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-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.

在本文中，我们研究了空间尺寸为\（n \ le 5 \）的导数半线性波动方程在\（H ^ {s} \）中局部适当摆放的最优正则性问题。我们表明该方程中，与功率\（2 \文件p \文件1 + 4 /（N-1）\） ，是（强）在病态\（H ^ {S} \）与\（S =（ n + 5）/ 4 \）。此外，当非线性为二次项时，我们根据正则性建立了非线性项结构的表征。作为副产品，我们提供了局部时间终结标度不变\（L_ {t} ^ {4 /（n-1）} L_ {x} ^ {\ infty} \）失败的替代证明。Strichartz估计。最后，作为一种应用，我们还证明了某些半线性半波方程的不适定结果。