  In this paper, we prove the global Hadamard well-posedness of strong
solutions to a non-isospectral two-component cubic Camassa-Holm system in the
critical Besov space $B_{2,1}^{\frac{1}{2}}(\mathbb{T})$. Our results shows
that in comparison with the well-known work for classic Camassa-Holm-type
equations, 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 key
ingredient of the proof hinges on the careful analysis of the mutual effect
among two component forms, the uniform bound of approximate solutions, and
several crucial estimates of cubic nonlinearities in low-regularity Besov
spaces via the Littlewood-Paley decomposition theory. A reduced case in our
results yields the global existence of solutions in a Besov space for two kinds
of well-known isospectral peakon system with weakly dissipative terms.}
Moreover, we derive two kinds of precise blow-up criteria for a strong solution
in both critical and non-critical Besov spaces, as well as providing specific
characterization for the lower bound of the blow-up time, which implies the
global existence with additional conditions on the time-dependent parameters
$\alpha(t)$ an $\gamma(t)$.
