Vol. 13, No. 8, 2020

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Propagation properties of reaction-diffusion equations in periodic domains

Romain Ducasse

Vol. 13 (2020), No. 8, 2259–2288
Abstract

We study the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question raised by Berestycki, Hamel and Nadirashvili concerning the connection between the speed of invasion and the critical speed of fronts. To do so, we extend the classical Freidlin–Gärtner formula to such equations and we derive some bounds on the speed of invasion using estimates on the heat kernel. We also give geometric conditions on the domain that ensure that the spreading occurs at the critical speed of fronts.

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Keywords
propagation, spreading, reaction-diffusion equations, heat kernel, domains with obstacles, periodic domains, parabolic equations, elliptic equations, speed of propagation, geometry of the domain
Mathematical Subject Classification
Primary: 35K57, 35B40, 35K05, 35B51, 35B06
Milestones
Received: 16 April 2018
Revised: 4 July 2019
Accepted: 7 October 2019
Published: 28 December 2020
Authors
Romain Ducasse
École des Hautes Études en Sciences Sociales
PSL Research University
Centre d’Analyse et Mathématiques Sociales
Paris, France