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. 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.

在本说明中，我们证明了在比例不变情况下具有阻尼和质量且具有微分类型非线性项的半线性波动模型的爆破结果。我们考虑单个方程和弱耦合系统。在第一种情况下，对于低于Glassey指数偏移的指数，我们会得到爆破结果。对于弱耦合系统，我们发现具有相同非线性的半线性波动方程的弱耦合系统的相应曲线的位移作为临界曲线。我们的方法遵循Zhou的相应经典波动方程式。特别地，在爆破参数中采用了通过使用Yagdjian的积分变换方法导出的线性比例尺不变波动方程的显式积分表示公式。