A Liouville Theorem For The Second-Order Camassa-Holm Equation

In recent years,with the wide application of shallow water wave equations in mechanics,economics and ecosystems,the properties of the solution of this type of equation has attracted the attention of many researchers.The higher-order CamassaHolm equation is an important type of shallow water wave equations.This thesis studies the properties of the second-order Camassa-Holm equation near the traveling wave Q in the space H2.Finally,the Liouville property of the second-order Camassa-Holm equation is obtained,that is,if the solution of the equation exists globally,then there exist translation invariant and scaling invariant,so that the solution is equivalent to the travelling wave solution up to scaling and translation;If the solution of the secondorder Camassa-Holm equation does not exist globally,the solution blows up in a finite time.Firstly,we use the pseudo-conformal transformation ?1/2(t)u(t,x+x(t))to decompose the solution of the second-order Camassa-Holm equation near the travelling wave Q into ?1/2(t)u x+x(t))=Q(x)+?(t,x),so as to get the governing equation of ?.And we study the related properties of the travelling wave Q and the remainder?.Then,it is demonstrated that when the initial value of ? can be controlled by a decaying exponential function,? can also be controlled by a decaying exponential function.And also,we use the comptraction mapping theorem to prove the wellposedness of the governing equation of ?,then it is obtained that there exists a unique solution ? of the governing equation that satisfies ?(0)=?0.Finally,in the space H2,if the solution of the second-order Camassa-Holm equation exists globally,we use the contradiction method to assume ?(?)0.We deform the governing equation by the decay of ? and Q and derive the contradiction by the orthogonality of ? and Q.Finally,we get ??0,that is,there exist parameters?0(t),x0(t)? C1,such that the solution of the second-order Camassa-Holm equation is equal to ?0-1/2(t)Q(x-xo(t)).If the solution of the second-order Camassa-Holm equation does not exist globally,we take the partial derivative of the equation on the variable x,and use the inequality to estimate it,so as to get that the solution blows up in a finite time.