Super and ultracontractive bounds for doubly nonlinear evolution equations

Abstract

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