The Cauchy problem for coupled system of the generalized Camassa-Holm equations

Local well-posedness for the Cauchy problem of coupled system of generalized Camassa-Holm equations in the Besov spaces is established by employing the Littlewood-Paley theory and a priori estimate of solution to transport equation. Furthermore, the blow-up criterion of solutions to the problem is illustrated. Our main new contribution is that the effects of dissipative coefficient $\lambda$ and exponent $b$ in the nonlinear terms to the solutions are analyzed. To the best of our knowledge, the results in Theorems 1.1 and 1.2 are new.