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 depends only on a discrete set of points. As a consequence, the long-time 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. 
Philippe Lauren\c{c}ot, Barbara Niethammer and Juan J.L. Vel\'azquez
math.AP