Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

TL;DR

This paper analyzes the asymptotic spreading speeds of solutions to Fisher-KPP reaction-diffusion equations with heterogeneous, shifting diffusion coefficients, revealing how the profile's monotony influences the spreading behavior and speeds.

Contribution

It characterizes the spreading speed as a function of the forcing speed and linear spreading speeds, especially highlighting the case of increasing diffusion profiles where faster spreading occurs.

Findings

Spreading speed depends on the monotony of the diffusion profile.

For increasing profiles, spreading can exceed the linear speed.

Constructs monotone traveling fronts with exponential decay near the unstable state.

Abstract

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotony of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems. Most notably, when the profile of the coefficient diffusion is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the coefficient diffusion is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.