IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i4p353-d496992.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/4/353/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/4/353/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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.
    2. Camacho, Carmen & Pérez-Barahona, Agustín, 2015. "Land use dynamics and the environment," Journal of Economic Dynamics and Control, Elsevier, vol. 52(C), pages 96-118.
    3. Boucekkine, Raouf & Camacho, Carmen & Zou, Benteng, 2009. "Bridging The Gap Between Growth Theory And The New Economic Geography: The Spatial Ramsey Model," Macroeconomic Dynamics, Cambridge University Press, vol. 13(1), pages 20-45, February.
    4. Giorgio FABBRI, 2014. "Ecological Barriers and Convergence: a Note on Geometry in Spatial Growth Models," LIDAM Discussion Papers IRES 2014014, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    5. Raouf Boucekkine & Carmen Camacho & Fabbri Giorgio, 2013. "On the optimal control of some parabolic partial differential equations arising in economics," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00973388, HAL.
    6. Augeraud-Veron, Emmanuelle & Boucekkine, Raouf & Gozzi, Fausto & Venditti, Alain & Zou, Benteng, 2024. "Fifty years of mathematical growth theory: Classical topics and new trends," Journal of Mathematical Economics, Elsevier, vol. 111(C).
    7. Fabbri, Giorgio, 2016. "Geographical structure and convergence: A note on geometry in spatial growth models," Journal of Economic Theory, Elsevier, vol. 162(C), pages 114-136.
    8. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "Growth and agglomeration in the heterogeneous space: a generalized AK approach," Journal of Economic Geography, Oxford University Press, vol. 19(6), pages 1287-1318.
    9. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," Working Papers halshs-01982243, HAL.
    10. Boucekkine, R. & Camacho, C. & Fabbri, G., 2013. "Spatial dynamics and convergence: The spatial AK model," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2719-2736.
    11. Boucekkine, Raouf & Fabbri, Giorgio & Federico, Salvatore & Gozzi, Fausto, 2022. "A dynamic theory of spatial externalities," Games and Economic Behavior, Elsevier, vol. 132(C), pages 133-165.
    12. Albeverio, Sergio & Mastrogiacomo, Elisa, 2022. "Large deviation principle for spatial economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    13. Camacho, Carmen & Zou, Benteng & Briani, Maya, 2008. "On the dynamics of capital accumulation across space," European Journal of Operational Research, Elsevier, vol. 186(2), pages 451-465, April.
    14. Breinlich, Holger & Ottaviano, Gianmarco I.P. & Temple, Jonathan R.W., 2014. "Regional Growth and Regional Decline," Handbook of Economic Growth, in: Philippe Aghion & Steven Durlauf (ed.), Handbook of Economic Growth, edition 1, volume 2, chapter 4, pages 683-779, Elsevier.
    15. Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2023. "Spatial growth theory: Optimality and spatial heterogeneity," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    16. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2020. "Optimal location of economic activity and population density: The role of the social welfare function," AMSE Working Papers 2003, Aix-Marseille School of Economics, France.
    17. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "A spatiotemporal framework for the analytical study of optimal growth under transboundary pollution," AMSE Working Papers 1926, Aix-Marseille School of Economics, France.
    18. Calvia, Alessandro & Gozzi, Fausto & Leocata, Marta & Papayiannis, Georgios I. & Xepapadeas, Anastasios & Yannacopoulos, Athanasios N., 2024. "An optimal control problem with state constraints in a spatio-temporal economic growth model on networks," Journal of Mathematical Economics, Elsevier, vol. 113(C).
    19. Brock, William & Xepapadeas, Anastasios, 2008. "Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control," Journal of Economic Dynamics and Control, Elsevier, vol. 32(9), pages 2745-2787, September.
    20. Brito, Paulo, 2011. "Global endogenous growth and distributional dynamics," MPRA Paper 41653, University Library of Munich, Germany.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:353-:d:496992. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.