In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network \(\mathcal {R}\), where the damping is localized on one branch at the infinity and the initial data are assumed to be in \(L^{2}(\mathcal {R})\). We use the fixed point argument and Strichartz estimates on a star-shaped network to obtain results of local and global well-posedness. The proof of the exponential decay is based on smoothing properties for Schrödinger equation, on the unique continuation and on the semigroup properties.

中文翻译：

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星形网络上非线性薛定ding方程的指数稳定性

本文证明了星型网络\（\ mathcal {R} \）上非线性耗散薛定ding方程解的指数稳定性，其中阻尼位于无穷大的一个分支上，初始数据为假定位于\（L ^ {2}（\ mathcal {R}）\）中。我们在星形网络上使用定点参数和Strichartz估计以获得局部和全局适定性的结果。指数衰减的证明是基于Schrödinger方程的平滑特性，唯一连续性和半群特性。