Abstract and Applied Analysis
Volume 2011 (2011), Article ID 647368, 15 pages
http://dx.doi.org/10.1155/2011/647368
Research Article

Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

Zaihong Jiang1 and Sevdzhan Hakkaev2

1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Faculty of Mathematics and Informatics, Shumen University, 9712 Shumen, Bulgaria

Received 24 June 2011; Revised 1 September 2011; Accepted 13 September 2011

Academic Editor: P. J. Y. Wong

Copyright © 2011 Zaihong Jiang and Sevdzhan Hakkaev. 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

We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solution u(t,x) with compactly supported initial datum u0(x) does not have compact x-support any longer in its lifespan.