ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM*

doi:10.1007/s10473-024-0508-1

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 1766-1786.doi: 10.1007/s10473-024-0508-1

ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*

Xiao SU^1,†, Shubin WANG²

1. College of Science, Henan University of Technology, Zhengzhou 450001, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

收稿日期:2023-02-20 修回日期:2024-05-16 出版日期:2024-10-25 发布日期:2024-10-22
通讯作者: †Xiao SU, E-mail,: suxiao@haut.edu.cn
作者简介:Shubin WANG, E-mail,: wangshubin@zzu.edu.cn
基金资助:
National Natural Science Foundation of China (12301272), the Natural Science Foundation of Henan (202300410109), the Cultivation Programme for Young Backbone Teachers in Henan University of Technology, and the Innovative Funds Plan of Henan University of Technology (2020ZKCJ09).

ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*

Xiao SU^1,†, Shubin WANG²

1. College of Science, Henan University of Technology, Zhengzhou 450001, China;
2. chool of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Received:2023-02-20 Revised:2024-05-16 Online:2024-10-25 Published:2024-10-22
Contact: †Xiao SU, E-mail,: suxiao@haut.edu.cn
About author:Shubin WANG, E-mail,: wangshubin@zzu.edu.cn
Supported by:
National Natural Science Foundation of China (12301272), the Natural Science Foundation of Henan (202300410109), the Cultivation Programme for Young Backbone Teachers in Henan University of Technology, and the Innovative Funds Plan of Henan University of Technology (2020ZKCJ09).

摘要/Abstract

摘要： This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

关键词: damped Boussinesq equation, Cauchy problem, global solutions, blow-up

Abstract: This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

Key words: damped Boussinesq equation, Cauchy problem, global solutions, blow-up

中图分类号:

35L30

Xiao SU, Shubin WANG. ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*[J]. 数学物理学报(英文版), 2024, 44(5): 1766-1786.

Xiao SU, Shubin WANG. ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*[J]. Acta mathematica scientia,Series B, 2024, 44(5): 1766-1786.

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ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*

ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*

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