Potential theory and optimal convergence rates in fast nonlinear diffusion

Abstract

A potential theoretic comparison technique is developed, which yields the conjectured optimal rate of convergence as t --> infinity for solutions of the fast diffusion equation u(t) = Delta(u(m)), (n - 2)(+)/n < m <= n/(n + 2), u, t >= 0, x is an element of R-n, n >= 1, to a spreading self-similar profile, starting from integrable initial data with sufficiently small tails. This 1/t rate is achieved uniformly in relative error, and in weaker norms such as L-1 (R-n). The range of permissible nonlinearities extends upwards towards m = 1 if the initial data shares enough of its moments with a specific self-similar profile. For example, in one space dimension, n = 1, the 1/t rate extends to the full range m is an element of]0, 1 [ of nonlinearities provided the data is correctly centered. (C) 2006 Elsevier SAS. All rights reserved.