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Autonomous and non-autonomous unbounded attractors under perturbations

Part of: Parabolic equations and systems Partial differential equations Topological dynamics

Published online by Cambridge University Press:  27 December 2018

Alexandre N. Carvalho  and
Juliana F. S. Pimentel
Alexandre N. Carvalho
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos - SP, Brazil ([email protected])
Juliana F. S. Pimentel
Affiliation:
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-580, Santo André - SP, Brazil ([email protected])
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Abstract

Pullback attractors with forwards unbounded behaviour are to be found in the literature, but not much is known about pullback attractors with each and every section being unbounded. In this paper, we introduce the concept of unbounded pullback attractor, for which the sections are not required to be compact. These objects are addressed in this paper in the context of a class of non-autonomous semilinear parabolic equations. The nonlinearities are assumed to be non-dissipative and, in addition, defined in such a way that the equation possesses unbounded solutions as the initial time goes to -∞, for each elapsed time. Distinct regimes for the non-autonomous term are taken into account. Namely, we address the small non-autonomous perturbation and the asymptotically autonomous cases.

Keywords

Global attractorsasymptotic behaviorsemilinear parabolic equationsnonautonomous equationsperturbation of attractors

MSC classification

Primary: 35B41: Attractors
Secondary: 35B20: Perturbations 35B40: Asymptotic behavior of solutions 35B44: Blow-up 37B55: Nonautonomous dynamical systems 35K58: Semilinear parabolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 877 - 903
Copyright
Copyright © Royal Society of Edinburgh 2018 

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