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In this paper, we are concerned with $N$-Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

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