## Abstract and Applied Analysis

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

#### 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 ${H}^{s}(R)$ with $s>3/2$. Although the ${H}^{1}$-norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower-order Sobolev space ${H}^{s}$ with $1\le s\le 3/2$ is proved under the assumptions ${u}_{0}\in {H}^{s}$ and $\parallel {u}_{0x}{\parallel }_{{L}^{\infty }}<\infty$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 872187, 15 pages.

Dates
First available in Project Euclid: 4 April 2013

https://projecteuclid.org/euclid.aaa/1365099943

Digital Object Identifier
doi:10.1155/2012/872187

Mathematical Reviews number (MathSciNet)
MR2926885

Zentralblatt MATH identifier
1242.35191

#### Citation

Lai, Shaoyong; Wang, Aiyin. The Well-Posedness of Solutions for a Generalized Shallow Water Wave Equation. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 872187, 15 pages. doi:10.1155/2012/872187. https://projecteuclid.org/euclid.aaa/1365099943

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