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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 iscompactly supported. We prove the existence of a Lipschitz function s of theunit 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 travellingfront with minimal speed c * = 2. This extends an earlier result ofG{\"a}rtner.

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