Sharp interface limit of the Fisher-KPP equation

TL;DR

This paper analyzes the behavior of solutions to the Fisher-KPP equation as the parameter approaches zero, demonstrating the emergence and movement of sharp interfaces at a constant speed and estimating transition layer thickness.

Contribution

It provides a rigorous analysis of the sharp interface limit of the Fisher-KPP equation, including interface generation, motion, and conditions affecting interface phenomena.

Findings

Interface moves at a constant speed equal to the minimal wave speed.

Transition layer thickness can be estimated.

Certain initial data prevent interface formation.

Abstract

We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $\Vert x \Vert \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. We obtain an estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.