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

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