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

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