Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

TL;DR

This paper establishes new local well-posedness results and ill-posedness thresholds for the inhomogeneous nonlinear Schrödinger equation, expanding understanding of solution behavior in Sobolev spaces across various parameters.

Contribution

It introduces an adapted fractional Leibniz rule to prove local well-posedness and analyzes the Duhamel operator to identify ill-posedness regimes for the inhomogeneous NLS.

Findings

New local well-posedness results in Sobolev spaces for the inhomogeneous NLS.

Ill-posedness results based on Duhamel operator analysis.

Extended the range of parameters where well-posedness holds.

Abstract

We consider the initial value problem associated to the inhomogeneous nonlinear Schr\"o\-din\-ger equation, \begin{equation} iu_t + \Delta u +\mu|x|^{-b}|u|^{\alpha}u=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N), \end{equation} with $\mu=\pm 1$, $b > 0$, $s\geq 0$ and $0 < \alpha \leq \frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator.