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.
ASJC Scopus subject areas
- 数学 (全般)