Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

TL;DR

This paper proves that solutions to the porous medium equation on manifolds with conical singularities exist globally, become instantly smooth, and have well-understood asymptotic behavior near singularities, using maximal L^q-regularity in Mellin-Sobolev spaces.

Contribution

It establishes the existence, smoothness, and asymptotic analysis of solutions to the porous medium equation on singular manifolds, extending classical results to conical singularities.

Findings

Solutions exist globally for positive initial data.

Solutions become instantly smooth in space and time.

Asymptotic behavior near singularities is precisely characterized.

Abstract

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in space and time, where the maximal $L^{q}$-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.