Some Properties Of Solution To The Weakly Dissipative Camassa-Holm Equation

In this paper, we consider the weakly dissipative Camassa-Holm equation. Firstly, we give a brief introduction about Camassa-Holm equation, which includes some results obtained by other authors in this field. Subsequently, we introduce the fundamental theorem on local well-posedness (Kato's theorem) and list some results for later use including Gronwall's inequality,Sobolev embedding theorem,Mean value theorem,Rademacher's theorem and the identity on H~1 norm. At the same time, we rewrite the equation to get a proper form we want so as to make us convenient in considering the problem. In the third part, we try to improve other authors' results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa-Holm equation.