# Sharp large time behaviour in N -dimensional Fisher-KPP equations

## TL;DR

This paper investigates the long-term behavior of solutions to the N-dimensional Fisher-KPP equation with compact initial data, revealing convergence to a specific traveling wave profile modulated by a Lipschitz function on the sphere.

## Contribution

It extends Gärtner's earlier results by establishing the existence of a Lipschitz function that describes the asymptotic shape of solutions in higher dimensions.

## Key findings

- Solutions converge to a traveling wave profile with a Lipschitz modulation.
- The asymptotic shape involves a Lipschitz function on the unit sphere.
- The results generalize previous one-dimensional findings to N dimensions.

## Abstract

We study the large time behaviour of the Fisher-KPP equation $\partial$ t u = $\Delta$u + u -- u 2 in spatial dimension N , when the initial datum is compactly supported. We prove the existence of a Lipschitz function s of the unit sphere, such that u(t, x) converges, as t goes to infinity, to U c * |x| -- c * t + N + 2 c * lnt + s $\infty$ x |x| , where U c * is the 1D travelling front with minimal speed c * = 2. This extends an earlier result of G{\"a}rtner.