Scattering for 3d cubic focusing NLS on the domain outside a convex obstacle revisited

Abstract

In this article, we consider the focusing cubic nonlinear Schr\"odinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, 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 the solution decays at a large time interval, which meets the scattering criterions.