In this note, we study the semilinear wave equation with power nonlinearity \(|u|^p\) on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any \(p>1\), under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.

中文翻译：

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紧李群上的半线性波动方程

在本笔记中，我们研究了紧李群上具有幂非线性\(|u|^p\)的半线性波动方程。首先，我们通过对紧李群的傅里叶分析证明了能量空间中的局部时间存在结果。然后，在初始数据的适当符号假设下，我们证明了任何\(p>1\)的半线性柯西问题的爆炸结果。此外，还导出了本地（及时）解决方案的精确寿命估计。