JournalsrmiVol. 22, No. 1pp. 111–129

Super and ultracontractive bounds for doubly nonlinear evolution equations

  • Gabriele Grillo

    Politecnica di Milano, Italy
  • Matteo Bonforte

    Universidad Autónoma de Madrid, Spain
Super and ultracontractive bounds for doubly nonlinear evolution equations cover
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Abstract

We use logarithmic Sobolev inequalities involving the pp--energy functional recently derived in [Del Pino, M. and Dolbeault, J.: The optimal euclidean Lp\mathrm{L}^p-Sobolev logarithmic inequality. J. Funct. Anal. 197 (2003), 151-161], [Gentil, I.: The general optimal Lp\mathrm{L}^p-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202 (2003), 591-599] to prove Lp^p-Lq^q smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form u˙=p(um)\dot u=\triangle_p(u^m) (with (m(p1)1(m(p-1)\ge 1) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bounds are of the form u(t)qCu0rγ/tβ\Vert u(t)\Vert_q\le C\Vert u_0\Vert_r^\gamma/t^\beta for any rq[1,+]r\le q\in[1,+\infty] and t>0t>0 and the exponents β,γ\beta,\gamma 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