论文标题
Fisher-KPP方程的渐近一维对称性
Asymptotic one-dimensional symmetry for the Fisher-KPP equation
论文作者
论文摘要
令$ u $为Fisher-kpp方程的解决方案$$ \ partial_t u =ΔU+f（u），\ quad t> 0，\ x \ in \ mathbb {r}^n。 $$我们解决以下问题：$ u $是否成为$ t \ to+\ infty $的本地平面？也就是说，$ u（t_n，x_n+\ cdot）$ $在本地统一，直到子序列，朝着一维函数，对于任何序列$（（（t_n，x_n））_ {n \ in \ mathbb {n}}} $ $ t_n \ to+\ infty $ as $ n \ to+\ infty $？这个问题本着de Giorgi的猜想，用于艾伦 - 卡纳方程的固定解决方案。答案取决于$ u $的初始基准$ u_0 $。众所周知，当$ u_0 $的支持是有限的，或者位于两个平行半空间之间时，它是肯定的。相反，当$ U_0 $的支持为“ V形”时，答案为负。我们在这里证明，当$ u_0 $的支撑是凸集（另外令人满意的内部球条件）时，或者更一般而言，当它在有限的Hausdorff距离距离集合时，$ u $是渐近的本地平面。实际上，我们根据$ u_0 $的支持，在更通用的几何假设下得出了结果。我们特别恢复了文献中已知的上述结果。我们进一步描述了$ u $在本地平面上渐近的指示集，我们表明渐近配置文件是单调的。当$ u_0 $的支持是具有消失的全局平均值的函数的子图时，我们的结果特别适用。
Let $u$ be a solution of the Fisher-KPP equation $$ \partial_t u=Δu+f(u),\quad t>0,\ x\in\mathbb{R}^N. $$ 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 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.
