An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation

TL;DR

This paper introduces an iterative method to estimate disturbances in semi-wavefronts of a delayed Fisher-KPP equation, establishing their exponential stability and uniqueness for certain wave speeds and delays.

Contribution

The paper presents a novel iterative estimation technique for disturbances and proves exponential stability and uniqueness of semi-wavefronts under specific conditions.

Findings

Semi-wavefronts are exponentially stable with an unbounded weight for speeds c > 2√2.

Uniqueness of semi-wavefronts is established for the same conditions.

The method applies to the delayed Fisher-KPP equation with delay h > 0.

Abstract

We give an iterative method to estimate the disturbance of semi-wavefronts of the equation: $\dot{u}(t,x) = u''(t,x) +u(t,x)(1-u(t-h,x)),$ $x \in \mathbb{R},\ t >0;$ where $h>0.$ As a consequence, we show the exponential stability, with an unbounded weight, of semi-wavefronts with speed $c>2\sqrt{2}$ and $h>0$. Under the same restriction of $c$ and $h$, the uniqueness of semi-wavefronts is obtained.