Blow-up of solution of an initial boundary value problem for a generalized Camassa–Holm equation

In this Letter, we study the following initial boundary value problem for a generalized Camassa–Holm equation{ut−uxxt+3uux−2uxuxx−uuxxx+k(u−uxx)x=0,t⩾0,x∈[0,1],u(0,t)=u(1,t)=ux(0,t)=ux(1,t)=0,t⩾0,u(0,x)=u0(x),x∈[0,1], where k is a real constant. We establish local well-posedness of this closed-loop system by using Kato's theorem for abstract quasilinear evolution equation of hyperbolic type. Then, by using multiplier technique, we obtain a conservation law which enable us to present a blow-up result.