Super and ultracontractive bounds for doubly nonlinear evolution equations

  • Gabriele Grillo

    Politecnica di Milano, Italy
  • Matteo Bonforte

    Universidad Autónoma de Madrid, Spain

Abstract

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.

Cite this article

Gabriele Grillo, Matteo Bonforte, Super and ultracontractive bounds for doubly nonlinear evolution equations. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 111–129

DOI 10.4171/RMI/451