Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

TL;DR

This paper advances understanding of the Fisher-KPP equation by precisely characterizing the propagation of solutions with slowly decaying initial data, extending previous results to broader classes of data and equations.

Contribution

It improves the precision of level set location estimates and characterizes the large-time profile for solutions with slowly decaying initial data.

Findings

Explicit highest order term in level set location

Broader class of initial data analyzed

Profile of solutions characterized for large times

Abstract

The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. Building on the results of Hamel and Roques '10, in this paper we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the level sets. In addition, we characterize the profile of the inhomogeneous problem for large times.