Abstract and Applied Analysis

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

Shaoyong Lai and Aiyin Wang

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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 s 3 / 2 is proved under the assumptions u 0 H s and u 0 x L < .

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

Permanent link to this document
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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