带立方非线性修正 Camassa-Holm 方程的初值问题

数学物理学报 ›› 2024, Vol. 44 ›› Issue (2): 376-383.

带立方非线性修正 Camassa-Holm 方程的初值问题

张欣¹(),吴兴龙^2,^*()

¹武汉理工大学数学系武汉 430070
²武汉理工大学数学科学研究中心武汉 430070

收稿日期:2023-04-29 修回日期:2023-09-05 出版日期:2024-04-26 发布日期:2024-04-07
通讯作者: * 吴兴龙, Email:wxl8758669@whut.edu.cn
作者简介:张欣,Email:zhangxin319548@whut.edu.cn
基金资助:
国家自然科学基金(11771442);国家自然科学基金(11971024)

The Initial Value Problem for a Modified Camassa-Holm Equation with Cubic Nonlinearity

Zhang Xin¹(),Wu Xinglong^2,^*()

¹Department of Mathematical, Wuhan University of Technology, Wuhan 430070
²Center for Mathematical Sciences, Wuhan University of Technology, Wuhan 430070

Received:2023-04-29 Revised:2023-09-05 Online:2024-04-26 Published:2024-04-07
Supported by:
NSFC(11771442);NSFC(11971024)

摘要/Abstract

摘要：

该文利用 Kato 半群理论和耗散算子的定义, 研究了带立方非线性修正 Camassa-Holm 方程的初值问题, 并在 Sobolev 空间 $ H^{s,p}(\mathbb{R}) $ ($ s\ge 1,\ p\in (1,\infty ) $) 中建立了其解的存在唯一性, 推广了 Fu 等在 Besov 空间 $ B_{p,r}^{s}(\mathbb{R}) $ $ (p,\ r\ge 1,\ s> \max \{2+\frac{1}{p},\frac{5}{2}\}) $ 中所得到的适定性结果 (J Differ Equations, 2013, 255: 1905-1938).

关键词: 修正 Camassa-Holm 方程, 适定性, 耗散算子, Kato 半群理论

Abstract:

In this paper, by Kato's theory of semigroup and the definition of the dissipative operator, we study the initial value problem of a modified Camassa-Holm equation with cubic nonlinearity, and establish the existence and uniqueness of its solutions in Sobolev space $ H^{s,p}(\mathbb{R}) $, $ s\ge 1 $, $ p\in (1,\infty ) $, which extend the well-posedness of its solutions in Besov space $ B_{p,r}^{s}(\mathbb{R}) $ $ (p,\ r\ge 1,\ s> \max \{2+\frac{1}{p},\frac{5}{2}\}) $ obtained by Fu et al. (J Differ Equations, 2013, 255: 1905-1938).

Key words: The modified Camassa-Holm equation, Well-posedness, Dissipative operator, Kato's theory of semigroup

中图分类号:

O175.2

张欣, 吴兴龙. 带立方非线性修正 Camassa-Holm 方程的初值问题[J]. 数学物理学报, 2024, 44(2): 376-383.

Zhang Xin, Wu Xinglong. The Initial Value Problem for a Modified Camassa-Holm Equation with Cubic Nonlinearity[J]. Acta mathematica scientia,Series A, 2024, 44(2): 376-383.

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