The Porous Medium Equation with Large Initial Data on Negatively Curved Riemannian Manifolds

We discuss 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 in?nity at a prescribed rate, that depends crucially on the curvature bounds. Furthermore, 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 in?nity too fast. Such results have been recently obtained jointly with G. Grillo and M. Muratori.