Qualitative analysis for a two-component Camassa-Holm system with high order nonlinearity

This paper is dedicated to a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Camassa-Holm equation. Firstly, the local well-posedness for the Cauchy problem of the system is established in the Sobolev-Besov spaces. Then, we construct the precise blow-up mechanism by using the transport equation theory. It is widely known that the classical methods for studying blow-up phenomena require the $H^1$-norm to control the velocity component. However, the system we consider no longer has the $H^1$-norm conservation law due to its high-order coupling property. Our approach is to utilize the fine structure of system, and then derive a new conservation law $H$. Furthermore, we deduce two different new blow-up results in finite time. Finally, peakon solutions are discussed as well.