Stability and relaxation of power-law distribution.

Abstract

We study the stability of the steady state in an aggregation system with injection for one dimension and the mean-field cases. We prove that for any initial perturbation the system is stable and eventually recovers the power-law steady state. The form of the relaxation follows either a stretched exponential (${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}\mathrm{\ensuremath{\alpha}}}$, \ensuremath{\alpha}=2,3) or a power law (${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$, \ensuremath{\beta}=1,3/4) depending on the type of injection.