The defocusing energy-supercritical inhomogeneous NLS in four space dimension

In this paper, we investigate the global well-posedness and scattering theory for the defocusing energy supcritical inhomogeneous nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u$ 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, $u\in L_t^{\mathrm{\infty }}(I;{\dot{H}}_x^{s_c}({\mathbb{R}}^4))$, 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.