, Volume 233, Issue 1, pp 75-91

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

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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.

Received September 25, 1998; in final form October 12, 1998