OSCILLATORY DYNAMICS IN SMOLUCHOWSKI'S COAGULATION EQUATION WITH DIAGONAL KERNEL

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity gamma < 1. Due to the property of the diagonal kernel, the value of a solution at a given cluster size depends only on a discrete set of points. As a consequence, the longtime behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.