双曲空间上拟线性波动方程的全局存在性
Global existence for quasilinear wave equations on hyperbolic space
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中文总结 AI 辅助
研究双曲空间上拟线性波动方程小振幅光滑初值的全局可解性，通过建立能量和局部能量估计实现，无需零条件等假设，还得到半线性问题低正则全局解及径向Glassey猜想类似结果。
中文摘要 AI 辅助
本文主要研究双曲空间上一类一般的拟线性平移波动方程对于小振幅光滑初始数据的全局可解性。与欧几里得空间情形不同，当空间维数为3时，无需假设如零条件等结构条件来确保全局存在性。为此，在\(\mathbb{R}\times \mathbb{H}^n\)上为扰动波动算子建立了能量和局部能量估计。这些估计允许依赖时间的度量扰动，且仅要求在径向变量\(r\)上适当小并具有多项式衰减。作为副产品，对于具有幂型非线性和径向数据的半线性问题，也得到了低正则性的全局解。特别地，证明了双曲空间上径向Glassey猜想的类似结果。
英文摘要
The main purpose of this paper is to study the global solvability for a general class of quasilinear shifted wave equations on hyperbolic spaces, for smooth initial data with small amplitude. In contrast to the case of Euclidean spaces, when the space dimension is three, we do not need to assume structural conditions like the null conditions to ensure global existence. To achieve this, we establish the energy and local energy estimates for perturbed wave operators on $\mathbb{R}\times \mathbb{H}^n$. These estimates allow time-dependent metric perturbations and require only suitable smallness together with polynomial decay in the radial variable $r$. As a byproduct, for semilinear problems with power-type nonlinearities and radial data, we also obtain global solutions with low-regularity. In particular, we prove an analog of the radial Glassey conjecture on hyperbolic space.
