Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold

Abstract In this paper, we consider the following nonlinear Schrödinger equation: i u t + Δ g u + i a ( x ) u − | u | p − 1 u = 0 ( x , t ) ∈ M × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( 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 ) −1 a( x ) u t and i a ( x ) ( − Δ ) 1 2 a ( 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.