On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type \begin{document}$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $\end{document} in anisotropic Sobolev spaces, where \begin{document}$ 1\le \alpha \le 1 $\end{document}, \begin{document}$ \mathscr{H} $\end{document} is the Hilbert transform and \begin{document}$ D_{x}^{\alpha} $\end{document} is the fractional derivative, both with respect to \begin{document}$ x $\end{document}.