Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

Yonghui Zhou; Shuguan Ji

doi:10.3934/cpaa.2021188

Article Contents

2022, Volume 21, Issue 2: 555-566. Doi: 10.3934/cpaa.2021188

This issue Previous Article Liouville type theorems for stable solutions of elliptic system involving the Grushin operator Next Article Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in

$ \mathbb{R}^3 $

nonlinear KGS system

Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

Yonghui Zhou^1,2, and
Shuguan Ji^1, ,

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
^* Corresponding author

Received: June 30, 2021

Revised: August 31, 2021

Early access: October 2021

Published: February 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A01, 35B44, 35Q35.

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