The porous medium equation with large initial data on negatively curved\n Riemannian manifolds

Abstract

We show existence and uniqueness of very weak solutions of the Cauchy problem\nfor the porous medium equation on Cartan-Hadamard manifolds satisfying suitable\nlower bounds on Ricci curvature, with initial data that can grow at infinity at\na prescribed rate, that depends crucially on the curvature bounds. The\ncurvature conditions we require are sharp for uniqueness in the sense that if\nthey are not satisfied then, in general, there can be infinitely many solutions\nof the Cauchy problem even for bounded data. Furthermore, under matching upper\nbounds on sectional curvatures, we give a precise estimate for the maximal\nexistence time, and we show that in general solutions do not exist if the\ninitial data grow at infinity too fast. This proves in particular that the\ngrowth rate of the data we consider is optimal for existence. Pointwise blow-up\nis also shown for a particular class of manifolds and of initial data.\n