The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds

TL;DR

This paper studies the existence, uniqueness, and blow-up behavior of very weak solutions to the porous medium equation on Cartan-Hadamard manifolds with general curvature bounds, identifying critical growth rates of initial data.

Contribution

It introduces a class of initial data growth rates linked to curvature bounds, providing sharp criteria for global existence and finite-time blow-up of solutions.

Findings

Solutions exist up to a finite time depending on initial data growth

Slower initial data growth ensures global solutions

Critical growth data can lead to finite-time blow-up

Abstract

We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on $[0,T]$ for a suitable $T>0$. The maximal existence time $T$ is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class.