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June  2008, 1(2): 329-338. doi: 10.3934/dcdss.2008.1.329

## Generalizations of logarithmic Sobolev inequalities

 1 Universität Rostock, Institut für Mathematik, Universitätsplatz 1, 18051 Rostock, Germany

Received  September 2006 Revised  April 2007 Published  March 2008

We generalize logarithmic Sobolev inequalities to logarithmic Gagliardo-Nirenberg inequalities, and apply these inequalities to prove ultracontractivity of the semigroup generated by the doubly nonlinear $p$-Laplacian

$\dot{u}=\Delta_p u^m.$

Our proof does not use Moser iteration, but shows that the time-dependent Lebesgue norm $\||u(t)|\|_{r(t)}$ stays bounded for a variable exponent $r(t)$ blowing up in arbitrary short time.

Citation: Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
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