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Article Contents

# Generalizations of logarithmic Sobolev inequalities

• 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.

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

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