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We study the spatial critical points of the solutions $u=u(x,t)$ of thefractional 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 datasatisfy a balance law of the form $\int_{\mathbb{S}^{N-1}} \omega u_0(r\omega)\, \mathrm{d} \omega=0$ for a. e. $r \ge 0$. Moreover, for the Dirichletinitial-boundary value problem, we prove two symmetry results: $\Omega$ is aball 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;$\Omega$ 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 sometheorems obtained by Magnanini and Sakaguchi in 1997-1999 for the (local) heatequation to the fractional context. These extensions are nontrivial because ofthe nonlocal nature of the fractional Laplacian. Among others, the proof of thecharacterization of a ball in the Dirichlet initial-boundary value problem forthe fractional heat flow not only works for the classical heat flow but alsogives a new insight into the problem.

The stationary critical points of the fractional heat flow