Existence and Characterization of Attractors for a Nonlocal Reaction–Diffusion Equation with an Energy Functional

Abstract

In this paper we study a nonlocal reaction–diffusion equation in which the diffusion depends on the gradient of the solution. Firstly, we prove the existence and uniqueness of regular and strong solutions. Secondly, we obtain the existence of global attractors in both situations under rather weak assumptions by defining a multivalued semiflow (which is a semigroup in the particular situation when uniqueness of the Cauchy problem is satisfied). Thirdly, we characterize the attractor either as the unstable manifold of the set of stationary points or as the stable one when we consider solutions only in the set of bounded complete trajectories.

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Acknowledgements

The first author is a fellow of the FPU program of the Spanish Ministry of Education, Culture and Sport, Reference FPU15/03080. This work has been partially supported by the Spanish Ministry of Science, Innovation and Universities, Project PGC2018-096540-B-I00, by the Spanish Ministry of Science and Innovation, Project PID2019-108654GB-I00, and by the Junta de Andalucía and FEDER, Project P18-FR-4509. We would like to thank the reviewer for his/her useful remarks.

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Correspondence to José Valero.

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Dedicated to the memory of Russell Johnson

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Caballero, R., Marín-Rubio, P. & Valero, J. Existence and Characterization of Attractors for a Nonlocal Reaction–Diffusion Equation with an Energy Functional. J Dyn Diff Equat (2021). https://doi.org/10.1007/s10884-020-09933-5

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Keywords

  • Reaction–diffusion equations
  • Nonlocal equations
  • Global attractors
  • Multivalued dynamical systems
  • Structure of the attractor

Mathematics Subject Classification

  • 35B40
  • 35B41
  • 35B51
  • 35K55
  • 35K57