Bounds For The Size Of The Support Of A Compactly Supported Momentum Density Of The Camassa-Holm Equation

In this paper, firstly, we consider the bounds for the size of the supportof a compactly supported momentum density of the Camassa-Holm equation onthe whole line. This is achieved by estimating the eigenvalue of the DirichletLaplacian operator, exploiting the relations between the eigenvalues and the ge-ometric properties of a domain. Secondly, we consider the bounds for the size ofthe compactly supported momentum density of the Camassa-Holm equation onthe half-line, We can get the same conclusion as above. Lastly, we extend theestimate of the size of the compactly supported momentum density to a type ofgeneralized Camassa-Holm equation.It is known that compactly support momentum density will remain com-pactly support. The estimate of the size of the support in this thesis furtherelaborates this result. Also the behavior of the solution outside the support ofthe momentum density is known. Hence we are estimating the size of the regionover which the solution is not that well understood.This paper is divided into five sections. In the Section 1, we give the back-ground of the topic we are studying and state the main theorems. In section 2,we give the notation, and the preliminaries to be used in the paper.In section 3,we prove the main theorem, giving an estimate of the size of the support of acompactly supported momentum density of the Camassa-Holm equation on thewhole line. In section 4, the same problem on the half-line is considered. Insection 5, we extend the conclusion of section 3 to a type of generalized Camassa-Holm equation. In section 6, we do a summary and point out some problems tobe studied.