On heat conductors with a stationary hot spot

We consider a convex heat conductor having initial constant temperature and zero boundary temperature at every time. A hot spot is a point at which temperature attains its maximum at each given time; for convex heat conductors it is known that there is only one hot spot. If the conductor has certain symmetries, it is easy to prove that the hot spot does not move with time and we say it is stationary. Thus, we suppose to know that the hot spot does not move with time we prove symmetry results for planar triangular and quadrangular conductors. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Then, we examine the case of a general conductor (possibly not convex) and, by a asymptotic formula, we prove that, if there is a stationary critical point (not necessarily the hot spot), then the conductor must satisfy a very general geometric condition. This condition is obtained by using a balance law for solutions of the heat equation, the short-time behavior of the solution and the properties of the distance function from the boundary of the conductor.