Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation Joachim EscherYue LiuZhaoyang Yin 35G2535L05the periodic Degasperis-Procesi equationperiodic peakonsperiodic shock wavesblow-up rateblow-up set 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 establish 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. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3040 10.1512/iumj.2007.56.3040 en Indiana Univ. Math. J. 56 (2007) 87 - 118 state-of-the-art mathematics http://iumj.org/access/