Wave-breaking and persistence properties in weighted \(L^p\) spaces for a Camassa–Holm type equation with quadratic and cubic nonlinearities

We consider the Cauchy problem of a Camassa–Holm type equation with quadratic and cubic nonlinearities. We establish a new sufficient condition on the initial data that leads to the wave-breaking for this equation. Moreover, we obtain the persistence results of solutions for the equation in weighted \(L^p\) $L^p$ spaces.