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, partial derivative(t) u = Delta(u(m))/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] x Omega, with Omega subset of R-d. 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 <= m(c) = (d - 2)/d. The boundedness statements are true even for m <= 0, while the positivity ones cannot be true in that range. (C) 2009 Elsevier Inc. All rights reserved.