The Redner - Ben-Avraham - Kahng cluster system

We consider a coagulation model first introduced by Redner, Ben-Avraham and Krapivsky in [Redner, Ben-Avraham, Kahng: Kinetics of ‘cluster eating’, J. Phys. A: Math. Gen., 20 (1987), 1231-1238], the main feature of which is that the reaction between a j-cluster and a k-cluster results in the creation of a |j-k|-cluster, and not, as in Smoluchowski’s model, of a (j+k)-cluster. In this paper we prove existence and uniqueness of solutions under reasonably general conditions on the coagulation coefficients, and we also establish differenciability properties and continuous dependence of solutions. Some interesting invariance properties are also proved. Finally, we study the long-time behaviour of solutions, and also present a preliminary analysis of their scaling behaviour.