Intrinsic Harnack inequality for local weak solutions to an anisotropic parabolic equation

We prove a Harnack inequality for non-negative solutions of a parabolic equation having an anisotropic slow diffusion. We study the propagation of support of solutions, through an iterative technique reminiscent of De Giorgi’s method and through the investigation of particular embeddings in anisotropic Sobolev spaces. At this point, we make an analysis of the natural scaling of the equation to reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution thanks to finite speed of propagation. Then we exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale-invariant Harnack inequality in an anisotropic intrinsic geometry, dictated by the powers of the diffusion coefficients. Finally we show some consequences as H¨older continuity of solutions, Liouville-type theorems and we formulate some open problems.