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Existence and uniqueness results for the porous medium equation with measure data on Cartan-Hadamard manifolds
Dr. Matteo Muratori, Universitą di Pavia
Sala conferenze IMATI-CNR - Martedì 10 Novembre 2015 h.15:00
Abstract. This talk deals with the Cauchy problem for the porous medium equation on complete, simply connected Riemannian manifolds M having nonpositive sectional curvatures, namely Cartan-Hadamard manifolds. In addition, we also require that the Ricci curvature is bounded from below by a negative constant times the square of the geodesic distance from a pole. We show existence of weak solutions whose initial datum is a finite Radon measure on M, not necessarily positive. Moreover, such solutions satisfy a suitable smoothing effect (in fact they are bounded for positive times) and conserve the mass. As for uniqueness, we first establish some results in potential analysis on nonparabolic manifolds, which concern the validity of a modified version of the mean-value inequality for superharmonic functions and properties of potentials of positive Radon measures. These tools turn out to be crucial in order to prove uniqueness in the class of nonnegative weak solutions.
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