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Stationary fronts and pulses for multistable equations with saturating diffusion

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Abstract

We deal with stationary solutions of a reaction-diffusion equation with flux-saturated diffusion and multistable reaction term, in dependence on a positive parameter \(\varepsilon \). Motivated by previous numerical results obtained by A. Kurganov and P. Rosenau (Nonlinearity, 2006), we investigate stationary solutions of front and pulse-type and discuss their qualitative features. We study the limit of such solutions for \(\varepsilon \rightarrow 0\), showing that, in spite of the wide variety of profiles that can be constructed, there is essentially a unique configuration in the limit for both stationary fronts and pulses. We finally discuss some variational features that include the case where the solutions having continuous energy may not be global minimizers of the associated action functional.

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Acknowledgements

The authors sincerely thank Prof. Pierpaolo Omari for helpful comments and the anonymous referee for his/her suggestions, which helped to improve a previous version of the manuscript.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

    Maurizio Garrione

  2. Dipartimento di Scienze e Metodi dell’Ingegneria, Università degli Studi di Modena e Reggio Emilia, Via Università 4, 41121, Modena, Italy

    Elisa Sovrano

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  1. Maurizio Garrione
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  2. Elisa Sovrano
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Correspondence to Elisa Sovrano.

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The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and acknowledge financial support from this institution. The first author has been supported by the PRIN project 201758MTR2: “Direct and inverse problems for partial differential equations: theoretical aspects and applications.” The second author has been supported by the Fondation Sciences Mathématiques de Paris (FSMP) through the project: “Reaction-diffusion equations in population genetics: a study of the influence of geographical barriers on traveling waves and non-constant stationary solutions”.

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Garrione, M., Sovrano, E. Stationary fronts and pulses for multistable equations with saturating diffusion. Nonlinear Differ. Equ. Appl. 30, 31 (2023). https://doi.org/10.1007/s00030-023-00842-2

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  • DOI: https://doi.org/10.1007/s00030-023-00842-2

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