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.

中文翻译：

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具有耗散和色散的广义Camassa-Holm模型的整体存在和传播速度

在本文中，我们研究了具有耗散和色散的广义Camassa-Holm（gCH）模型，该模型具有（\（N + 1 \））阶非线性，并包括以下三个可积分方程：Camassa-Holm，Degasperis –Procesi和Novikov方程，作为其简化。我们首先给出gCH方程的柯西问题的局部适定性和精确的爆炸情形。然后，我们提供了几个足以保证gCH方程强解整体存在的条件。最后，当紧凑地支持初始数据时，我们研究了gCH方程的传播速度。