Ill-posedness in the critical Sobolev space for the Fokas-Olver-Rosenau-Qiao equation

Abstract

This article proves norm inflation in the critical Sobolev space $H^{5/2}(\mathbb{R})$ for the Fokas-Olver-Rosenau-Qiao equation, which is a modified Camassa-Holm-type equation with cubic nonlinearity. This result complements the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^s(\mathbb{R})$ for $s>5/2$. The proof relies on the construction of explicit initial data satisfying the previously known blow-up criteria for the Fokas-Olver-Rosenau-Qiao equation, a step that appears to be of independent interest.