Some regularity results on the ‘relativistic’ heat equation

We prove some partial regularity results for the entropy solution u of the so-called relativistic heat equation. In particular, under some assumptions on the initial condition u0, we prove that ut(t) is a Radon measure in RN. Moreover, if u0 is log-concave inside its support Ω, Ω being a convex set, then we show the solution u(t) is also log-concave in its support Ω(t). This implies its smoothness in Ω(t). In that case we can give a simpler characterization of the notion of entropy solution.