Oscillatory and peak solutions in coagulation-fragmentation equations
Smoluchowski’s classical mean-field model for coagulation is used to describe cluster formation and growth in a large variety of applications. Of particular relevance is the so-called scaling hypothesis, which suggests that the long-time behaviour is universal and described by self-similar solutions. This conjecture has however been rigorously confirmed only for particular cases, such as solvable kernels and perturbations of them.
In this talk I will report on results that provide evidence that in fact the long-time behaviour is in general not self-similar but solutions converge for a certain class of kernels towards time-periodic peak solutions whose details depend sensitively on the initial data. We also found related non-universal long-time behaviour for coagulation-fragmentation equations.
(based on joint work with M. Bonacini, M. Herrmann, P. Laurencot and J.J.L. Velazquez)