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 <= 5. We show this equation, with power 2 <= p <= 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 (Lt4/(n-1)Lx infinity) Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.