Analysis on the blow-up of the generalized modified Camassa-Holm equation

This study investigates the Cauchy problem associated with a generalized modified Camassa-Holm (gmCH) equation, incorporating a dissipative term (λ>0) to account for unavoidable energy loss and a dispersive term (γ∈R) to facilitate wave spreading and maintain the smoothness of solutions. The local well-posedness for the equation is first established, followed by the derivation of a blow-up criterion in Besov spaces. Our analysis of singularity formation proceeds in two regimes. In the dispersionless case (γ=0), we focus on the interplay between the local nonlinearity and the nonlocal term, employing a refined analysis of how the ratio ux/u evolves over time. In the general case (γ≠0), the dissipative term disrupts the conservation laws typically used for estimation. To overcome this, we utilize a Morawetz-type identity to control the ‖ux‖L4, which exhibits exponential growth with respect to time. Consequently, an a priori restriction on the lifespan is imposed to close the estimates. Furthermore, by leveraging the fact that u and ux vary monotonically to evaluate how m evolves, we derive a key Riccati-type differential inequality, ultimately establishing finite-time blow-up results across three diverse scenarios.