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 $\dot{u}={△}_p(u^m)$ (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.