THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION*

doi:10.1007/s10473-023-0220-6

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 736-750.doi: 10.1007/s10473-023-0220-6

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*

Mingxuan Zhu¹, Zaihong Jiang^2,†

1. School of Mathematical Sciences, Qufu Normal University, Qufu 273100, China;
2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

收稿日期:2021-11-07 修回日期:2022-04-05 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Zaihong Jiang, E-mail: jzhong@zjnu.cn.
作者简介:Mingxuan Zhu, E-mail: mxzhu@qfnu.edu.cn
基金资助:
This work was partially supported by the National Natural Science Foundation of China (12071439), the Zhejiang Provincial Natural Science Foundation of China (LY19A010016) and the Natural Science Foundation of Jiangxi Province (20212BAB201016).

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*

Mingxuan Zhu¹, Zaihong Jiang^2,†

1. School of Mathematical Sciences, Qufu Normal University, Qufu 273100, China;
2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received:2021-11-07 Revised:2022-04-05 Online:2023-03-25 Published:2023-04-12
Contact: †Zaihong Jiang, E-mail: jzhong@zjnu.cn.
About author:Mingxuan Zhu, E-mail: mxzhu@qfnu.edu.cn
Supported by:
This work was partially supported by the National Natural Science Foundation of China (12071439), the Zhejiang Provincial Natural Science Foundation of China (LY19A010016) and the Natural Science Foundation of Jiangxi Province (20212BAB201016).

摘要/Abstract

摘要： In this paper, we consider the Cauchy problem for the Camassa-Holm-Novikov equation. First, we establish the local well-posedness and the blow-up scenario. Second, infinite propagation speed is obtained as the nontrivial solution $u(x,t)$ does not have compact $x$-support for any $t>0$ in its lifespan, although the corresponding $u_0(x)$ is compactly supported. Then, the global existence and large time behavior for the support of the momentum density are considered. Finally, we study the persistence property of the solution in weighted Sobolev spaces.

关键词: Camassa-Holm-Novikov equation, local well-posedness, blow-up scenario, infinite propagation speed, global existence, large time behavior, persistence property

Abstract: In this paper, we consider the Cauchy problem for the Camassa-Holm-Novikov equation. First, we establish the local well-posedness and the blow-up scenario. Second, infinite propagation speed is obtained as the nontrivial solution $u(x,t)$ does not have compact $x$-support for any $t>0$ in its lifespan, although the corresponding $u_0(x)$ is compactly supported. Then, the global existence and large time behavior for the support of the momentum density are considered. Finally, we study the persistence property of the solution in weighted Sobolev spaces.

Key words: Camassa-Holm-Novikov equation, local well-posedness, blow-up scenario, infinite propagation speed, global existence, large time behavior, persistence property

Mingxuan Zhu, Zaihong Jiang. THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*[J]. 数学物理学报(英文版), 2023, 43(2): 736-750.

Mingxuan Zhu, Zaihong Jiang. THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 736-750.

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THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*

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