Time decay for nonlinear dissipative Schrödinger equations in optical fields (Q1796543)

Summary: We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearity \(\lambda \left|u\right|^{p - 1} u\) of order \(p \left(n\right) < p \leq 1 + 2 / n\) for arbitrarily large initial data, where the lower bound \(p \left(n\right)\) is a positive root of \(\left(n + 2\right) p^2 - 6 p - n = 0\) for \(n \geq 2\) and \(p \left(1\right) = 1 + \sqrt{2}\) for \(n = 1 \). Our purpose is to extend the previous results for higher space dimensions concerning \(\mathbf{L}^2\)-time decay and to improve the lower bound of \(p\) under the same dissipative condition on \(\lambda \in \mathbb{C}\): \(\operatorname{Im} \; \lambda < 0\) and \(\left|\operatorname{Im} \; \lambda\right| > \left(\left(p - 1\right) / 2 \sqrt{p}\right) \left|\operatorname{R} \operatorname{e} \; \lambda\right|\) as in the previous works.