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

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.