Multidimensional transition fronts for Fisher-KPP reactions

TL;DR

This paper characterizes multidimensional transition fronts and solutions with bounded width for Fisher-KPP reaction-diffusion equations, especially those exceeding the classical propagation speed, under certain smoothness and concavity conditions.

Contribution

It provides an almost complete classification of transition fronts in higher dimensions for Fisher-KPP equations with super-critical speeds, extending previous understanding.

Findings

Characterization of transition fronts exceeding classical speed

Conditions under which solutions have bounded width

Extension of known results to multidimensional settings

Abstract

We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for each $c<2\sqrt{f'(0)}\,$,} \] and we consider here those satisfying \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for some $c>2\sqrt{f'(0)}\,$.} \] When $f$ is $C^2$ and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions.