带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破

数学物理学报 ›› 2024, Vol. 44 ›› Issue (6): 1550-1562.

带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破

石金诚¹(),刘炎^2,^*()

¹广州华商学院应用数学系广州 511300
²广东金融学院应用数学系广州 510521

收稿日期:2023-08-31 修回日期:2024-04-29 出版日期:2024-12-26 发布日期:2024-11-22
通讯作者: ^*刘炎, Email: ly801221@163.com
作者简介:石金诚, Email: hning0818@163.com
基金资助:
国家自然科学基金(11223344);广东省自然基金(2023A1515012044);广州华商学院科研团队(2021HSKT01);广州华商学院(2024HSTS09)

Global Existence and Blow-Up for Semilinear Third Order Evolution Equation with Different Power Nonlinearities

Shi Jincheng¹(),Liu Yan^2,^*()

¹Department of Apllied Mathematics, Guangzhou Huashang College, Guangzhou 511300
²Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521

Received:2023-08-31 Revised:2024-04-29 Online:2024-12-26 Published:2024-11-22
Supported by:
NSFC(11223344);Guangdong Natural Science foundation(2023A1515012044);Scientific Research Team Funding of Guangzhou Huashang College(2021HSKT01);Science foundation of Guangzhou Huashang College (Qualitative Study of Thermoelastic Equations)(2024HSTS09)

摘要/Abstract

摘要：

该文研究了一类带不同幂次型非线性项的半线性三阶发展方程的 Cauchy 问题, 其线性化模型来自于考虑 Fourier 法则的经典热弹性板方程组. 首先, 通过适当的远离渐近线的 $L^r$--$L^q$ 估计并结合 Banach 不动点原理, 得到了在小初值条件下整体解的存在性; 其次, 对于满足特定条件的非线性项, 利用检验函数法证明了解的爆破; 最后, 基于这些研究结果, 得出该半线性三阶模型的一些临界指标.

关键词: 整体解的存在性, 半线性三阶发展方程, 爆破, 热弹性板方程组

Abstract:

This paper studies the Cauchy problem of a class of semilinear third-order evolution equations with different power-type nonlinear terms. Its linearized model is derived from the classical thermoelastic plate equations considering Fourier's law. Firstly, by using the appropriate $L^r\!-\!L^q$ estimation away from the asymptote and combining with the Banach fixed point theorem, the existence of the global solution under small initial conditions is obtained. Secondly, for the nonlinear terms that satisfy specific conditions, the explosion of the solution is proved by the test function method. Finally, based on these research results, some critical indicators of the semilinear third-order model are obtained.

Key words: Global existence of solution, Semilinear third order evolution equation, Blow-up, Thermoelastic plate equations

中图分类号:

0175.2

石金诚, 刘炎. 带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破[J]. 数学物理学报, 2024, 44(6): 1550-1562.

Shi Jincheng, Liu Yan. Global Existence and Blow-Up for Semilinear Third Order Evolution Equation with Different Power Nonlinearities[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1550-1562.

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