Transition fronts for the Fisher-KPP equation

TL;DR

This paper studies the complex dynamics of transition fronts in Fisher-KPP reaction-diffusion equations, characterizing their speeds, profiles, and acceleration properties, and classifying them within superpositions of traveling fronts.

Contribution

It provides a comprehensive description and classification of transition fronts, including their asymptotic speeds, profiles, and the fact that they can only accelerate.

Findings

Characterized admissible asymptotic speeds and profiles of transition fronts.

Proved that transition fronts can only accelerate.

Classified transition fronts as superpositions of standard traveling fronts.

Abstract

This paper is concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts, but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich. In the paper, we describe the class of transition fronts and we study their qualitative dynamical properties. In particular, we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate. We also classify the transition fronts in the class of measurable superpositions of standard traveling fronts.