Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation. (English) Zbl 1124.35041

The authors study the problem of the development of singularities for solutions to the periodic Degasperis-Procesi equation. It is shown that the first blow-up of strong solution to the equation must occur only in the form of wave breaking and shock waves possibly appear afterwards. Two new blow-up results are found. The blow-up rate for all non-global strong solutions and the blow-up set of blowing-up strong solutions to the equation for a large class of initial data are found. Finally an explicit example of weak solutions to the equation is given. This may be considered as periodic shock waves.


35L67 Shocks and singularities for hyperbolic equations
35G25 Initial value problems for nonlinear higher-order PDEs
35L05 Wave equation
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