  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.
