The focusing cubic NLS on exterior domains in three dimensions (Q2800168)

This paper considers the focusing, cubic nonlinear Schrödinger equation on the exterior of a convex smooth obstacle in \(\mathbb R^3\). The purpose of the paper is to study global existence and scattering questions for large data. When the Dirichlet boundary condition is enforced, the authors prove that the threshold for global well-posedness and scattering is the same as for the problem posed on the whole Euclidean space \(\mathbb R^3\). More precisely, let \(E\) (energy) and \(M\) (mass) are conservation laws for the NLS equation and \(Q\) denotes the unique, positive, spherically symmetric, decaying solution to \(\Delta Q-Q+Q^3=0\). The main result of this paper states that if the initial data \(u_0\) satisfy NEWLINE\[NEWLINEE(u_0)M(u_0)<E(Q)M(Q),\| \nabla u_0\|_2\| u_0\|_2<\| \nabla Q\|_2\| Q\|_2,NEWLINE\]NEWLINE then the corresponding solution to the associated initial-value problem with Dirichlet boundary conditions exists globally and scatters in both time directions.