Ill-posedness in the critical Sobolev space for the hyperelastic rod equation

This article is the third in a series investigating ill-posedness in critical Sobolev spaces for Camassa-Holm-type equations. Following results for the $b$-Novikov and Fokas-Olver-Rosenau-Qiao equations, we study the hyperelastic rod equation at the critical Sobolev regularity $H^{3\mathrm{/}2}(\mathbb{R})$. We prove ill-posedness at this endpoint, complementing the local well-posedness theory in $H^s(\mathbb{R})$ for $s>3\mathrm{/}2$ and the recently established norm-inflation result for $1<s<3\mathrm{/}2$. The nature of the critical ill-posedness depends on the parameter $\gamma$. For $0<\gamma \le 3$, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For $\gamma <0$ or $\gamma >3$, we construct solutions with arbitrarily small $H^{3\mathrm{/}2}$ initial data whose $H^{3\mathrm{/}2}$ norm becomes arbitrarily large in arbitrarily short time.