Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations (Q1047665)

The authors investigate qualitative properties of local solutions \(u(t,x)\geqslant 0\) to the fast diffusion equation, \(\partial _tu=\Delta (u^m)/m\) with \(m<1\). For such a kind of equation it is known that intrinsic Harnack inequality does not hold for low \(m\) in the so-called very fast diffusion range, precisely for all \(m\leqslant m_c=(d - 2)/d\). Their main results are quantitative positivity and boundedness estimates for locally defined solutions that they combine into forms of interesting new Harnack-like inequalities. The boundedness statements are true even for \(m\leqslant 0\).