Study On The Solutions For The Two-component Camassa-holm Equation With Zero Order Dissipation

In this paper, we study the two-component Camassa-Holm equation with the zero order dissipative term λu. 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 of the solutions for the Cauchy problem. Third,we study global existence of the solutions for the Cauchy problem.The existence of the global weak solutions for the Cauchy problem are studied ?nally.In the ?rst chapter, we introduce the research background of the two-component Camassa-Holm equation with dissipation and state the main results. We also give some symbols related to the context in this chapter.In the second chapter, it is shown that the Cauchy problem is locally well-posed for initial data z0=u0ρ0∈ Hs × Hs-1, s ≥ 2.In the third chapter, the blow-up mechanism of solutions for the Cauchy problem is proved, and two su?cient conditions that lead to blow-up of the solutions are given.In the forth chapter, we prove the global existence of solutions for the Cauchy problem.In the ?fth chapter,we prove the existence of the global weak solutions for the Cauchy problem.