The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

TL;DR

This paper studies the long-term behavior of solutions to the porous medium equation on negatively curved Riemannian manifolds, classifying geometric cases and establishing bounds, smoothing effects, and free boundary estimates.

Contribution

It provides sharp bounds and a classification of the asymptotic behavior of PME solutions on manifolds with unbounded negative curvature, introducing a key variable change technique.

Findings

Sharp upper and lower bounds on solution behavior

Classification into quasi-hyperbolic, quasi-Euclidean, and critical cases

Establishment of a global Harnack principle

Abstract

We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such manifolds. We also estimate the location of the free boundary. A global Harnack principle follows. We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.