变系数临界半线性波动方程小初值Cauchy问题解的破裂机制及生命跨度估计

In this project we will study the Cauchy problem of critical semiliner wave equations with variable coefficients, assuming the initial data is small, and futhermore study the critical semilinear wave equations on Schwarzschild spacetime. Wave equation is one of the most important evolution equations, since many physical phenomena and laws can be modeled by nonlinear wave equations. As is konwn to all, for most nonlinear wave equations, the solutions will blow up at finite time. The time, at which the solutions fail to be smooth, is the so called "lifespan". The small data problem of semilinear wave equations with constant coefficients have been understood clearly by the work of many mathematicians. However, there is no result about small data Cauchy problem of critical semiliner wave equations with variable coefficients. Our goals are: 1) establish the blowup mechanism and lifespan estimate for small data Cauchy problem of critical semiliner wave equations with variable coefficients in three dimension; 2) establish the blowup mechanism for small data Cauchy problem of critical semiliner wave equations with variable coefficients in high dimensions; 3) study the small data Cauchy problem of critical semilinear wave equations on Schwarzschild spacetime.

本项目旨在研究变系数临界半线性波动方程小初值Cauchy问题解的性质，进而研究Schwarzschild时空中带临界指标半线性波动方程小初值Cauchy问题解的性质。波动方程是一类重要的发展方程，许多物理现象都可以由非线性波动方程来描述。非线性波动方程的解往往会在有限时间内破裂，而与解的破裂密切相关的一个概念是解的生命跨度，即光滑解存在的最大时间。对于常系数半线性波动方程小初值Cauchy问题，无论是次临界还是临界指标，都已经有了非常完善的结果。然而，对于变系数临界半线性波动方程小初值Cauchy问题，目前还没有结果。我们将研究：1）三维空间变系数临界半线性波动方程小初值Cauchy问题解的破裂机制及其生命跨度估计；2）高维空间变系数临界半线性波动方程小初值Cauchy问题解的破裂性质；3）Schwarzschild时空中带临界指标半线性波动方程小初值Cauchy问题解的破裂机制。

项目主持人及项目组成员通过查阅相关文献、与同行专家讨论及参加相关学术研讨会，完成了项目研究计划，达到预期目标。发表SCI论文两篇，接收一篇，投稿两篇。其中项目负责人与周忆教授合作《An elementary proof of Strauss conjecture》，得到了Strauss猜想的一个初等证明，并进一步给出了高维（大于等于4维）次临界半线性波动方程解的生命跨度下界估计，结果发表在杂志《Journal of Functional Analysis》。该篇论文中首先证明了一个加权的能量估计，进一步得到加权的Strichartz估计，从而给出了Strauss猜想的一个初等证明。并且，利用该方法我们得到了高维（大于等于4维）次临界半线性波动方程解的生命跨度精确下界估计。另外，项目主持人研究了Schwarzschild时空中聚焦型半线性波动方程的一个控制问题：考虑径向对称解，则方程变为一维，然后利用势阱理论得到了Schwarzschild时空中3次方聚焦型半线性波动方程解的精确边界能控性，结果发表在杂志《Communications on Pure and Applied Analysis》。