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 = Δ ( u m ) , ( n − 2 ) + / n < m ⩽ n / ( n + 2 ) , u , t ⩾ 0 , x ∈ 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 ) . To cite this article: Y.J. Kim, R.J. McCann, C. R. Acad. Sci. Paris, Ser. I 341 (2005).