Concerning ill-posedness for semilinear wave equations

Abstract

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.