Entire Solution in Cylinder-Like Domains for a Bistable Reaction–Diffusion Equation

Antoine Pauthier

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3 Scopus citations

Abstract

We construct nontrivial entire solutions for a bistable reaction–diffusion equation in a class of domains that are unbounded in one direction. The motivation comes from recent results of Berestycki et al. (Calc Var Partial Differ Equ 55(3):1–32, 2016) concerning propagation and blocking phenomena in infinite domains. A key assumption in their study was the “cylinder-like” assumption: their domains are supposed to be straight cylinders in a half space. The purpose of this paper is to consider domains that tend to a straight cylinder in one direction. We need a different approach based on the long time stability of the bistable wave in heterogeneous media. We also prove the existence of an entire solution for a one-dimensional problem with a non-homogeneous linear term.

Original languageEnglish (US)
Pages (from-to)1273-1293
Number of pages21
JournalJournal of Dynamics and Differential Equations
Volume30
Issue number3
DOIs
StatePublished - Sep 1 2018

Bibliographical note

Funding Information:
The research leading to these results has received funding from the European Research Council under the European Union?s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 321186?ReaDi?Reaction?Diffusion Equations, Propagation and Modelling. This work was also partially supported by the French National Research Agency (ANR), within the project NONLOCAL ANR-14-CE25-0013. I am grateful to Henri Berestycki and Jean-Michel Roquejoffre for suggesting me the model and many fruitful conversations. I also would like to thank the anonymous referee for many helpful comments.

Keywords

  • Bistable equations
  • Invasion fronts
  • Reaction-diffusion equations

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