The focusing cubic NLS on exterior domains in three dimensions

Abstract

We consider the focusing cubic NLS in the exterior $\Omega$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\Vert \nabla u_0{\Vert }_2\Vert u_0{\Vert }_2<\Vert \nabla Q{\Vert }_2\Vert Q{\Vert }_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubic NLS in ${\mathbb{R}}^3$.

References in corpus (2)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Quintic NLS in the exterior of a strictly convex obstacle