Semilinear damped wave equation on a compact Lie group with a non-autonomous forcing term

In the present note, we consider a semilinear damped wave equation on a compact Lie group with a non-autonomous nonlinearity $\varphi(t)|u|^p$. We are interested in describing how the nonnegative time-dependent factor $\varphi$ affects the global in time prolongability of a local solution. In particular, the summability of the function $\varphi$ 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 $\varphi\not\in L^1([0,+\infty)))$ and satisfies a certain scaling condition, that we named uniform upper scaling condition.