Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds

TL;DR

This paper studies the behavior of solutions to the porous medium equation with reaction terms on negatively curved manifolds, revealing conditions for global existence or finite-time blow-up depending on parameters and initial data.

Contribution

It extends the analysis of the porous medium equation with reaction to Cartan-Hadamard manifolds, highlighting differences from Euclidean cases due to negative curvature effects.

Findings

Small data lead to global solutions when p > m.

Large data cause finite-time blow-up for p > m.

Negative curvature accelerates diffusion, altering blow-up and existence regimes.

Abstract

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If $p<m$, large data blow up at worst in infinite time, and under the stronger restriction $p\in(1,(1+m)/2]$ all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation.