A one-dimensional symmetry result for entire solutions to the Fisher-KPP equation

TL;DR

This paper proves that entire solutions to the Fisher-KPP equation with exponential decay similar to a traveling wave must be that wave, establishing a one-dimensional symmetry result for such solutions.

Contribution

It establishes a symmetry result showing solutions with specific decay properties are necessarily planar traveling waves, extending understanding of solution structure in Fisher-KPP equations.

Findings

Solutions with exponential decay matching a traveling wave are identical to that wave.

The result applies to solutions with decay properties for all times, positive and negative.

Provides a characterization of entire solutions based on decay behavior.

Abstract

We consider the Fisher-KPP reaction-diffusion equation in the whole space. We prove that if a solution has, to main order and for all times (positive and negative), the same exponential decay as a planar traveling wave with speed larger than the minimal one at its leading edge, then it has to coincide with the aforementioned traveling wave.