  In this note, we study the ill-posedness of nonlinear wave equations (NLW).
Namely, we show that NLW experiences norm inflation at every initial data in
negative Sobolev spaces. This result covers a gap left open in a paper of
Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and
the second author (2019) to non-cubic integer nonlinearities. In particular,
for some low dimensional cases, we obtain norm inflation above the scaling
critical regularity. We also prove ill-posedness for NLW, via norm inflation at
general initial data, in negative regularity Fourier-Lebesgue and
Fourier-amalgam spaces.
