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We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions

Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation