本项目计划研究Schwarzschild时空中半线性波动方程Cauchy问题解的大时间性态，包括小初值问题和大初值问题。对于小初值问题，当初值的支集靠近黑洞视界时，我们拟证明带次临界和临界指标（非线性项的指数）的半线性波动方程Cauchy问题的解将在有限时间内破裂，并分别建立解的生命跨度精确上界估计。对于大初值问题，本项目计划研究Schwarzschild时空中带能量临界指标的焦散型半线性波动方程Cauchy问题，预期建立光滑解的整体存在性。Schwarzschild时空中线性和非线性波动方程的研究，是广义相对论学界公认的解决Schwarzschild时空非线性稳定性问题的先决步骤。因此，本项目的研究具有很强的物理背景和理论意义。

In this project we consider the long time behavior to Cauchy problem of semilinear wave equations in Schwarzschild spacetime, including both small initial data and large initial data problem. For small data problem, we plan to establish blow up result of subcritical and critical semilinear wave equations in Schwarzschild spacetime, when the supports of the data are close to the event horizon. Furthermore, the sharp upper bound of the lifespan will be obtained. For the Cauchy problem with large initial data, we consider energy critical semilinear wave equation in Schwarzschild spacetime and global existence of classical solution will be established. The study of linear and nonlinear wave equations in Schwarzschild spacetime is the key step to solve the nonlinear stability of Schwarzschild spacetime, which is the famous open problem in general relativity, and hence it admits strong physical background and theoretical significance.