Front location determines convergence rate to traveling waves

TL;DR

This paper introduces a new method to determine how quickly solutions to reaction-diffusion equations approach traveling waves, revealing that convergence speed depends on the front location difference and unifying known convergence rates.

Contribution

The paper presents a novel approach based on the shape defect function to analyze convergence rates, simplifying proofs and unifying different cases of traveling wave convergence.

Findings

Convergence rate is controlled by the distance between phantom and true front locations.

The approach simplifies proofs in the Fisher-KPP case.

Provides a unified explanation for algebraic and exponential convergence rates.

Abstract

We propose a novel method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in [2]. It turns out that the convergence rate is controlled by the distance between the ``phantom front location'' for the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave itself has a pulled nature, regardless of whether the traveling wave is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach simplifies dramatically the proof in the Fisher-KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher-KPP case and the exponential rates in the pushed case.