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

We study the asymptotic behaviour of nonnegative solutions to: ut = ∆_p u^m using an entropy estimate based on a sub-family of the Gagliardo-Nirenberg inequalities – or, in the limit case m=(p−1)^−1,on a logarithmic Sobolev inequality in W^1,p –for which optimal functions are known.