  In this paper, we study the Cauchy problem of a weakly dissipative $\mu$HS
equation. We first establish the local well-posedness for the weakly
dissipative $\mu$HS equation by Kato's semigroup theory. Then, we derive the
precise blow-up scenario for strong solutions to the equation. Moreover, we
present some blow-up results for strong solutions to the equation. Finally, we
give two global existence results to the equation.
