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Generalizations of logarithmic Sobolev inequalities. (English) Zbl 1152.35326
Summary: 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.

##### MSC:
 35B45 A priori estimates in context of PDEs 35K55 Nonlinear parabolic equations 35B35 Stability in context of PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35K65 Degenerate parabolic equations
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