A remark on norm inflation for nonlinear wave equations

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.

Keywords

nonlinear wave equation, ill-posedness, norm inflation

2010 Mathematics Subject Classification

35B30, 35L05

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J. F. was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. Both authors also acknowledge support from Tadahiro Oh’s ERC starting grant no. 637995 “ProbDynDispEq”. M.O. was supported by JSPS KAKENHI Grant numbers JP16K17624 and JP20K14342.

Received 2 June 2020

Published 16 November 2020