Uniqueness of fast travelling fronts in reaction-diffusion equations with delay

TL;DR

This paper proves the uniqueness of positive traveling fronts in delayed reaction-diffusion equations for large velocities, using Lyapunov-Schmidt reduction and a small parameter approach.

Contribution

It introduces a novel method employing Lyapunov-Schmidt reduction with a small parameter to establish uniqueness for large velocities in delayed reaction-diffusion equations.

Findings

Uniqueness of positive traveling fronts for large velocities

Application of Lyapunov-Schmidt reduction in this context

Method applicable to monostable reaction-diffusion equations

Abstract

We consider positive travelling fronts of the time-delayed reaction-diffusion equation with the monostable birth function. Our main result says that for every fixed and sufficiently large velocity c, the positive travelling front is unique (modulo translations). To prove the uniqueness, we introduce a small parameter 1/c and realize the Lyapunov-Schmidt reduction in a scale of Banach spaces.