数学物理学报 ›› 2025, Vol. 45 ›› Issue (3): 726-747.

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一类半线性波动方程弱耦合系统解的破裂

冯振东1(),郭飞2,3,*(),李岳群2()   

  1. 1丽水职业技术学院智能制造学院 浙江丽水 323000
    2南京师范大学数学科学学院 南京 210023
    3南京师范大学大规模复杂系统数值模拟教育部重点实验室 南京 210023
  • 收稿日期:2024-01-25 修回日期:2024-10-15 出版日期:2025-06-26 发布日期:2025-06-20
  • 通讯作者: *郭飞, E-mail: guof@njnu.edu.cn
  • 作者简介:冯振东, E-mail: 492024816@qq.com;李岳群yqli1214@163.com
  • 基金资助:
    国家自然科学基金(11731007);江苏省高校优势学科建设工程;江苏省自然科学基金(BK20221320);和江苏省研究生科研与实践创新计划项目(KYCX24_1786)

Breakdown of Solutions to a Weakly Coupled System of Semilinear Wave Equations

Feng Zhendong1(),Guo Fei2,3,*(),Li Yuequn2()   

  1. 1School of Intelligent Manufacturing, Lishui Vocational and Technical College, Zhejiang Lishui 323000
    2School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023
    3Key Laboratory of NSLSCS (NNU), Ministry of Education, Nanjing 210023
  • Received:2024-01-25 Revised:2024-10-15 Online:2025-06-26 Published:2025-06-20
  • Supported by:
    NSFC(11731007);Priority Academic Program Development of Jiangsu Higher Education Institutions, the NSF of Jiangsu Province(BK20221320);Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX24_1786)

摘要:

该文关注了一类带有尺度不变阻尼项, 质量项和一般的非线性记忆项的半线性波动方程弱耦合系统的 Cauchy 问题. 首先, 在指数 p,q 和系数 μ1,μ2 的适当假设下, 利用 Banach 不动点定理建立了该问题的局部解. 这里, pq 为非线性记忆项的指数, μ1μ2 分别为阻尼项和质量项的系数. Palmieri 对相应线性齐次方程解的衰减估计在证明局部解过程中起了重要作用. 之后, 采用迭代技术并结合试验函数方法, 得到了能量解的破裂.

关键词: 半线性波动方程, 弱耦合系统, 破裂, 试验函数, 迭代技术

Abstract:

This paper addresses the Cauchy problem for a weakly coupled system of semilinear wave equations with scale-invariant dampings, mass, and general nonlinear memory terms. Firstly, a local (in time) existence result for this problem is established using Banach's fixed point theorem, subject to suitable assumptions on the exponents p,q and coefficients μ1,μ2.Here, p and q represent the powers of the nonlinear memory terms, while μ1 and μ2 denote the coefficients of the dampings and mass terms, respectively. It is noteworthy that Palmieri's decay estimates for the solution to the corresponding linear homogeneous equation play a crucial role in proving the local well-posedness result. Subsequently, employing an iteration technique in conjunction with the test function method, we obtain a blowup result for energy solutions.

Key words: semilinear wave equation, weakly coupled system, blowup, test function method, iteration skill

中图分类号: 

  • O175.27