Abstract In this paper, we study the defocusing nonlinear Schrödinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L2-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively. The distinctive feature of the paper is the monotonicity approach, which makes the analysis independent from the commonly used Strichartz estimates and allows us to work without artificial smoothing terms inserted into the main equation. We in addition implement a precise and efficient algorithm for studying the exponential decay established in the first part of the paper numerically. Our simulations illustrate the efficacy of the proposed control design.

中文翻译：

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具有局部分布阻尼的非线性薛定谔方程的指数稳定性

摘要 在本文中，我们研究了在光滑有界域以及整个空间和外部域上具有局部分布阻尼的散焦非线性薛定谔方程。我们首先使用单调算子理论构造近似解。我们表明，通过使用乘法器技术和独特的连续性，近似解在 L2 意义上以指数方式快速衰减。然后，我们分别通过传递到极限和使用弱下半连续性论证来证明原始模型的全局存在性和 L2 衰减。该论文的显着特点是单调性方法，它使分析独立于常用的 Strichartz 估计，并允许我们在不将人工平滑项插入主方程的情况下工作。此外，我们还实现了一种精确有效的算法，用于数值研究论文第一部分中建立的指数衰减。我们的模拟说明了所提出的控制设计的有效性。