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

Abstract

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