Abstract.
We study the cubic, focusing nonlinear Schrödinger 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).
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Submitted: February 2002, Final version: September 2002.
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Burq, N., Gérard, P. & Tzetkov, N. Two singular dynamics of the nonlinear Schrödinger equation on a plane domain. GAFA, Geom. funct. anal. 13, 1–19 (2003). https://doi.org/10.1007/s000390300000
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DOI: https://doi.org/10.1007/s000390300000