A characterization of a hyperplane in two-phase heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one has temperature 0 and the other has temperature 1. Suppose that the interface is uniformly of class $C^6$. We show that if the interface has a time-invariant constant temperature, then it must be a hyperplane.
Comment: 19 pages. arXiv admin note: text overlap with arXiv:1905.12380