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Both authors funded by Project MTM2005-08760-C01 and -C02 (Spain) and European Science Foundation Programme "Global And Geometric Aspects of Nonlinear Partial Differential Equations".

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Reyes, GuillermoCorresponding AuthorVazquez, Juan LuisAuthor

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November 13, 2024
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The inhomogeneous PME in several space dimensions.: Existence and uniqueness of finite energy solutions

Publicated to:COMMUNICATIONS ON PURE AND APPLIED ANALYSIS. 7 (6): 1275-1294 - 2008-11-01 7(6), DOI: 10.3934/cpaa.2008.7.1275

Authors: Reyes, Guillermo; Vazquez, Juan Luis

Affiliations

Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain - Author
Univ Politecn Madrid, ETSI Caminos Canales & Puertos, Dept Matemat Aplicada, E-28040 Madrid, Spain - Author

Abstract

We study the questions of existence and uniqueness of non-negative solutions to the Cauchy problem {rho(x)partial derivative(u)(t) = Delta u(m) in Q := R-n x R+ u(x, 0) = u(0) in dimensions n >= 3. We deal with a class of solutions having finite energy E(t) = integral(Rn) rho(x)u(x, t) dx for all t >= 0. We assume that m > 1 (slow diffusion) and the density rho(x) is positive, bounded and smooth. We prove existence of weak solutions starting from data u(0) >= 0 with finite energy. We show that uniqueness takes place if rho has a moderate decay as vertical bar x vertical bar -> infinity that essentially amounts to the condition rho is not an element of L-1(R-n). We also identify conditions on the density that guarantee finite speed of propagation and energy conservation, E(t) = const. Our results are based on a new a priori estimate of the solutions.

Keywords

Cauchy problemCauchy-problemClasses of uniquenessDiffusionFiltration equationInfinitInhomogeneous porous medium flowParabolic equationsSemigroup solutioSemigroup solution

Quality index

Bibliometric impact. Analysis of the contribution and dissemination channel

From a relative perspective, and based on the normalized impact indicator calculated from the Field Citation Ratio (FCR) of the Dimensions source, it yields a value of: 5.19, which 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: Dimensions Jul 2025)

Specifically, and according to different indexing agencies, this work has accumulated citations as of 2025-07-03, the following number of citations:

  • WoS: 26
  • Scopus: 28

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 2025-07-03:

  • 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).

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 (REYES COPA, GUILLERMO) and Last Author (VAZQUEZ SUAREZ, JUAN LUIS).

the author responsible for correspondence tasks has been REYES COPA, GUILLERMO.