Infinite propagation speed and asymptotic behavior for a generalized Camassa-Holm equation
This paper is devoted to the Cauchy problem for a generalized Camassa-Holm equation. First, we prove that the solution <i>u</i>(<i>x, t</i>) to the generalized Camassa-Holm equation with compactly supported initial data <i>u</i><sub>0</sub>(<i>x&...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Academic Journals Center of Shanghai Normal University
2018-06-01
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| Series: | Journal of Shanghai Normal University (Natural Sciences) |
| Subjects: | |
| Online Access: | http://qktg.shnu.edu.cn/zrb/shsfqkszrb/ch/reader/view_abstract.aspx?file_no=20180304 |
| Summary: | This paper is devoted to the Cauchy problem for a generalized Camassa-Holm equation. First, we prove that the solution <i>u</i>(<i>x, t</i>) to the generalized Camassa-Holm equation with compactly supported initial data <i>u</i><sub>0</sub>(<i>x</i>) instantly loses compact support. In this sense, the localized disturbance represented by <i>u</i><sub>0</sub> propagates with an infinite speed. We further prove that the solution <i>u</i>(<i>x, t</i>) to the generalized Camassa-Holm equation has an exponential decay as |<i>x</i>| goes to infinity. Moreover, the asymptotic behaviors of the solution at infinity are investigated as the initial data decays exponentially or algebraically. |
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| ISSN: | 1000-5137 1000-5137 |