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The speed of propagation for KPP type problems. II: General domains

Authors: Henri Berestycki, François Hamel and Nikolai Nadirashvili
Journal: J. Amer. Math. Soc. 23 (2010), 1-34
MSC (2000): Primary 35A08, 35B30, 35K05, 35K57; Secondary 35B40, 35K15
Published electronically: July 6, 2009
MathSciNet review: 2552247
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Abstract: This paper is devoted to nonlinear propagation phenomena in general unbounded domains of $\mathbb {R}^N$, for reaction-diffusion equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. This article is the second in a series of two and it is the follow-up of the paper The speed of propagation for KPP type problems. I - Periodic framework, by the authors, which dealt which the case of periodic domains. This paper is concerned with general domains, and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data. We study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition. Some a priori bounds are proved for large classes of domains. The case of exterior domains is also discussed in detail. Lastly, some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited; the construction is based on some new heat kernel estimates in such domains.

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Additional Information

Henri Berestycki
Affiliation: EHESS, Centre d’Analyse et Mathématique Sociales, 54 Boulevard Raspail, F-75006 Paris, France
MR Author ID: 35060
ORCID: 0000-0003-1724-2279

François Hamel
Affiliation: Université Aix-Marseille III, Laboratoire d’Analyse, Topologie, Probabilités, Faculté des Sciences et Techniques, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France

Nikolai Nadirashvili
Affiliation: CNRS, Laboratoire d’Analyse, Topologie, Probabilités, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex 13, France

Keywords: Propagation, spreading, reaction-diffusion equations, heat kernel
Received by editor(s): March 26, 2007
Published electronically: July 6, 2009
Additional Notes: Part of this work was carried out during a visit by the first author to the Department of Mathematics of the University of Chicago, the hospitality of which is thankfully acknowledged.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.