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arXiv · 2402.18706

The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach

Gabriele Grillo·Dario D. Monticelli·Fabio Punzo
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Abstract

We consider the porous medium equation (PME) on complete noncompact manifolds $M$ of nonnegative Ricci curvature. We require nonparabolicity of the manifold and construct a natural space $X$ of functions, strictly larger than $L^1$, in which the Green function on $M$ appears as a weight, such that the PME admits a solution in the weak dual (i.e. potential) sense whenever the initial datum $u_0$ is nonnegative and belongs to $X$. Smoothing estimates are also proved to hold both for $L^1$ data, where they take into account the volume growth of Riemannian balls giving rise to bounds which are shown to be sharp in a suitable sense, and for data belonging to $X$ as well.

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Gabriele Grillo, Dario D. Monticelli, Fabio Punzo. 2024-02-28. The porous medium equation on noncompact manifolds with nonnegative Ricci curvature: a Green function approach. https://arxiv.org/abs/2402.18706

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