The Properties Of Solutions For Several Kinds Of Camassa-Holm Type Equations

This paper mainly studies several models related to Camassa-Holm equation with high order nonlinear terms and multiple branches,which originate from the practical fields of fluid mechanics,lattice dynamics,elasticity,communication and natural disaster prediction.This main contents are as follows:The second chapter 2,we consider the Cauchy problem for the generalized two-component Camassa-Holm equation.Firstly,the local well-posedness of the system in nonhomogeneous Besov spaces is established by using Littlewood-Paley decomposition and transport equation.Then we mainly study the blow up criteria of the generalized two-component Camassa-Holm equations in the Sobolev spaces,that is,the blow up occurs for this system only in the form of breaking waves.Finally,we calculate the periodic and non-periodic peakon solutions and the multiple peakon solutions.?The main results of this chapter are published in Nonlinear Differential Equations and Applications,2018,25?4?:37.?In the third chapter,we investigate the persistence properties of the solutions of two-component Novikov equation.More precisely,by using a proper truncated weighting function and energy estimation method,the persistence properties of the solutions of two-component Novikov equation in weighted L_?^pspaces are obtained.Our results extend the persistence properties of solutions of Camassa-Holm equation to more general system with cubic nonlinearity and interaction between the two component.?The main results of this chapter are accepted by Applicable Analysis.?In the fourth chapter,we study the continuation of solutions to the Novikov equation beyond wave breaking.Our method is based on the characteristic and by introducing a new set of variables,and we can transform the Novikov equation to a semilinear system?these new variables so that all singularities are resolved due to possible wave breaking?.Then the local existence of the solution of the semi-linear system is obtained by the fixed point theorem.Since the solution of the semi-linear system obtained is continuous after collision,these solutions of the original equation can be extended to the case of wave breaking.By this transformation,a dissipative solution is given when the energy dissipates.Returning to the original variables,we obtain a semigroup of global dissipative solutions,which depends continuously on the initial data.?The main results of this chapter are published in Communications in Mathematical Sciences,6?2018?,1615-1633.?In the fifth chapter,we investigate the uniqueness of global conservative solutions to the generalized Camassa-Holm equation based on the characteristics.From a given conservative solution u=u?t,x?,introduce a new set of variables and prove the existence of a unique characteristic curve through each initial point.Then obtain a semilinear ODE system,by analyzing the evolution of the quantities u and v=2 arctan u_x along each characteristic,it is obtained that the Cauchy problem with general initial data u₀?H¹?R?has a unique global conservative solution.?The main results of this chapter are published in Discrete and Continuous Dynamical Systems,10?2018?,5205-5220.?...