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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.
MSC:
35L67Shocks and singularities
35G25Initial value problems for nonlinear higher-order PDE
35L05Wave equation (hyperbolic PDE)