Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows

Olexiy V. Kapustyan; Pavlo O. Kasyanov; José Valero

doi:10.3934/cpaa.2020096

Article Contents

2020, Volume 19, Issue 4: 2197-2217. Doi: 10.3934/cpaa.2020096

This issue Previous Article PDE problems with concentrating terms near the boundary Next Article Attractors for semilinear wave equations with localized damping and external forces

Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows

1.
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
2.
Institute for Applied System Analysis, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
3.
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

Dedicated to Prof. Tomás Caraballo on the occasion of his 60-th birthday

Received: June 30, 2019

Revised: August 31, 2019

Published: January 2020

The first two authors were partially supported by the State Fund for Fundamental Research of Ukraine. The third author was partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, project MTM2016-74921-P, and by Spanish Ministry of Science, Innovation and Universities, project PGC2018-096540-B-I00

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Abstract

We study properties of $ \omega $-limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains.

First, we prove that under certain conditions the $ \omega $-limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side.

Second, we give conditions ensuring that the $ \omega $-limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the $ \omega $-limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

Keywords:

Mathematics Subject Classification: 35B40, 35B41, 35K55, 37B20, 37B25, 37D15, 58C06.

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