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

We study the large time behaviour of the Fisher-KPP equation $\mathrm{\partial }$ t u = $\mathrm{\Delta }$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 $\mathrm{\infty }$ x |x| , where U c * is the 1D travelling front with minimal speed c * = 2. This extends an earlier result of G{"a}rtner.