The porous medium equation with measure data on negatively curved Riemannian manifolds (Q1990883)

The paper deals with existence and uniqueness of weak solutions of Cauchy problems for the porous medium equation on Riemannian manifolds of the type \[ \begin{cases} u_t=\Delta(u^m) & \text{ in }\;M\times(0,\infty),\\ u=\mu & \text{ on }\;M\times \{0\}, \end{cases} \] where \(M\) is an \(N\)-dimensional complete, simply connected Riemannian manifold with nonpositive sectional curvatures (that is, a Cartan-Hadamard manifold), \(\Delta\) is the Laplace-Beltrami operator on \(M\), \(m>1\) and \(\mu\) is a finite, not necessarily positive, Radon measure on \(M\). The authors prove that any weak solution necessarily has a finite Radon measure as initial trace. The results are obtained via potential analysis on manifolds, showing the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures.