Well-posedness and non-uniform dependence for a generalized two-component Camassa-Holm-type system with high-order nonlinearities and multi-peakons in Besov spaces

Considered herein is a generalized two-component Camassa-Holm-type system with high-order nonlinearities and multi-peakons, which contains some integrable shallow water wave equations, such as Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and Geng-Xue system. At first, the results with respect to the local well-posedness of the solutions of Cauchy problem of this system in Besov spaces are demonstrated in detail. Then, the non-uniformly continuous dependence on initial data of the solutions of this problem in Besov spaces on the torus and line is established by constructing new appropriate approximate solutions and initial data. In the process of proof, we need to rely upon Littlewood-Paley decomposition theory and overcome the difficulties resulting from the high-order nonlinearities of the system.