In this paper we mainly study the problem of the development of singularities for solutions to the periodic Degasperis-Procesi equation. Firstly, we show 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. Secondly, we established two new blow-up results. Thirdly, we investigate the blow-up rate for all non-global strong solutions and determine the blow-up set of blowing-up strong solutions to the equation for a large class of initial data. We finally give an explicit example of weak solutions to the equation, which may be considered as periodic shock waves.

This paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the Degasperis-Procesi equation on the line. Global strong solutions to the equation are determined for a class of initial profiles. On the other hand, it is shown that the first blow-up can occur only in the form of wavebreaking. A new wave-breaking mechanism for solutions is described in detail and two results of blow-up solutions with certain initial profiles are established.

We prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions.