一类半线性波动方程的适定性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1568-1589.

一类半线性波动方程的适定性

肖常旺¹(),郭飞^1,^2,*()

¹ 南京师范大学数学科学学院南京 210023
² 江苏省“大规模复杂系统数值模拟”重点实验室南京 210023

收稿日期:2019-10-25 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 郭飞 E-mail:15996269522@163.com;guof@njnu.edu.cn
作者简介:肖常旺, E-mail: 15996269522@163.com
基金资助:
江苏省自然科学基金(BK20181381);江苏省高校自然科学研究重大项目(17KJA110002);江苏省青蓝工程

Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces

Changwang Xiao¹(),Fei Guo^1,^2,*()

¹ School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023
² School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023

Received:2019-10-25 Online:2020-12-26 Published:2020-12-29
Contact: Fei Guo E-mail:15996269522@163.com;guof@njnu.edu.cn
Supported by:
Priority Academic Program Development of Jiangsu Higher Education Institutions, the NSF of Jiangsu Province(BK20181381);the NSF of the Jiangsu Higher Education Institutions(17KJA110002);the Qing Lan Projects of Jiangsu Province

摘要/Abstract

摘要：

该文研究了一类带有效时变阻尼与变质量项的半线性波动方程弱解的整体存在性和有限时刻爆破现象.当非线性项指数$p>p_F (N)=1+\frac{2}{N}$时, 证明了在指数加权能量空间中, 此问题的小初值的解整体存在; 而当<p\leq p_F (\alpha, n)=1+\frac{2(1+\alpha)}{N (1+\alpha)-2\alpha}~(0<\alpha<1)$时, 对于一些方程中特定的参数和满足积分符号条件的初值, 证明了此问题的解必在有限时间内爆破.

关键词: 半线性波动方程, 整体解, 爆破, Fujita指数

Abstract:

We consider the global small data solutions and blowup to the Cauchy problem for a semilinear wave equation with time-dependent damping and mass term as well as power nonlinearity. On one hand, if the power of the nonlinearity $p >p_F(N)=1+ \frac 2N$, it is proved that solutions with small initial data exist for all time in exponentially weighted energy spaces. On the other hand, if the power satisfies <p\leq p_F(\alpha, n)=1+\frac{2(1+\alpha)}{N(1+\alpha)-2\alpha}~(0<\alpha<1)$, for some special chosen parameters it is shown that solutions must blow up in finite time provided that the initial data satisfy some integral sign conditions.

Key words: Semilinear wave equation, Global solution, Blow up, Fujita exponent

中图分类号:

O175.27

肖常旺,郭飞. 一类半线性波动方程的适定性[J]. 数学物理学报, 2020, 40(6): 1568-1589.

Changwang Xiao,Fei Guo. Global Existence and Blowup Phenomena for a Semilinear Wave Equation with Time-Dependent Damping and Mass in Exponentially Weighted Spaces[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1568-1589.

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