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

Generalizations of logarithmic Sobolev inequalities

Jochen Merker 1,

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.

Keywords: ultracontractive semigroups, doubly nonlinear evolution equations, logarithmic Gagliardo-Nirenberg. inequalities, $p$-Laplacian.
Mathematics Subject Classification: Primary: 35K65, 35B35; Secondary: 46E35, 35B4.
Citation: Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
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