The Property of the Solution about Cauchy Problem for Fourth-Order Schrödinger Equation with Critical Time-Oscillating Nonlinearity

Abstract

We study the property of the solution in Sobolev spaces for the Cauchy problem of the following fourth-order Schrödinger equation with critical time-oscillating nonlinearity $i{u}_t+{\mathrm{\Delta }}^{2}u+\theta (\omega t)|u{|}^{8/(n-4)}u=0$, where $\omega ,t\in R$, $x\in R^n$, and $\theta$ is a periodic function. We obtain the asymptotic property of the solution for the above equation as $\left|\omega \right|\to \infty$ under some conditions.