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 u = ∆pu 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.
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