Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar\'e inequalities

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.