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

Abstract

In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in ℝ3 with 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 Schrödinger 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.