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This work was partially funded by Projects MTM2014-52240-P and MTM2017-85757-P (Spanish Government), and by the E.U. H2020 MSCA programme, grant agreement 777822. N.S. was partially funded by the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Spain). We would like to warmly thank the anonymous reviewer for an extraordinary work that helped us a lot to improve the paper to its current status.

Analysis of institutional authors

Bonforte, MatteoAuthorSimonov, NikitaAuthor

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Quantitative a priori estimates for fast diffusion equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and Holder continuity

Publicated to:ADVANCES IN MATHEMATICS. 345 1075-1161 - 2019-03-17 345(), DOI: 10.1016/j.aim.2019.01.018

Authors: Bonforte, Matteo; Simonov, Nikita;

Affiliations

Univ Autonoma Madrid, Dept Matemat, Campus Cantoblanco, E-28049 Madrid, Spain - Author

Abstract

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation u(t) = vertical bar x vertical bar(gamma)del . (vertical bar x vertical bar(-beta)del u(m)), with 0 < m < 1 posed on cylinders of (0, T) x R-N. The weights vertical bar x vertical bar(gamma) and vertical bar x vertical bar(-beta), with gamma < N and gamma - 2 < beta <= gamma(N - 2)/N can be both degenerate and singular and need not belong to the class A(2), a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain Holder continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, m = 1, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights. (C) 2019 Elsevier Inc. All rights reserved.

Keywords

Caffarelli–kohn–nirenberg inequalitiesCaffarolli-kohn-nirenberg inequalitiesDegenerate parabolic equationsFast diffusion with weightsFiltration equationHarnack inequalitiesHolder continuityInhomogeneous pmeLong-time behaviorNonnegative solutionsParabolic regularityPoincare inequalitiesPorous-medium equationPositive cauchy-problemSharp asymptotic ratesSmoothing effectsSymmetry-breaking

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

The work has been published in the journal ADVANCES IN MATHEMATICS due to its progression and the good impact it has achieved in recent years, according to the agency WoS (JCR), it has become a reference in its field. In the year of publication of the work, 2019, it was in position 48/325, thus managing to position itself as a Q1 (Primer Cuartil), in the category Mathematics.

From a relative perspective, and based on the normalized impact indicator calculated from World Citations provided by WoS (ESI, Clarivate), it yields a value for the citation normalization relative to the expected citation rate of: 2.46. This indicates that, compared to works in the same discipline and in the same year of publication, it ranks as a work cited above average. (source consulted: ESI Nov 14, 2024)

This information is reinforced by other indicators of the same type, which, although dynamic over time and dependent on the set of average global citations at the time of their calculation, consistently position the work at some point among the top 50% most cited in its field:

  • Weighted Average of Normalized Impact by the Scopus agency: 2.67 (source consulted: FECYT Feb 2024)
  • Field Citation Ratio (FCR) from Dimensions: 8.5 (source consulted: Dimensions Jun 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-06-27, the following number of citations:

  • WoS: 11
  • Scopus: 16
  • Google Scholar: 23

Impact and social visibility

From the perspective of influence or social adoption, and based on metrics associated with mentions and interactions provided by agencies specializing in calculating the so-called "Alternative or Social Metrics," we can highlight as of 2025-06-27:

  • The use of this contribution in bookmarks, code forks, additions to favorite lists for recurrent reading, as well as general views, indicates that someone is using the publication as a basis for their current work. This may be a notable indicator of future more formal and academic citations. This claim is supported by the result of the "Capture" indicator, which yields a total of: 4 (PlumX).

It is essential to present evidence supporting full alignment with institutional principles and guidelines on Open Science and the Conservation and Dissemination of Intellectual Heritage. A clear example of this is:

  • The work has been submitted to a journal whose editorial policy allows open Open Access publication.

Leadership analysis of institutional authors

There is a significant leadership presence as some of the institution’s authors appear as the first or last signer, detailed as follows: First Author (BONFORTE -, MATTEO) and Last Author (SIMONOV ., NIKITA).