Asymptotic behavior of nonlinear diffusions

Abstract

We describe the asymptotic behavior as $t\to \infty$ of the solution of $u_t=\Delta_p u$ in $\R^N$, for $(2N+1)/(N+1)\le p < N$ and non-negative, integrable initial data. Optimal rates in $L^q$, $q=2-1/(p-1)$ 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.