Properties Of Solutions Near The Traveling Waves Of The Second-order Camassa-Holm Eqaution

This paper studies the decay property and blow-up of solutions near the traveling waves for Cauchy problem of the second-order Camassa-Holm（CH）equation by the extended pseudo-conformal transformation method and decomposition method.First,according to the translation invariance（if u（t,,x）is a solution of the equation,then u（t,χ+χ₀）is also a solution）and dilatation invariance(if u（t,x）is a solution of the equation,then λ₀^1/2u(λ₀^1/2 t,x)is also a solution)of the second-order CH equation,the solution of Cauchy problem for the equation is transformed by the extended pseudo-conformal transformation and decomposed into λ^1/2（t）u(t,y+ χ（t）=ε（t,y）+Q（y）Next,by the Banach’s fixed point theorem and contradiction,the solution is controlled by the decaying function with exponential speed,corresponding to the initial data and its second derivative with exponential decay.Finally,a sufficient condition for the existence of blow-up solution is obtained,corresponding to the initial data with negative slope.A equivalent proposition of the solution breaking in finite time around the traveling waves is constructed and the relation is established between the blow-up time and rate of the solution and the residuals’ by discussing the blow-up of ε（t,·）The structure of this paper is organized as follows:In the first section,the research background,current situation,research contents and main results of this paper are introducedIn the second section,some basic concepts,related theorems and several inequalities involved in this paper are givenIn the third section,the stability of the residual ε（t）in H² is studied according to the boundedness of modulation parameters λ（t）,x（t）and their derivative.The decay property of the residual ε（t）is discussed by applying Banach’s fixed point theorem and contradiction.In the fourth section,the blow-up of solution near the traveling waves for the second-order CH equation is investigated by discussing the blow-up of ε（t）.In the fifth section,the summary and outlook is provided.