Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions u(t,x)⩾0 to the fast diffusion equation, ∂tu=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m⩽mc=(d−2)/d. The boundedness statements are true even for m⩽0, while the positivity ones cannot be true in that range.