Entire Solution in Cylinder-Like Domains for a Bistable Reaction–Diffusion Equation
- 116 Downloads
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.
- 1.Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In Partial Differential Equations and Related Topics (Program, Tulane University, New Orleans, LA, 1974). Lecture Notes in Mathematics, vol. 446, pp 5–49. Springer, Berlin (1975)Google Scholar
- 3.Berestycki, H., Bouhours, J., Chapuisat, G.: Front blocking and propagation in cylinders with varying cross section. Calc. Var. Partial Differ. Equ. 55(3), 1–32 (2016)Google Scholar
- 12.Kolmogorov, A., Petrovsky, I., Piskounov, N.: Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Etat Moscou 1, 1–26 (1937)Google Scholar
- 13.Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)Google Scholar
- 15.Matano, H.: Talk presented at IHP, Paris, September 2002Google Scholar
- 20.Zlatoš, A.: Existence and non-existence of transition fronts for bistable and ignition reactions. Ann. Inst. H. Poincaré Anal. Non Linéaire (2017)Google Scholar