Study On The Solutions For A Dissipative Two-component Camassa-holm Equation

In this paper,we study the two-component Camassa-Holm equation with dis-sipative term λ(u-uxx).Firstly,we prove by Kato’s theorem that the Cauchy problem of the equation is locally well-posed. Secondly,we study the blow-up phenomena and global existence of the solutions for the Cauchy problem.The ex-istence of global weak solutions for the Cauchy problem of the equation are studied finally.The paper is divided into four chapters.In the first chapter,we introduce the research background of the two-componemt Camassa-Holm equation with dissipation and state the main results of the paperIn the second chapter,it is shown that the Cauchy problem of the equation is locally well-posed for initial value z0=(uo,ρ0)∈Hs×Hs-1 with s≥2.In the third chapter,we study the blow-up phenomina and global existence of the solution for the Cauchy problem.The blow-up mechanism of solutions is proved and two necessary sufficient conditions of blow-up are established,and we obtain a new global existence resultIn the forth chapter,the existence of global weak solution for the Cauchy problem of the equation is proved.