Abstract and Applied Analysis
Volume 2011 (2011), Article ID 285040, 15 pages
http://dx.doi.org/10.1155/2011/285040
Research Article

The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

Shaoyong Lai

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

Received 17 January 2011; Revised 3 August 2011; Accepted 8 August 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 Shaoyong Lai. 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

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in space 𝐶 ( [ 0 , 𝑇 ) , 𝐻 𝑠 ( 𝑅 ) ) ∩ 𝐶 1 ( [ 0 , 𝑇 ) , 𝐻 𝑠 − 1 ( 𝑅 ) ) with 𝑠 > 3 / 2 . In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space 𝐻 𝑠 ( 𝑅 ) with 1 ≤ 𝑠 ≤ 3 / 2 is developed.