Advances in Differential Equations

The speed of propagation for KPP reaction-diffusion equations within large drift

Mohammad El Smaily and Stéphane Kirsch

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This paper is devoted to the study of the asymptotic behaviors of the minimal speed of propagation of pulsating travelling fronts solving the Fisher-KPP reaction-advection-diffusion equation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. We consider a periodic heterogenous framework and we use the formula of Berestycki, Hamel, and Nadirashvili [3] for the minimal speed of propagation to prove the asymptotics in any space dimension $N.$ We express the limits as the maxima of certain variational quantities over the family of ``first integrals'' of the advection field. Then, we perform a detailed study in the case $N=2$ which leads to a necessary and sufficient condition for the positivity of the asymptotic limit of the minimal speed within a large drift.

Article information

Adv. Differential Equations Volume 16, Number 3/4 (2011), 361-400.

First available in Project Euclid: 18 December 2012

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Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35Q80: PDEs in connection with classical thermodynamics and heat transfer 35Q92: PDEs in connection with biology and other natural sciences 37C10: Vector fields, flows, ordinary differential equations 80A32: Chemically reacting flows [See also 92C45, 92E20]


El Smaily, Mohammad; Kirsch, Stéphane. The speed of propagation for KPP reaction-diffusion equations within large drift. Adv. Differential Equations 16 (2011), no. 3/4, 361--400.

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