Rigorous results for the speed of Kolmogorov--Petrovskii--Piscounov fronts with a cutoff

TL;DR

This paper rigorously analyzes how a cut-off affects the speed of reaction-diffusion fronts, providing bounds and validity ranges for the Brunet--Derrida approximation across various reaction terms.

Contribution

It introduces rigorous bounds on front speed with a cut-off and assesses the Brunet--Derrida formula's applicability for different reaction functions.

Findings

Established upper and lower bounds on front speed based on cut-off parameter epsilon.

Quantified the validity range of the Brunet--Derrida formula for general reaction terms.

Provided insights into the influence of cut-offs on reaction-diffusion front propagation.

Abstract

We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the range of validity of the Brunet--Derrida formula for a general class of reaction terms.