Wave-breaking phenomena and Gevrey regularity for the weakly dissipative generalized Camassa-Holm equation

This paper aims to establish a mechanism for the blow-up on a class of weakly dissipative shallow-water equations with a vavariable dipersion term, which is related to the integrable systems: the Camassa-Holm equation and the Novikov equation. Our blow-up analysis commences with the consideration of two cases. In the first case, the linear dispersion parameter is ? E R, while in the second case, ? is equal to zero. The approach is to extract the true blow-up component and instead trace its dynamics to ensures the occurrence of wave-breaking in finite time before the other component undergoes degeneration. To address the issue of non-conservation in the previous functional which is caused by weak linear dispersion and the loss of the conservation law H-1[u] = ?(R)(u(2) +u(x )(2)) dx due to the presence of a weakly dissipative term, we propose alternative methods. These methods include making a modified functional J (t) (see Lemma 3.2) and establishing an energy inequality. Moreover, we investigate the formation of singularities by tracing the whole blow-up quantity. Lastly, we examine the Gevrey regularity and analyticity of solutions to the system in the Gevrey-Sobolev spaces by utilizing the generalized Ovsyannikov theorem and show the continuity of the data-to-solution mapping.