The $H^1$-subcritical and critical NLS with time-oscillating coefficients

We consider the $H^1$--subcritical and critical nonlinear Schr"odinger equation $$ i \partial_{t} u_\omega + \gamma (\omega t) \Delta u_\omega + \theta (\omega t) | u_\omega|^{\alpha} u_\omega = 0 $$ in $\mathbb R^N$. Here, the coefficients $\gamma$ and $\theta\in C^1(\mathbb R)$ are periodic functions and $\gamma$ is further assumed to be one sign, bounded, and bounded away from zero. We prove for a given initial data in $H^1(\mathbb R^N)$, the solution $u_\omega$ converges to the solution of the limiting equation as $|\omega|$ goes to $\infty$. Furthermore, we show that if the limiting solution is global and has a certain decay property, then $u_\omega$ is also global for $|\omega|$ sufficiently large. This work improves our previous result [7]. The method is inspired by Fang and Han [10] which considered NLS with time--oscillating nonlinearity.