Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation

Abstract

In this paper, we study the ill-posedness issue for the generalized improved Boussinesq equation. In particular we prove there is norm inflation with infinite loss of regularity at general initial data in $⟨\nabla {⟩}^{-s}(L^2\cap L^{\infty })(\mathbb{R})$ for any $s<0$. This result is sharp in the $L^2$-based Sobolev scale in view of the well-posedness in $L^2(\mathbb{R})\cap L^{\infty }(\mathbb{R})$. We also show that the same result applies to the multi-dimensional generalized improved Boussinesq equation. Finally, we extend our norm inflation result to Fourier-Lebesgue, modulation and Wiener amalgam spaces.

21 pages