  In this article, we investigate a class of improved modified Boussinesq
equations, for which we provide first an alternate proof of local
well-posedness in the space $(H^s\cap L^\infty)\times (H^s\cap
L^\infty)(\mathbb{R})$ ($s\geq 0$) to the one obtained by Constantin and
Molinet. Secondly, we show that the associated flow map is not smooth when
considered from $H^s\times H^s(\mathbb{R})$ into $H^s(\mathbb{R})$ for $s<0$,
thus providing a threshold for the regularity needed to perform a Picard
iteration for these equations.
