Robustness of dynamically gradient multivalued dynamical systems

Rubén Caballero; Alexandre N. Carvalho; Pedro Marín-Rubio; José Valero

doi:10.3934/dcdsb.2019006

Article Contents

2019, Volume 24, Issue 3: 1049-1077. Doi: 10.3934/dcdsb.2019006

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Robustness of dynamically gradient multivalued dynamical systems

1.
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain
2.
Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil
3.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain
4.
Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

Received: January 2018

Revised: May 2018

Early access: January 7, 2019

Published online: January 7, 2019

The authors of this work have been partially funded by the following projects: R. Caballero is a fellow of Programa de FPU del Ministerio de Educación, Cultura y Deporte, reference FPU15/03080; P. Marín-Rubio was supported by projects PHB2010-0002-PC (Ministerio de Educación-DGPU), MTM2015-63723-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía); A. N. Carvalho was supported by grant 2018/10997-6 from FAPESP-Brazil and grant 303929/2015-4 from CNPq-Brazil; and J.Valero by projects MTM2015-63723-P, MTM2016-74921-P (MINECO/FEDER, EU) and P12-FQM-1492 (Junta de Andalucía).

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Abstract

In this paper we study the robustness of dynamically gradient multivalued semiflows. As an application, we describe the dynamical properties of a family of Chafee-Infante problems approximating a differential inclusion studied in [3], proving that the weak solutions of these problems generate a dynamically gradient multivalued semiflow with respect to suitable Morse sets.

Keywords:

Attractors,
reaction-diffusion equations,
stability,
dynamically gradient multivalued semiflows,
Morse decomposition,
set-valued dynamical systems.

Mathematics Subject Classification: 37B25, 35B40, 35B41, 35K55, 37L30, 58C06.

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References

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