José M Arrieta Alexandre N Carvalho José A Langa Aníbal Rodriguez-Bernal |
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Institution: | 1. Departamento de Matem??tica Aplicada, Facultad de Matem??ticas, Universidad Complutense de Madrid, 28040, Madrid, Spain 2. Instituto de Ci??ncias Matem??ticas e de Computa?ao, Universidade de S?o Paulo, Campus de S?o Carlos, Caixa Postal 668, S?o Carlos, SP, 13560-970, Brazil 3. Departamento de Ecuaciones Diferenciales y An??lisis Num??rico, Universidad de Sevilla, Apdo. de Correos 1160, Sevilla, 41012, Spain
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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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