Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

• Main Authors: Lianbing She, Yangrong Li, Renhai Wang
• Format: Article
• Language: English
• Published: Hindawi Limited 2018-01-01
• Series: Discrete Dynamics in Nature and Society
• Online Access: http://dx.doi.org/10.1155/2018/2139792

This paper deals with pullback dynamics for the weakly damped Schrödinger equation with time-dependent forcing. An increasing, bounded, and pullback absorbing set is obtained if the forcing and its time-derivative are backward uniformly integrable. Also, we obtain the forward absorption, which is only used to deduce the backward compact-decay decomposition according to high and low frequencies. Based on a new existence theorem of a backward compact pullback attractor, we show that the nonautonomous Schrödinger equation has a pullback attractor which is compact in the past. The method of energy, high-low frequency decomposition, Sobolev embedding, and interpolation are quite involved in calculating a priori pullback or forward bound.