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In this paper, we study the Cauchy problem of a periodic 2-component$\mu$-Hunter-Saxton system. We first establish the local well-posedness for theperiodic 2-component $\mu$-Hunter-Saxton system by Kato's semigroup theory.Then, we derive precise blow-up scenarios for strong solutions to the system.Moreover, we present a blow-up result for strong solutions to the system.Finally, we give a global existence result to the system.

On the Cauchy problem of a periodic 2-component $\mu$-Hunter-Saxton equation