The Freidlin-Gartner formula for general reaction terms

TL;DR

This paper introduces a geometric approach to analyze the propagation speeds in various reaction-diffusion equations, extending classical formulas to more general and complex cases with almost periodic and multistable dynamics.

Contribution

It provides a new geometric method that generalizes the Freidlin-Gartner formula to a wider class of reaction-diffusion equations, including non-periodic and multistable cases.

Findings

Proved the classical Freidlin-Gartner formula for periodic Fisher-KPP equations.

Extended the formula to monostable, combustion, and bistable cases.

Established the existence of asymptotic spreading speeds in almost periodic equations.

Abstract

We devise a new geometric approach to study the propagation of disturbance - compactly supported data - in reaction diffusion equations. The method builds a bridge between the propagation of disturbance and of almost planar solutions. It applies to very general reaction-diffusion equations. The main consequences we derive in this paper are: a new proof of the classical Freidlin-Gartner formula for the asymptotic speed of spreading for periodic Fisher-KPP equations, extension of the formula to the monostable, combustion and bistable cases, existence of the asymptotic speed of spreading for equations with almost periodic temporal dependence, multilevel propagation for multistable equations.