Sharp threshold of scattering versus non-scattering for some mass-sub-critical inhomogeneous NLS

TL;DR

This work addresses the long-time behavior of mass-subcritical inhomogeneous NLS and Hartree-type equations with singular inhomogeneity $|x|^{-b}$. By leveraging the Duhamel formulation, conservation laws, dispersive and Hardy-Littlewood-Sobolev estimates, and virial-type identities, the authors establish a sharp $L^2$ threshold that separates non-scattering from scattering in both models. Specifically, for the NLS with $0<b<1$, no scattering occurs when $1<q<1+\frac{2-2b}{N}$, while scattering occurs for $1+\frac{2-2b}{N}\le q<1+\frac{2(2-b)}{N}$ (data in $\Sigma$); the Hartree equation exhibits an analogous dichotomy with thresholds depending on $(b,\alpha,q)$. In both cases, the results furnish a complete dichotomy in the inhomogeneous regime, clarifying the asymptotic behavior of finite-mass data and filling a gap in the literature on inhomogeneous NLS dynamics.

Abstract

This work investigates the long time asymptotic behavior of some inhomogeneous non-linear Schrödinger type equations. We give sharp a threshold of scattering versus non-scattering of mass solutions, depending on the source term. This work complements the literature to the inhomogeneous regime.

Theorems & Definitions (1)

• proof : proof of the claim