On the Cauchy problem for a combined mCH-Novikov integrable equation with linear dispersion

This paper aims to understand a blow-up mechanism on a family of shallow-water models with linear dispersion, which are linked with the modified Camassa-Holm equation and the Novikov equation. We first demonstrate the local well-posedness of the model equation in Besov spaces. Our blow-up analysis begins with two cases where the first case is $2k_1+3k_2\ne 0$ and then we deduce the results on the curvature blow-up in finite time. To overcome the lack of conservation in the functional due to weak linear dispersion, we can determine a suitable alternative via a slight modification to conserved quantity $H_2\left[u\right]$ (see Lemma 4.1). Furthermore, we explore the formation of singularities in another case when nonlocal terms are absent. Lastly, we investigate the Gevrey regularity and analyticity of solutions for Cauchy problem within a specified range of Gevrey-Sobolev spaces by employing the generalized Ovsyannikov theorem and study the continuity of the data-to-solution mapping.