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

TL;DR

This paper shows that the blow-up of solutions in scale-invariant wave equations with damping and mass occurs regardless of the sign of the discriminant, challenging previous restrictions based on this parameter.

Contribution

It demonstrates that the sign of the discriminant is not essential for blow-up, extending the understanding of blow-up regions beyond classical restrictions.

Findings

Blow-up regions are invariant even when the discriminant is negative.

The critical exponent aligns with the Glassey-type in the shifted dimension.

Classical restrictions on the discriminant are due to technical limitations, not fundamental properties.

Abstract

This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\frac{\mu}{1+t} \partial_t u$, the mass term $\frac{\nu^2}{(1+t)^2} u$, and a time-derivative nonlinearity $| \partial_t u |^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $\delta = (\mu-1)^2 - 4\nu^2$ 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.