A note on the well-posedness in the energy space for the generalized ZK equation posed on \(\mathbb {R}\times \mathbb {T}\)

In this note, we prove the local well-posedness in the energy space of the k-generalized Zakharov–Kuznetsov equation posed on \( \mathbb {R}\times \mathbb {T}\) $\mathbb{R}\times \mathbb{T}$ for any power non-linearity \( k\ge 2\) $k\ge 2$ . Moreover, we obtain global solutions under a precise smallness assumption on the initial data by proving a sharp Gagliardo Nirenberg type inequality.