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Global well-posedness and infinite propagation speed for the N − abc family of Camassa–Holm type equation with both dissipation and dispersion

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)].