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 $\phi (t)|u{|}^p$. We are interested in describing how the nonnegative time-dependent factor $\phi$ affects the global in time prolongability of a local solution. In particular, the summability of the function $\phi$ 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 $\phi \notin L^1([0,+\infty )))$ and satisfies a certain scaling condition, that we named uniform upper scaling condition.