New blow-up result for the weakly coupled wave equations with a scale-invariant damping and time derivative nonlinearity

We consider in this article the weakly coupled system of wave equations in the $\mathit{\text{scale-invariant}}\mathit{\text{case}}$ and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the delimitation of the blow-up region by obtaining a new candidate for the critical curve. More precisely, we enhance the results obtained in \cite{Palmieri} 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 the solution.