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Blocking and invasion for reaction–diffusion equations in periodic media

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Abstract

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.

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Notes

  1. Initial data and solutions are always understood to be bounded in order to avoid non-uniqueness issues.

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Authors and Affiliations

  1. Centre d’Analyse et Mathématiques Sociales, CNRS, École des Hautes Études en Sciences Sociales, PSL Research University, 54 boulevard Raspail, 75006, Paris, France

    Romain Ducasse & Luca Rossi

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  1. Romain Ducasse
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  2. Luca Rossi
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Correspondence to Luca Rossi.

Additional information

Communicated by P. Rabinowitz.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP/2007–2013)/ERC Grant Agreement No. 321186—ReaDi—Reaction–Diffusion Equations, Propagation and Modeling, and from the French National Research Agency (ANR), within project NONLOCAL ANR-14-CE25-0013. During this research, Luca Rossi was on academic leave from the University of Padova.

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Ducasse, R., Rossi, L. Blocking and invasion for reaction–diffusion equations in periodic media. Calc. Var. 57, 142 (2018). https://doi.org/10.1007/s00526-018-1412-0

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