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

Iwabuchi, Tsukasa; Ogawa, Takayoshi

doi:10.1090/S0002-9947-2014-06000-5

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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