A remark on norm inflation for nonlinear wave equations

TL;DR

This paper demonstrates that nonlinear wave equations exhibit norm inflation and ill-posedness in negative Sobolev and related spaces, extending previous results to more general nonlinearities and spaces.

Contribution

It extends the understanding of ill-posedness for NLW by covering more nonlinearities and function spaces, filling gaps in prior research.

Findings

Norm inflation occurs at every initial data in negative Sobolev spaces.

Ill-posedness is established in Fourier-Lebesgue and Fourier-amalgam spaces.

Results include cases above the scaling critical regularity.

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 Fourier-amalgam spaces.