On the blow-up rate and the blow-up set of breaking waves for a shallow water equation

We consider the problem of the development of singularities for classical s olutions to a new periodic shallow water equation. Blow-up can occur only i n the form of wave-breaking, i.e. the solution remains bounded but its slop e becomes unbounded in finite time. A quite detailed description of the wav e-breaking phenomenon is given: there is at least a point (in general depen ding on time) where the slope becomes infinite exactly at breaking time. Th e precise blow-up rate is established and for a large class of initial data we also determine the blow-up set.