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An improvement of [18] on the blow-up region and the lifespan estimate of aweakly coupled system of wave equations with damping and mass in thescale-invariant case and with time-derivative nonlinearity is obtained in thisarticle. Indeed, thanks to a better understanding of the dynamics of thesolutions, we give here a better characterization of the blow-up region.Furthermore, the techniques used in this article may be extended to othersystems and interestingly they simplify the proof of the blow-up result in [3]which is concerned with the single wave equation in the same context as in thepresent work.

Improvement on the blow-up for a weakly coupled wave equations with scale-invariant damping and mass and time derivative nonlinearity