Blow-up phenomena and global existence for a two-component Camassa-Holm system with an arbitrary smooth function

In the paper, we mainly investigate blow-up phenomena and global existence of strong solutions to a two-component Camassa-Holm systems with an arbitrary smooth function H. For three types of smooth functions H, by using a conservation law and the sign-preserving property of strong solutions, we obtain two new blow-up results and a new global existence result for the two-component system. Our obtained results generalize and cover the recent results in Yan et al. (2015), Zhang and Yin (2015, 2016).