Existence and Non-existence of Fisher-KPP Transition Fronts

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.

Cited by in corpus (8)

• Spreading speeds for one-dimensional monostable reaction-diffusion equations
• Slowdown for time inhomogeneous branching Brownian motion
• Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence
• Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations
• Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero
• Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity
• Multidimensional transition fronts for Fisher-KPP reactions
• Existence and non-existence of transition fronts in mixed ignition-monostable media