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X-WR-CALNAME:Dipartimento di Matematica UNIPV
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UID:MEC-71ee911dd06428a96c143a0b135041a4@matematica.unipv.it
DTSTART:20200116T150000Z
DTEND:20200116T160000Z
DTSTAMP:20201030T102600Z
CREATED:20201030
LAST-MODIFIED:20210114
PRIORITY:5
TRANSP:OPAQUE
SUMMARY:Global Harnack Principle for a class of fast diffusion equations
DESCRIPTION:Nikita Simonov, Universidad Autonoma de Madrid\nAula Beltrami, Dipartimento di Matematica – Giovedì 16 Gennaio 2020 h.16:00\n \nAbstract. We study global properties of non-negative, integrable solutions to the Cauchy problem of the weighted fast diffusion equation u_t = |x|^s div(|x|^{-r} ∇u^m ) with (d − 2 − r)/(d − s) < m < 1. The weights |x|^s and |x|^ {−r} , with s < d and s − 2 < r ≤ s(d − 2)/d can be both degenerate and singular and need not belong to the class A_2 , this range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities.\nWe characterize the largest class of data for which the so called Global Harnack Principle (GHP) holds\n(a global lower and upper bound in terms of suitable Barenblatt solutions). As a consequence of the GHP,\nwe prove convergence of the uniform relative error, namely |(u − B)/B| → 0 as t → ∞ uniformly in Rd,\nwhere B is a suitable Barenblatt solution. In the case with no weights (s = r = 0) and for a special\nclass of data, we give (almost) sharp rates of convergence to the Barenblatt profile in the L^1 and the L^∞\ntopologies, in the radial case we give sharp rates.\nWe extend some of the results to non-negative, integrable solutions to the Cauchy problem of the\np-Laplace evolution equation u_t = ∆_p(u), where ∆_p(w) := div(|∇w|^(p−2) ∇w), with 2d/(d + 1) < p < 2.\nThe above results were obtained in collaboration with Prof. M. Bonforte and D. Stan.\n
URL:https://matematica.unipv.it/events/global-harnack-principle-for-a-class-of-fast-diffusion-equations-2/
ORGANIZER;CN=Nikita Simonov:MAILTO:
CATEGORIES:Appuntamenti
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