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Article Dans Une Revue Journal of the European Mathematical Society Année : 2024

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.
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Dates et versions

hal-03715695 , version 1 (11-07-2022)
hal-03715695 , version 2 (02-07-2024)

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François Hamel, Luca Rossi. Asymptotic one-dimensional symmetry for the Fisher-KPP equation. Journal of the European Mathematical Society, In press. ⟨hal-03715695v2⟩
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