Asymptotic one-dimensional symmetry for the Fisher–KPP equation

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Let u be a solution of the Fisher–KPP equation \partial_{t} u=\Delta u+f(u) , t>0 , x\in\mathbb{R}^{N} , with an initial datum u_{0} . We address the following question: does u become locally planar as t\to+\infty ? Namely, does u(t_{n},x_{n}+\cdot) converge locally uniformly, up to subsequences, towards a one-dimensional function, for any sequence ((t_{n},x_{n}))_{n\in\mathbb{N}} in (0,+\infty)\times\mathbb{R}^{N} such that t_{n}\to+\infty as n\to+\infty ? This question is in the spirit of the celebrated De Giorgi’s conjecture concerning stationary solutions of the Allen–Cahn equation. Some affirmative answers to the above question are known in the literature: when the support of the initial datum 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.