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We study weighted porous media equations on domains Ω ⊆ R N , either with Dirichlet or with Neumann homogeneous boundary conditions when Ω = R N . Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, L q 0 -L ̺ smoothing effects (1 ≤ q0 < ̺ < ∞) are discussed for short time, in connection with the validity of a Poincaré inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain L q 0 -L ̺ smoothing effects and suitable weighted Poincaré-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case Ω = R N when the corresponding weight makes its measure finite, so that solutions converge to their weighted mean value instead than to zero. Examples are given in terms of wide classes of weights.

We show here that decay estimates can be derived simply by integral inequalities. This result allows us to prove these kind of estimates, with an unified proof, for different nonlinear problems, thus obtaining both well known results (for example for the p-Laplacian equation and the porous medium equation) and new decay estimates

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Abstract. In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype iś