Porous medium equations on manifolds with critical negative curvature: unbounded initial data

TL;DR

This paper studies the existence, uniqueness, and blow-up behavior of solutions to the porous medium equation on certain negatively curved manifolds, allowing for unbounded initial data and critical curvature conditions.

Contribution

It extends previous results by handling unbounded initial data and solutions under critical negative curvature conditions on manifolds.

Findings

Existence and uniqueness of solutions with unbounded initial data.

Blow-up results for solutions with rapidly growing initial data.

Analysis under critical curvature divergence at infinity.

Abstract

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.