Invariants and wave breaking analysis of a Camassa-Holm type equation with quadratic and cubic non-linearities (Q2189766)

This work is devoted to the study of generalized Camassa-Holm equation with dissipation \[ u_t-u_{txx}+\lambda (u-u_{xx})+3u^2u_x=uu_{xxx}+2u_xu_{xx}. \tag{1}\] The Cauchy problem is considered using Kato's theory for abstract quasilinear evolution equations. The authors prove that the equation (1) is locally well-posed in the Sobolev space \(H^s(\mathbb{R})\) for \(s > \frac{3}{2}\). Moreover, they give the precise blow-up scenario of strong solution of the equation with certain initial data. To prove such result, they firstly construct conserved currents for the equation and from them, conserved quantities.