### Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $p$-Laplacian

Manuel Del Pino & Jean Dolbeault
We study the asymptotic behaviour of nonnegative solutions to: ut = ∆pum using an entropy estimate based on a sub-family of the Gagliardo-Nirenberg inequalities – or, in the limitcasem=(p−1)−1,onalogarithmicSobolevinequalityinW1,p –forwhichoptimal functions are known.
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