  The article is devoted to investigating the initial boundary value problem
for the damped wave equation in the scale-invariant case with time-dependent
speed of propagation on the exterior domain. By presenting suitable multipliers
and applying the test-function technique, we study the blow-up and the lifespan
of the solutions to the problem with derivative-type nonlinearity
  $ \d u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t=|u_t|^p, \quad \mbox{in}\
\Omega^{c}\times[1,\infty),$ that we associate with appropriate small initial
data.
