Remark on the ill-posedness for KdV-Burgers equation in Fourier amalgam spaces

TL;DR

This paper demonstrates ill-posedness of the KdV-Burgers equation in Fourier amalgam spaces for certain parameters, extending previous results and covering new function space cases.

Contribution

It establishes ill-posedness of the KdV-Burgers equation in Fourier amalgam spaces with s<-1, including new cases like Fourier Lebesgue and modulation spaces.

Findings

Ill-posedness in Fourier amalgam spaces for s<-1

Extension of previous results to new function spaces

Ill-posedness also holds in Fourier Lebesgue and modulation spaces

Abstract

We have established (a weak form of) ill-posedness for the KdV-Burgers equation on a real line in Fourier amalgam spaces $\widehat{w}_s^{p,q}$ with $s<-1$. The particular case $p=q=2$ recovers the result of L. Molinet and F. Ribaud [Int. Math. Res. Not., (2002), pp. 1979-2005]. The result is new even in Fourier Lebesgue space $\mathcal{F}L_s^q$ which corresponds to the case $p=q(\neq 2)$ and in modulation space $M_s^{2,q}$ which corresponds to the case $p=2,q\neq 2$.