Abstract and Applied Analysis
Volume 2012 (2012), Article ID 532369, 26 pages
http://dx.doi.org/10.1155/2012/532369
Research Article

The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

Nan Li, Shaoyong Lai, Shuang Li, and Meng Wu

Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 13 October 2011; Accepted 13 January 2012

Academic Editor: Sergey Piskarev

Copyright © 2012 Nan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev space 𝐻 𝑆 ( 𝑅 ) with 𝑠 > 3 / 2 is established via a limiting procedure. Provided that the initial value 𝑢 0 satisfies the sign condition and 𝑢 0 ∈ 𝐻 𝑠 ( 𝑅 ) ( 𝑠 > 3 / 2 ) , it is shown that there exists a unique global solution for the equation in space 𝐶 ( [ 0 , ∞ ) ; 𝐻 𝑠 ( 𝑅 ) ) ∩ 𝐶 1 ( [ 0 , ∞ ) ; 𝐻 𝑠 − 1 ( 𝑅 ) ) .