  In the present manuscript, we determine the critical condition for the
nonlinearity in a semilinear wave equation with a derivative-type nonlinearity.
More precisely, we consider a nonlinear term depending on the time derivative
of the solution, which is the product of a power nonlinearity with critical
Glassey exponent and a modulus of continuity. By employing Zhou's approach
along a certain characteristic line, we prove the blow-up in finite time for
classical solutions (under a suitable sign condition for the Cauchy data) and
we derive upper bound estimates for the lifespan for a not Dini continuous
modulus of continuity. Furthermore, in the 3-dimensional and radially symmetric
case, by using weighted $L^{\infty}$ estimates, we establish the global
existence of small data solutions for a Dini continuous modulus of continuity,
and lower bound estimates for the lifespan in the not Dini continuous case.
These results provide the regularity threshold (i.e. the Dini condition) for
the modulus of continuity in the nonlinearity.
