您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2021, Vol. 56 ›› Issue (9): 59-65.doi: 10.6040/j.issn.1671-9352.0.2020.675

• • 上一篇    

一类具有空变系数的非线性项的半线性双波动方程解的全局非存在性

欧阳柏平1,肖胜中2   

  1. 1.广州华商学院数据科学学院, 广东 广州 511300;2.广东农工商职业技术学院, 广东 广州 510507
  • 发布日期:2021-09-13
  • 作者简介:欧阳柏平(1979— ),男,硕士,讲师,研究方向为偏微分方程. E-mail:oytengfei79@tom.com
  • 基金资助:
    广东省普通高校创新团队资助项目(2020WCXTD008);广东财经大学华商学院校内资助项目(2020HSDS01);广州华商学院科研团队资助项目(2021HSKT01)

Global nonexistence of solutions to a class of semilinear double-wave equations with space-dependent coefficients on the nonlinearity

OUYANG Bai-ping1, XIAO Sheng-zhong2   

  1. 1. College of Data Science, Guangzhou Huashang College, Guangzhou 511300, Guangdong, China;
    2. Guangdong AIB Polytechnic College, Guangzhou 510507, Guangdong, China
  • Published:2021-09-13

PDF (PC)

286

摘要： 考虑了一类具有空变系数的非线性项的半线性双波动方程解的爆破问题。运用微分不等式方法和迭代方法证明了半线性双波动方程柯西问题在非临界情况下解的全局非存在性,且给出了生命区间的上界估计。进一步推广了波动方程在高阶上柯西问题的有关结果。

关键词: 非线性项, 半线性双波动方程, 全局非存在性

Abstract: Blow-up of solutions to a class of semilinear double-wave equations with space-dependent coefficients on the nonlinearity is considered. By using the methods of differential inequalities and an iteration argument, the global nonexistence and the upper bound of the lifespan of solutions to the Cauchy problem for semilinear double-wave equations in the subcritical case are obtained, which generalize further the facts on the Cauchy problem for wave equations of high orders.

Key words: nonlinearity, semilinear double-wave equation, global nonexistence

中图分类号: 

  • O175.27
[1] KATO T. Blow-up of solutions of some nonlinear hyperbolic equations[J]. Communications on Pure and Applied Mathematics, 1980, 33(4):501-505.
[2] JOHN F. Blow-up of solutions of nonlinear wave equations in three space dimensions[J]. Manuscripta Mathematica, 1979, 28(1/2/3):235-268.
[3] STRAUSS W A. Nonlinear scattering theory at low energy[J]. Journal of Functional Analysis, 1981, 41(1):110-133.
[4] GLASSEY R T. Finite-time blow-up for solutions of nonlinear wave equations[J]. Mathematische Zeitschrift, 1981, 177(3):323-340.
[5] SIDERIS T C. Nonexistence of global solutions to semilinear wave equations in high dimensions[J]. Journal of Differential Equations, 1984, 52(3):378-406.
[6] SCHAEER J. The equation utt-Δu=|u|p for the critical value of p[J]. Proceedings of the Royal Society of Edinburgh: Section A, 1985, 101(1/2):31-44.
[7] TAKAMURA H, WAKASA K. The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions[J]. Journal of Differential Equations, 2011, 251(4/5):1157-1171.
[8] ZHOU Yi. Life span of classical solutions to utt-u=|u|1+α[J]. Chinese Annals of Mathematics: Series B, 1992, 13(2):230-243.
[9] TAKAMURA H. Improved Katos lemma on ordinary differential inequality and its application to semilinear wave equations[J]. Nonlinear Analysis, 2015, 125:227-240.
[10] ZHOU Yi, HAN Wei. Life-span of solutions to critical semilinear wave equations[J]. Communications in Partial Differential Equations, 2014, 39(3):439-451.
[11] CHEN Wenhui. Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms[J]. Nonlinear Analysis, 2021, 202:112160.
[12] CHEN W H, PALMIERI A. Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case[J]. Discrete and Continuous Dynamical Systems, 2020, 40(9):5513-5540.
[13] CHEN W H, REISSIG M. Blow-up of solutions to Nakaos problem via an iteration argument[J]. Journal of Differential Equations, 2021, 275:733-756.
[14] CHEN W H, PALMIERI A. Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2019, 71(2):67.
[15] LAI N G, TAKAMURA H. Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glasseys conjecture[J]. Differential Integral Equations, 2019, 32(1/2):37-48.
[16] LAI N G, TAKAMURA H, WAKASA K. Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent[J]. Journal of Differential Equations, 2017, 263(9):5377-5394.
[17] PALMIERI A, TAKAMURA H. Blow-up for a weekly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities[J]. Nonlinear Analysis, 2019, 187:467-492.
[18] CHEN W H, PALMIERI A. A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case[J/OL]. Evolution Equations and Control Theory, 2020[2020-08-01]. http://dx.doi.org/10.3934/eect.2020085.
[19] LIU Yan. Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition[J]. Computers and Mathematics with Applications, 2013, 66(10):2092-2095.
[20] LI Yuanfei, LIU Yan, LIN Changhao. Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions[J]. Nonlinear Analysis: Real World Applications, 2010, 11(5):3815-3823.
[21] LIU Yan, LUO Shiguang, YE Yunhua. Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions[J]. Computers and Mathematics with Applications, 2013, 65(8):1194-1199.
[22] CHEN Wenhui, LIU Yan. Lower bound for the blow-up time for some nonlinear parabolic equations[J]. Boundary Value Problems, 2016, 2016(1):161.
[23] FANG Zhongbo, WANG Yuxiang. Blow-up analysis for a semi-linear parabolic equation with time-dependent coefficients under nonlinear boundary flux[J]. Zeitschrift fur Angewandte Mathematik und Physik, 2015, 66(5):1-17.
[24] YORDANOV B T, ZHANG Q S. Finite time blow up for critical wave equations in high dimensions[J]. Journal of Functional Analysis, 2006, 231(2):361-374.
[1] 张申贵. 一类非自治二阶系统的多重周期解[J]. J4, 2011, 46(11): 64-69.
[2] 牛艳艳,邢伟. 含非线性项的不确定离散切换系统的容错控制[J]. J4, 2010, 45(5): 52-57.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发