Potential theory and optimal convergence rates in fast nonlinear diffusion
- Kim, Yong Jung
- McCann, Robert J.
DOIAbstract
AbstractA potential theoretic comparison technique is developed, which yields the conjectured optimal rate of convergence as t→∞ for solutions of the fast diffusion equationut=Δ(um),(n−2)+/n<m⩽n/(n+2),u,t⩾0,x∈Rn,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 L1(Rn). 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
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