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

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