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

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