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

By Matteo Bonforte, Juan Luis V'azquez. 2008

95 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d9ed021337e512b21986231f742071f4b3a1e7dd

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Citations in the corpus

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation (2009, background): “See precise expression in (4.2).”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, methodology): “our works [6, 9, 10, 11]; the above estimates are surprisingly valid for m > 1 and to our knowledge they have never been observed before.”