Communications in Partial Differential Equations

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Communications in Partial Differential Equations Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lpde20

Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains a

César A. Bortot & Marcelo M. Cavalcanti a

b

Engineering Center of Mobility , Federal University of Santa Catarina , Joinville , Brazil

b

Department of Mathematics , State University of Maringá , Maringá , Brazil Published online: 08 Aug 2014.

To cite this article: César A. Bortot & Marcelo M. Cavalcanti (2014) Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains, Communications in Partial Differential Equations, 39:9, 1791-1820, DOI: 10.1080/03605302.2014.908390 To link to this article: http://dx.doi.org/10.1080/03605302.2014.908390

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Communications in Partial Differential Equations, 39: 1791–1820, 2014 Copyright © Taylor & Francis Group, LLC ISSN 0360-5302 print/1532-4133 online DOI: 10.1080/03605302.2014.908390

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Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains CÉSAR A. BORTOT1 AND MARCELO M. CAVALCANTI2 1

Engineering Center of Mobility, Federal University of Santa Catarina, Joinville, Brazil 2 Department of Mathematics, State University of Maringá, Maringá, Brazil The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold g is considered. Assuming that g is nontrapping and, in addition, that the damping term is effective in \, where ⊂⊂ is an open bounded and connected subset with smooth boundary , such that is a compact set, exponential and uniform decay rates of the L2 −level energy are established. The main ingredients in the proof of the exponential stability are: (A) an unique continuation property for the linear problem; and (B) a local smoothing effect for the linear and nonhomogeneous associated problem. Keywords Exponential stability; Non-compact manifolds; Schröedinger equation. Mathematics Subject Classification 58Jxx.

1. Introduction This paper addresses the well-posedness as well as exponential decay rate estimates of the energy Et = 21 ux t2 dx related to the Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete and noncompact n dimensional Riemannian manifold g:

iut + u + iaxgu = 0 in × 0 + ux 0 = u0 x x ∈

(1.1)

Received August 22, 2013; Accepted February 20, 2014 Address correspondence to Marcelo M. Cavalcanti, Department of Mathematics, State University of Maringá, Av. Colombo 5790, Maringá 87020-900, Brazil; E-mail: mmcavalcanti@ uem.br

1791


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Bortot and Cavalcanti

where denotes the Laplace-Beltrami operator. We shall divide our study into two fold: (i) when g is an exterior domain of n endowed with the Euclidean metric. More precisely, let ⊂ n , n ≥ 2 be a compact smooth obstacle. Denote by the complementary of , that is, = n \. In the first part of the present paper we shall suppose that the obstacle is nontrapping, which means that any light ray reflecting on the boundary of according to the laws of the geometric optics leaves any compact set in finite time. In this case, Dirichlet boundary condition is assumed on the boundary of , namely, u = 0 on M × 0 ; (ii) when g is a noncompact, “nontrapping” n-dimensional Riemannian manifold, n ≥ 2, simply connected, orientable and without boundary endowed by a Riemannian metric g· · = · ·. In addition, we suppose that g is complete of class C . When a ≡ 0 in (1.1) and = n \ it is well known (see Tsutsumi [23]) that if the domain is nontrapping in the geometrical optics sense, then the L2 localized energy, u· tL2 BR (where BR means a ball of radius R > 0) decays like t−n/2 . In the trapping case, no uniform decay rate is possible (see J. V. Ralston [19]). All these results are obtained through estimates of the resolvent + −1 . More recently, when a = 0, a ≥ 0, that is, in presence of a linear damping term iaxu, Aloui and Khenissi [1] proved a similar polynomial decay rate estimate as in [23] by relaxing the nontrapping condition as follows: We say that n \ satisfies an EGC (exterior geometric control) condition, if each trapped ray meets the set x ax > 0. Their proof proceeds via the established scheme; namely, L2 → L2 estimates for the cut-off resolvents. The present paper isconcerned the exponential decay rate estimate for the energy Et = 21 n \ ux t2 dx, instead of the local one Eloc t = full (total) 2 ux t dx, as considered in the references mentioned above. For this purpose BR the additional damping term iaxgu is fundamental, since if a = 0 the full energy is conserved, namely, Et = E0 for all t ≥ 0 and no decay is expected. In the first part of the paper, we shall assume that ax ≥ a0 > 0 in , where ⊂ is defined in the following form: Considering = n \, let R > 0 such that ⊂ BR = x ∈ n x < R, thus = \BR (see Figure 1). In the second part of the present article, we shall assume (roughly speaking) that ax ≥ a0 > 0 in \, where ⊂⊂ is an open bounded and connected subset with smooth boundary , such that is a compact set. The precise assumption is stated in Assumption 5.1 (we hope

Figure 1. The damping term is effective in = n \\BR .

Noncompact Riemannian Manifolds

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Figure 2. The black region inside possesses measure arbitrarily small while the white one possesses measure arbitrarily large. However, both are totaly distributed.

Figure 3. It is possible to avoid put damping in radially symmetric disjoint regions inside the mesh.

that Figures 2, 3, and 4 will contribute towards better understanding of this fact. In any case this will be clarified during the proof). Our main result reads as follows: Theorem 1.1. Let u be a weak solution to problem (3.16) (respectively (5.70)), with the energy defined as in (3.18) (respectively (5.72)). Then, under Assumption 3.1 and 3.2 (respectively Assumption 3.1, 5.1, and 5.2) there exist positive constants T0 , C0 and 0 such that Et ≤ C0 e−0 t E0 ∀t ≥ T0 provided the initial data are taken in bounded sets of L2 . The main ingredients in the proof of the exponential stability to problem (1.1) are: (A) an unique continuation property for the linear problem (as in Triggiani and Xu [21]); and (B) a local smoothing effect for the linear and nonhomogeneous

Figure 4. When Kg ≤ 0 the region is free of damping.


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associated problem (as in Burq, Gerard, and Tzvetkov [9]). In fact, the main goal of the present paper is to prove that conditions (A) and (B) are sufficient to establish the exponential decay of the total energy Et = 21 ux t2 dx to problem (1.1). This is will be explained during the proof. Whether conditions (A) and (B) are necessary or not seems to remain an open question. It is important to observe that in order to employ the unique continuation property considered in [21], it is essential the existence of a strictly convex function satisfying some properties and the construction of such function can be found in Cavalcanti et al. (see [12]). Regarding condition (B) it is worth mentioning that que once trapped geodesics breaks the smoothing effect or, in other words, in view of the necessity of the nontrapping condition for the plain smoothing effect (see Doi [14] or Burq [7]), it is crucial to consider “nontrapping” n-dimensional Riemannian manifolds in the present context. At this moment, it is important to mention the work [2] where the authors prove that the geometric control condition is not necessary to obtain the smoothing effect and the uniform stabilization for the strongly dissipative Schrödinger equation posed in a bounded domain of n , which is not the case of the present manuscript. Finally, we would like to mention other papers in connection to the stabilization of Schrödinger equation subject to locally distributed damping posed in unbounded domains, as, for instance, [3], [10], [11] and references therein and we would like to mention that the present paper is an extension of previous results due to Bortot et al. [6] from the compact setting to the noncompact one.

2. Preliminaries: Geometric Riemannian Tools Let n g be a n-dimensional complete Riemannian manifold, n ≥ 2 orientable, simply connected and without boundary, induced by the Riemannian metric g· · = · ·, of class C . We shall denote by gij n×n the matrix n × n in connection with the metric g. The tangent space at in p ∈ will be denoted by Tp ≡ n . Let f ∈ C 2 , and let us define the Laplace-Beltrami operator of f , as f = divf

(2.2)

where f denotes the gradient of f in the metric g, that is, for all vector field X in f X = Xf

(2.3)

and div denotes the divergent operator, namely, if X is a vector field in , divXp = trace of the linear map Yp → Y Xp, p ∈ . From the definitions and notations above we have the following lemma: Lemma 2.1. Let p ∈ . Let us consider f ∈ C 1 and H a vector field in . Then, the following identity hold (see [17] p. 21): 1 f Hf = Hf f + divf 2 H − f 2 divH 2 where H is the differential covariant derivative defined by HX Y = X H Y .


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Finally we shall define the Hessian of f ∈ C 2 as the symmetric tensor of order two in , namely, HessfX Y = 2 fX Y = fX Y = Y f X

(2.4)

for all X and Y vector fields in . Remark 1. In order to simplify the notation, we denote the L2 -norm, without distinguishing whether the argument of the norm is a function or tensor field of type 0 m.


Let k ∈ e p ≥ 1. We define the space Ckp as p j p Ck = u ∈ C u d < ∀j = 0 1 k

(2.5)

where j u denotes the j th differential covariant derivative of u 0 u = u 1 u = u). Thus, we define the Sobolev space Hkp as the closure of Ckp with respect to the topology upH p = k

k j=0

j up d

(2.6)

From the above, we deduce: i) L2 = H02 is the closure of C02 with respect to the tolopogy u2L2 M = u2 d

(2.7)

ii) H 1 = H12 is the closure of C12 with respect to the topology u2H 1 = u2 d + u2 d

(2.8)

iii) H 2 = H22 is the closure of C22 with respect to the topology u2H 2 = 2 u2 d + u2 d + u2 d

(2.9)

Remark 2. From the above definitions we have the following chain of continuous embedding H 2 → H 1 → L2

(2.10)

Furthermore, by Hebey ([16], Theorem 2.7, p. 13), it follows that H01 = H 1

H 1 , where H01 = , in other words, the space of infinitely differentiable functions with compact support is dense in H 1 . So, from the above and making use of density arguments we can extend the formulas presented previously to Sobolev spaces. In the sequel, we shall announce three theorems that will play an important role in the present work (see [20]).

1796


Theorem 2.1 (Gauss Divergent Theorem). Let n a Riemannian manifold, orientable, with smooth boundary , X ∈ H 1 n a vector filed and the normal unitary vector field point towards , thus

divX d =

X d

(2.11)

Theorem 2.2 (Green Theorem 1). Let n a Riemannian orientable manifold, with smooth boundary , X ∈ H 1 n a vector field, q ∈ H 1 and the normal unitary vector field point towards , then


divXq d = −

X q d +

X q d

(2.12)

Theorem 2.3 (Green Theorem 2). Let n a orientable Riemannian Manifold, with smooth boundary , f ∈ H 2 , q ∈ H 1 and the normal unitary vector field point towards , then

fq d = −

f q d +

fq d

(2.13)

Remark 3. Let g be a Riemannian manifold and u → a smooth function. Then, we have Re u → and Im u → consequently we can talk about Re u and Im u defined intrinsically as in (2.3). Let X be a complex vector field over , that is, X = Y + iZ, where Y e Z are real vector fields. We shall denote by u X = Re u Y g − Im u Zg + iRe u Zg + Im u Y g Consequently, u u¯ = Re u Re ug + Im u Im ug = u2 2.1. Preliminaries: Partial Differential Equations Let X be a real Banach space and X its topological dual. For all x ∈ X, we associate the set Fx = f ∈ X f xX X = x2 = f 2 and we define s X × X → by x ys = sup f yX X f ∈ Fx Remark 4. If X = H, where H is a Hilbert space, it results from Riesz representation theorem that Fx is an unitary set and from the identification H ≡ H we deduce that s = H .


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2.2. Homogeneous Equation Let us consider the following problem: ut t = Tut + But t ∈ 0 u0 = u0

(2.14)

posed in a Banach space X.


Definition 2.1. A map u 0 → X is called a regular solution to problem (2.14) if u is continuous in 0 , Lipschitz continuous in every compact set contained in 0 , u0 = u0 , u is differentiable almost everywhere in 0 , ut ∈ DT + B a. e. in 0 and satisfies ut t = Tut + But a. e. 0 . Definition 2.2. The map u 0 → X is called a generalized solution to problem (2.14) if u is continuous in 0 , u0 = u0 and satisfies the following inequality for each T > 0 t ut − v2X ≤ us − v2X + 2 Tv + Bu u − vs d (2.15) s

∀v ∈ DT e 0 ≤ s ≤ t ≤ T .

3. Schrödinger Equation in Exterior Domains In what follows we shall omit some variables in order to make easier the notation, and we will denote the Laplace operator by . We shall study following damped problem:   iut + u + iaxgu = 0 in × 0 (3.16) ux t = 0 in × 0   u0 = u0 in where is an exterior domain of n , that is, = n \ where is an compact and connected subset of n with smooth boundary. In addition, we suppose that is nontrapping, namely, any light ray reflecting on the boundary of according to the laws of the geometric optics leaves any compact set in finite time. We observe that = , thus is smooth. Find in Figure 5 a counter example, where is a trapping obstacle. Assumption 3.1. Hypotheses on the function g → (i) (ii) (iii) (iv)

gz is continuous, g0 = 0; Re gz − gw¯z − w ¯ ≥ 0, ∀ z w ∈ ; Im gz¯z = 0, ∀ z ∈ ; There exist positive constants c1 and c2 , such that c1 z2 ≤ gs¯z ≤ c2 z2 , ∀z ∈ .

1798


Figure 5. Example of a subset such that its boundary does not satisfy the nontrapping condition.


We observe that taking (ii) and (iii) into account, we deduce that gz¯z = gz¯z. Remark 5. Find in the Appendix of this manuscript some examples of functions g satisfying Assumption 3.1. Assumption 3.2. Hypotheses on the function a → i) ax ∈ L is nonnegative function; ii) ax ≥ a0 > 0 em , where ⊂ is defined as follows: Let R > 0 such that ⊂ BR = x ∈ n x < R, then = \BR , according to Figure 1. It is important to observe that we shall work with complex-valued functions, so that, in order that the spaces L2 , as well as, H m , m ∈ , become real Hilbert spaces, we define w vL2 = Re

w¯vdx

Finally, we shall denote by H01 the Hilbert space H01 = w ∈ H 1 w = 0 3.1. Existence and Uniqueness of Solutions: Exterior Domains Problem (3.16) can be rewritten as   ut − iu + axgu = 0 in × 0 ux t = 0 in × 0   u0 = u0 in

(3.17)

The energy associated with problem (3.16) is defined by: Et =

1 ux t2 dx 2

(3.18)


1799

We define the operators A DA ⊂ L2 −→ L2 u −→ Au = −iu and, B DB ⊂ L2 −→ L2


u −→ Bu = agu Thus, DA = H01 ∩ H 2 and DB = L2 . Our next purpose is to prove that A + B is a maximal monotone operator. Initially, we note that it is not difficult to verify that A is maximal monotone. In the sequel we shall prove some properties associated with the operator B. • B takes bounded sets in bounded sets. Indeed, let u ∈ L2 such that u2L2 ≤ R. Thus, accordingly to the Assumption (3.1) iv we infer Bu2L2 ≤ a2L gux2 dx 2 2 ≤ aL c2 ux2 dx ≤ Ra2L c22

• B is monotone. Indeed, let u1 u2 ∈ L2 . Then, from Assumption 3.1 ii we obtain Bu1 − Bu2 u1 − u2 L2 = ax Re gu1 − gu2 u1 − u2 dx ≥ 0

• B is hemicontinuous. In fact, we have to prove that given an arbitrary sequence tn ⊂ such that tn → 0 then lim Bu + tn v wL2 = Bu wL2 ∀ u v w ∈ L2

n→

¯ Thus, For this purpose we define fn = agu + tn vw. fn x = axgux + tn vxwx ≤ c2 axux + tn vxwx ≤ c2 aL uxwx + c2 c3 aL vxwx almost everywhere in , where c3 is such that tn ≤ c3 Since u v w ∈ L2 , it results that fn ∈ L1 , for all n ∈ . Furthermore, if h is the function defined by hx = c2 aL uxwx + c2 c3 aL vxwx it follows that h ∈ L1 and fn x ≤ hx almost everywhere in .

1800


We observe that, from the continuity of g, we deduce ¯ = axguxwx ¯ lim axgux + tn vxwx

n→

Thus, from Lebesgue dominated convergence theorem we conclude that axgux + tn vxwx ¯ − axguxwxdx ¯ → 0

Then, ¯ − axguxwxdx ¯ → 0 axgux + tn vxwx


and, consequently, ¯ → Re axguxwxdx ¯ Re axgux + tn vxwxdx

that is, lim Bu + tn v wL2 = Bu wL2

n→

From the above we proved that A is maximal monotone, B is monotone, hemicontinuous and take bounded sets in bounded set. Thus, from Barbu ([4], Corol. 1.1, p. 39) we deduce that A + B DA + B ⊂ L2 → L2 is maximal monotone. Consequently, for each u0 ∈ DA + B = DA = H01 ∩ H 2 there exists, taking Brézis ([5], Theorem 3.1, p. 54) into account, a unique function u 0 → L2 which is the regular solution to problem (3.16). In addition, for all u0 ∈ DA = L2 there exists, considering Barbu ([4], Theorem 3.1, p. 152), a unique map u 0 → L2 such that is the unique weak solution to problem (3.16).

4. Stability Result: Exterior Domains Before presenting the proof of the stability result, let us consider an important identity, namely, the identity of the energy. Let u be a regular solution to problem (3.16). Multiplying the equation given in (5.71) by u¯ and integrating by parts, we deduce: t2 axgux tux tdxdt (4.19) Et2 − Et1 = − t1

for all t2 > t1 ≥ 0, which remains valid for weak solutions by standard density arguments. Our main task is to prove the following inequality: 0

T

Etdt ≤ C

T

axgu¯udxdt

0

where C is a positive constant. It is sufficient to work with regular solutions to problem (3.16), since the above exponential decay rate estimate of the energy Et


1801

remains valid for weak solutions by using arguments of density. So, let u a regular solution to problem (3.16). We observe that

T

2 0

Etdt =

T 0

≤

T

0

≤

T

0

\

\

\

ux t2 dxdt +

T

0

ux t2 dxdt

ux t2 dxdt + a−1 0

T 0

−1 ux t2 dxdt + a−1 0 c1

axux t2 dxdt

T

axgu¯udxdt

0

(4.20)


T Thus, it remains to estimate 0 \ ux t2 dxdt in terms of the “damping term.” For this purpose, we consider the following lemma: Lemma 4.1. Let u be a regular solution to problem (3.16), with u0 L2 ≤ L L > 0. Then, for all T > 0, there exists a constant C = CT > 0, such that 0

T

\

ux t2 dxdt ≤ C

T

axgu¯udxdt

0

(4.21)

Proof. We argue by contradiction. For simplicity we shall denote u = ut as well as = \ . Assume that (4.21) does not hold. Then, there exists a sequence of initial data

u0k k∈ ⊂ DA, such that the corresponding regular solutions, uk k∈ , of problem (3.16), with Ek 0 =

1 0 2 u 2 ≤ L 2 k L

for all k ∈ , verify T 0

uk t2L2 dt

axguk ¯uk dxdt

lim T

k→

0

= +

(4.22)

that is, T lim

k→

0

T 0

axguk ¯uk dxdt uk t2L2 dt

= 0

(4.23)

We know, according to (4.19), that the energy is a non increasing function on the parameter t, thus Ek t ≤ Ek 0 ≤ L. Consequently, we obtain uk L 0T L2 ≤

√ 2L

(4.24)

for all k ∈ . Then, there exists a subsequence of uk , still denoted by the same notation, and u ∈ L 0 T L2 such that

uk u weak star in L 0 T L2

(4.25)

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Since uk is bounded in L 0 T L2 → L2 0 T L2 , uk tL2 ≤ uk tL2 , for almost everywhere t ∈ 0 T and taking (4.23) into account, we infer

T

0

axguk ¯uk dxdt =

uk 2L2 0T L2 uk 2L2 0T L2

T

0

axguk ¯uk dxdt → 0

(4.26)

when k → . Then, from (4.26), assumption (3.1) and assumption (3.2), we conclude that T T T uk 2 dxdt ≤ a−1 axuk 2 dxdt ≤ a−1 axuk 2 dxdt 0 0 0

0


−1 ≤ a−1 0 c1

T 0

0

axguk ¯uk dxdt −→ 0

(4.27)

Consequently, taking (4.25) and (4.27) into account, we infer, for almost everywhere t ∈ 0 T, ut in (4.28) ut = 0 in At this point we shall divide our proof into two cases, namely: u = 0 and u = 0. (i) u = 0 Let us consider the sequence of problems   uk − iuk + axguk = 0 in × 0 in × 0 uk x t = 0   0 uk 0 = uk in

(4.29)

We observe that from (4.26) and considering the assumptions (3.1) and (3.2) we can write T axguk 2 dxdt aguk 2L2 0T L2 ≤ aL 0

≤ aL c22

T 0

≤ aL c22 c1−1

T 0

axuk 2 dxdt

axguk ¯uk dxdt −→ 0

that is, aguk −→ 0 strongly in L2 0 T L2

(4.30)

Our objective is to pass to the limit in (4.29). Initially, we recall the following embedded chain: L2 0 T H01 → L2 0 T L2 ≡ L2 Q → L2 0 T H −1 → Q (4.31)


1803

where Q = × 0 T. From (4.25) we have, uk u weakly in L2 0 T L2

(4.32)

Let us consider ∈ 0 T and → tal que ∈ . Multiplying the first equation of (4.29) by and integrating over Q, we arrive at 0

T

uk x ttx − iuk x ttx + axguk x ttxdxdt = 0 (4.33)

Integrating by parts and taking the real part it follows that


− Re

T

0

+ Re

T

0

uk x t txdxdt − Re

0

T

iuk x ttxdxdt

axguk x ttxdxdt = 0

(4.34)

Taking k → in (4.34) and taking (4.30) and (4.32) into account, we conclude that −

T

0

ut tL2 dt −

T 0

iut tL2 dt = 0

(4.35)

From the density of ∈ D0 T and ∈ in Q it results that T T − ut tL2 dt − iut tL2 dt = 0 (4.36) 0

0

for all ∈ Q namely, u − iu = 0 in Q

(4.37)

where u ∈ L 0 T L2 and, from (4.28), ux t = 0 ae em × 0 T

(4.38)

Let us consider, now, the ball centered at the origin and radius 2R, B2R , and let us define = \B2R ⊂ keeping in mind that = \BR and = \ . Thus, from (4.37) and (4.38) we deduce u − iu = 0 in × 0 T (4.39) ux t = 0 in \ × 0 T Employing Holmgren’s uniqueness theorem, we conclude that u = 0 a.e. in × 0 T, and, consequently, u = 0 a.e. in × 0 T which is a contradiction. (ii) u = 0 Denoting ck = uk L2 0T L2 and vk =

uk ck

(4.40)

1804


we deduce vk L2 0T L2 = 1

(4.41)

Dividing (4.29) by ck we obtain  axguk  vk − ivk + ck = 0 in × 0 on × 0 vk x t = 0   u0k 0 vk 0 = vk = c in

(4.42)


k

We observe that T T ck2 vk 2L2 0T L2 uk 2L2 dt uk 2L2 dt ck2 0 0 = 2 T = T T ck 0 axguk ¯uk dxdt axguk ¯uk dxdt axguk ¯uk dxdt 0 0 vk 2L2 0T L2 vk 2L2 0T L2 = T ≤ c 1 T axguk uc¯ 2k dxdt axvk 2 dxdt 0 0

(4.43)

k

Thus, taking (4.22) and (4.43) into account, we have vk 2L2 0T L2 = + lim T k→ axvk 2 dxdt 0

(4.44)

and since vk 2L2 0T L2 = 1, it results that

T

lim

k→ 0

axvk 2 dxdt = 0

(4.45)

From the fact that ax ≥ a0 > 0 em , we conclude that

T

0

vk 2 dxdt ≤ a−1 0 ≤ a−1 0

T 0

T 0

axvk 2 dxdt axvk 2 dxdt −→ 0

that is, vk −→ 0 strongly in L2 0 T L2

(4.46)

On the other hand, taking (4.20) into consideration, we infer 2 0

T

Ek tdt ≤

0

T

−1 uk 2 dxdt + a−1 0 c1

T 0

axguk ¯uk dxdt

(4.47)

and since Ek T ≤ Ek t for all T ≥ t ≥ 0, it follows that 2TEk T ≤ 2

0

T

Ek tdt ≤

0

T

−1 uk 2 dxdt + a−1 0 c1

T 0

axguk ¯uk dxdt


1805

Consequently, Ek T ≤ C

T

0

uk dxdt + 2

T

0

axguk ¯uk dxdt

(4.48)

Making use of the identity of the energy established in (4.19), we arrive at Ek T − Ek 0 = −

T

0

axguk ¯uk dxdt

so that,


Ek 0 =

T

0

axguk ¯uk dxdt + Ek T

Then, from (4.48) and (4.49) we conclude that T

T C uk 2 dxdt + axguk ¯uk dxdt Ek 0 ≤ 0

0

(4.49)

(4.50)

Dividing (4.50) by ck2 and having in mind that the energy is a nonincreasing function on the parameter t, we have, for all t ≥ 0, that T Ek t Ek 0 0 axguk ¯uk dxdt ≤ ≤C +1 (4.51) ck2 ck2 ck2 and from (4.23) and (4.51) we guarantee the existence of a constante M > 0 such that vk0 2L2 =

u0k 2L ck2

=

2Ek 0 ≤ M ck2

(4.52)

for all k ∈ , and, therefore,

vk0 is bounded in L2

(4.53)

Since vk is a regular solution to problem (4.42), we have that vk satisfies the integral equation vk t = Stvk0 −

t 0

St − s

a guk sds ck

(4.54)

where St is the semigroup generated by −i. Employing the local smoothing effect due to Burq, Gérard and Tzvetkov (see [9], Prop. 2.7, p. 302), we deduce, for all T > 0, and for all ∈ C0 n , that 2 wk L2 0T H 1 ≤ C1 fk L2 0T L2 where wk t =

0

t

St − sfk sds and fk s =

a guk s ck

(4.55)

1806


In addition, Stvk0

1

L2 0T HD2

≤ C2 vk0 L2

1

(4.56)

n where HD2 is the domain of the operator + I 4 . Considering ∈ 0 such n that = 1 in and 0 ≤ ≤ 1 in , we obtain, taking (4.55) and (4.56) into account, that 1

2 wk L2 0T H 1 ≤ 2 wk L2 0T H 1 ≤ C1 fk L2 0T L2 ≤ C3

(4.57)

since, from (4.23), is valid that


T

0

axgu 2 T axgu 2 k k dxdt ≤ a dxdt L ck2 ck2 0 T axu 2 k ≤ aL c22 dxdt ck2 0 T axguk ¯uk dxdt 2 −1 0 −→ 0 ≤ aL c2 c1 T uk 2 dxdt 0

that is, aguk −→ 0 strongly in L2 0 T L2 ck and since = 1 in we deduce that wk L2 0T H 1 ≤ C3 1

Furthermore, H 1 → H 2 , then wk

1

L2 0T H 2

≤ C4

(4.58)

Finally, using the properties of , considering the inequality (4.56) as well as (4.53), we conclude that Stvk0

1

L2 0T H 2

≤ C5

(4.59)

Then, from (4.54), (4.58), and (4.59), we deduce 1

vk is bounded in L2 0 T H 2

(4.60)

In what follows, we shall estimate the term vk . Since Ek t ≤ Ek 0 for all t ≥ 0, we have that vk tDA =

uk tDA ck

≤ C6

uk tL2 ck

≤ C6

uk 0L2 ck


1807

where C6 is a positive constant which does not depend on k. From the above inequality and taking (4.52) into account we deduce that

vk is bounded in L 0 T DA and since DA = H01 ∩ H 2 and w w ≥ sup = wH01 ∩H 2 1 ∈DA ∈H0 ∩H 2

wDA = sup we infer

vk is bounded in L 0 T H01 ∩ H 2


Therefore, from (4.42), it results that

vk is bounded in L2 0 T H01 ∩ H 2 once

aguk ck

(4.61)

is bounded in L2 0 T L2

Now, making use of the embedded chain c

1

H 2 → L2 → H01 ∩ H 2 it follows from the boundness (4.60) and (4.61) and employing Aubin-Lions theorem, that there exists a subsequence of vk , still denote by the same form such that, vk −→ v˜ strongly in L2 0 T L2

(4.62)

Again, considering the fact that the energy is a nonincreasing function on the parameter t from (4.51), it results that there exists v ∈ L 0 T L2 such that

vk v weak star in L 0 T L2

(4.63)

and from (4.46) and (4.62) we deduce that vt =

v˜ t in = \ 0 in

(4.64)

Thus, taking k → in (4.42) we conclude that

v − iv = 0 in × 0 T vx t = 0 in × 0 T

(4.65)

where v ∈ L 0 T L2 . Applying the same arguments used in case i, jointly with Holgren’s uniqueness theorem we conclude that v = 0 a. e. in × 0 T, which

1808


is a contradiction, since from (4.41) vk L2 0T L2 = 1 and from (4.62) vk −→ 0 in L2 0 T L2

which concludes the proof of the lemma.

We observe that taking (4.20) into account and considering Lemma (4.1), we obtain the desired inequality, namely, T T Etdt ≤ C axgu¯udxdt 0

0


where C is a positive constant. Proof of Theorem 1.1: Exterior Domains. Let u be a regular solution to problem (3.16). According to Lemma (4.1), from the inequality (4.20) and considering the assumption (3.1) we have that, for all T > 0, there exists C = CT > 0 such that T T Etdt ≤ C axgu¯udxdt 0

0

≤ C

T

0

axu2 dxdt

(4.66)

Since the energy is a nonincreasing function on the parameter t, we obtain, for all T ≥ T0 , T0 > 0, and t ∈ 0 T0 that T0 Etdt ET ≤ ET0 ≤ Et ⇒ T0 ET ≤ 0

On the other hand, from (4.66), we can write T0 T0 Etdt ≤ CT0 axu2 dxdt T0 ET ≤ 0

≤ CT0

T

0

0

axu2 dxdt

thus, ET ≤ C

T

0

axu2 dxdt

(4.67)

for all T ≥ T0 , where C is a constant such that C = CT0 > 0. Employing the identity of the energy and taking the assumption (3.1) into account, we infer E0 = ET +

0

T

axgu¯udxdt ≥ ET + c1

T 0

axu2 dxdt

that is, ET − E0 ≤ −c1

T 0

axu2 dxdt

(4.68)


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From (4.67) it results that −ET ≥ −C

T

0

axu2 dxdt

(4.69)

and, consequently, multiplying (4.68) by C and taking (4.69) into consideration we conclude, for all T ≥ T0 , that CET − E0 ≤ c1 −C

T

0

axu dxdt ≤ −c1 ET 2

namely,


ET ≤

1 C E0 E0 = C + c1 1+ C

where C = cC > 0. 1 Repeating the procedure for nT , n ∈ , we deduce EnT ≤

1 1 + Cn

E0

for all T ≥ T0 . Let us consider, now, t ≥ T0 , then t = nT0 + r 0 ≤ r < T0 . Thus, Et ≤ Et − r = EnT0 ≤ r

Setting C0 = e T0

ln1+ C

and 0 =

1 1 + Cn

ln1+ C T0

E0 =

1 t−r E0 1 + C T0

> 0 we obtain

Et ≤ C0 e−0 t E0 ∀t ≥ T0 which proves the exponential decay for regular solutions to problem (3.16). From standard arguments of density the exponential decay holds for weak solutions as well.

5. Schrödinger Equation on Noncompact Manifolds In what follows we shall omit some variables in order to make easier the notation and we will denote the Laplace-Beltrami operator by . We shall study following damped problem: iut + u + iaxgu = 0 in × 0 (5.70) in u0 = u0 where n g a noncompact, nontrapping n-dimensional Rimannian manifold, n ≥ 2, simply connected, orientable and without boundary endowed by a Riemannian metric g· · = · ·. In addition, assume that g is complete of class C .


1810


Figure 6. A trapped geodesics breaks the local smoothing effect.

A Riemannian manifold is nontrapping when, roughly speaking, there is no geodesic completely contained in a compact subset of . Remember that trapped geodesics breaks the local smoothing effect (see [9], [14]) which plays an important role in the proof (see Figure 6). The assumptions on the function g → are the same announced in Assumption 3.1. Assumption 5.1. Hypotheses on the function a → i) ax ∈ L is nonnegative function; ii) ax ≥ a0 > 0 em \ ∪ ∗ , where ⊂⊂ is an open, connected, and bounded subset with smooth boundary , and such that is a compact set. The set ∗ ⊂ is an open set as considered in Cavalcanti et al. [12] (see Section 6, p. 945), namely, ∗ ⊃ \V , where V = ∪ki=1 Vi ⊂ is an open set with boundary V = 1 V ∪ 2 V regular, such that 1 V intercepts transversally, measV ≥ meas − and meas2 V ≥ − , for all > 0. In addition, for each i = 1 k, there exists a strictly convex function fi Vi → + . It is worth mentioning that also in the present case we shall work with complexvalued functions so that in order that the spaces L2 , s well as, H 1 , become real Hilbert spaces, we shall define w vL2 = Re

w¯v d

5.1. Existence and Uniqueness of Solutions: Noncompact Manifolds Problem (5.70) can be rewritten as

ut − iu + axgu = 0 in × 0 u0 = u0 in

(5.71)

Noncompact Riemannian Manifolds The energy associated with problem (5.70) is, again, defined as: 1 Et = ux t2 d 2

1811

(5.72)

Defining A DA ⊂ L2 −→ L2 u −→ Au = −iu and, B DB ⊂ L2 −→ L2


u −→ Bu = agu we can show, exactly as considered previously in Section 3.1 that A is maximal monotone and B is monotone, hemicontinuous and take bounded sets in bounded sets. It is important to observe that in this case we have DA = w ∈ H 1 w ∈ L2 . Then, according to Barbu ([4], Corol. 1.1, p. 39) it results that A + B DA + B ⊂ L2 → L2 is maximal monotone. Consequently, for all initial data u0 ∈ DA + B = DA there exists, considering Brézis ([5], Theorem 3.1, p. 54), an unique function u 0 → L2 which is the regular solution to problem (5.70). In addition, for all initial data u0 ∈ L2 there exists, accordingly to Barbu ([4], Theorem 3.1, p. 152), an unique map u 0 → L2 which is the weak solution to problem (5.70). 5.2. Stability Result: Noncompact Manifolds In order to establish the exponential decay rate desired to problem (5.70), we strongly need two results, namely: 1) A unique continuation principle for the linear and homogeneous Schrödinger equation. 2) A local smoothing effect for the linear and nonhomogeneous Schrödinger equation. The first result can be found in the previous literature in a general setting, see, for instance, Triggiani e Xu ([21]). In fact, this result substitutes the traditional Holgren’s uniqueness theorem. However, the second result, posed in a general setting, will be assumed, that is, we shall impose: Assumption 5.2. Let u0 ∈ L2 , F ∈ L1 0 T L2 , for all T > 0. Then the solution u of problem iut + u = F in × 0 (5.73) in u0 = u0 1

2 belongs to the class u ∈ L2 0 T Hloc . In addition, for all ∈ C0 × such that supp ⊂ × 0 T, one has 1 (5.74) ≤ C u0 L2 + F L1 0T L2 u 2 2

L 0T H

1812


Remark 6. • When = n endowed with the Euclidean topology, Hypothesis 5.2 is no longer necessary, since this result is proved in Counstatin e Saut (see [13], Theorem 3.1, p. 425.). • Furthermore, endowing n with a Riemannian metric g such that the Riemannian manifold n g is nontrapping, the result assumed in the Hypothesis 5.2 remains valid. This is a direct consequence on the arguments presented in Burq [8], combined with those ones established in Burq, Gérard, and Tzvetkov in [9]. In this direction we would like to thank professor Nicolas Burq for fruitful discussion regarding this issue.


Before considering the stability result, let us recall the identity of the energy which will play an important role in the proof, namely, Et2 − Et1 = −

t2

t1

axgux tux t ddt

(5.75)

for all t2 > t1 ≥ 0. The above identity proved initially for regular solutions remains valid for weak ones for standard density arguments. 5.3. Unique Continuation Principle Let ∗ ⊂ an open, connected and bounded set with regular boundary such that ⊂⊂ ∗ and ∗ ⊂ ∗ , where and ∗ are those ones defined according to Assumption 5.1. Assume that problem  ∗  iut + u = 0 in × 0 T ux t = 0 in ∗ \ ∪ ∗ × 0 T   u0 = u0 ∈ L2 ∗

(5.76)

admits a solution u in the class u ∈ L 0 T L2 ∗ T > 0. Our objective is to prove that u = 0 in ∗ × 0 T by exploiting the unique continuation principle due to Triggini and Xu [21]. For this purpose, let us consider the following problem  ∗  ivt + v = 0 in × 0 T vx t = 0 in ∗ × 0 T   v0 = u0 ∈ L2 ∗

(5.77)

Problem (5.77) admits an unique solution v belonging to the class v ∈ C0 T L2 ∗ However, u is also a solution to problem (5.77), thus u = v almost everywhere. Consequently u ∈ C0 T L2 ∗


1813

and therefore, ux 0 = u0 x = 0 almost everywhere in ∗ \ ∪ ∗ , that is, u ∈ C0 T H and u0 ∈ H where H = w ∈ L2 ∗ w = 0 qs in ∗ \ ∪ ∗ . Setting H∗1 ∗ = w ∈ H01 ∗ w = 0 qs in ∗ \ ∪ ∗ and V = H∗1 ∗ ∩ H 2 ∗ , it results that V possesses a continuous and dense embedding in H. Since u0 ∈ H, there exists a sequence u0m ⊂ V such that u0m −→ u0 in H


For each m ∈ let us consider the problem:  ∗  ium + um = 0 in × 0 T in ∗ \ ∪ ∗ × 0 T um x t = 0   0 um 0 = um ∈ V

(5.78)

According to the semigroup theory, for each m ∈ , problem (5.78) admits a unique solutions um in the class um ∈ C0 T V ∩ C 1 0 T H where V = D−i, Consequently

where

we

are

considering

(5.79) −i D−i ⊂ H → H.

um ∈ H 1 0 T L2 ∗ ∩ L2 0 T H 1 ∗

(5.80)

In addition, employing Lummer-Philips theorem, −i is the infinitesimal generator of a contraction semigroup St, and, therefore, um t − un tL2 ∗ = Stu0m − Stu0n L2 ∗ ≤ u0m − u0n L2 ∗ which proves that um is a Cauchy sequence in C0 T H. Thus, there exists w ∈ C0 T H such that um −→ w strongly in C0 T H So, it results w0 = lim um 0 = lim u0m = u0 in H m→

m→

Then, taking m → in (5.78), we deduce that w is a weak solution to problem  ∗  iwt + w = 0 in × 0 T wx t = 0 in ∗ \ ∪ ∗ × 0 T   ∈ H w0 = u0

(5.81)

1814


and, by uniqueness of solution, we conclude that w = u, that is, um −→ u in C0 T H

(5.82)

The next step is to employ the unique continuation principle due to Triggiani and Xu (see [21], Theorem 2.3.1, p. 348), in the sequence of problems (5.78).


Remark 7. It is important to mention that in [12], the function f is constructed in a compact Riemannian manifold with boundary. In the present paper ∗ is just a compact set of . However, in spite of this, the construction is analogous and consequently it will be omitted. So, we are in a position to employ the unique continuation property above mentioned, since, for all m ∈ , we have um ∈ H 1 0 T L2 ∗ ∩ L2 0 T H 1 ∗ is a solution of (5.78) which satisfies: • um = 0 in ∗ \ ∪ ∗ × 0 T where, as previously mentioned, ∗ is an open subset of which contains \V and V = ∪ki=1 Vi . • Since \V ⊂ ∗ , for each i = 1 k, there exist open, bounded and connected sets Ai and Ci such that Ai ⊂ Ci ⊂⊂ Vi and um = 0 in Vi \Ai × 0 T. Thus, um = 0 in Ci × 0 T for all i = 1 k. • According to the construction of the functions fi Vi → + given in Cavalcanti et al. [12], for each i = 1 k, fi is strictly convex in Vi , consequently in Ci for all i = 1 k. Then, making use of the unique continuation principle due to Triggiani and Xu (see [21], Theorema 2.3.1, p. 348), it results that um = 0 in Ci × 0 T, for all i = 1 k, that is, um = 0 in Vi , for all i = 1 k, which proves that um = 0 in ∗ × 0 T. Consequently, according to (5.82), u = 0 in ∗ × 0 T where u is the solution of (5.76). From the above we can announce the following theorem: Theorem 5.1. Let ⊂⊂ ∗ ⊂ , two open, bounded and connected subsets of with smooth boundaries ∗ and such that and ∗ are compact sets. Let u be a solution to problem  ∗  iut + u = 0 in × 0 T ux t = 0 in ∗ \ × 0 T   ∈ L2 ∗ u0 = u0 belonging to the class u ∈ L 0 T L2 ∗ T > 0. Then u=0 in ∗ × 0 T. Relevant Comments. For each i = 1 k, fi is strictly convex in each component Vi of V so that we have to put damping in a subset inside with measure arbitrarily small according to the Figure 2.


1815

Following similar arguments as in [12], we can avoid put damping in radially symmetric disjoint regions as well (see Figure 3): There exist some Riemannian manifolds M g such that they admit a strongly convex function f , namely, there exists a positive constant c such that: HessfX X ≥ c gX X ∀ X ∈ TM


This happens, in general, when one has sectional curvature Kg non positive. In this case it is well known that (see [15], [24]): (a) M is noncompact. In addition, if inf f > −, then: (b) f has a (unique) minimum point; (c) f is proper; (d) M is diffeomorphic to n . In the above particular manifolds, namely, g such that Kg ≤ 0 it is possible to void put damping inside the whole (see Figure 4). 5.4. Controlling the Equation Analogously, our main task is to prove the following inequality:

T

Etdt ≤ C

0

0

T

axgu¯u ddt

where C is a positive constant, since proceeding analogously as we have done previously we obtain the exponential decay of the energy. It is sufficient to work with regular solutions to problem (5.70) since the decay for weak solutions is recovered by using arguments of density. So, let u be a regular solution to problem (5.70). We observe that, from the hypotheses: Assumption 3.1 and Assumption 5.1, we deduce T T T 2 Etdt = ux t2 ddt + ux t2 ddt 0

0

≤ ≤

0

0

T

\

0

T

ux t2 ddt + a−1 0

T 0

−1 ux t2 ddt + a−1 0 c1

\ T 0

axux t2 ddt

axgu¯u ddt

(5.83)

T Therefore, it remains to estimate 0 ux t2 ddt in terms of the damping term. For this purpose we shall announce the following lemma whose proof is identical to Lemma 4.1 and consequently it will be omitted. Lemma 5.1. Let u be a regular solution to problema (5.70), with initial data u0 such that u0 L2 ≤ L L > 0. Then, for all T > 0, there exists a constant C = CT > 0, such that T T ux t2 ddt ≤ C axgu¯u ddt (5.84) 0

0

1816


Proof. Is verbatim the same of Lemma 4.1. The main ingredients are the Assumption 5.2 and Theorem 5.1 and the rest of the proof is exactly the same as in Lemma 4.1. We observe that from (5.83) and applying Lemma (5.1) we have proved the desired inequality:

T 0

Etdt ≤ C

T

0

axgu¯udxdt

where C is a positive constant.


Proof of Theorem 1.1: Noncompact and nontrapping Manifold. Analogous proof of exterior domains and consequently will be omitted.

to

the

5.5. Nontrapping Riemannian Manifolds This section is devoted to some examples of nontrapping Riemannian manifolds. The second nontrivial example we borrow from the work due to Thorbergsson [22]. 1) The simplest example is the case where = n endowed with the Euclidean metric, where the geodesics are straight lines. A more refined one is when n is endowed with a metric g such that n g is nontrapping. 2) Let be an arbitrary simply connected Riemannian manifold endowed by a complete Riemannian metric g ∗ . We define in × the following metric: X Y = xy + er g ∗ X ∗ Y ∗ X = x X ∗ Y = y Y ∗ ∈ Trp × The Riemannian metric is clearly complete. We shall prove that × endowed with the metric defined above is nontrapping. Indeed, let ct = rt ut be a geodesic in × . We shall prove that the function rt does not possess maximum, which implies that there are no closed geodesics in × . Let us assume that there exists t0 maximum of rt. Let U u1 un a local coordinate system in a neighborhood of ut0 and let gik∗ be the local representation of g ∗ in U . In the local coordinate system × U Id u1 un the Riemannian metric possesses the following form: g00 = 1 gi0 = 0 for i ≥ 1 gik = er gik∗ for i k ≥ 1 The Christoffel symbols of the differential equation associated with rt are 0 = 0 for k ≥ 0 0k

ik0 = −

er ∗ g for i k ≥ 1 2 ik


1817

The differential equation corresponding to rt is r¨ t +

ij0 u˙ i tu˙ j t = 0

ij≥1

or, still, r¨ t −

er ∗ i g u˙ tu˙ j t = 0 2 ij≥1 ij

where r˙ = dr . dt Thus, we conclude that


r¨ t =

er ∗ ˙ ut ˙ g ut 2

˙ 0 = 0. Consequently r¨ t0 > 0, which implies Since r˙ t0 = 0, it results that ut that t0 is not a maximum point of rt, which is a contradiction. This ends the proof.

Appendix We start this section with the useful lemma: Lemma 6.1. With s0 > 0, let 0 s0 → + be a continuous function of a real variable, 0 = lim s ≥ 0 s > 0 for s > 0 such that s → ss is monotone increasing. s↓0

(6.85) Let us define the continuous function gz by gz = zz z ∈ and assume that g0 = 0 so that gs is increasing on 0 s0 . Then gz = zz satisfies the Assumption 3.1(ii) and (iii). Proof. See [18], Lemma 2.3, p. 494.

Remark 8. In the present context we are not taking into account the case such that 0 = +, since we are in a different scenario, namely, we are considering unbounded sets. As a consequence we need that c1 z2 ≤ gzz ≤ c1 z2 , for all z ∈ , instead of z ≥ 1 as considered in [18]. The following examples were borrowed from Lasiecka e Triggiani (see [18]), adapted to the present context: 1) The simplest example is when the dissipative term possesses a linear character, that is, gz = z. We observe that the Assumption 3.1 is trivially satisfied. 2) Let r C > 0. We consider s = sr + Cs with s ∈ 0 1

1818


Thus, gz = zr z + Cz with z ∈ 0 1 If one considers z ≥ 1, we define gz = C + 1z Employing Lemma 6.1, we guarantee that gz satisfies Assumption 3.1(ii) and (iii). We observe that g is continuous. In addition, if z ∈ 0 1, we deduce, gz ≤ zr+1 + Cz ≤ C + 1z


and, gz = zr + Cz ≥ Cz For z ≥ 1 the required inequalities in Assumption 3.1(iv) follow immediately. 3) Let C > 0. Let us consider 0 = 0 and 1

s = s2 e− s2 + Cs with s ∈ 0 1 Then g0 = 0 and gz = ze

−

1 z2

z + Cz with z ∈ 0 1

For z ≥ 1, we define gz = C + 1z Applying Lemma 6.1, we guarantee that gz satisfies Assumption 3.1(ii) and (iii). We note that g is continuous and g0 = 0 Furthermore, if z ∈ 0 1, we infer, gz ≤ z2 e

−

1 z2

+ Cz ≤ e−1 + Cz

and, gz ≥ z2 e

−

1 z2

+ Cz ≥ Cz

For z ≥ 1 the inequalities required in Assumption 3.1(iv) follow easily.

Acknowledgments The authors would like to thank Professor Nicolas Burq for fruitful discussions during the preparation of this manuscript and the anonymous referee for careful reading the manuscript and for his(her) comments and suggestions which resulted in the present version of it.


1819

Funding Research of César Augusto Bortot partially supported by the CNPq Grant 141122/2011-0. Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0.


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