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

• Volume: 007, Issue: 2, page 173-213
• ISSN: 1435-9855

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## Abstract

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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 [7]. 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.

## How to cite

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Berestycki, Henry, Hamel, François, and Nadirashvili, Nikolai. "The speed of propagation for KPP type problems. I: Periodic framework." Journal of the European Mathematical Society 007.2 (2005): 173-213. <http://eudml.org/doc/277219>.

@article{Berestycki2005,
abstract = {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 [7]. 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.},
author = {Berestycki, Henry, Hamel, François, Nadirashvili, Nikolai},
journal = {Journal of the European Mathematical Society},
keywords = {reaction-diffusion equations; travelling fronts; propagation; periodic media; eigenvalue problems},
language = {eng},
number = {2},
pages = {173-213},
publisher = {European Mathematical Society Publishing House},
title = {The speed of propagation for KPP type problems. I: Periodic framework},
url = {http://eudml.org/doc/277219},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Berestycki, Henry
AU - Hamel, François
TI - The speed of propagation for KPP type problems. I: Periodic framework
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 2
SP - 173
EP - 213
AB - 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 [7]. 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.
LA - eng
KW - reaction-diffusion equations; travelling fronts; propagation; periodic media; eigenvalue problems
UR - http://eudml.org/doc/277219
ER -

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