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Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions


Authors: Tsukasa Iwabuchi and Takayoshi Ogawa
Journal: Trans. Amer. Math. Soc. 367 (2015), 2613-2630
MSC (2010): Primary 35Q55
DOI: https://doi.org/10.1090/S0002-9947-2014-06000-5
Published electronically: December 4, 2014
MathSciNet review: 3301875
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Abstract: We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time $ T_N\to 0$ and solution $ u_N(t)$ such that $ u_N(T_N)\to \infty $ in the Besov space $ B_{2,\sigma }^{-1}(\mathbb{R})$ ($ \sigma >2$) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space $ H^{-1}(\mathbb{R}^2)$. We systematically utilize the modulation space $ M_{2,1}^0$ for one dimension and the scaled modulation space $ (M_{2,1}^0)_N$ for two dimensions.


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Additional Information

Tsukasa Iwabuchi
Affiliation: Department of Mathematics, Chuo University, Kasuga, Bunkyoku, Tokyo, 112-8551 Japan

Takayoshi Ogawa
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

DOI: https://doi.org/10.1090/S0002-9947-2014-06000-5
Received by editor(s): May 24, 2012
Received by editor(s) in revised form: October 22, 2012
Published electronically: December 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.