Morse decomposition of global attractors with infinite components

Tomás Caraballo; Juan C. Jara; José A. Langa; José Valero

doi:10.3934/dcds.2015.35.2845

Article Contents

2015, Volume 35, Issue 7: 2845-2861. Doi: 10.3934/dcds.2015.35.2845

This issue Previous Article From one-sided dichotomies to two-sided dichotomies Next Article Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

Morse decomposition of global attractors with infinite components

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3.
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received: November 30, 2013

Revised: November 30, 2014

Published: May 2017

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Abstract

In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.

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Mathematics Subject Classification: 37B25, 37L99, 35B40, 35B41.

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