  In the present paper, we study small data blow-up of the semi-linear wave
equation with a scattering dissipation term and a time-dependent mass term from
the aspect of wave-like behavior. The Strauss type critical exponent is
determined and blow-up results are obtained to both sub-critical and critical
cases with corresponding upper bound lifespan estimates. For the sub-critical
case, our argument does not rely on the sign condition of dissipation and mass,
which gives the extension of the result in \cite{Lai-Sch-Taka18}. Moreover, we
show the blow-up result for the critical case which is a new result.
