Positivity, local smoothing and Harnack inequalities for very fast diffusion equations
نویسندگان
چکیده
We investigate qualitative properties of local solutions u(t, x) ≥ 0 to the fast diffusion equation, ∂tu = ∆(u )/m with m < 1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions. 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 ≤ mc = (d− 2)/d. The boundedness statements are true even for m ≤ 0, while the positivity ones cannot be true in that range.
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