Propagation properties of reaction-diffusion equations in periodic domains

TL;DR

This paper investigates invasion phenomena in heterogeneous reaction-diffusion equations within periodic domains, extending classical formulas to connect invasion speed with front speed, using geometric and heat kernel estimates.

Contribution

It extends the Freidlin-Gartner formula to heterogeneous periodic domains and clarifies the relationship between invasion and front speeds.

Findings

Extended Freidlin-Gartner formula for periodic domains

Derived bounds on front speeds using heat kernel estimates

Connected invasion speed with front speed in heterogeneous media

Abstract

This paper studies 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 rised by Berestycki, Hamel and Nadirashvili in [5] concerning the connection between the speed of invasion and the speed of fronts. To do so, we extend the classical Freidlin-Gartner formula to such equations, using a geometrical argument devised by Rossi in [17], and derive some bounds on the speed of fronts using estimates on the heat kernel.