Asymptotic one-dimensional symmetry for the Fisher–KPP equation

Abstract

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.