Concerning ill-posedness for semilinear wave equations

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.