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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.