Self-similar asymptotic behavior for the solutions of a linear coagulation equation

Abstract

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law v−σ. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent σ>3, but the model can be expected to be valid, on heuristic grounds, for σ>53. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for 53<σ<2 the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.