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The fractional porous medium equation on noncompact Riemannian manifolds

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Abstract

We study nonnegative solutions to the fractional porous medium equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual \(L^1\) space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold includes both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involves a class of data which is larger than the previously known one.

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Acknowledgements

The first, third and fourth authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA, Italy) of the Istituto Nazionale di Alta Matematica (INdAM, Italy) and are partially supported by the PRIN project 201758MTR2: “Direct and Inverse Problems for Partial Differential Equations: Theoretical Aspects and Applications” (Italy). The second author is partially supported by the Project PID2020-113596GB-I00 (Spain) and acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R &D” (CEX2019-000904-S) and by the E.U. H2020 MSCA programme, grant agreement 777822.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Elvise Berchio

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, and ICMAT-Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049, Madrid, Spain

    Matteo Bonforte

  3. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Gabriele Grillo & Matteo Muratori

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  1. Elvise Berchio
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  4. Matteo Muratori
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Correspondence to Matteo Bonforte.

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Dedicated to the Memory of Marek Fila, Mathematician and Friend.

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Berchio, E., Bonforte, M., Grillo, G. et al. The fractional porous medium equation on noncompact Riemannian manifolds. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02731-6

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