  In this paper, we consider the Cauchy problem for semilinear classical wave
equations \begin{equation*} u_{tt}-\Delta u=|u|^{p_S(n)}\mu(|u|)
\end{equation*} with the Strauss exponent $p_S(n)$ and a modulus of continuity
$\mu=\mu(\tau)$, which provides an additional regularity of nonlinearities in
$u=0$ comparing with the power nonlinearity $|u|^{p_S(n)}$. We obtain a sharp
condition on $\mu$ as a threshold between global (in time) existence of small
data radial solutions by deriving polynomial-logarithmic type weighted
$L^{\infty}_tL^{\infty}_r$ estimates, and blow-up of solutions in finite time
even for small data by applying iteration methods with slicing procedure. These
results imply the critical regularity of source nonlinearities for semilinear
classical wave equations.
