一类半线性波动方程Cauchy问题破裂的新证法  

A new proof for blow-up to the Cauchy problem of a semi-linear wave equation

作　　者：杨姗姗 蒋红标[2] YANG Shanshan;JIANG Hongbiao(School of Sciences,Zhejiang Sci-Tech University,Hangzhou 310018,China;Department of Mathematics,Lishui University,Lishui 323000,China)

机构地区：[1]浙江理工大学理学院,杭州310018 [2]丽水学院工学院,丽水323000

出　　处：《浙江理工大学学报（自然科学版）》2020年第3期368-372,共5页Journal of Zhejiang Sci-Tech University(Natural Sciences)

基　　金：浙江省自然科学基金项目(LY18A010008)。

摘　　要：研究n维空间中带导数非线性项的半线性波动方程小初值Cauchy问题(Glassey猜想),证明当■为非线性指标)时,能量解将在有限时间破裂,并进一步建立了破裂解的生命跨度上界估计。在证明过程中,利用截断函数和线性波动方程的一个特殊解,构造了一个自身为负但其关于时间的一阶导数为非负的试探函数,用一种简洁明了的新方法得到了结论,简化了前人的证明。In this paper, we study the Cauchy problem(Glassey guess) of small initial value of the semi-linear wave equations with derivative nonlinear term in n-dimension space. We prove that the energy solution will blow up in a finite time when ■(where p is a nonlinear indicator),and what is more,the upper bound estimate of the lifespan is established. By using the truncation function and a special solution of the linear wave equation, we construct a test function, which is negative itself but has nonnegative derivative with respect to the time variable. We draw the conclusions by a simple new method and simplify the predecessors’ proof.

关 键 词：半线性波动方程 破裂 能量解 生命跨度 试探函数 

分 类 号：O175.2[理学—数学]