EXPONENTIAL STABILITY FOR THE LOCALLY DAMPED DEFOCUSING SCHR¨ODINGER EQUATION ON COMPACT MANIFOLD

. In this paper we study the asymptotic dynamics for semilinear defocusing Schr¨odinger equation subject to a damping locally distributed on a n-dimentional compact Riemannian manifold M n without boundary. The proofs are based on a result of unique continuation property, in the construction of a function f whose Hessian is positive deﬁnite and ∆ f = C 0 in some region contained in M and about the smoothing eﬀect due to Aloui adapted to the present context.


1.
Introduction. In general, if one consider the Schrödinger equation posed on a compact Riemannian manifold without boundary we get a reasonable number of contributors regarding well-posedness results. In fact, Burq, Gérard and Tzvetkov in [8] (see also [9,10]) established Strichartz estimates and consequently the solvability for a general class of nonlinearities related to the following Schrödinger equation, where P : R + → R is a polynomial. Those estimates allowed them to establish global existence results for regular as well as mild solutions of the nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities and, moreover, on three-manifolds in case of defocusing nonlinearities. Similar considerations can be found also in Dehman, Gérard and Lebeau [17] (see Proposition 7). It is worth quoting other papers in connection with compact manifolds: [3,9,18]. Thomann in [32], has considered supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (M, g) of dimension d ≥ 3, given by, in M, and established results of ill-posedness in energy norms. Here, the metric g is analytic and with either ω = −1 (defocusing equation) or ω = 1 (focusing equation).
Using an analytic WKB method, it was possible to construct an Ansatz for the semiclassical equation for times independent of the small parameter. In Euclidean context, we have a large number of results which treat about ill-posedness phenomena for equation above, specially when the critical power is considered, see for instance, [27,26].
Regarding results of stabilization and exact controllability for the Schrödinger equation posed on a compact Riemannian manifold without boundary we do not have a large numbers of contributors in the current literature. However, we can mention a nice work due to Dehman, Gérard and Lebeau in [17]. These authors established the stability and exact control in H 1 − norm for the following equation, where M is 2−dimensional and P is a polynomial function with real coefficients, satisfying P (0) = 0 and the following defocusing assumption, P (r) → +∞, as r → +∞.
The approach is based on a result of propagation of singularities and recent dispersion estimates (Strichartz type inequalities) due to Burq, Gérard and Tzvetkov in [8].
Next, when the Euclidean setting is considered, we have a recent work due to Cavalcanti, Domingos Cavalcanti, Fukuoka and Natali in [13] which establishes the exponential stability associated to equation iu t + ∆u − |u| 2 u + ia(x)u = 0, in R 2 × (0, +∞).
By using classical results of well-posedness on H 3 −norm in bounded domains, the authors proved a result of unique continuation for the well-known Schrödinger equation with defocusing cubic nonlinearity and consequently the exponential stability in L 2 -norm. In addition, Bortot and Corrêa [6] also establish the exponential decay in level L 2 −norm to the similar problem in a Euclidean domain.
This paper is concerned the study of the unique continuation property associated with the defocusing semilinear Schrödinger equation where (M, g) is a compact, connected, orientable Riemannian manifold without boundary. The ∆ denotes the Laplace-Beltrami operator associated with Riemannian metric g. Even more, we get exponential stability of the energy for the strong locally distributed dissipation problem The damping term i a(x)(1−∆) 1/2 a(x) y is specially fitted to the use the smoothing effect given in Aloui [1,2].
We consider the standard assumptions: (H1) Assumptions about the function a : M → R: (i) a is nonnegative function and a ∈ C ∞ (M ); (ii) a(x) ≥ a 0 ≥ 0 on M * , where M * contains properly the complementary part V strategically chosen in M (see comments before figure 1). (H2) Assumptions about the function h : R + → R: The assumption (H2) are inspired by Brézis (see [7], Theorem 1, page 31). Some examples of functions that satisfies (H2) can be found in [6].
The main goal of the present paper is a unique continuation property to mild solutions of the problem (1) in the class y ∈ L ∞ (0, T ; L 2 (M )).
In addition, we obtain exponential decay rates of the energy in L 2 -norm, that is, there exist positive constants C, γ such that , for all regular solutions to problem (2) provided that the initial data y 0 is taken in bounded sets of L 2 (Ω).
In order to obtain the exponential stabilization, we need strongly a function f , a subset V of M and a constant C > 0 such that We need ∆f ≡ C 0 > 0 in V and meas(V) > meas(M ) − . The dissipative effect is considered effective in the complementary of V (see Figure 1).
Recently Cavalcanti et al. [16] obtain relevant results about this our problem on surfaces. In what follows, we would like the relevance of this paper compared to Cavalcanti. Indeed, in [16], the authors obtains the exponential employing assumptions and results patterned from Dehman, Gérard and Lebeau [17]. On the other hand, this paper presents the proof of its own unique continuation result. Although this proof comes from known multiplier techniques, to this end, we need of the tools from Riemannian geometry to construct the function f given in (3). The main difference between [16] and the present article is the unique continuation property, here has proved the property based on contruction of function f satisfying the property (3), while which [16] assume unique continuation property (due Dehman, Gérard and Lebeau [17]).
There is some relevant works in connection with the subject of the present paper, see [5,15,21,22,23,25,30,33]. In this paper, we will focus on the energy decay rates of the problem (2) in light of [6]. The main features of our work are summarized as follows.
1. We prove the existence of global unique solution for the semilinear defocusing Schrödinger equation on compact manifold. In [6] was investigated on a bounded domain. 2. For the problem (1) we can get unique continuation result in two steps regular and mild solutions. From Riemaniann geometry arguments and important results we construct the function f whose Hessian is positive definite to conclude the unique continuation. 3. The main result, is the proof of uniform decay rates using energy methods. Our paper is organized as follows. Section 2 is notations and some preliminaries. In Section 3 we present the corresponding results for the well-posedness of Problem (1) and (2). In Section 4 we establish a unique continuation property in connection with problem (1). In Section 5 we present the construction of f satisfying (3), and finally, in Section 6 the proofs of the exponential stability.
2. Notation and organization. Here we will show some results and statements for differential calculus of tensor fields and Sobolev spaces on Riemannian manifolds, for more detail see [24,29] and [31]. First, we consider the space L 2 (M ) of complex valued function on M , with the following real inner product and norm and H with the standart L 2 -norm. Note that, from Poincaré's inequality, where λ 1 is the first eigenvalue of the Laplace-Beltrami operator. Thus it is possible to show that the norms in H 1 (M ) and V are equivalent. Consider the following Hilbert space Then, In addition if f : M → R is a sufficiently regular real function and X 1 = Y 1 + iZ 1 is a complex vector field on M we have and therefore using the following identity (see [14], p.22) When X = ∇y and y : M → C, we can write Hess f (∇y, ∇y) = ∇(Re y), ∇ ∇f, ∇(Re y) where |∇y| 2 = |∇(Re y)| 2 + |∇(Im y)| 2 . Observe that Re ∇y, ∇ ∇f, ∇y = ∇(Re y), ∇ ∇f, ∇(Re y) + ∇(Im y), ∇ ∇f, ∇(Im y) .
Therefore we conclude that Re Hess f (∇y, ∇y) = Re ∇y, ∇ ∇f, ∇y Replacing ∇f in the identity (6) by an arbitrary real vector field H sufficiently regular, have the translation of the identity (5) for the complex case, namely: The following result is also true (see [11], Section 4.1): Let M be a compact Riemannian manifold without boundary y : M → C ∈ H 1 (M ) and X a vector field of class Consequently, if y ∈ H 1 (M ) such that ∆y ∈ L 2 (M ) and w ∈ H 1 (M ) then the following identity is valid: 3. Well-posedness. The well-posedness of the problems (1) and (2) was studied in [6], through semigroup theory, in the case of locally distributed damping on a bounded domain case. The similar result with additional the smoothing effect ia(x)(1 − ∆) 1/2 a(x)y given by Aloui will follow with the same arguments. For the sake of completeness, we show the main steps of the proofs.
First, we write problem (2) as an equivalent Cauchy problem Note that the operator A is maximal dissipative in D(A) and F is globally Lipschitz in L 2 (M ). We can see the similar result in [6]. So using standard arguments from Pazy [28] follows next Theorem.
Theorem 3.1. Assume the hypotheses (H1) and (H2). Then, given y 0 ∈ D(A), the problem (2) has a unique regular solution y for the problem satisfying We know that D(1 − ∆) = H 2 (M ) ∩ V and such operator can be extended to V. According to Lion-Magenes [24] we have D((1 − ∆) In fact, observe that the operator (1 − ∆) 1/2 is positive and multiplying (2) by y, we infer that Thus Consequently we obtain (9).
So, studying the problem (1), note that the energy is given by Thanks to (H2)(iv), we get Thus, multiplying (1) by ∂ t y and considering the real part, we have Consequently y V remains bounded for any t > 0. After these observations the next theorem guarantees the existence, uniqueness of the problem (1).
Then, problem (1) possesses a unique solution which belongs to The proof of Theorem 3.2 is similar to Theorem 3.1. Just note that operator (1) is equivalent to an Cauchy problem, given by dy dt = By + F (y), Consequently we have Theorem 3.2.
Remark 4. By Lummer-Phillip's Theorem the operador B is the infinitesimal generator of C 0 semigroup contractions S(t) in H. So, given y 0 ∈ H the problem (1) have a unique mild solution y in class L ∞ (0, T ; L 2 (M )) which satisfies the follow integral equation 4. Unique continuation. In this section we establish the unique continuation property of the problem (1). Before we need proof a technical lemma on the regular solutions.
Lemma 4.1. Let M n be a compact Riemannian manifold , without boundary, orientable and conected. Let q ∈ C 2 (M ) n be a real vector field. Then, for all regular solution of problem (1) the following identity holds where G is given in (10).
Proof. Indeed, multiplying equation (1) by q, ∇y and integrating over M × (0, T ), we obtain Integrating I 1 by parts, On the other hand, from (8) we obtain an identity for the second term of (14), q, ∇y y t dM dt. Then However, knowing that Now, using (15), we get Taking the real part of (13), having in mind the real part of (16), and observing that Re(z) = Re(z), for all z ∈ C, we deduce that In what follows let us analyze some terms on right-hand side of (17). According to (7) and (8) we get Observe that 2 Re h(|y| 2 )y q, ∇y = h(|y| 2 ) q, ∇(|y| 2 ) = q, h(|y| 2 )∇(|y| 2 ) = q, ∇G(|y| 2 ) .
The next theorem is a crucial result to obtain the exponential decay.
Remark 5. In the next Section we find a subset V of M and a constant C > 0 such that Hess f (∇y, ∇y) dM dt.

If Remark 5 is valid, we obtain
From (21), (22), (23) and (24), we get Using the initial assumptions, Consequently, That is, where Remember that So, according to (11) and (27) we get the following estimate where C 7 = C 7 (λ 1 , f, h). From (26) and (28), we have From (H2)(ii) and (4), Combining (29) and (30) we conclude where C = C(h, f ). Thus Then, for all T > 1 2C + M 1 Cλ 1 we have the following inverse inequality The proof will be done in several steps. The objects on M will be denoted by usual symbols. The objects on the tangent space will be denoted by caligraphic symbols.

5.2.
General idea of the proof. Let B(p, r) ⊂ M be a geodesic ball inside the injectivity radius of p. The exponential map exp : B(0, r) → B(p, r) induces normal coordinates in B(p, r). Consider the homothetyh r : B(0, 1) → B(0, r) and h r : B r (p, 1) → B(p, r), where B r (p, 1) is the 1/r Riemannian dilation of B(p, r). Denote by exp r : B(0, 1) → B r (p, 1) the exponential map such that exp •h r = h r • exp r . In this section we always use these coordinate systems on B(p, r) and B r (p, 1). The symobols ∇ r , ∆ r and Hess r denote respectively the gradient, Laplacian and the Hessian of B r (p, 1). We identify f : B(p, r) → R with f • exp when there is no possibility of misunderstandings.
For each r, we consider the problem We prove that as r → 0, the solution f r of this problem converges C 2 to the solution of the Euclidean problem, which is Notice that f 0 satisfies all the required conditions stated in Lemma 5.1 for the Euclidean setting. Therefore there exist a positive number r > 0 such that (31) and the other conditions stated in Lemma 5.1 are satisfied. Finally, we conclude that the function f : B(p, r) → R defined as f (x) = f r (x/r) satisfies the conditions stated in Lemma 5.1.

5.3.
Some formulas in a coordinate system. Let M be a Riemannian manifold and let U ⊂ M be an open set. Let (x 1 , . . . , x n ) be a coordinate system on U . Denote the components of the Riemannian metric with respect to this coordinate system by g ij . We denote the components of the inverse matrix of (g ij ) by g ij . Given f ∈ C 1 (U ), the gradient of f is given by Hess f is given by are the Christoffel symbols of (U, (x 1 , . . . , x n ), g). Finally, the Laplacian of f is the trace of the Hessian with respect to the metric g and it is given by 5.4. Equation (31) in a coordinate system. Consider B r (p, 1) and B(p, r) with their coordinate systems exp r and exp respectively. If we denote their metric matrix by (g r ) ij and g ij respectively, then (g r ) ij (x) = g ij (rx). It also follows that (Γ r ) k ij (x) = rΓ k ij (rx). Observe that in normal coordinates we have that lim x→0 Γ k ij (x) = 0 for every i, j and k. Let κ be a positive number strictly smaller than the injectivity radius of M at p. For every r ∈ (0, κ] the problems (31) with respect to the parametrization exp r are given by where L r are given by (32). Therefore lim r→0 L r f = L 0 f for every f ∈ C 2 (B(0, 1)), where L 0 is the Laplacian in the Euclidean space.

5.5.
Solutions and a priori estimates. Here we remember some classical a priori estimates of solutions of elliptic differential equations Lφ = ϕ defined on a domain Ω ⊂ R n with smooth boundary. The elliptic operator is given by where a ij , b i , c, ϕ ∈ C 0 (Ω) and there exist positive real numbers λ and Λ such that the coefficients satisfy 0 < λ|ξ| ≤ a ij ξ i ξ j ≤ Λ|ξ|, ∀ξ ∈ R n \{0}. The theorems stated below are valid under much more general conditions, but we present them with these conditions for the sake of simplicity. For more general cases, see [20].

5.6.
Convergence of the one parameter family of solutions. Consider the family of Dirichlet problems (33). We prove in this subsection that lim First of all, we consider an equivalent problem making the transformationf r = f r − 3x 1 . The resulting family of equations is given by whereφ r = 2n − L r (3x 1 ). For each r, (34) has a unique solution due to Lemma 5.4. Moreover due to Lemma 5.2 and 5.3, we have that for every f ∈ C 2 (B(0, 1)) that vanishes on the boudary. Here C = C(n, sup r λ, sup r Θ, sup r |b| C 0 ) (notice that C does depend on r). We claim that for every ε > 0, there exist δ > 0 such that f r1 −f r2 C 2 < ε whenever r 1 , r 2 ∈ [0, δ]. In particular we have that lim r→0 f r −f 0 C 2 = 0. Consider r 1 , r 2 ≥ 0. Then Due to Lemma 5.2 there exist a constant C 1 such that f r C 0 ≤ C 1 for every r ∈ [0, κ] (Use L = L r for each r). Using Lemma 5.3 we have that f r C 2 ≤ C 2 for every r. Moreover notice that the indices of L r varies continuously with respect to r. Then we can find a δ > 0 such that if r < δ, then f r −f 0 C 2 < ε and the claim is proved. Now it is immediate that lim The C 2 convergence of f r implies that there exist a ε > 0 such that 1. (∆ r )f r = 2n for every r ∈ [0, κ]; 2. Hess r f r is positive definite for every r ∈ [0, ε]; 3. (∇ r )f r ≥ C > 0 for every r ∈ [0, ε].

5.7.
Proof of Lemma 5.1. Consider the geodesic balls B(p, r) and its 1/r Riemannian dilation B r (p, 1) as in Subsection 5.2. We use the coordinate systems induced by exp and exp r on B(p, r) and B r (p, 1) respectively. If f r : B r (p, 1) → R is a smooth function, then the function f : B(p, r) → R defined as f (x) = f r (x/r) satisfies the following equalities We begin proving some preliminary results. The following lemma is classical and can be found in [34] (See the proof of Lemma 1.9).
Lemma 5.6. Let M be a topological space which is locally compact, Hausdorff and has countable basis. Then there exist a increasing sequence of open sets (V i ) i∈N such that: The function η is clearly C ∞ . We normalize η and get η(x, y, ε) = η(x, y, ε) M η(x, y, ε)dM (y) .
Notice that η is also smooth. We define the mollifier smoothing f ε : M → R by Lemma 5.7. Let M be a compact Riemannian manifold, f : M → R be a locally summable function and ε ∈ (0, inj(M )) be a strictly positive number. Then the mollifier smoothing f ε : M → R defined by (36) is a smooth function.
Proof. The theorem holds because a Riemannian manifold behaves like Euclidean domains inside the injectivity radius. For the complete proof, see [19].
Notice that f is smooth because f | Wi = f i ρ i is smooth and f | Wi = 0 near B for every i = 1, . . . , k. Moreover f | V satisfies Items 2, 3 and 4 because ρ| V ≡ 1 and f | Vi = f i for every i = 1, . . . , k. Finally |∇f | is bounded on M because it is continuous and M is compact. 6. Stabilization. Let's start the main section of this work with a technical Lemma whose the similar proof can be found in [6] and [4]. In fact we only join our ideas. In this step we need to use the smoothing effect due Aloui [1]. From the assumption (H2)(ii) we get F (y k ) = ih(|y k | 2 )y k ∈ L 2 (0, T ; L 2 (M )), noting that E k 0 (t) := y k (t) 2 L 2 (M ) is a non-increasing function and the sequence is uniform bounded. So smoothing effect mentioned above give us the following estimate: (θ y k )(t) L 2 (0,T ;H 1 2 (M )) ≤ C y 0 k L 2 (M ) + F (y k (t)) L 2 (0,T ;L 2 (M )) ≤ C y 0 k L 2 (M ) + M 1 y k (t) L 2 (0,T ;L 2 (M )) ≤ C L (1 + M 1 T ) , ∀ θ ∈ C ∞ 0 (0, T ) .
Thus, y k −→ỹ strongly in L 2 (0, T ; L 2 (M )), wherẽ y = y, a.e. in M \M * 0, a.e. in M * Therefore the lemma follows in two cases, when y = 0 and when y = 0, using Theorem 4.2 (Unique Continuation Property). Below is the main result and a sketch proof.
Proof. Denote by β the positive constant of embedding D((1 − ∆) 1/4 ) → L 2 (M ). Then from (9) and (H1)(ii) we have From Lemma 6.1 we deduce that where C is a positive constant. Therefore the main result follows from semigroup property (see [4]).