Sharp short and long time L∞ bounds for solutions to porous media equations with homogeneous Neumann boundary conditions

We study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains Ω⊆RN, N⩾3, with homogeneous Neumann boundary conditions. We improve the known results in such model case, proving sharp Lq0–L∞ regularizing properties of the evolution for short time and sharp long time L∞ bounds for convergence of solutions to their mean value. The generality of the discussion allows to consider, almost at the same time, weighted versions of the above equation provided an appropriate weighted Sobolev inequality holds. In fact, we show that the validity of such weighted Sobolev inequality is equivalent to the validity of a suitable Lq0–L∞ bound for solutions to the associated weighted porous media equation. The long time asymptotic analysis relies 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.