Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 16, pp. 1-17.

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

Authors: Adan J. Corcho  (Univ. de Cordoba, Spain)
         Lindolfo P. Mallqui (Univ. Federal do Rio de Janeiro, Brazil)
  
Abstract:
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.   

Submitted August 7, 2024. Published February 21, 2025.
Math Subject Classifications: 35Q55, 35Q35, 35Q60.
Key Words: Elliptic/hyperbolic cubic nonlinear Schrodinger equation; Cauchy problem; well-poseddness.