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

Abstract

We study the large time behaviour of the reaction-diffsuion equation ∂ t u = ∆u + f (u) in spatial dimension N , when the nonlinear 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 R N , as t goes to infinity, to, where U c * is the unique 1D travelling profile.This extends earlier results that identified the locations of the level sets of the solutions with o t→+∞ (t) precision, or identified precisely the level sets locations for almost radial initial data.

Global behaviour of solutions stable at infinity for gradient systems in higher space dimension: the no invasion case

“…For general solutions (without the radial symmetry hypothesis) stable at infinity of system (1.1), little seems to be known. This is in strike contrast with the scalar case d st equals 1, which is the subject of a large amount of literature: for extinction/invasion (threshold) results in relation with the initial condition and the reaction term see for instance [3,11,33,34,46], for local convergence and quasi-convergence results at large time see for instance [11,13,20,26,28,37], and for further estimates on the location and shape at large times of the level sets see for instance [12,19,20,23,24,[43][44][45].…”