Gradient estimates and symmetrization for Fisher-KPP front propagation with fractional diffusion

TL;DR

This paper establishes exponential decay estimates for the gradient of solutions to the fractional Fisher-KPP equation and demonstrates that the reaction front becomes circular over time, indicating flattening and symmetry.

Contribution

It provides the first gradient decay estimates for solutions with fractional diffusion and uses these to prove front symmetrization in the Fisher-KPP equation.

Findings

Gradient of solutions decays exponentially over time

Reaction front flattens and becomes circular

New symmetrization result for fractional Fisher-KPP

Abstract

In this paper, we study gradient decay estimates for solutions to the multi-dimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flattens and quantifiably circularizes, losing its initial structure.