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 \textit{scale-invariant 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.