# Thin front limit of an integro--differential Fisher--KPP equation with fat--tailed kernels

## TL;DR

This paper analyzes the asymptotic behavior of solutions to a Fisher-KPP integro-differential equation with fat-tailed kernels, revealing sharp bounds on spreading rates and Hamilton-Jacobi limits in different regimes.

## Contribution

It introduces new rescaling techniques and derives precise asymptotic bounds for solutions with fat-tailed dispersal kernels in Fisher-KPP equations.

## Key findings

- Sharp bounds on super-linear spreading rates
- Hamilton-Jacobi limits in long-time regimes
- Characterization of mutation effects in asymptotic behavior

## Abstract

We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit.