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

We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spatial infinity like |x|-gamma , for gamma epsilon [0, 2), and is allowed to be singular at the origin. In particular we show local-intime 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 L1-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-Benilan inequality and using the Benilan-Crandall inequality instead, which may be of independent interest since the latter holds in much more general settings. (c) 2023 Elsevier Inc. All rights reserved.