On The Well-posedness And Blow-up Theorem For A Class Of Higher-order Shallow Water Wave Equations

In this paper, we study the well-posedness and blow-up theorem for a class of higher-order shallow water wave equations.The global existence of solution to the higher-order Camassa-Holm equation is established.We obtain the global existence of solution and blow-up phenomenon for higher-order two-component Camassa-Holm equation. We also study the well-posedness of higher-order Dullin-Gottwald-Holm equations.We mainly use Sobolev inequalities, some PDE knowledge and Kato's theory to solve problems.The local well-posedness of higher-order Camassa-Holm equation and higher-order two-component Camassa-Holm equation is proved by Kato's theorem. We get conservation laws and prior estimates of each equation. Based on the above discussion, we get the global existence of solutions to the two equations.In addition, we get the blow-up result for higher-order Camassa-Holm equation. From the higher-order Camassa-Holm equations, we construct the higher-order DGH equations.We derive the existence and uniqueness of the solutions by means of norm estimates.There are five sections in this paper:In the first section, we introduce the background and actuality.In the second section, we will introduce the basic theory, basic concepts needed in the studies.In the third section, we introduce the global existence theory for higher-order Camassa-Holm equation.In the fourth section, we introduce the global existence and blow-up phenomena for higher-order two-component Camassa-Holm equation.In the fifth section, we introduce the well-posedness of higher-order DGH equations.