On solutions to the Degasperis–Procesi equation

In this paper we consider a new integrable equation (the Degasperis–Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa–Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis–Procesi equation is proved in the following sense: the corresponding solution u ( x , t ) with compactly supported initial datum u 0 ( x ) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t > 0 in its lifespan, the corresponding solution u ( x , t ) behaves as: u ( x , t ) = L ( t ) e − x for x ≫ 1 , and u ( x , t ) = l ( t ) e x for x ≪ − 1 , with a strictly increasing function L ( t ) > 0 and a strictly decreasing function l ( t ) < 0 respectively.