The spatial critical points not moving along the heat flow (Q1361069)

The authors consider the heat equation \(u_t= \Delta u\) in \(\mathbb{R}^N \times (0, \infty)\), \(u(x,0) =\varphi (x)\), and the spatial critical set of the solutions \(C(t)= \{x\in \mathbb{R}^N \mid \nabla u(t,x) =0\}\). The basic problem in this paper is the following: If the spatial critical point does not move in \(t\), then what can we say about the solution? The author proves a certain balance law on the initial condition \(\varphi\). That is if the origin 0 satisfies \(\nabla u(0,t) =0\) in some time interval \((a,b)\), then \(\int_{S^{N-1}} \omega\varphi (r\omega) d\omega =0\) holds for almost all \(r\geq 0\). The case of a bounded domain is also considered, and several kinds of balance laws and their relation to the symmetry of the situation are studied.