On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction

This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\frac{\mu }{1+t}{\mathrm{\partial }}_{t}u$, the mass term $\frac{{\nu }^2}{(1+t{)}^2}u$, and a time-derivative nonlinearity $\mathrm{\mid }{\mathrm{\partial }}_{t}u{\mathrm{\mid }}^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $\delta =(\mu -1{)}^2-4{\nu }^2$ is not a structural prerequisite for determining the blow-up range. Indeed, we show that even in the regime $\delta <0$, the blow-up region remains invariant and is uniquely determined by the shifted dimension $n+\mu$, aligning with the Glassey-type critical exponent. Our result suggest that the classical restriction $\delta \ge 0$ is due to a technical tool rather than an intrinsic feature of the blow-up mechanism.