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

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 $\text{d}{u}_{tt}-t^{2m}\mathrm{\Delta }u+\frac{\mu }{t}{u}_t=\mathrm{\mid }{u}_t{\mathrm{\mid }}^p,in{\mathrm{\Omega }}^c\times [1,\mathrm{\infty }),$ that we associate with appropriate small initial data.