Fractional porous media equations: existence and uniqueness of weak solutions with measure data

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• 作者：Gabriele Grillo ; Matteo Muratori…
• 关键词：35K15 ; 35K55 ; 35K65
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：54
• 期：3
• 页码：3303-3335
• 全文大小：809 KB
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• 作者单位：Gabriele Grillo (1)
  Matteo Muratori (1)
  Fabio Punzo (2)
  
  1. Dipartimento di Matematica “F. Brioschi- Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy
  2. Dipartimento di Matematica “F. Enriques- Università degli studi di Milano, via Cesare Saldini 50, 20133, Milan, Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space \({\mathbb R}^d\). For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (power-like). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in Grillo et al. (Discret Contin Dyn Syst 35:5927-962, 2015), where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in Vázquez (J Eur Math Soc 16:769-03, 2014). Mathematics Subject Classification 35K15 35K55 35K65