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 → ∞ for solutions of the fast diffusion equation u t = Δ ( u m ) , ( n − 2 ) + / n < m ⩽ n / ( n + 2 ) , u , t ⩾ 0 , x ∈ 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 ∈ ] 0 , 1 [ of nonlinearities provided the data is correctly centered.