The Persistence Property Of A Kind Of Higher-order Camassa-Holm Equations

In this thesis,we consider the persistence property of a kind of higher-order Camassa-Holm equations.Persistence property is one of long-time behaviours of solution.The classical method to prove persistence property is to select the appropriate weight function according to the decay of the initial value,then act on both sides of the equation,finally do energy estimates for this equation.We call it the weight function estimate method.The method was firstly used to study the persistence property of Camassa-Holm equation,then the method has been used to study the persistence property of other nonlinear evolution equations(or system).In this thesis,we firstly apply the Kato theory to prove the local well-posedness of the higher-order Camassa-Holm equations(k=2,3);secondly,we use the weight function estimate method to obtain the persistence property of the higher-order Camassa-Holm equations(k=2,3).The content of this paper is arranged as follows:In first chapter,we briefly introduce the concept,method and progress of the persistence property and the background,progress of higher-order Camassa-Holm equation.In second chapter,the related symbols,the difinitions and the preliminary theories required to solve the problem are described concretely.In third chapter,we apply Kato theory to prove the local well-posedness of the higher-order Camassa-Holm equations(k=2,3).In fourth chapter,by using the weight function estimate method,we obtain the persistence property of the higher-order Camassa-Holm equations(k=2,3).