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. (C) 2011 Elsevier Inc. All rights reserved.