Journal of the European Mathematical Society

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Volume 7, Issue 2, 2005, pp. 173–213
DOI: 10.4171/JEMS/26

The speed of propagation for KPP type problems. I: Periodic framework

Henry Berestycki[1], François Hamel[2] and Nikolai Nadirashvili[3]

(1) CAMS - EHESS, Ecole des hautes études en sciences sociales, 190-198, avenue de France, 75244, PARIS Cedex 13, FRANCE
(2) Institut de Mathématiques de Marseille, UMR 7373, Université d'Aix-Marseille, 39 rue Frédéric Joliot-Curie, 13453, MARSEILLE CEDEX 13, FRANCE
(3) Department of Mathematics, University of Chicago, 5734 S. University Avenue, IL 60637, CHICAGO, UNITED STATES

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article \cite{bh}. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain, of the reaction, advection and diffusion coefficients are given. The last section deals with the notion of asymptotic spreading speed. The main properties of the spreading speed are given. Some of them are based on some new Liouville type results for nonlinear elliptic equations in unbounded domains.

Keywords: Reaction-diffusion equations, travelling fronts, propagation, periodic media, eigenvalue problems

Berestycki Henry, Hamel François, Nadirashvili Nikolai: The speed of propagation for KPP type problems. I: Periodic framework. J. Eur. Math. Soc. 7 (2005), 173-213. doi: 10.4171/JEMS/26