Codice  QDD 123 
Titolo  Porous media equations with two weights: existence, uniqueness, smoothing and decay properties of energy solutions via PoincarĂ© inequalities 
Data  20120514 
Autore/i  Grillo, G.; Muratori, M.; Porzio, M.M. 
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Abstract  We study weighted porous media equations on Euclidean domains, either with Dirichlet or with Neumann homogeneous boundary conditions. Existence of weak solutions and uniqueness in a suitable class is studied in detail. Moreover, short time smoothing effects in Lebesgue spaces are discussed for short time, in connection with the validity of a PoincarĂ© inequality in appropriate weighted Sobolev spaces, and the longtime asymptotic behaviour is also studied. In fact, we prove full equivalence between certain smoothing effects and suitable weighted PoincarĂ©type inequalities. Particular emphasis is given to the Neumann problem, which is much less studied in the literature, as well as to the case of the whole Euclidean case 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. 
