Well-posedness and blow-up for a two-component Degasperis-Procesi equation with infinitely fast propagating solutions

In this paper, a two-component variant of the Degasperis-Procesi equation on the real line is discussed. Applying Kato's theory, we first prove the local well-posedness for the equation under consideration in $H^s\times H^{s-1}$, for $s\geq 2$. Second we establish the precise blow-up scenario. For compactly supported initial data, we show that the associated solution does not have compact support for any positive time; the localized initial disturbance propagates with an infinite speed. Although the solution is no longer compactly supported we prove that it decays at an exponentially fast rate for the duration of its existence.