  In this paper, we consider blow-up behavior of weak solutions to a weakly
coupled system for a semilinear damped wave equation and a semilinear wave
equation in $\mathbb{R}^n$. This problem is part of the so-called Nakao's
problem proposed by Professor Mitsuhiro Nakao (Kyushu University) for a
critical relation between the exponents $p$ and $q$. By applying an iteration
method for unbounded multipliers with a slicing procedure, we prove blow-up of
weak solutions for Nakao's problem even for small data. We improve the blow-up
result and upper bound estimates for lifespan comparing with the previous
research, especially, in higher dimensional cases.
