# A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

## TL;DR

This paper establishes blow-up results for semilinear wave equations with scale-invariant damping and mass, including derivative nonlinearities, extending classical results to more complex models with explicit integral representations.

## Contribution

It provides new blow-up criteria for both single and coupled wave models with scale-invariant damping and derivative nonlinearities, using explicit integral formulas and iterative methods.

## Key findings

- Blow-up occurs below a shifted Glassey exponent for single equations.
- Critical curves are identified for weakly coupled systems.
- Explicit integral representation formulas are employed in the analysis.

## Abstract

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.