Existence and Non-existence of Fisher-KPP Transition Fronts

TL;DR

This paper investigates how localized inhomogeneities in reaction rates affect the existence of transition fronts in Fisher-KPP reaction-diffusion equations, revealing conditions that prevent or allow such fronts.

Contribution

It provides the first example of a medium where no reaction-diffusion transition front exists due to localized inhomogeneity.

Findings

Strong localized inhomogeneity prevents transition front existence.

Weaker inhomogeneity allows transition fronts only within certain speed ranges.

Contrasts with known results in homogeneous and ignition-type media.

Abstract

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion transition front exists. A weaker localized inhomogeneity leads to existence of transition fronts but only in a finite range of speeds. These results are in contrast with both Fisher-KPP reactions in homogeneous media as well as ignition-type reactions in inhomogeneous media.