Propagation properties of reaction-diffusion equations in periodic domains

Ducasse, Romain

doi:10.2140/apde.2020.13.2259

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Propagation properties of reaction-diffusion equations in periodic domains

Romain Ducasse

Vol. 13 (2020), No. 8, 2259–2288

DOI: 10.2140/apde.2020.13.2259

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.

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

Vol. 13, No. 8, 2020