AI 中文总结
研究紧致李群上带非自治强迫项的半线性阻尼波动方程，通过分析非负时间依赖因子$\varphi$，利用其可和性区分解的情况，在特定条件下推导局部时间解的精确寿命估计。
AI 中文摘要
在本笔记中，我们考虑紧致李群上具有非自治非线性项$\varphi(t)|u|^p$的半线性阻尼波动方程。我们关注非负时间依赖因子$\varphi$如何影响局部解的全局时间可延拓性。特别地，函数$\varphi$的可和性提供了区分有限时间爆破和小数据全局存在性的准则。最后，当$\varphi\notin L^1([0,+\infty))$且满足特定缩放条件（我们称为均匀上缩放条件）时，我们推导了局部时间解的精确寿命估计。
英文摘要
In the present note, we consider a semilinear damped wave equation on a compact Lie group with a non-autonomous nonlinearity $φ(t)|u|^p$. We are interested in describing how the nonnegative time-dependent factor $φ$ affects the global in time prolongability of a local solution. In particular, the summability of the function $φ$ provides a criterion to distinguish between the blow-up in finite time and global existence of small data. Finally, we derive sharp lifespan estimates for local in time solutions when $φ\not\in L^1([0,+\infty)))$ and satisfies a certain scaling condition, that we named uniform upper scaling condition.
