Asymptotic one-dimensional symmetry for the Fisher-KPP equation
Résumé
Let u be a solution of the Fisher-KPP equation ∂ t u = ∆u + f (u), t > 0, x ∈ R N. We address the following question: does u become locally planar as t → +∞? Namely, does u(t n , x n + •) converge locally uniformly, up to subsequences, towards a one-dimensional function, for any sequence ((t n , x n)) n∈N in (0, +∞) × R N such that t n → +∞ as n → +∞? This question is in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. The answer depends on the initial datum u 0 of u. It is known to be affirmative when the support of u 0 is bounded or when it lies between two parallel half-spaces. Instead, the answer is negative when the support of u 0 is "V-shaped". We prove here that u is asymptotically locally planar when the support of u 0 is a convex set (satisfying in addition a uniform interior ball condition), or, more generally, when it is at finite Hausdorff distance from a convex set. We actually derive the result under an even more general geometric hypothesis on the support of u 0. We recover in particular the aforementioned results known in the literature. We further characterize the set of directions in which u is asymptotically locally planar, and we show that the asymptotic profiles are monotone. Our results apply in particular when the support of u 0 is the subgraph of a function with vanishing global mean.
Origine | Fichiers produits par l'(les) auteur(s) |
---|