  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.
