Inhomogeneous NLS in negative Sobolev spaces via Fourier-Lebesgue and modulation spaces

Abstract

The purpose of this work is to provide a broader framework for analyzing the 1D mass-subcritical inhomogeneous nonlinear Schrödinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{α-1}u=0, \quad 1<α< 5-2b\; \text{and}\; 0< b\leq 1/4.\] in Sobolev spaces of negative order, reaching almost critical Sobolev spaces by working in Fourier-Lebesgue and modulation spaces. More precisely, we establish low-regularity well-posedness in the Fourier-Lebesgue ${\hat{L}}^p$ spaces for $4/3<p<8$. Primarily, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces, with generalized Strichartz estimates for $p<2$ and classical Strichartz estimates for $p>2$. Furthermore, we obtain analogous results for (INLS) in modulation spaces $M^{p,p^{\prime }}$ for $4/3<p<2$.

arXiv comment28 pages, 1 figure, Revised