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

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

2008-05-30

arxiv.org/abs/0805.4823v2

Mathematics

Analysis of PDEs

36 pages, 1 figure

Scientific paper

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ 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 $[0,T]\times\RR^d$. 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 $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0$, while the positivity ones cannot be true in that range.

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