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

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 − ∆)a(x)ut and ia(x)(−∆) 1 2 a(x)u, 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) byMorawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.


Introduction . Notations
Suppose that (M, g) is a smooth complete compact Riemannian manifold of dimension n(n = , ) without boundary.
(1.1) Finally, we set div g , ∇g and ∆g as the divergence operator, the gradient operator and the Laplace−Beltrami operator of (M, g), respectively.

. Nonlinear Schrödinger equation
We consider the following system: where < p < +∞ for dimension n = , < p ≤ for dimension n = . And a(x) ∈ C (M) is a nonnegative real function.
When dimension n = , the exponential stability of the following system has been proved by [19] and the exponential stability of the following system iu t + ∆u − f (|u| )u + ia(x)(−∆) a(x)u = (x, t) ∈ M × ( , +∞), has been obtained in [13]. When dimension n = , the exponential stability of the following system: has been established in [24]. In fact, from the physical point of view, it would be more important to consider the damping term like ia(x)u. When dimension n = , asymptotic stability of the system (1.3) has been proved by [13]. It is said that the energy of system (1.3) goes to zero as time goes to in nity. However, the exact stability (especially the exponential stability) of system (1.3) is still an open problem. In this paper, under suitable geometric assumptions, we obtain the exponential stability of system (1.3) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.
Our paper is organized as follows. In Section 2, we will state our main results. Then, multiplier identities and key lemmas are presented in Section 3. Finally, we prove the exponential stability of the nonlinear Schrödinger equation in Section 4.

Main results
When n = , the well-posedness of system (1.3) has been proved by Theorem 1.4 in [13]. When n = , the unique global weak solution of system (1.3) has been given by [8] and the well-posedness of system (1.3) in Bourgain spaces has been proved by [3]. Throughout the paper, we assume that the system (1.
The followings are the main assumptions of this paper.

Assumption (A)
There exists a C vector eld H on M such that where δ > is a constant and Ω ⊂ M is an open set with smooth boundary. Moreover, a(x) satis es where a > is a constant.
Remark 2.1. The vector eld given by assumption (A) is called escape vector eld and it was introduced by Yao [33] for the controllability of the wave equation with variable coe cients, which is also a useful condition for the controllability and the stabilization of the quasilinear wave equation (see [17,34,36] [35] to nd out escape vector eld. The explicit expression of D r ∂ ∂r in Riemannian manifold (R n , g) is given by [25,26].

Assumption (B) (Unique continuation)
Let Ω ⊂ M be an open set with smooth boundary and ω ⊂ Ω be an open subset. Assume that ω satis es the geometric control condition: (GCC) There exists constant T > such that for any x ∈ Ω and any unit-speed geodesic γ(t) of (M, g) starting at x, there exists t < T such that γ(t) ⊂ ω. As a consequence, for every T > , the only solution in C([ , T], H (Ω)) to the system is the trivial one u ≡ , where b (x, t) and b (x, t) ∈ L ∞ ([ , T], L (Ω)). [24]. It can also be proved in particular cases by Carleman estimates in Euclidean space, see [22,23,31].

Remark 2.2. Let H = Dφ, where H is given by (2.2) and φ is a strictly convex function, then assumption (B) follows from Proposition B.3 in
(2.5)

Multiplier Identities and Key Lemmas
We need to establish several multiplier identities, which are useful for our problem.

Lemma 3.1. Suppose that u(x, t) solves the following equation:
Let H be a C vector eld de ned on M. Then Moreover, assume that the real function P ∈ C (M). Then Proof. Multiplying the Schrödinger equation in (3.1) by H(ū) and then integrating over M × ( , T), we deduce that and The equality (3.2) follows from Green's formula.
The following lemma shows the relationship between the metric g and the geometric control condition.
where δ > is a constant. Then, for any x ∈ Ω and any unit-speed geodesic γ(t) starting at x, if

Exponential stability of the nonlinear Schrödinger equation
From Lemma 3.2, the following lemma holds true. (4.1) The following lemmas follow from assumption (B). is the trivial one u ≡ .

Lemma 4.3. (Unique continuation) Let assumption (B) hold true. Let Ω ⊂ M be an open set with smooth boundary and ω ⊂ Ω be an open subset. Assume that ω satis es the geometric control condition:
(GCC) There exists constant T > such that for any x ∈ Ω and any unit-speed geodesic γ(t) of (M, g) starting at x, there exists t < T such that γ(t) ⊂ ω.
Therefore, for every T > , the only solution in C([ , T], H (Ω)) to the system is the trivial one u ≡ . It follows from assumption (B) that u ≡ .