Two singular dynamics of the nonlinear Schrödinger equation on a plane domain

Abstract

We study the cubic, focusing nonlinear Schrodinger equation (NLS) posed on a bounded domain of $ \mathbb{R}^{2} $ with Dirichlet boundary conditions. We describe two types of nonlinear evolutions. First we obtain solutions which blow up with a minimal L 2 norm in .nite time at a .xed point of the interior of the domain. The argument can be performed equally well for the cubic NLS posed on the .at torus $ \mathbb{T}^{2} $ . In the case when the domain is a disc, we also prove that the Cauchy problem is ill posed in the following sense: the .ow map is not uniformly continuous on bounded sets of the Sobolev space Hs, s> 1/3, contrary to what is known on the square (recall that the scale invariant Sobolev space for the cubic NLS in 2D is L 2).