Space-time fractional NLS equation

TL;DR

This paper establishes local well-posedness for a space-time fractional nonlinear Schrödinger equation with nonlocal effects, and demonstrates ill-posedness in the supercritical regime, advancing understanding of fractional PDEs.

Contribution

It proves local well-posedness for a novel space-time fractional NLS and analyzes ill-posedness in supercritical cases, using smoothing effects and handling nonlocal behavior.

Findings

Proved local well-posedness for the fractional NLS.

Identified ill-posedness in supercritical cases.

Developed a proof strategy exploiting smoothing effects.

Abstract

In this paper we prove local well-posedness of a space-time fractional generalization of the nonlinear Schr\"odinger equation with a power-type nonlinearity. The linear equation coincides with a model proposed by Naber, and displays a nonlocal behavior both in space and time which accounts for long-range interactions and a so-called memory effect. Because of a loss of derivatives produced by the latter and the lack of semigroup structure of the solution operator, we employ a strategy of proof based on exploiting some smoothing effect similar to that used by Kenig, Ponce and Vega for the KdV equation. Finally, we prove analytic ill-posedness of the data-to-solution map in the supercritical case.