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

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- by Tsukasa Iwabuchi and Takayoshi Ogawa PDF
- Trans. Amer. Math. Soc.
**367**(2015), 2613-2630 Request permission

## 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.## References

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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
- MR Author ID: 289654
- Received by editor(s): May 24, 2012
- Received by editor(s) in revised form: October 22, 2012
- Published electronically: December 4, 2014
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**367**(2015), 2613-2630 - MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/S0002-9947-2014-06000-5
- MathSciNet review: 3301875