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

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