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 iut+Δ2u+θ(ωt)|u|8/(n-4)u=0, where ω,t∈R, x∈Rn, and θ is a periodic function. We obtain the asymptotic property of the solution for the above equation as ω→∞ under some conditions.