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 TN → 0 and solution uN(t) such that uN(TN)→∞ in the Besov space (σ > 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(ℝ2). We systematically utilize the modulation space for one dimension and the scaled modulation space for two dimensions.
Original language | English |
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Pages (from-to) | 2613-2630 |
Number of pages | 18 |
Journal | Transactions of the American Mathematical Society |
Volume | 367 |
Issue number | 4 |
DOIs | |
Publication status | Published - 2015 Apr 1 |
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics