  We study weighted porous media equations on domains $\Omega\subseteq{\mathbb
R}^N$, either with Dirichlet or with Neumann homogeneous boundary conditions
when $\Omega\not={\mathbb R}^N$. Existence of weak solutions and uniqueness in
a suitable class is studied in detail. Moreover, $L^{q_0}$-$L^\varrho$
smoothing effects ($1\leq q_0<\varrho<\infty$) are discussed for short time, in
connection with the validity of a Poincar\'e inequality in appropriate weighted
Sobolev spaces, and the long-time asymptotic behaviour is also studied.
Particular emphasis is given to the Neumann problem, which is much less studied
in the literature, as well as to the case $\Omega={\mathbb R}^N$ when the
corresponding weight makes its measure finite, so that solutions converge to
their weighted average instead than to zero. Examples are given in terms of
wide classes of weights.
