Wave-breaking phenomena for a new weakly dissipative quasilinear shallow-water waves equation

In this paper, we mainly study a new weakly dissipative quasilinear shallow-water waves equation, which can be formally derived from a model with the effect of underlying shear flow from the incompressible rotational two-dimensional shallow water in the moderately nonlinear regime by Wang, Kang and Liu (Appl Math Lett 124:107607, 2022). Considering the dissipative effect, the local well-posedness of the solution to this equation is first obtained by using Kato’s semigroup theory. We then establish the precise blow-up criterion by using the transport equation theory and Moser-type estimates. Moreover, some sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied according to the different real-valued intervals in which the dispersive parameter \(\theta \) $\theta$ being located. It is noteworthy that we need to overcome the difficulty induced by complicated nonlocal nonlinear structure and different dispersive parameter ranges to get corresponding convolution estimates.