Geodesic
The porous medium equation with measure data on negatively curved Riemannian manifolds
Journal of the European Mathematical Society, Tome 20 (2018) no. 11, pp. 2769-2812

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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, under a quadratic lower bound on the Ricci curvature. 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. Those results are new and of independent interest, and are crucial for our approach.
DOI : 10.4171/jems/824
Classification : 35-XX, 31-XX, 46-XX, 58-XX
Keywords: Porous medium equation, Sobolev inequalities, Green function, potential analysis, superharmonic functions, nonlinear diffusion equations, smoothing effect, asymptotics of solutions 
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     author = {Gabriele Grillo and Matteo Muratori and Fabio Punzo},
     title = {The porous medium equation with measure data on negatively curved {Riemannian} manifolds},
     journal = {Journal of the European Mathematical Society},
     pages = {2769--2812},
     publisher = {mathdoc},
     volume = {20},
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     year = {2018},
     doi = {10.4171/jems/824},
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Gabriele Grillo; Matteo Muratori; Fabio Punzo. The porous medium equation with measure data on negatively curved Riemannian manifolds. Journal of the European Mathematical Society, Tome 20 (2018) no. 11, pp. 2769-2812. doi: 10.4171/jems/824

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