On nonlinear Schrödinger equations with almost periodic initial data

Abstract

We consider the Cauchy problem of nonlinear Schrödinger equations (NLS) with almost periodic functions as initial data. We first prove that, given a frequency set $\pmbω =\{ω_j\}_{j = 1}^\infty$, NLS is local well-posed in the algebra $\mathcal{A}_{\pmbω}(\mathbb R)$ of almost periodic functions with absolutely convergent Fourier series. Then, we prove a finite time blowup result for NLS with a nonlinearity $|u|^p$, $p \in 2\mathbb{N}$. This elementary argument presents the first instance of finite time blowup solutions to NLS with generic almost periodic initial data.

18 pages. References updated. To appear in SIAM J. Math. Anal