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. 1 Introduction and main result Let us consider a non-negative solution u(x, t) of the p-Laplace flow in IRN, namely the equation ut = ∇ · (|∇u| p−2∇u) (1