Entire Solution in Cylinder-Like Domains for a Bistable Reaction–Diffusion Equation
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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.
KeywordsReaction-diffusion equations Invasion fronts Bistable equations
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.
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