Sharp decay rates for the fastest conservative diffusions

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

$u_t=\mathrm{\Delta }(u^m),{(n?2)}_{+}/n<m?n/(n+2),u,t?0,\mathbf{x}\in {\mathbf{R}}^n,$

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 $L^1({\mathbf{R}}^n)$. To cite this article: Y.J. Kim, R.J. McCann, C. R. Acad. Sci. Paris, Ser. I 341 (2005).