Study On The Solutions For A Dissipative Camassa-Holm Equation

In this paper, we study the Camassa-Holm equation with dissipative term λuxx. First-ly, we prove by Kato’s theorem that the initial data problem of the equation is locallywell-posed. Secondly, we study the blow-up phenomena of the solutions for the initial da-ta problem. Persistence properties of the solutions for the initial data problem are studiedfnally. The dissertation is divided into four chapters.In the frst chapter, we introduce the research background of the Camassa-Holmequation with dissipation and state the main results. We also give some symbols relatedto the context in this chapter.In the second chapter, it is shown that initial data problem is locally well-posed forinitial data u0∈Hs(R) with s>32.In the third chapter, the blow-up mechanism of solutions for the initial data problemis proved, and two sufcient conditions for solutions of blow-up are established. It isproved that blow-up rate of the blowuping solutions to the equation is-2.In the forth chapter, we discuss the persistence properties of solutions for the initialdata problem, and reveal mainly some relations between the decay of the solution and thedecay of initial data u0(x) as xâ†'∞.