arXiv 2207.051472022-07-13math.AP

Fisher-KPP方程的渐近一维对称性

Asymptotic one-dimensional symmetry for the Fisher-KPP equation

François Hamel, Luca Rossi

备注：: arXiv admin note: text overlap with arXiv:2105.08344

摘要：

设$u$是Fisher-KPP方程的解$$ \partial_t u=Δu+f(u),\quad t>0,\ x\in\mathbb{R}^N. $$，我们解决以下问题：$u$是否成为局部平面的$t\to+\infty$？也就是说，$u(t_n,x_n+\cdot)$是否局部一致收敛，直到收敛，朝向一维函数，对于$(0,+\infty)\times\mathbb{R}^N$中的任何序列$((t_n,x_n))_{n\in\mathbb{N}}$，使得$t_n\to+\infty$作为$n\to+\infty$？这个问题是在德乔吉猜想的精神，固定解决方案的艾伦-卡恩方程。答案取决于$u$的初始数据$u_0$。已知当$u_0$的支集有界或位于两个平行半空间之间时，它是肯定的。相反，当$u_0$的支持呈“V”形时，答案是否定的。我们证明了当$u_0$的支撑是一个凸集（另外满足一个均匀的内部球条件），或者更一般地，当它是在一个凸集的有限Hausdorff距离时，$u$是渐近局部平面的.我们实际上是在$u_0$的支持下，在一个更一般的几何假设下得出这个结果的。我们恢复，特别是在文献中已知的上述结果。进一步刻画了$u$渐近局部平面的方向集，并证明了渐近轮廓是单调的。当$u_0$的支持是一个函数的子图与消失的全球平均值时，我们的结果特别适用。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文