A Time-Oscillating Hartree-Type Schrödinger Equation

Abstract

We consider the time-oscillating Hartree-type Schrödinger equation ${iu}_t+\Delta u+\theta \left(\omega t\right)\left({\left|x\right|}^{-\gamma }*{\left|u\right|}^2\right)u=0$, where $\theta$ is a periodic function. For the mean value $I(\theta )$ of $\theta$, we show that the solution $u_{\omega }$ converges to the solution of ${iU}_t+\Delta U+I\left(\theta \right)\left({\left|x\right|}^{-\gamma }*{\left|U\right|}^2\right)U=0$ for their local well-posedness and global well-posedness.