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

TL;DR

The paper analyzes the porous medium equation $\partial_t u - \Delta(u^m)=0$ on complete noncompact manifolds with nonnegative Ricci curvature, assuming nonparabolicity and introducing a Green-function–weighted space $L^1_G(M)$. It proves the existence of Weak Dual Solutions for nonnegative initial data in $L^1_G(M)$ via monotone limits of mild $L^1\cap L^\infty$ solutions, and develops smoothing estimates that reflect ambient volume growth, including sharp large-time behavior for $L^1$ data and smoothing for $L^1_G(M)$ data. The analysis relies on Green-function techniques rather than global heat-kernel/Faber–Krahn inequalities, allowing geometry to enter through ball-volume growth conditions. Applications across manifolds with varying volume profiles yield optimality results for large-time decay in several natural geometric settings, highlighting the robustness of the Green-function approach in nonlinear diffusion on curved spaces.

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.

Theorems & Definitions (41)

• Definition 1.1
• Theorem 1.2: Existence of WDS for nonnegative initial data in $L^1_G(M)$
• Remark 1.3
• Theorem 1.4: Smoothing for $L^1$ data
• Remark 1.5
• Corollary 1.6
• Remark 1.7
• Theorem 1.8: Smoothing for $L^1_G(M)$ data
• Remark 2.1
• Remark 2.2