Sharp decay rates for the fastest conservative diffusions

Abstract

In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features - such as size and location - will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Here this rate is addressed for the fastest conservative nonlinearities in the singular diffusion equation u(t) = Delta(u(m)). (n-2)(+)/n < m <= n/(n-2), u, t >= 0, X is an element of R-n, which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. A potential theoretic comparison technique is outlined below which establishes the sharp 1/t conjectured power law rate of decay uniformly in relative error, and in weaker norms such as L-1(R-n).