The Study Of Solutions To Higher-Order Camassa-Holm Equations

In this paper, we consider the Cauchy problem of higher-order Camassa-Holm equations, and mainly we discuss conservation solutions of these equations. In order to get conservation solutions, we first use the small viscosity method to get solutions of local frequency forms of original equations. Then, under certain initial conditions, we obtain conservation solutions as the limit of local solutions.In addition, we also study the local well-posedness of higher-order Camassa-Holm equations, and blow-up solutions of periodic higher-order Camassa-Holm equations.These results, which added new contents to the family of Camassa--Holm equation, deepen our understanding of the structure of series of these equations, and further broaden the corresponding theoretical knowledge.There are four sections in this paper:In first section, we introduce the background and the actuality of this paper and the main results of my paper.The second section is used to discuss the local existence results of the Cauchy problem of the higher-order Camassa-Holm equations. By using a classical existence result and some priori estimate we establish the local well-posedness. In the third section, we study the global solutions of higher-order Camassa-Holm equations, and prove that the global solutions are satisfied the conservation law, i.e. these solutions are conservation solutions.Finally, a criteria guaranteeing the development of singularities in finite time for strong solutions with regular initial data is obtained and also the blow-up rate is obtained.