Abstract In this paper, we consider the following nonlinear Schrödinger equation: iut+Δgu+ia(x)u−|u|p−1u=0(x,t)∈M×(0,+∞),u(x,0)=u0(x)x∈M, $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g) is a smooth complete compact Riemannian manifold of dimension n(n = 2, 3) without boundary. For the damping terms −a(x)(1 − Δ)−1a(x)ut and ia(x)(−Δ)12a(x)u, $\begin{array}{} \displaystyle ia(x)(-{\it\Delta})^{\frac12}a(x)u, \end{array}$ the exponential stability results of system (0.1) have been proved by Dehman et al. (Math Z 254(4): 729-749, 2006), Laurent. (SIAM J. Math. Anal. 42(2): 785-832, 2010) and Cavalcanti et al. (Math Phys 69(4): 100, 2018). However, from the physical point of view, it would be more important to consider the stability of system (0.1) with the damping term ia(x)u, which is still an open problem. In this paper, we obtain the exponential stability of system (0.1) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.

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在紧黎曼流形上的指数稳定性

摘要 在本文中，我们考虑以下非线性薛定谔方程： iut+Δgu+ia(x)u−|u|p−1u=0(x,t)∈M×(0,+∞),u(x, 0)=u0(x)x∈M, $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p- 1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{ M},\end{cases} \end{array}$$(0.1) 其中 (𝓜, g) 是一个无边界的维度为 n(n = 2, 3) 的光滑完全紧致黎曼流形。对于阻尼项 −a(x)(1 − Δ)−1a(x)ut 和 ia(x)(−Δ)12a(x)u, $\begin{array}{} \displaystyle ia(x)( -{\it\Delta})^{\frac12}a(x)u, \end{array}$ 系统 (0.1) 的指数稳定性结果已被 Dehman 等人证明。(Math Z 254(4): 729-749, 2006), Laurent。(SIAM J. Math. Anal. 42(2): 785-832, 2010) 和 Cavalcanti 等人。（数学物理 69(4): 100, 2018）。但是，从物理角度来看，使用阻尼项 ia(x)u 来考虑系统 (0.1) 的稳定性会更重要，这仍然是一个悬而未决的问题。在本文中，我们通过非欧几何中的 Morawetz 乘子和紧凑性-唯一性参数获得系统 (0.1) 的指数稳定性。