  In this paper, we investigate the problem of optimal regularity for
derivative semilinear wave equations to be locally well-posed in $H^{s}$ with
spatial dimension $n \leq 5$. We show this equation, with power $2\le p\le
1+4/(n-1)$, is (strongly) ill-posed in $H^{s}$ with $s = (n+5)/4$ in general.
Moreover, when the nonlinearity is quadratic we establish a characterization of
the structure of nonlinear terms in terms of the regularity. As a byproduct, we
give an alternative proof of the failure of the local in time endpoint
scale-invariant $L_{t}^{4/(n-1)}L_{x}^{\infty}$ Strichartz estimates. Finally,
as an application, we also prove ill-posed results for some semilinear half
wave equations.
