6 nov 2017 -- 14:15
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
We investigate the pointwise behaviour of nonnegative solutions of the porous medium equation on Cartan-Hadamard manifolds, namely complete and simply connected Riemannian manifolds with nonpositive sectional curvatures. Such manifolds are diffeomorphic to the Euclidean space due to the Cartan-Hadamard Theorem. Our main focus is on compactly supported initial data, since they are the simplest nontrivial data which give rise to a free boundary that grows. The kind of manifolds we deal with can have unbounded negative curvature at spatial infinity. We divide the corresponding geometrical frameworks into quasi-Euclidean, quasi-hyperbolic, super-hyperbolic and borderline critical cases, according to the allowed behaviour of the curvature. We establish upper and lower pointwise estimates on the solutions, which only depend on the power-type upper bound on the sectional curvatures and lower bound on the Ricci curvature, respectively. If the two bounds match our estimates are sharp, and a so-called global Harnack principle follows. In particular, we can approximately locate the free boundary and estimate the growth rate of the support. In the super-hyperbolic range we also obtain a convergence result towards a separable solution involving the solution of a sublinear elliptic equation, which we study separately. This talk is based on joint works with G. Grillo and J.L. Vazquez.