Norm inflation with infinite loss of regularity at general initial data for nonlinear wave equations in Wiener amalgam and Fourier amalgam spaces

TL;DR

This paper demonstrates strong ill-posedness, specifically norm inflation with infinite loss of regularity, for nonlinear wave equations in various Wiener and Fourier amalgam spaces, extending and complementing previous results.

Contribution

It establishes sharp ill-posedness results in Wiener and Fourier amalgam spaces, including infinite regularity loss, for nonlinear wave equations on both Euclidean space and torus.

Findings

Proves norm inflation with infinite loss of regularity in Wiener and Fourier amalgam spaces.

Extends ill-posedness results to a broad class of initial data and spaces.

Complements and sharpens previous well-posedness and ill-posedness results.

Abstract

We study the strong ill-posedness (norm inflation with infinite loss of regularity) for the nonlinear wave equation at every initial data in Wiener amalgam and Fourier amalgam spaces with negative regularity. In particular these spaces contain Fourier-Lebesgue, Sobolev and some modulation spaces. The equations are posed on $\mathbb R^d$ and on torus $\mathbb T^d$ and involve a smooth power nonlinearity. Our results are sharp with respect to well-posedness results of B\'enyi and Okoudjou (2009) and Cordero and Nicola (2009) in the Wiener amalgam and modulation space cases. In particular, we also complement norm inflation result of Christ, Colliander and Tao (2003) and Forlano and Okamoto (2020) by establishing infinite loss of regularity in the aforesaid spaces.