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 $n\le 5$. We show this equation, with power $2\le p\le 1+4/(n-1)$,is(strongly)ill-posedin$H^{s}$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 $L_t^{4\mathrm{/}(n-1)}{L}_x^{\mathrm{\infty }}$ Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.