The defocusing energy-supercritical inhomogeneous NLS in four space dimension
Xuan Liu, Chengbin Xu
TL;DR
This work proves global well-posedness and scattering for the defocusing inhomogeneous NLS in four spatial dimensions under a priori control of the critical Sobolev norm $u\in L_t^\infty\dot H^{s_c}_x$, with $s_c=2-\frac{2-b}{\alpha}\in(1,2)$ and $0<b<\min\{(s_c-1)^2+1,3-s_c\}$. The authors adapt the Kenig–Merle concentration-compactness/rigidity framework to the energy-supercritical INLS, augmented by long-time Strichartz estimates and both spatially and frequency-localized Morawetz inequalities, to preclude almost periodic blow-up scenarios. They treat separately the case $1<s_c<3/2$ via a space-localized Morawetz argument, and the range $\tfrac{3}{2}\le s_c<2$ via long-time Strichartz bounds plus a dichotomy between rapid frequency cascade and quasi-soliton, ruling out both. A notable advance is the removal of radial symmetry for $s_c\ge\tfrac{3}{2}$, enabled by the decay $|x|^{-b}$ in the nonlinearity, and a careful stability analysis that hinges on the condition $b<(s_c-1)^2+1$.
Abstract
In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schrödinger equation $iu_t + Δu =|x|^{-b} |u|^αu$ in four space dimension, where $s_c := 2- \frac{2-b}α \in (1, 2)$ 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.