Smoothing effects for the porous medium equation on Cartan-Hadamard manifolds

TL;DR

This paper establishes sharp bounds and smoothing effects for solutions to the porous medium equation on Cartan-Hadamard manifolds, extending known results from hyperbolic space and Euclidean domains, with implications for weighted equations and functional inequalities.

Contribution

It introduces new sharp bounds and smoothing effects for porous medium equations on Cartan-Hadamard manifolds, including weighted cases and the role of functional inequalities.

Findings

Proves a smoothing effect on Cartan-Hadamard manifolds with negatively bounded curvature.

Establishes bounds interpolating between hyperbolic and Euclidean cases under Sobolev inequalities.

Shows that sub-Poincaré inequalities do not hold on Cartan-Hadamard manifolds.

Abstract

We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan-Hadamard manifold whose sectional curvatures are bounded above by a strictly negative constant. This bound includes as a special case the sharp smoothing effect recently proved by V\'azquez on the hyperbolic space in \cite{V}, which is similar to the absolute bound valid in the case of bounded Euclidean domains but has a logarithmic correction. Secondly we prove a bound which interpolates between such smoothing effect and the Euclidean one, supposing a suitable quantitative Sobolev inequality holds, showing that it is sharp by means of explicit examples. Finally, assuming a stronger functional inequality of sub-Poincar\'e type, we prove that the above mentioned (sharp) absolute bound holds, and provide examples of weighted porous media equations on manifolds of infinite volume in which it holds, in contrast with the non-weighted Euclidean situation. It is also shown that sub-Poincar\'e inequalities cannot hold on Cartan-Hadamard manifolds.