The porous medium equation with large initial data on negatively curved Riemannian manifolds

TL;DR

This paper establishes existence, uniqueness, and non-existence results for the porous medium equation on negatively curved Riemannian manifolds with large initial data, highlighting the sharpness of curvature conditions and growth rates.

Contribution

It provides the first sharp conditions on Ricci curvature and initial data growth rates for well-posedness of the porous medium equation on Cartan-Hadamard manifolds.

Findings

Existence and uniqueness of very weak solutions under specific curvature bounds.

Non-existence of solutions if initial data grow too fast at infinity.

Pointwise blow-up for certain manifolds and initial data.

Abstract

We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise estimate for the maximal existence time, and we show that in general solutions do not exist if the initial data grow at infinity too fast. This proves in particular that the growth rate of the data we consider is optimal for existence. Pointwise blow-up is also shown for a particular class of manifolds and of initial data.