A remark on norm inflation for nonlinear wave equations

Abstract

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 Fourieramalgam spaces.

Topics & Concepts