Wavefront's stability with asymptotic phase in the delayed monostable equations

TL;DR

This paper extends initial condition classes for delayed reaction-diffusion equations, enabling phase distortion computation of traveling waves, with specific results for Mackey-Glass and KPP-Fisher models.

Contribution

It introduces a method to compute phase distortion in delayed reaction-diffusion equations, improving stability analysis for certain models.

Findings

Phase distortion $\alpha$ can be computed in the moving frame.

$\alpha=0$ for KPP-Fisher delayed equation, indicating no phase shift.

The approach generalizes initial conditions for convergence to traveling waves.

Abstract

We extend the class of initial conditions for scalar delayed reaction-diffusion equations $u_t (t,x)=u_{xx}(t,x)+f(u(t, x), u(t-h, x))$ which evolve in solutions converging to monostable traveling waves. Our approach allows to compute, in the moving reference frame, the phase distortion $\alpha$ of the limiting travelling wave with respect to the position of solution at the initial moment $t=0$. In general, $\alpha\not=0$ for the Mackey-Glass type diffusive equation. Nevertheless, $\alpha=0$ for the KPP-Fisher delayed equation: the related theorem also improves existing stability conditions for this model.