In this paper, we consider the Cauchy problem for the N − abc family of the Camassa–Holm type equation with both dissipation and dispersion. First, we establish the global well-posedness of the strong solutions under certain conditions on the initial datum. Then, we investigate the propagation speed with compactly supported initial data. This result improves earlier ones reported in the literature, such as those by Novruzov et al. [J. Differ. Equations 257, 4525–4541 (2014)], Hwang and Moon [Electron. Res. Arch. 28(1), 15–25 (2020)], and Himonas and Thompson [J. Math. Phys. 55, 091503 (2014)].

中文翻译：

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具有耗散和色散的 Camassa-Holm 型方程 N − abc 族的全局适定性和无限传播速度

在本文中，我们考虑了具有耗散和色散的 Camassa-Holm 型方程的 N - abc 族的柯西问题。首先，我们在初始数据上建立一定条件下强解的全局适定性。然后，我们使用紧凑支持的初始数据研究传播速度。这一结果改进了文献中报道的早期结果，例如 Novruzov 等人的结果。[J. 不同。方程 257, 4525–4541 (2014)], Hwang 和 Moon [Electron. 水库 拱。28(1), 15–25 (2020)]，以及 Himonas 和 Thompson [J. 数学。物理。55, 091503 (2014)]。