L^2 Solutions for cubic NLS equation with higher order fractional elliptic/hyperbolic operators on R cross T and  R^2

In this work, we consider the Cauchy problem for the cubic Schrodinger equation posed on cylinder \(\mathbb{R}\times\mathbb{T}\) with fractional derivatives \((-\partial_y^2)^{\alpha}\), \(\alpha >0\), in the periodic direction. The spatial operator includes elliptic and hyperbolic regimes. We prove \(L^2\) global well-posedness results when \(\alpha \ge 1\) by proving a \(L^4\)-\(L^2\) Strichartz inequality for the linear equation, following the ideas in [19], where it was considered the elliptical case with \(\alpha=1\). Further, these results remain valid on the Euclidean environment \(\mathbb{R}^2\), so well-posedness in \(L^2\) are also achieved in this case. Our proof in the elliptic (hyperbolic) case does not work in the small directional dispersion case \(0<\alpha <1\) (\(0<\alpha \leq 1\)), respectively. For more information see https://ejde.math.txstate.edu/Volumes/2025/16/abstr.html