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arXiv · 0805.4823

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

Matteo Bonforte·Juan Luis Vazquez
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Abstract

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =Δ(u^m)/m$ 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 $[0,T]\times\RR^d$. 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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Matteo Bonforte, Juan Luis Vazquez. 2008-12-01. Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations. https://arxiv.org/abs/0805.4823

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