Schwarzschild黑洞时空中带导数非线性项半线性波动方程解的生命跨度估计  

Lifespan estimate for the semilinear wave equation with a derivative nonlinear term in Schwarzschild spacetime

作　　者：赖宁安 周忆 Ning'an Lai;Yi Zhou

机构地区：[1]丽水学院非线性分析研究所,数学系,丽水323000 [2]复旦大学数学科学学院,上海200433

出　　处：《中国科学：数学》2021年第6期957-970,共14页Scientia Sinica：Mathematica

基　　金：浙江省自然科学基金(批准号:LY18A010008);国家自然科学基金(批准号:11771194和11421061);非线性数学模型与方法教育部重点实验室(复旦大学);上海市现代应用数学重点实验室(复旦大学数学科学学院)资助项目。

摘　　要：本文研究Schwarzschild黑洞时空中带导数非线性项的半线性波动方程小初值Cauchy问题□_(gS)φ=|∂tφ|^(p),其中□_(gS)表示Schwarzschild度量下的波动算子;证明当1<p≤2时径向小初值解将在有限时间内爆破,并且建立生命跨度上界估计.这里比较有意义的是不需要假设初值的支集远离黑洞视界,与平坦Minkowski时空中相应问题相比,证明有限时间爆破结果需要克服靠近黑洞时带来的困难∂.In this paper, we study the Cauchy problem of the semilinear wave equation with a derivative nonlinear term and small initial data □_(gS)φ=|∂tφ|^(p),, where □_(gs) denotes the wave operator with the Schwarzschild metric. We show the blow-up result for the radial solution if p ∈(1, 2], and furthermore the lifespan estimate from above is established. The interesting thing is that we do not have to assume that the support of the initial data should be far away from the event horizon, which means that we have to overcome the difficulty caused by the black hole, compared with the corresponding problem in flat Minkowski spacetime.

关 键 词：SCHWARZSCHILD黑洞 波动方程 生命跨度 

分 类 号：P145.8[天文地球—天体物理] O175[天文地球—天文学]