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

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 at constant speed in a random distribution of fixed particles. The volumes $v$ of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{-\sigma }$. The validity of the equation has been rigorously proved in \cite{NoV} for values of the exponent $\sigma >3$. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent $\sigma$. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.