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

Abstract

We investigate qualitative properties of local solutions u ( t , x ) ⩾ 0 to the fast diffusion equation, ∂ t u = Δ ( 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 ] × Ω , with Ω ⊆ 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.