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The article is devoted to investigating the initial boundary value problemfor the damped wave equation in the scale-invariant case with time-dependentspeed of propagation on the exterior domain. By presenting suitable multipliersand applying the test-function technique, we study the blow-up and the lifespanof 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 initialdata.

Blow-up and lifespan estimate for the generalized tricomi equation with  the scale-invariant damping and time derivative nonlinearity on exterior  domain