Front location determines convergence rate to traveling waves

2023/07/18 by An, Jing, Henderson, Christopher, Ryzhik, Lenya · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.09523

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.

Cited by

• Finite propagation and saturation in reaction-diffusion-advection equations governed by p-Laplacian operator
• Fronts and patterns with a dynamic parameter ramp

Related

• Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems
• Front propagation in both directions and coexistence of traveling fronts and pulses
• Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts
• Traveling waves for a Fisher-type reaction-diffusion equation with a flux in divergence form
• The Evolution of Travelling Waves in a KPP Reaction-Diffusion Model with cut-off Reaction Rate. II. Evolution of Travelling Waves