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

Berestycki, Henri; Hamel, François; Nadirashvili, Nikolai

doi:10.1090/S0894-0347-09-00633-X

The speed of propagation for KPP type problems. II: General domains
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by Henri Berestycki, François Hamel and Nikolai Nadirashvili;

J. Amer. Math. Soc. 23 (2010), 1-34

DOI: https://doi.org/10.1090/S0894-0347-09-00633-X

Published electronically: July 6, 2009

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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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