  In this paper, we first establish the local well-posednesss for the Cauchy
problem of a $N$-peakon system in the sense of Hadamard in both critical Besov
spaces and supercritical Besov spaces. Second, we gain a blow-up criterion.
According to blow-up criterion wemreach a precise blow-up criterion. Under a
sign condition, we reach the existence of global solution. Finally, based on
the first-order difference method, we give a simulation example of the blow-up
of the equation and the properties of the global solution.
