一类 mCH-CH 方程的持久性和传播速度

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 609-620.

一类 mCH-CH 方程的持久性和传播速度

李耀红^1,²(),田守富^2,^*()

1.中国矿业大学数学学院江苏徐州 221116
2.宿州学院数学与统计学院安徽宿州 234000

收稿日期:2022-10-26 修回日期:2023-12-19 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *田守富, Email: sftian@cumt.edu.cn
作者简介:李耀红, Email:liz.zhanghy@163.com
基金资助:
安徽省教育厅自然科学研究项目(KJ2021ZD0136);安徽省教育厅自然科学研究项目(KJ2021A1102)

Persistence Property and Propagation Speed for the mCH-CH Equation

Li Yaohong^1,²(),Tian Shoufu^2,^*()

1. School of Mathematics, China University of Mining and Technology, Jiangsu Xuzhou 221116
2. School of Mathematics and Statistics, Suzhou University, Anhui Suzhou 234000

Received:2022-10-26 Revised:2023-12-19 Online:2024-06-26 Published:2024-05-17
Supported by:
NSF of Anhui Provincial Education Department(KJ2021ZD0136);NSF of Anhui Provincial Education Department(KJ2021A1102)

摘要/Abstract

摘要：

研究了一类具有平方和立方非线性项的 mCH-CH 方程初值问题, 该方程是利用双哈密顿对偶方法对 Gardner 方程约化得到的一类重要的可积方程. 首先, 通过构建权函数, 利用能量方法, 结合 Gronwall 不等式, 获得了该方程具有指数或代数衰减初值时强解的持久性. 其次, 证明了方程初始值 $ m_{0}, u_{0} $ 有紧支集时, 解 $ m(x, t) $ 有紧支集, 非平凡解 $ u $ 不再具有紧支集, 但在无穷远处有指数衰减性质.

关键词: mCH-CH 方程, 持久性, 传播速度

Abstract:

Initial value problem of the mCH-CH equation with cubic and quadratic nonlinearities is studied, which is an important integrable equations obtained by applying the bi-Hamiltonian duality approach to reduce the Gardner equation. Firstly, by constructing the weight function and using the energy method and Gronwall's inequality, the persistence property of the strong solution at infinity was obtained when the initial data decays exponentially or algebraically. Secondly, we prove that the strong solution $ m(x, t) $ has compact support when the initial data $ m_{0}, u_{0} $ has compact support, and the nontrivial solution $ u(x, t) $ no longer has compact support, but has exponential decay property at infinity.

Key words: mCH-CH equation, Persistence property, Propagation speed

中图分类号:

O175.29

李耀红, 田守富. 一类 mCH-CH 方程的持久性和传播速度[J]. 数学物理学报, 2024, 44(3): 609-620.

Li Yaohong, Tian Shoufu. Persistence Property and Propagation Speed for the mCH-CH Equation[J]. Acta mathematica scientia,Series A, 2024, 44(3): 609-620.

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