Generalizations of logarithmic Sobolev inequalities

Jochen Merker

doi:10.3934/dcdss.2008.1.329

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2008, Volume 1, Issue 2: 329-338. Doi: 10.3934/dcdss.2008.1.329

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Generalizations of logarithmic Sobolev inequalities

Jochen Merker^1,

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

Received: August 31, 2006

Revised: March 31, 2007

Published: May 2008

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Abstract

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:

$p$-Laplacian,
doubly nonlinear evolution equations,
ultracontractive semigroups,
logarithmic Gagliardo-Nirenberg inequalities.

Mathematics Subject Classification: Primary: 35K65, 35B35; Secondary: 46E35, 35B45.

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