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

TL;DR

This paper investigates the long-term behavior of solutions to the Porous Medium Equation on negatively curved Riemannian manifolds, revealing unexpected variable separation and establishing existence and uniqueness results for related elliptic problems.

Contribution

It extends previous work by analyzing superquadratic curvature effects on the Porous Medium Equation, including existence, uniqueness, and comparison principles, and connects these to weighted Euclidean problems.

Findings

Solutions exhibit separate-variable behavior similar to bounded Euclidean domains.

Existence of solutions to a related sublinear elliptic problem is proved.

Comparison principles and uniqueness results are established for solutions.

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.