In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (N+1)-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported.
Hu, Qiaoyi and Qiao, Zhijun, "Global existence and propagation speed for a generalized Camassa-Holm model with both dissipation and dispersion" (2019). Mathematical and Statistical Sciences Faculty Publications and Presentations. 38.
Journal of Evolution Equations