Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle

被引:8
作者
Duyckaerts, Thomas [1 ,2 ]
Landoulsi, Oussama [1 ]
Roudenko, Svetlana [3 ]
机构
[1] Univ Sorbonne Paris Nord, Inst Galilee, LAGA, UMR 7539, Villetaneuse, France
[2] Inst Univ France, Villetaneuse, France
[3] Florida Int Univ, Dept Math & Stat, Miami, FL 33199 USA
基金
美国国家科学基金会;
关键词
Focusing NLS equation; Exterior domain; Global existence and scattering; GLOBAL WELL-POSEDNESS; SCHRODINGER-EQUATION; BLOW-UP; EXTERIOR; SCATTERING; COMPACTNESS; EXISTENCE; DECAY;
D O I
10.1016/j.jfa.2021.109326
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We study the dynamics of the focusing 3d cubic nonlinear Schrodinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when E-Omega[u(0)]M-Omega[u(0)] = E-R3 [Q]M-R3 [Q] and ||del u(0)||(L2(Omega))||u(0)||(L2(Omega)) < ||del Q||(L2(R3))||Q||(L2(R3)), where u0 is an element of H-0(1)(Omega) is the initial data, Q is the ground state on the Euclidean space, E is the energy and M is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on the energy-critical problem) have proved the existence of a specific global solution that scatters for negative times and converges to the soliton in positive times. We prove that these heteroclinic orbits do not exist for the problem in the exterior domain and that all solutions at the threshold are globally defined and scatter. This is the first step in the study of the global dynamics of the equation above the ground-state threshold. The main difficulty is to control the position of the center of mass of the solution for large time without the momentum conservation law and the Galilean transformation which are not available for this equation. (c) 2021 Published by Elsevier Inc.
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页数:55
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