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About the Structure of Attractors for a Nonlocal Chafee-Infante Problem

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  • Rubén Caballero

    (Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avenida Universidad s/n, 03202 Elche, Spain)

  • Alexandre N. Carvalho

    (Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, Brazil)

  • Pedro Marín-Rubio

    (Departamento Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/Tarfia, 41012 Sevilla, Spain)

  • José Valero

    (Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avenida Universidad s/n, 03202 Elche, Spain)

Abstract

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

Suggested Citation

  • Rubén Caballero & Alexandre N. Carvalho & Pedro Marín-Rubio & José Valero, 2021. "About the Structure of Attractors for a Nonlocal Chafee-Infante Problem," Mathematics, MDPI, vol. 9(4), pages 1-36, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:353-:d:496992
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    References listed on IDEAS

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    1. Paulo B. Brito, 2022. "The dynamics of growth and distribution in a spatially heterogeneous world," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 21(3), pages 311-350, September.
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    Cited by:

    1. Carmen Ionescu & Radu Constantinescu, 2022. "Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method," Mathematics, MDPI, vol. 10(15), pages 1-14, August.

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