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In this paper, we prove the global Hadamard well-posedness of strongsolutions to a non-isospectral two-component cubic Camassa-Holm system in thecritical Besov space $B_{2,1}^{\frac{1}{2}}(\mathbb{T})$. Our results showsthat in comparison with the well-known work for classic Camassa-Holm-typeequations, the existence of global solution only relies on the$L^1$-integrability of the variable coefficients $\alpha(t)$ and $\gamma(t)$,but nothing to do with the shape or smoothness of the initial data. The keyingredient of the proof hinges on the careful analysis of the mutual effectamong two component forms, the uniform bound of approximate solutions, andseveral crucial estimates of cubic nonlinearities in low-regularity Besovspaces via the Littlewood-Paley decomposition theory. A reduced case in ourresults yields the global existence of solutions in a Besov space for two kindsof well-known isospectral peakon system with weakly dissipative terms.}Moreover, we derive two kinds of precise blow-up criteria for a strong solutionin both critical and non-critical Besov spaces, as well as providing specificcharacterization for the lower bound of the blow-up time, which implies theglobal existence with additional conditions on the time-dependent parameters$\alpha(t)$ an $\gamma(t)$.

Global-in-time solvability and blow-up for a non-isospectral two-component cubic Camassa-Holm system in a critical Besov space