Shattering and non-uniqueness in fragmentation models-an analytic approach

Fragmentation equations have features which are rather unexpected in such a relatively simple linear framework. These are shattering, attributed to a phase transition creating a dust of 'zero' size particles, and the existence of multiple solutions. The existence of these phenomena indicates that the fragmentation equation alone does not offer a full description of the dynamics of the problem. In this paper we present a survey of recent results explaining both features from a unified dynamical systems point of view. We also use the developed theory to discuss a number of conjectures and observations made in the literature of the subject.