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Unbounded Sturm attractors for quasilinear parabolic equations

Part of: Partial differential equations Local and nonlocal bifurcation theory Parabolic equations and systems

Published online by Cambridge University Press:  08 March 2024

Phillipo Lappicy Open the ORCID record for Phillipo Lappicy [Opens in a new window]  and
Juliana Fernandes
Phillipo Lappicy
Affiliation:
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. trabalhador são-carlense, São Carlos, Brazil (lappicy@hotmail.com) Depto. Análisis Matemático y Mat. Aplicada, Universidad Complutense de Madrid, Pl. de las Ciencas 3, Madrid, Spain
Juliana Fernandes
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos, Rio de Janeiro, Brazil (jfernandes@im.ufrj.br)
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Abstract

We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

Keywords

quasilinear parabolic equationsinfinite dimensional dynamical systemsgrow upinfinite time blow upunbounded global attractor

MSC classification

Primary: 35K10: Second-order parabolic equations 35B40: Asymptotic behavior of solutions 35B41: Attractors 35B44: Blow-up 37G35: Attractors and their bifurcations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 24
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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