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.
Categories: math.AP