POROUS MEDIA EQUATIONS WITH TWO WEIGHTS: SMOOTHING AND DECAY PROPERTIES OF ENERGY SOLUTIONS VIA POINCARE INEQUALITIES

We study weighted porous media equations on domains Omega subset of R-N, either with Dirichlet or with Neumann homogeneous boundary conditions when Omega not equal R-N. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, L-q0-L-rho smoothing effects (1 <= q(0) < rho < infinity) are discussed for short time, in connection with the validity of a Poincare inequality in appropriate weighted Sobolev spaces, and the long-time asymptotic behaviour is also studied. In fact, we prove full equivalence between certain L-q0-L-rho smoothing effects and suitable weighted Poincare-type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case Omega = 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.