An inhomogeneous porous medium equation with large data: well-posedness

TL;DR

This paper investigates the well-posedness of solutions to an inhomogeneous porous medium equation with large initial data, establishing local existence, uniqueness, and conditions for global existence or blow-up, using novel smoothing techniques.

Contribution

It introduces new existence and uniqueness results for large data solutions of a weighted porous medium equation with singular weights and develops a novel smoothing approach avoiding classical inequalities.

Findings

Established local-in-time existence and uniqueness for large initial data.

Identified conditions for global existence and blow-up of solutions.

Developed a new local smoothing effect using the Bénilan-Crandall inequality.

Abstract

We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spacial infinity like $|x|^{-\gamma}$, for $\gamma\in [0,2)$, and is allowed to be singular at the origin. In particular we show local-in-time existence and uniqueness for a class of large initial data which includes as "endpoints" those growing at a rate of $ |x|^{(2-\gamma)/(m-1)}$, in a weighted $L^1$-average sense. We also identify global-existence and blow-up classes, whose respective forms strongly support the claim that such a growth rate is optimal, at least for positive solutions. As a crucial step in our existence proof we establish a local smoothing effect for large data without resorting to the classical Aronson-B\'{e}nilan inequality and using the B\'{e}nilan-Crandall inequality instead, which may be of independent interest since the latter holds in much more general settings.