We use logarithmic Sobolev inequalities involving the --energy functional recently derived in [Del Pino, M. and Dolbeault, J.: The optimal euclidean -Sobolev logarithmic inequality. J. Funct. Anal. 197 (2003), 151-161], [Gentil, I.: The general optimal -Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202 (2003), 591-599] to prove L-L smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form (with ) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bounds are of the form for any and and the exponents are shown to be the only possible for a bound of such type.
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Gabriele Grillo, Matteo Bonforte, Super and ultracontractive bounds for doubly nonlinear evolution equations. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 111–129DOI 10.4171/RMI/451