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

Abstract

We study the dynamics of the focusing 3d cubic nonlinear Schrödinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when EΩ[u0]MΩ[u0]=ER3[Q]MR3[Q] and ‖∇u0‖L2(Ω)‖u0‖L2(Ω)<‖∇Q‖L2(R3)‖Q‖L2(R3), where u0∈H01(Ω) 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.