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 $Delta_{mathbb{H}^{N}}$, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1−L^infty$ 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.

The fractional porous medium equation on the hyperbolic space

Grillo, Gabriele
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 $Delta_{mathbb{H}^{N}}$, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1−L^infty$ 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.
File in questo prodotto:
File Dimensione Formato  
2020.07.14_BBGG-FPME-Hyperbolic.pdf

Accesso riservato

Descrizione: file del lavoro accettato
: Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione 632.25 kB
Formato Adobe PDF
632.25 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1147737
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 2
social impact