We consider the Cauchy problem for the dissipative nonlinear Schrodinger equation with a cubic nonlinear term $$\lambda |u|^2u$$, where $$\lambda \in {\mathbb {C}}$$ with Im $$\lambda < 0$$. We prove the global existence of a unique solution and obtain the uniform estimate in the Gevrey class. Using the uniform regularity estimate, we show the $$L^2$$-decay rate for the solution which has the Gevrey regularity.

中文翻译：

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Gevrey 类中耗散非线性薛定谔方程的 $$L^2$$-衰减估计

我们考虑具有三次非线性项 $$\lambda |u|^2u$$ 的耗散非线性薛定谔方程的柯西问题，其中 $$\lambda \in {\mathbb {C}}$$ with Im $$\lambda < 0$$。我们证明了唯一解的全局存在性，并在 Gevrey 类中获得了统一估计。使用均匀正则性估计，我们显示了具有 Gevrey 正则性的解决方案的 $$L^2$$-衰减率。