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In this article, we consider the damped wave equation in the\textit{scale-invariant case} with time-dependent speed of propagation, massterm and time derivative nonlinearity. More precisely, we study the blow-up ofthe solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}\Deltau+\frac{\mu}{t}u_t+\frac{\nu^2}{t^2}u=|u_t|^p, \quad \mbox{in}\\mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data.Assuming some assumptions on the mass and damping coefficients, $\nu$ and$\mu>0$, respectively, that the blow-up region and the lifespan bound of thesolution of $(E)$ remain the same as the ones obtained for the case withoutmass, {\it i.e.} $\nu=0$ in $(E)$. The latter case constitutes, in fact, ashift of the dimension $N$ by $\frac{\mu}{1+m}$ compared to the problem withoutdamping and mass. Finally, we think that the new bound for $p$ is a seriouscandidate to the critical exponent which characterizes the threshold betweenthe blow-up and the global existence regions.

Nonexistence result for the generalized Tricomi equation with the scale-invariant damping, mass term and time derivative nonlinearity