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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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