Sharp large time behaviour in n-dimensional Fisher-KPP equations

We study the large time behaviour of the Fisher-KPP equation ∂tu = ∆u+u−u2 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) approaches, as t goes to infinity, the function Uc∗ ( |x| − c∗t + Nc+∗2 lnt + s∞(|xx| )) , where Uc∗ is the 1D travelling front with minimal speed c∗ = 2. This extends an earlier result of Gärtner