Scattering for 3D Cubic Focusing NLS on the Domain Outside a Convex Obstacle Revisited  

作　　者：Cheng Bin XU Teng Fei ZHAO Ji Qiang ZHENG 

机构地区：[1]The Graduate School of China Academy of Engineering Physics,Beijing 100088,P.R.China [2]School of Mathematics and Physics,University of Science and Technology Beijing,Beijing 100083,P.R.China [3]Institute of Applied Physics and Computational Mathematics,P.O.Box 8009,Beijing 100088,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2022年第6期1054-1068,共15页数学学报（英文版）

基　　金：Supported by Beijing Natural Science Foundation 1222019,PFCAEP project(Grant No.YZJJLX201901);NSFC(Grant No.11901041,12101040);NSAF(Grant No.U1530401)。

摘　　要：In this article,we consider the focusing cubic nonlinear Schr?dinger equation(NLS)in the exterior domain outside of a convex obstacle in R3with Dirichlet boundary conditions.We revisit the scattering result below ground state in Killip-Visan-Zhang[The focusing cubic NLS on exterior domains in three dimensions.Appl.Math.Res.Express.AMRX,1,146-180(2016)]by utilizing the method of Dodson and Murphy[A new proof of scattering below the ground state for the 3d radial focusing cubic NLS.Proc.Amer.Math.Soc.,145,4859-4867(2017)]and the dispersive estimate in Ivanovici and Lebeau[Dispersion for the wave and the Schr?dinger equations outside strictly convex obstacles and counterexamples.Comp.Rend.Math.,355,774-779(2017)],which avoids using the concentration compactness.We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary.Based on this effect and the interaction Morawetz estimates,we prove that the solution decays at a large time interval,which meets the scattering criterion.

关 键 词：Schrodinger equation exterior domain global well-posedness scattering criterion 

分 类 号：O175[理学—数学]