Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearity λ\u\p - 1 u of order p n < p ≤ 1 + 2 / n for arbitrarily large initial data, where the lower bound p n is a positive root of n + 2 p 2 - 6 p - n = 0 for n ≥ 2 and p 1 = 1 + 2 for n = 1. Our purpose is to extend the previous results for higher space dimensions concerning L 2 -time decay and to improve the lower bound of p under the same dissipative condition on λ € C: Im λ < 0 and Im /Im λ > p - 1 / 2 p R e λ| as in the previous works.