半线性波动方程Cauchy问题解的生命跨度估计

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1146-1157.

半线性波动方程Cauchy问题解的生命跨度估计

蒋红标¹(),汪海航^2,*()

¹ 丽水学院数学系浙江丽水 323000
² 浙江理工大学理学院杭州 310018

收稿日期:2018-09-27 出版日期:2019-10-26 发布日期:2019-11-08
通讯作者: 汪海航 E-mail:hbj@126.com;1196443777@qq.com
作者简介:蒋红标, E-mail:hbj@126.com
基金资助:
浙江省自然科学基金(LY18A010008)

Lifespan Estimation of Solutions to Cauchy Problem of Semilinear Wave Equation

Hongbiao Jiang¹(),Haihang Wang^2,*()

¹ Department of Mathematics, Lishui University, Zhejiang Lishui 323000
² School of Science, Zhejiang Sci-Tech University, Hangzhou 310018

Received:2018-09-27 Online:2019-10-26 Published:2019-11-08
Contact: Haihang Wang E-mail:hbj@126.com;1196443777@qq.com
Supported by:
the Natural Science Foundation of Zhejiang Province(LY18A010008)

摘要/Abstract

摘要：

该文研究了${\mathbb{R}}$ⁿ中半线性波动方程u_tt-△u=（1+|x|²）^α|u|^p的小初值Cauchy问题解的生命跨度估计.主要利用了改进的Kato型引理，得到了当n=2，1 < p ≤ 2时及n=1，p > 1时改进的生命跨度上界估计.

关键词: 半线性波动方程, 初值问题, 常微分不等式, 生命跨度

Abstract:

In this paper, the lifespan estimate to the Cauchy problem of the semi-linear wave equation utt u_tt-△u=(1+|x|²)^α|u|^p in ${\mathbb{R}}$ⁿ is studied. The upper bound of lifespan is improved for the cases n=2, 1 < p ≤ 2 and n=1, p > 1, by using the improved Kato's type lemma.

Key words: Semilinear wave equations, Initial value problems, Ordinary differential inequality, Lifespan

中图分类号:

O175

蒋红标,汪海航. 半线性波动方程Cauchy问题解的生命跨度估计[J]. 数学物理学报, 2019, 39(5): 1146-1157.

Hongbiao Jiang,Haihang Wang. Lifespan Estimation of Solutions to Cauchy Problem of Semilinear Wave Equation[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1146-1157.

参考文献 13

1	Sideris T C . Nonexistence of global solutions to semilinear wave equations in high dimensions. Journal of Differential Equations, 1984, 52 (3): 378- 406 doi: 10.1016/0022-0396(84)90169-4
2	Georgiev V , Takamura H , Yi Z . The lifespan of solutions to nonlinear systems of a high-dimensional wave equation. Nonlinear Analysis, 2006, 64 (10): 2215- 2250 doi: 10.1016/j.na.2005.08.012
3	Kato T . Blow up of solutions of some nonlinear hyperbolic equations. Communications on Pure & Applied Mathematics, 2010, 33 (4): 501- 505
4	Strauss W A . Nonlinear scattering theory at low energy. Journal of Functional Analysis, 1981, 43 (3): 281- 293 doi: 10.1016/0022-1236(81)90019-7
5	Zhou Y . Life span of classical solutions to u_tt + u_xx=\|u\|^1+α. Chinese Annals of Mathematics, 1992, (2): 230- 243
6	Lindblad H . Blow-up for solutions of □u=\|u\|^p with small initial data. Communications in Partial Differential Equations, 1990, 15 (6): 757- 821 doi: 10.1080/03605309908820708
7	Zhou Y , Han W . Blow-up of solutions to semilinear wave equations with variable coefficients and boundary. Journal of Mathematical Analysis and Applications, 2011, 374 (2): 585- 601 doi: 10.1016/j.jmaa.2010.08.052
8	Takamura H . Improved Kato's lemma on ordinary differential inequality and its application to semilinear wave equations. Nonlinear Analysis:Theory, Methods & Applications, 2015, 125: 227- 240 doi: 10.1016/j.na.2015.05.024
9	李卓然.一类半线性波动方程解的爆破性质[D].杭州:浙江大学, 2017
	Li Z R. Blow up Properties of Solutions for a Class of Semilinear Wave Equations[D]. Hangzhou: Zhejiang University, 2017
10	Zhou Y . Life span of classical solutions to □u=\|u\|^p in two space dimensions. Chinese Annals of Mathematics, 1993, 14 (2): 225- 236
11	Lai N A , Zhou Y , Lai N A , et al. An elementary proof of Strauss conjecture. Journal of Functional Analysis, 2014, 267 (5): 1364- 1381 doi: 10.1016/j.jfa.2014.05.020
12	Lindblad H , Sogge C D . Long-time existence for small amplitude semilinear wave equations. American Journal of Mathematics, 1996, 118 (5): 1047- 1135 doi: 10.1353/ajm.1996.0042
13	Takamura H , Wakasa K . The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions. Journal of Differential Equations, 2011, 251 (4/5): 1157- 1171

[1]	张淑琴, 胡雷. 含有变量阶导数及一个参数的非线性分数阶微分方程初值问题[J]. 数学物理学报, 2018, 38(2): 291-303.
[2]	耿金波, 杨珍珍, 赖宁安. 半无界区域上半线性薛定谔方程初边值问题解的破裂及其生命跨度的估计[J]. 数学物理学报, 2016, 36(6): 1186-1195.
[3]	陈翔英. 具有弱阻尼项的广义IMBq方程初值问题解的整体存在性[J]. 数学物理学报, 2015, 35(1): 68-82.
[4]	李中平, 徐思, 杜宛娟. 快速扩散方程的第二临界指标及解的生命跨度[J]. 数学物理学报, 2012, 32(5): 904-913.
[5]	赵永祥, 肖爱国. 两步W -方法关于时滞奇异摄动初值问题的误差分析[J]. 数学物理学报, 2011, 31(5): 1239-1252.
[6]	张海, 郑祖庥, 蒋威. 非线性分数阶泛函微分方程解的存在性[J]. 数学物理学报, 2011, 31(2): 289-297.
[7]	张新光;刘立山. Banach 空间中一阶脉冲积分-微分方程初值问题整体解的存在性[J]. 数学物理学报, 2008, 28(2): 383-392.
[8]	谢胜利. Banach空间二阶非线性脉冲积分-微分方程初值问题[J]. 数学物理学报, 2007, 27(1): 138-145.
[9]	李天然;陈传淼. 非线性常微分方程初值问题的间断有限元[J]. 数学物理学报, 2006, 26(1): 63-068.
[10]	陶双平, 崔尚斌. 七阶非线性色散方程初值问题解的局部和整体存在性[J]. 数学物理学报, 2005, 25(4): 451-460.
[11]	徐玉梅. Banach空间中二阶非线性脉冲微分积分方程初值问题的整体解[J]. 数学物理学报, 2005, 25(1): 47-56.
[12]	尚亚东, 郭柏灵. 耗散的广义对称正则长波方程周期初值问题的整体吸引子[J]. 数学物理学报, 2003, 23(6): 745-757.
[13]	李天然, 陈传淼. 二阶常微初值问题的单步格式[J]. 数学物理学报, 2003, 23(3): 379-384.
[14]	郭定辉. 一类广义层流方程组的初值问题的局部适定性[J]. 数学物理学报, 2001, 21(4): 433-438.
[15]	许兴业. 在环形区域上的半线性椭圆方程边值问题[J]. 数学物理学报, 2000, 20(zk): 675-683.