  In this paper we consider the blow-up of solutions to a weakly coupled system
of semilinear damped wave equations in the scattering case with nonlinearities
of mixed type, namely, in one equation a power nonlinearity and in the other a
semilinear term of derivative type. The proof of the blow-up results is based
on an iteration argument. As expected, due to the assumptions on the
coefficients of the damping terms, we find as critical curve in the p-q plane
for the pair of exponents (p,q) in the nonlinear terms the same one found by
Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled
system of semilinear wave equations with the same kind of nonlinearities. In
the critical and not-damped case we provide a different approach from the test
function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the
solution on the critical curve, improving in some cases the upper bound
estimate for the lifespan. More precisely, we combine an iteration argument
with the so-called slicing method to show the blow-up dynamic of a weighted
version of the functionals used in the subcritical case.
