一类半线性波动方程的适定性  

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

作　　者：肖常旺 郭飞[1,2] Xiao Changwang;Guo Fei(School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023;School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS,Nanjing Normal University,Nanjing 210023)

机构地区：[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

出　　处：《数学物理学报（A辑）》2020年第6期1568-1589,共22页Acta Mathematica Scientia

基　　金：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。

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>pF(N)=1+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1<p≤pp(α,n)=1+2(1+α)/N(1+α)-2α(0<α<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.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>pF(N)=1+2/N,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 1<p≤pF(a,n)=1+2(1+α)/N(1+α1)-2Ω(0<α<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.

关 键 词：半线性波动方程 整体解 爆破 Fujita指数 

分 类 号：O175.27[理学—数学]