  This work is devoted to the nonexistence of global-in-time energy solutions
of nonlinear wave equation of derivative type with weak time-dependent damping
in the scattering and scale invariant range. By introducing some multipliers to
absorb the damping term, we succeed in establishing the same upper bound of the
lifespan for the scattering damping as the non-damped case, which is a part of
so-called Glassey conjecture on nonlinear wave equations. We also study an
upper bound of the lifespan for the scale invariant damping with the same
method.
