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dc.contributor.authorBonforte, Matteo
dc.contributor.authorSimonov, Nikita
dc.date.accessioned2021-03-24T10:51:03Z
dc.date.available2021-03-24T10:51:03Z
dc.date.issued2019
dc.identifier.issn0001-8708
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/21657
dc.language.isoenen
dc.subjectFast diffusion with weightsen
dc.subjectParabolic regularityen
dc.subjectSmoothing effectsen
dc.subjectHarnack inequalitiesen
dc.subjectHölder continuityen
dc.subjectCaffarelli–Kohn–Nirenberg inequalitiesen
dc.subject.ddc515en
dc.titleQuantitative a priori estimates for fast diffusion equations with Caffarelli–Kohn–Nirenberg weights. Harnack inequalities and Hölder continuityen
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenWe investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) ut = |x|γdiv|x|−β∇um) posed on (0, +∞)×Rd, with d ≥ 3, in the so-called good fast diffusion range mc < m < 1, within the range of parameters γ, β which is optimal for the validity of the so-called Caffarelli-Kohn-Nirenberg inequalities. It is a natural question to ask in which sense such solutions behave like the Barenblatt B (fundamental solution): for instance, asymptotic convergence, i.e. ku(t) − B(t)kLp(Rd) t→∞ −−−→ 0, is well known for all 1 ≤ p ≤ ∞, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data X ⊂ L1+(Rd) that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (UREC), i.e. d((t))=ku(t)/B(t) − 1kL∞(Rd)t→∞ −−−→ 0. Such characterization is in terms of an integral condition on u(t = 0). To the best of our knowledge, analogous issues for the linear heat equation m = 1, do not possess such clear answers, only partial results. Our characterization is also new for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to B in different norms. Such rates are almost optimal in the non weighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in L1+(Rd) \ X, preserve the same “fat” spatial tail for all times, hence UREC fails and d∞(u(t))=∞, even if ku(t) − B(t)kL1(Rd)t→∞ −−−→ 0en
dc.relation.isversionofjnlnameAdvances in Mathematics
dc.relation.isversionofjnlvol345en
dc.relation.isversionofjnldate2019-03
dc.relation.isversionofjnlpages1075-1161en
dc.relation.isversionofdoi10.1016/j.aim.2019.01.018en
dc.relation.isversionofjnlpublisherElsevieren
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.relation.forthcomingprintnonen
dc.description.ssrncandidatenonen
dc.description.halcandidatenonen
dc.description.readershiprechercheen
dc.description.audienceInternationalen
dc.relation.Isversionofjnlpeerreviewedouien
dc.relation.Isversionofjnlpeerreviewedouien
dc.date.updated2021-03-24T10:48:00Z
hal.person.labIds82433
hal.person.labIds60


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