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

TL;DR

This paper studies the existence and uniqueness of weak solutions to the porous medium equation on negatively curved Riemannian manifolds, extending the theory to measure initial data and exploring potential analysis aspects.

Contribution

It establishes existence for measure initial data, uniqueness for nonnegative solutions, and introduces new potential analysis results on negatively curved manifolds.

Findings

Existence of solutions with Radon measure initial data

Uniqueness of nonnegative solutions

New potential analysis results on manifolds

Abstract

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative solutions. On the other hand, we prove that any weak solution of the porous medium equation necessarily takes on as initial datum a finite Radon measure. In addition, we obtain some results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions, and properties of potentials of positive Radon measures. Such results are new and of independent interest, and are crucial for our approach.