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Discontinuous stationary solutions to certain reaction-diffusion systems

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Abstract

Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the Brusselator model, the Gray-Scott model, the Oregonator model and a certain predator-prey model. It is shown that the considered systems have the both smooth and discontinuous stationary solutions, however, only discontinuous ones can be stable.

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Acknowledgements

S. Cygan acknowledges a support by the Polish NCN grant 2016/23/B/ST1/00434. The work of A. Marciniak-Czochra was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster and SFB1324 (B05).

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Authors and Affiliations

  1. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Szymon Cygan & Grzegorz Karch

  2. Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing (IWR) and BIOQUANT, University of Heidelberg, 69120, Heidelberg, Germany

    Anna Marciniak-Czochra

Authors
  1. Szymon Cygan
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  2. Anna Marciniak-Czochra
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  3. Grzegorz Karch
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Correspondence to Grzegorz Karch.

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Dedicated to Eiji Yanagida on the occasion of his birthday.

This article is part of the topical collection “Qualitative properties of solutions to nonlinear parabolic problems: In honor of Professor Eiji Yanagida on the occasion of his 65th birthday” edited by Senjo Shimizu, Tohru Ozawa, Kazuhiro Ishige, Marek Fila (Guest Editor)

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Cygan, S., Marciniak-Czochra, A. & Karch, G. Discontinuous stationary solutions to certain reaction-diffusion systems. Partial Differ. Equ. Appl. 3, 49 (2022). https://doi.org/10.1007/s42985-022-00188-x

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  • DOI: https://doi.org/10.1007/s42985-022-00188-x

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