Sharp large time behaviour in N-dimensional reaction-diffusion equations of bistable type

We study the large time behaviour of reaction-diffusion equation ∂tu=Δu+f(u) in spatial dimension N, when non-linear term is bistable and initial datum compactly supported. prove existence a Lipschitz function s∞ unit sphere, such that u(t,x) converges uniformly RN, as t goes to infinity, Uc⁎(|x|−c⁎t+N−1c⁎lnt+s∞(x|x|)), where Uc⁎ unique 1D travelling profile. This extends earlier results identified locations level sets solutions with ot→+∞(t) precision, or precisely for almost radial data.