源于非线性层晶格模型的一耦合Boussinesq型广义方程组的Cauchy问题

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1552-1567.

源于非线性层晶格模型的一耦合Boussinesq型广义方程组的Cauchy问题

陈翔英^1,*(),陈国旺²()

¹ 郑州电力高等专科学校公共教学部郑州 450000
² 郑州大学数学与统计学院郑州 450052

收稿日期:2019-10-25 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 陈翔英 E-mail:chenxiangying@126.com;chenguowang@zzu.edu.cn
作者简介:陈国旺, E-mail: chenguowang@zzu.edu.cn
基金资助:
国家自然科学基金(11671367);国家自然科学基金(11171311)

Cauchy Problem for a Generalized System of Coupled Boussinesq Type Equations Arising from Nonlinear Layered Lattice Model

Xiangying Chen^1,*(),Guowang Chen²()

¹ Common Teaching Department, Zhengzhou Electric Power College, Zhengzhou 450000
² School of Mathematics and Satistics, Zhengzhou University, Zhengzhou 450052

Received:2019-10-25 Online:2020-12-26 Published:2020-12-29
Contact: Xiangying Chen E-mail:chenxiangying@126.com;chenguowang@zzu.edu.cn
Supported by:
the NSFC(11671367);the NSFC(11171311)

摘要/Abstract

摘要：

该文证明一源于非线性层晶格模型的耦合Boussinesq型方程组 $ \begin{eqnarray} &&u_{tt}-a_{1}u_{xx}-a_{2}u_{xxtt}+a_{3}(u-w)=f(u_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0, \;\;\;\;(0.1)\&&w_{tt}-b_{1}w_{xx}-b_{2}w_{xxtt}+b_{3}(w-u)=g(w_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0\;\;\;\;(0.2)\end{eqnarray} $ 在$C([0, \infty);H^s(\ {\Bbb R})\times H^s(\ {\Bbb R}))\ (s\geq2$是一实数)中存在唯一的整体广义解, 在$C^2([0, \infty);$ $C_B^2(\ {\Bbb R})\times C_B^2(\ {\Bbb R}))(s>\frac{5}{2})$中存在唯一的整体古典解.对于Cauchy问题(0.1)——(0.2)解的爆破给出了充分条件.

关键词: 耦合Boussinesq型方程的广义方程组, Cauchy问题, 整体解, 解的爆破

Abstract:

In this paper, we prove that the Cauchy problem for a generalized system of the coupled Boussinesq-type equations arising from nonlinear layered lattice model $\begin{eqnarray} &u_{tt}-a_{1}u_{xx}-a_{2}u_{xxtt}+a_{3}(u-w)=f(u_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0, \nonumber\ &w_{tt}-b_{1}w_{xx}-b_{2}w_{xxtt}+b_{3}(w-u)=g(w_{x})_{x}, \qquad x\in\ {\Bbb R}, \ t>0\nonumber \end{eqnarray} $ has a unique global generalized solution in $C([0, \infty);H^s(\ {\Bbb R})\times H^s(\ {\Bbb R}))(s\geq2$ is a real number) and a unique global classical solution in $C^2([0, \infty);C_B^2(\ {\Bbb R})\times C_B^2(\ {\Bbb R}))(s>\frac{5}{2})$. The sufficient conditions for the blow up of the solution to the Cauchy problem above are given.

Key words: Generalized system of coupled Boussinesq-type equations, Cauchy problem, Global solution, Blow up of solution

中图分类号:

O175.2

陈翔英,陈国旺. 源于非线性层晶格模型的一耦合Boussinesq型广义方程组的Cauchy问题[J]. 数学物理学报, 2020, 40(6): 1552-1567.

Xiangying Chen,Guowang Chen. Cauchy Problem for a Generalized System of Coupled Boussinesq Type Equations Arising from Nonlinear Layered Lattice Model[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1552-1567.

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