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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 872187, 15 pages
http://dx.doi.org/10.1155/2012/872187
Research Article

The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation

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

Received 9 January 2012; Accepted 28 March 2012

Academic Editor: Yonghong Wu

Copyright © 2012 Shaoyong Lai and Aiyin Wang. 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 partial differential equation containing the famous Camassa-Holm and Degasperis-Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well-posedness of solutions for the equation in the Sobolev space with . Although the -norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space with is proved under the assumptions and .