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

Abstract

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