The fractional porous medium equation on the hyperbolic space
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Calculus of Variations
The fractional porous medium equation on the hyperbolic space Elvise Berchio1 · Matteo Bonforte2,3 · Debdip Ganguly4 · Gabriele Grillo5 Received: 3 May 2020 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual L p spaces or to larger (weighted) spaces determined either in terms of a ground state of H N , or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative L 1 − L ∞ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples. Mathematics Subject Classification Primary 35R01 · Secondary 35K65 · 35A01 · 35R11 · 58J35
Communicated by M.Del Pino.
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Matteo Bonforte [email protected] Elvise Berchio [email protected] Debdip Ganguly [email protected] Gabriele Grillo [email protected]
1
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
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Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
3
ICMAT-Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid, Spain
4
Department of Mathematics, Indian Institute of Technology Delhi, IIT Campus, Hauz Khas, New Delhi, Delhi 110016, India
5
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy 0123456789().: V,-vol
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of weak dual solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fundamental estimates for weak dual solutions (WDS) . . . . . . . . . . . . . . . . . . . . . . . . . 4 Boundedness of WDS: proof of the smoothing effects . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Semigroup theory in Banach and Hilbert spaces: existence, uniqueness, contractivity and comparison 5.1 Nonlinear semigroup in L 1 (H N ): mild versus weak dual solutions . . . . . . . . . . . . . . . . 5.2 Nonlinear se
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