Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay

TL;DR

This paper analyzes the sharp interface limit of the Fisher-KPP equation with initial data exhibiting slow exponential decay, revealing how the interface speed depends on initial tail behavior and providing detailed layer thickness estimates.

Contribution

It introduces a detailed analysis of the interface dynamics for the Fisher-KPP equation with slow exponential decay initial data, highlighting the dependence of interface speed on initial tail decay.

Findings

Interface moves with a constant speed influenced by initial tail decay

New estimates for the thickness of transition layers

Behavior differs significantly from compact support initial data

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 perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.