PARABOLIC HARNACK ESTIMATES FOR ANISOTROPIC SLOW DIFFUSION

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. 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 geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Holder continuity, an elliptic Harnack inequality and a Liouville theorem.

Automated summary

We establish a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. By identifying the natural scalings and reducing the problem to a Fokker-Planck equation, we construct a self-similar Barenblatt solution. Utilizing translation invariance and a self-iteration method, we obtain positivity near the origin and derive a sharp anisotropic expansion. This ultimately leads to a scale-invariant Harnack inequality in an anisotropic geometry determined by the diffusion coefficients' speed. As a consequence, Holder continuity, an elliptic Harnack inequality, and a Liouville theorem are inferred.