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In this paper, a two-component variant of the Degasperis-Procesi equation onthe real line is discussed. Applying Kato's theory, we first prove the localwell-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 compactlysupported initial data, we show that the associated solution does not havecompact support for any positive time; the localized initial disturbancepropagates with an infinite speed. Although the solution is no longer compactlysupported we prove that it decays at an exponentially fast rate for theduration of its existence.

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