American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021188
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Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

 1 School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China 2 School of Mathematics and Statistics, Hexi University, Zhangye 734000, China

* Corresponding author

Received  July 2021 Revised  September 2021 Early access November 2021

Fund Project: This work is partially supported by NSFC Grants (nos. 12071065 and 11871140) and the National Key Research and Development Program of China (nos. 2020YFA0713602 and 2020YFC1808301)

In this paper, we mainly study several problems on the weakly dissipative generalized Camassa-Holm equation. We first establish the local well-posedness of solutions by Kato's semigroup theory. We then derive the necessary and sufficient condition of the blow-up of solutions and a criteria to guarantee occurrence of wave breaking. Moreover, when the solution blows up, we obtain the precise blow-up rate. We finally show that the equation has a unique global solution provided the momentum density associated with their initial datum satisfies appropriate sign conditions.

Citation: Yonghui Zhou, Shuguan Ji. Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021188
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References:
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