Improvement on the Blow-Up for the Weakly Coupled Wave Equations with Scale-Invariant Damping and Time Derivative Nonlinearity

We consider in this article the weakly coupled system of wave equations in the scale-invariant case and with time-derivative nonlinearities. Under the assumption of small initial data, we obtain a better characterization of the delimitation of the blow-up region by deriving a new candidate for the critical curve. More precisely, we enhance the results obtained in Palmieri and Tu (Calc Var 60:72, 2021, https://doi.org/10.1007/s00526-021-01948-0) for the system under consideration in the present work. We believe that our result is optimal in the sense that beyond the blow-up region obtained here we may conjecture the global existence of small data solutions.