Global positivity estimates and Harnack inequalities for the fast diffusion equation

Abstract

We investigate local and global properties of positive solutions to the fast diffusion equation u t = Δ u m in the range ( d − 2 ) + / d < m < 1 , corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space R d we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; we use them to derive sharp global positivity estimates and a global Harnack principle. For the mixed initial and boundary value problem posed in a bounded domain of R d with homogeneous Dirichlet condition, we prove weak and elliptic Harnack inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.