  We study a class of nonlinear diffusion equations whose model is the
classical porous media equation on domains $\Omega\subseteq{\mathbb R}^N$,
$N\ge3$, with homogeneous Neumann boundary conditions. Firstly we improve some
known results in such model case, both as concerns sharp $L^{q_0}$-$L^\infty$
regularizing properties of the evolution for short time and as concerns sharp
long time asymptotics in the sense of $L^\infty$ convergence of solutions to
their mean value. The generality of the discussion allows to consider, almost
at the same time, also weighted versions of the above equation provided an
appropriate weighted Sobolev inequality is required to hold. \normalcolor In
fact, we show that the validity of a slightly weaker functional inequality is
equivalent to the validity of a suitable $L^{q_0}$-$L^\infty$ bound for
solutions to the associated weighted porous media equation. The long time
asymptotic analysis relies as well on the assumed weighted Sobolev inequality
only, and allows to prove uniform convergence to the mean value, with the rate
predicted by linearization, in such generality. This fact was not known even
for the explicit classes of weights previously considered in the literature.

Analysis of PDEs