We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan–Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the porous medium equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan–Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted porous medium equations in the Euclidean space having special weights.

The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

Grillo, Gabriele;Muratori, Matteo;
2019

Abstract

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan–Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is also shown, along with comparison principles, both in the parabolic and in the elliptic case. Our results complete previous analyses of the porous medium equation flow on negatively curved Riemannian manifolds, which were carried out first for the hyperbolic space and then for general Cartan–Hadamard manifolds with a negative curvature having at most quadratic growth. We point out that no similar analysis seems to exist for the linear heat flow. We also translate such results into some weighted porous medium equations in the Euclidean space having special weights.
Nonlinear diffusions, Riemannian manifolds, Nonlinear elliptic problems, asymptotics
File in questo prodotto:
File Dimensione Formato  
MA 2019.pdf

Accesso riservato

Descrizione: file della rivista
: Publisher’s version
Dimensione 456.15 kB
Formato Adobe PDF
456.15 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/11311/1082849
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 7
social impact