Study On Dynamical Behavior Of One Generalized Two-component Camassa-Holm Equation

This paper studies the local well-posedness,blow-up criterion,the blow-up of strong solutions,the property of blow-up points set,global existence of solutions of the Cauchy problem for generalized two-component Camassa-Holm equation with generalized weak dissipation in the case of the non-periodic and periodic.For the Cauchy problem for two-component Camassa-Holm equation with a generalized weak dissipation in the case of non-periodic.We establish the local well-posedness for the equation by utilizing Kato's theorem and establish the blow-up criteria and the blow-up rate by using energy method and monotonicity.Then,we estimated the blow-up rate of solutions of the equation is-2.Finally,the property of the blow-up points set is characterized.For the Cauchy problem for two-component Camassa-Holm equation with a generalized weak dissipation in the case of periodic.By Kato's theorem,we investigate the local well-posedness of Cauchy problem for any initial data z₀?28??u₀,?₀-?1?H^s?H^s-1,s?2.Then we also present a blow-up scenario,several blow-up results and the blow-up rate of solutions of the equation when??29?0.Then we obtain a sufficient condition for the existence of global solution of the equation when 0?27???27?2 by using Lyapunov function.