  In this paper we consider the blow-up for solutions to a weakly coupled
system of semilinear damped wave equations of derivative type in the scattering
case. After introducing suitable functionals proposed by Lai-Takamura for the
corresponding single semilinear equation, we employ Kato's lemma to derive the
blow-up result in the subcritical case. On the other hand, in the critical case
an iteration procedure based on the slicing method is employed. Let us point
out that we find as critical curve in the p-q plane for the pair of exponents
(p, q) in the nonlinear terms the same one as for the weakly coupled system of
semilinear not-damped wave equations with the same kind of nonlinearities.
