Asymptotic behavior of nonlinear diffusions

We describe the asymptotic behavior as t → ∞ of the solution of ut = ∆pu in IRN, for (2N +1)/(N +1) ≤ p < N and non-negative, integrable initial data. Optimal rates in Lq , q = 2 − 1/(p − 1) for the convergence towards a self-similar profile corresponding to a solution with Dirac distribution initial data are found. They are connected with optimal constants for a Gagliardo-Nirenberg inequality.