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

Matteo Muratori; Troy Petitt

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arXiv · 2206.09832

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

Matteo Muratori·Troy Petitt

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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.

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Matteo Muratori, Troy Petitt. 2022-06-20. An inhomogeneous porous medium equation with large data: well-posedness. https://arxiv.org/abs/2206.09832

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