论文标题
分数热流的固定临界点
The stationary critical points of the fractional heat flow
论文作者
论文摘要
我们研究了分数热方程的解决方案$ u = u（x，t）$的空间临界点。对于Cauchy问题，我们表明原点$ 0 $满足$ \ nabla_x u（0，t）= 0 $ for $ t> 0 $，并且仅当初始数据满足$ \ int _ {\ int _ {\ mathbb {s} s}^{n-1}} {n-1}}}}ωu_0（rω）的平衡定律。 e。 $ r \ ge 0 $。此外，对于Dirichlet初始边界价值问题，我们证明了两个对称结果：$ω$是一个以原点为中心的球，并且仅当$ \ nabla_x u（0，t）= 0 $ for $ t> 0 $ for $ t> 0 $时，其初始数据规定，初始数据满足上述余额法律； $ω$是且仅当$ \ nabla_x u（0，t）= 0 $ for $ t> 0 $的情况下，提供初始数据是centrosymmetric的。这些结果将Magnanini和Sakaguchi在1997 - 1999年将（局部）热方程式获得的一些定理扩展到了分数环境。这些延伸是不繁琐的，因为分数laplacian的非本地性质。除其他外，在Dirichlet的初始边界值问题中表征球的表征不仅适用于经典的热流，而且还可以对该问题有了新的了解。
We study the spatial critical points of the solutions $u=u(x,t)$ of the fractional heat equation. For the Cauchy problem, we show that the origin $0$ satisfies $\nabla_x u(0,t) = 0$ for $t>0$ if and only if the initial data satisfy a balance law of the form $\int_{\mathbb{S}^{N-1}} ωu_0(rω) \, \mathrm{d} ω=0$ for a. e. $r \ge 0$. Moreover, for the Dirichlet initial-boundary value problem, we prove two symmetry results: $Ω$ is a ball centered at the origin if and only if $\nabla_x u(0,t) = 0$ for $t>0$ provided that the initial data satisfies the above-mentioned balance law; $Ω$ is centrosymmetric if and only if $\nabla_x u(0,t) = 0$ for $t>0$ provided that the initial data is centrosymmetric. These results extend some theorems obtained by Magnanini and Sakaguchi in 1997-1999 for the (local) heat equation to the fractional context. These extensions are nontrivial because of the nonlocal nature of the fractional Laplacian. Among others, the proof of the characterization of a ball in the Dirichlet initial-boundary value problem for the fractional heat flow not only works for the classical heat flow but also gives a new insight into the problem.
