This paper is mainly concerned with the global existence of strong solutions for two generalized two-component Camassa–Holm (CH) shallow water systems derived under the KdV regime and CH regime, respectively. For the first system, we apply a different method from the one of Lyapunov functions to obtain the same global existence result in (R. M. Chen and Y. Liu, Int. Math. Res. Not., 2011, 1381–1416). It is also observed that this method we used applies equally well to the second system, and hence the corresponding result extends the previous global existence in T. Hu and Y. Liu, Physica D, 391, 87–110 (2019). Moreover, we establish a new wave breaking result for the second system.
Zhang et al. (Thu,) studied this question.