  In the present paper, we investigate blow-up and lifespan estimates for a
class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with
$n\geqslant 1$, which is part of the so-called Nakao's type problem weakly
coupled a semilinear damped wave equation with a semilinear wave equation with
nonlinearities of derivative type. By constructing two time-dependent
functionals and employing an iteration method for unbounded multiplier with
slicing procedure, the results of blow-up and upper bound estimates for the
lifespan of energy solutions are derived. The model seems to be hyperbolic-like
instead of parabolic-like. Particularly, the blow-up result for one dimensional
case is optimal.
