The defocusing energy-supercritical inhomogeneous NLS in four space dimension

TL;DR

This paper proves that solutions to a defocusing energy-supercritical inhomogeneous nonlinear Schrödinger equation in four dimensions are global and scatter if they are bounded in the critical Sobolev space, using concentration-compactness and Morawetz estimates.

Contribution

It establishes global well-posedness and scattering for the inhomogeneous NLS in four dimensions under critical Sobolev bounds, extending previous results to energy-supercritical cases.

Findings

Solutions are global if bounded in the critical Sobolev space.

Solutions scatter under the given conditions.

The proof combines concentration-compactness with Morawetz estimates.

Abstract

In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schr\"odinger equation $iu_t + \Delta u =|x|^{-b} |u|^\alpha u$ in four space dimension, where $s_c := 2- \frac{2-b}{\alpha} \in (1, 2)$ and $0<b<\min \{ (s_c-1)^2+1,3-s_c\}$. We prove that if the solution has a prior bound in the critical Sobolev space, that is, $u \in L_t^\infty(I; \dot{H}_x^{s_c}(\mathbb{R}^4))$, then $u$ is global and scatters. The proof of the main results is based on the concentration-compactness/rigidity framework developed by Kenig and Merle [Invent. Math. 166 (2006)], together with a long-time Strichartz estimate, a spatially localized Morawetz estimate, and a frequency-localized Morawetz estimate.