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

This article was downloaded by: [York University Libraries]On: 12 August 2014, At: 15:46Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

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

Asymptotic Stability for the Damped SchrödingerEquation on Noncompact Riemannian Manifolds andExterior DomainsCésar A. Bortot a & Marcelo M. Cavalcanti ba Engineering Center of Mobility , Federal University of Santa Catarina , Joinville , Brazilb Department of Mathematics , State University of Maringá , Maringá , BrazilPublished online: 08 Aug 2014.

To cite this article: César A. Bortot & Marcelo M. Cavalcanti (2014) Asymptotic Stability for the Damped SchrödingerEquation 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

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/lpde20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/03605302.2014.908390

http://dx.doi.org/10.1080/03605302.2014.908390

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

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

Asymptotic Stability for the Damped SchrödingerEquation on Noncompact RiemannianManifolds and

Exterior Domains

CÉSAR A. BORTOT1 ANDMARCELO M. CAVALCANTI2

1Engineering Center of Mobility, Federal University of Santa Catarina,Joinville, Brazil2Department 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 Riemannianmanifold �� g� is considered. Assuming that �� g� is nontrapping and, in addition,that the damping term is effective in �\�, where � ⊂⊂ � is an open boundedand connected subset with smooth boundary ��, such that � is a compact set,exponential and uniform decay rates of the L2−level energy are established. Themain ingredients in the proof of the exponential stability are: (A) an uniquecontinuation property for the linear problem; and (B) a local smoothing effect forthe 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 estimatesof the energy E�t� �= 1

2

∫� �u�x� t��2 dx related to the Schrödinger equation subject

to a nonlinear and locally distributed damping, posed in a connected, complete andnoncompact n dimensional Riemannian manifold �� g�:

{iut + �u+ ia�x�g�u� = 0� in �× �0�+��

u�x� 0� = u0�x� x ∈ ��(1.1)

Received August 22, 2013; Accepted February 20, 2014Address correspondence to Marcelo M. Cavalcanti, Department of Mathematics, State

University ofMaringá, Av. Colombo 5790,Maringá 87020-900, Brazil; E-mail: [email protected]

1791

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014

1792 Bortot and Cavalcanti

where � denotes the Laplace-Beltrami operator. We shall divide our study intotwo fold: (i) when �� g� is an exterior domain of �n endowed with the Euclideanmetric. 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 paperwe shall suppose that the obstacle � is nontrapping, which means that any lightray reflecting on the boundary of � according to the laws of the geometric opticsleaves any compact set in finite time. In this case, Dirichlet boundary conditionis 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 Riemannianmetric 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]) thatif the domain is nontrapping in the geometrical optics sense, then the L2 localizedenergy, ��u�·� t��L2�BR�

(where BR means a ball of radius R > 0) decays like t−n/2. Inthe trapping case, no uniform decay rate is possible (see J. V. Ralston [19]). All theseresults are obtained through estimates of the resolvent �+ ��−1. More recently,when a = 0, a ≥ 0, that is, in presence of a linear damping term ia�x�u, Aloui andKhenissi [1] proved a similar polynomial decay rate estimate as in [23] by relaxingthe nontrapping condition as follows: We say that �n\� satisfies an EGC (exteriorgeometric control) condition, if each trapped ray meets the set x � a�x� > 0�. Theirproof proceeds via the established scheme; namely, L2 → L2 estimates for the cut-offresolvents. The present paper is concerned the exponential decay rate estimate forthe full (total) energy E�t� �= 1

2

∫�n\� �u�x� t��2 dx, instead of the local one Eloc�t� =∫

BR�u�x� t��2 dx, as considered in the references mentioned above. For this purpose

the additional damping term ia�x�g�u� is fundamental, since if a = 0 the full energyis conserved, namely, E�t� = E�0� for all t ≥ 0 and no decay is expected. In the firstpart of the paper, we shall assume that a�x� ≥ a0 > 0 in �, where � ⊂ � is definedin 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 presentarticle, we shall assume (roughly speaking) that a�x� ≥ 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.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014

Noncompact Riemannian Manifolds 1793

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

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

that Figures 2, 3, and 4 will contribute towards better understanding of this fact. Inany 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)), withthe energy defined as in (3.18) (respectively (5.72)). Then, under Assumption 3.1 and3.2 (respectively Assumption 3.1, 5.1, and 5.2) there exist positive constants T0, C0 and�0 such that

E�t� ≤ C0e−�0tE�0� ∀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 Triggianiand Xu [21]); and (B) a local smoothing effect for the linear and nonhomogeneous

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

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


associated problem (as in Burq, Gerard, and Tzvetkov [9]). In fact, the maingoal of the present paper is to prove that conditions (A) and (B) are sufficientto establish the exponential decay of the total energy E�t� �= 1

2

∫� �u�x� t��2 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 importantto observe that in order to employ the unique continuation property consideredin [21], it is essential the existence of a strictly convex function satisfying someproperties 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 trappedgeodesics breaks the smoothing effect or, in other words, in view of the necessityof the nontrapping condition for the plain smoothing effect (see Doi [14] or Burq[7]), it is crucial to consider “nontrapping” n-dimensional Riemannian manifoldsin 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 toobtain the smoothing effect and the uniform stabilization for the strongly dissipativeSchrödinger equation posed in a bounded domain of �n, which is not the case of thepresent manuscript. Finally, we would like to mention other papers in connectionto the stabilization of Schrödinger equation subject to locally distributed dampingposed in unbounded domains, as, for instance, [3], [10], [11] and references thereinand we would like to mention that the present paper is an extension of previousresults 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 metricg�·� ·� = �·� ·�, of class C�. We shall denote by �gij�n×n the matrix n× n inconnection with the metric g. The tangent space at � in p ∈ � will be denoted byTp� ≡ �n.

Let f ∈ C2��, and let us define the Laplace-Beltrami operator of f , as

�f = div��f�� (2.2)

where �f denotes the gradient of f in the metric g, that is, for all vector field Xin �

��f�X� = X�f�� (2.3)

and div denotes the divergent operator, namely, if X is a vector field in �,divX�p� �= trace of the linear map Y�p� �→ �YX�p�, p ∈ �.

From the definitions and notations above we have the following lemma:

Lemma 2.1. Let p ∈ �. Let us consider f ∈ C1�� and H a vector field in �. Then,the following identity hold (see [17] p. 21):

��f� ��H�f�� = �H��f� �f�+ 12�div��f �2H�− ��f �2divH��

where �H is the differential covariant derivative defined by �H�X� Y� = ��XH� Y �.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Finally we shall define the Hessian of f ∈ C2�� as the symmetric tensor oforder two in �, namely,

Hess�f��X� Y� = �2f�X� Y� �= ��f��X� 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, withoutdistinguishing 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 Cpk �� as

Cpk �� =

{u ∈ C��

∫��ju�p d� < ��∀j = 0� 1� � � � � k

}� (2.5)

where �ju denotes the jth differential covariant derivative of u� ��0u = u� �1u = �u).Thus, we define the Sobolev space H

pk �� as the closure of Cp

k �� with respectto the topology

�u�pH

pk ��

=k∑

j=0

∫��ju�p d�� (2.6)

From the above, we deduce:

i) L2�� = H20 �� is the closure of C2

0�� with respect to the tolopogy

�u�2L2�M� =∫��u�2 d�� (2.7)

ii) H1�� = H21 �� is the closure of C2

1�� with respect to the topology

�u�2H1�� =∫��u�2 d�+

∫��u�2 d�� (2.8)

iii) H2�� = H22 �� is the closure of C2

2�� with respect to the topology

�u�2H2�� =∫��2u�2 d�+

∫��u�2 d�+

∫��u�2 d�� (2.9)

Remark 2. From the above definitions we have the following chain of continuousembedding

H2�� ↪→ H1�� ↪→ L2�� (2.10)

Furthermore, by Hebey ([16], Theorem 2.7, p. 13), it follows that H10 �� =

H1��, where H10 �� = ��

H1��, in other words, the space of infinitely

differentiable functions with compact support is dense in H1��.

So, from the above and making use of density arguments we can extend theformulas presented previously to Sobolev spaces. In the sequel, we shall announcethree theorems that will play an important role in the present work (see [20]).

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Theorem 2.1 (Gauss Divergent Theorem). Let �n a Riemannian manifold, orientable,with smooth boundary ��, X ∈ �H1��n a vector filed and � the normal unitary vectorfield point towards ��, thus ∫

�divX d� =

∫��X� ��d�� (2.11)

Theorem 2.2 (Green Theorem 1). Let �n a Riemannian orientable manifold, withsmooth boundary ��, X ∈ �H1��n a vector field, q ∈ H1�� and � the normal unitaryvector field point towards ��, then∫

��divX�q d� = −

∫��X��q�d�+

∫��X� ��q d�� (2.12)

Theorem 2.3 (Green Theorem 2). Let �n a orientable Riemannian Manifold, withsmooth boundary ��, f ∈ �H2��, q ∈ H1�� and � the normal unitary vector fieldpoint towards ��, then∫

��f�q d� = −

∫��f� �q�d�+

∫��f�q d�� (2.13)

Remark 3. Let �� g� be a Riemannian manifold and u � � → � a smoothfunction. Then, we have Re u � � → � and Im u � � → �� consequently we cantalk 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 arereal vector fields. We shall denote by

��u�X� = ��Re u�� Y �g − ��Im u�� Z�g + i��Re u�� Z�g + ��Im u�� Y �g�

Consequently,

��u� �u� = ��Re u�� Re u��g + ��Im u�� Im u��g = ��u�2

2.1. Preliminaries: Partial Differential Equations

Let X be a real Banach space and X′ its topological dual. For all x ∈ X, we associatethe set

F�x� = f ∈ X′ �f� x�X′�X = �x�2 = �f�2��

and we define � � �s � X × X → � by

�x� y�s = sup �f� y�X′�X f ∈ F�x��

Remark 4. If X = H , where H is a Hilbert space, it results from Rieszrepresentation theorem that F�x� is an unitary set and from the identification H ≡H ′

we deduce that � � �s = � � �H .

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


2.2. Homogeneous Equation

Let us consider the following problem:{ut�t� = Tu�t�+ Bu�t�� t ∈ �0��

u�0� = 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��, u�0� = u0, u is differentiable almost everywhere in �0��, u�t� ∈ D�T + B�a. e. in �0�� and satisfies

ut�t� = Tu�t�+ Bu�t��

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��, u�0� = u0 and satisfies the following inequalityfor each T > 0

�u�t�− v�2X ≤ �u�s�− v�2X + 2∫ t

s�Tv+ Bu�� u��− v�s d� (2.15)

∀v ∈ D�T� 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 dampedproblem:

iut + �u+ ia�x�g�u� = 0 in �× �0��

u�x� t� = 0 in ��× �0��

u�0� = u0 in ��

(3.16)

where � is an exterior domain of �n, that is, � = �n\�� where � is an compactand connected subset of �n with smooth boundary. In addition, we suppose that �is nontrapping, namely, any light ray reflecting on the boundary of � according tothe laws of the geometric optics leaves any compact set in finite time. We observethat �� = ��, thus �� is smooth. Find in Figure 5 a counter example, where � isa trapping obstacle.

Assumption 3.1. Hypotheses on the function g � � → � �

(i) g�z� is continuous, g�0� = 0;(ii) Re�g�z�− g�w��z− w�� ≥ 0, ∀ z� w ∈ �;(iii) Img�z�z� = 0, ∀ z ∈ �;(iv) There exist positive constants c1 and c2, such that c1�z�2 ≤ �g�s�z� ≤ c2�z�2,

∀z ∈ �.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Figure 5. Example of a subset � such that its boundary does not satisfy the nontrappingcondition.

We observe that taking (ii) and (iii) into account, we deduce that g�z�z = �g�z�z�.

Remark 5. Find in the Appendix of this manuscript some examples of functions gsatisfying Assumption 3.1.

Assumption 3.2. Hypotheses on the function a � � → � �

i) a�x� ∈ L�� is nonnegative function;ii) a�x� ≥ a0 > 0 em �, where � ⊂ � is defined as follows:

Let R > 0 such that �� ⊂ BR = x ∈ �n �x� < R�, then � �= �\BR, accordingto Figure 1.

It is important to observe that we shall work with complex-valued functions, sothat, in order that the spaces L2��, as well as, Hm��, m ∈ �, become real Hilbertspaces, we define

�w� v�L2�� = Re∫�wvdx�

Finally, we shall denote by H10 �� the Hilbert space

H10 �� = w ∈ H1�� w�� = 0��

3.1. Existence and Uniqueness of Solutions: Exterior Domains

Problem (3.16) can be rewritten asut − i�u+ a�x�g�u� = 0 in �× �0��

u�x� t� = 0 in ��× �0��

u�0� = u0 in ��

(3.17)

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

E�t� = 12

∫��u�x� t��2dx� (3.18)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


We define the operators

A � D�A� ⊂ L2�� −→ L2��

u �−→ Au = −i�u�

and,

B � D�B� ⊂ L2�� −→ L2��

u �−→ Bu = ag�u��

Thus, D�A� = H10 �� ∩H2�� and D�B� = 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 �u�2

L2��≤ R. Thus, accordingly to the

Assumption (3.1) �iv� we infer

�Bu�2L2�� ≤ �a�2L��

∫��g�u�x��2dx

≤ �a�2L��c22

∫��u�x��2dx ≤ R�a�2L��c

22�

• B is monotone.Indeed, let u1� u2 ∈ L2��. Then, from Assumption 3.1 �ii� we obtain

�Bu1 − Bu2� u1 − u2�L2�� =∫�a�x� Re�g�u1�− g�u2��u1 − u2��dx ≥ 0�

• B is hemicontinuous.In fact, we have to prove that given an arbitrary sequence �tn� ⊂ � such thattn → 0� then

limn→��B�u+ tnv�� w�L2�� = �Bu�w�L2�� ∀ u� v� w ∈ L2��

For this purpose we define fn �= ag�u+ tnv�w. Thus,

�fn�x�� = a�x��g�u�x�+ tnv�x��w�x��≤ c2a�x��u�x�+ tnv�x��w�x��≤ c2�a�L��u�x��w�x�� + c2c3�a�L��v�x��w�x��

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

h�x� �= c2�a�L��u�x��w�x�� + c2c3�a�L��v�x��w�x��

it follows that h ∈ L1�� and fn�x� ≤ h�x�� almost everywhere in �.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


We observe that, from the continuity of g, we deduce

limn→� a�x�g�u�x�+ tnv�x��w�x� = a�x�g�u�x��w�x��

Thus, from Lebesgue dominated convergence theorem we conclude that∫��a�x�g�u�x�+ tnv�x��w�x�− a�x�g�u�x��w�x��dx → 0�

Then, ∣∣∣∫�a�x�g�u�x�+ tnv�x��w�x�− a�x�g�u�x��w�x�dx

∣∣∣ → 0�

and, consequently,

Re∫�a�x�g�u�x�+ tnv�x��w�x�dx → Re

∫�a�x�g�u�x��w�x�dx�

that is,

limn→��B�u+ tnv�� w�L2�� = �Bu�w�L2��

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 � D�A+ B� ⊂ L2�� → L2�� is maximalmonotone. Consequently, for each u0 ∈ D�A+ B� = D�A� = H1

0 �� ∩H2�� thereexists, taking Brézis ([5], Theorem 3.1, p. 54) into account, a unique functionu � �0�� → L2�� which is the regular solution to problem (3.16). In addition, forall u0 ∈ D�A� = L2�� there exists, considering Barbu ([4], Theorem 3.1, p. 152), aunique 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 importantidentity, 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, wededuce:

E�t2�− E�t1� = −∫ t2

t1

∫�a�x�g�u�x� t��u�x� t�dxdt� (4.19)

for all t2 > t1 ≥ 0, which remains valid for weak solutions by standard densityarguments.

Our main task is to prove the following inequality:∫ T

0E�t�dt ≤ C

∫ T

0

∫�a�x�g�u�udxdt�

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

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


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

2∫ T

0E�t�dt =

∫ T

0

∫�\�

�u�x� t��2dxdt +∫ T

0

∫��u�x� t��2dxdt

≤∫ T

0

∫�\�

�u�x� t��2dxdt + a−10

∫ T

0

∫�a�x��u�x� t��2dxdt

≤∫ T

0

∫�\�

�u�x� t��2dxdt + a−10 c−1

1

∫ T

0

∫�a�x�g�u�udxdt� (4.20)

Thus, it remains to estimate∫ T

0

∫�\� �u�x� t��2dxdt 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 = C�T� > 0, such that∫ T

0

∫�\�

�u�x� t��2dxdt ≤ C∫ T

0

∫�a�x�g�u�udxdt� (4.21)

Proof. We argue by contradiction. For simplicity we shall denote u′ �= ut as wellas �′ �= �\�.

Assume that (4.21) does not hold. Then, there exists a sequence of initial datau0

k�k∈� ⊂ D�A�, such that the corresponding regular solutions, uk�k∈�, of problem(3.16), with

Ek�0� =12�u0

k�2L2�� ≤ L�

for all k ∈ �, verify

limk→�

∫ T

0 �uk�t��2L2��′�dt∫ T

0

∫�a�x�g�uk�ukdxdt

= +�� (4.22)

that is,

limk→�

∫ T

0

∫�a�x�g�uk�ukdxdt∫ T

0 �uk�t��2L2��′�dt= 0� (4.23)

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

�uk�L��0�T L2�� ≤√2L� (4.24)

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

uk

�⇀ u weak star in L��0� T L2�� (4.25)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


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

∫ T

0

∫�a�x�g�uk�ukdxdt =

�uk�2L2�0�T L2��′��

�uk�2L2�0�T L2��′��

∫ T

0

∫�a�x�g�uk�ukdxdt → 0� (4.26)

when k → �. Then, from (4.26), assumption (3.1) and assumption (3.2), weconclude that∫ T

0

∫��uk�2dxdt ≤ a−1

0

∫ T

0

∫�a�x��uk�2dxdt ≤ a−1

0

∫ T

0

∫�a�x��uk�2dxdt

≤ a−10 c−1

1

∫ T

0

∫�a�x�g�uk�ukdxdt −→ 0� (4.27)

Consequently, taking (4.25) and (4.27) into account, we infer, for almosteverywhere t ∈ �0� T�,

u�t� ={u�t� in �′�0 in ��

(4.28)

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 problemsu′k − i�uk + a�x�g�uk� = 0 in �× �0��

uk�x� t� = 0 in ��× �0��

uk�0� = u0k in ��

(4.29)

We observe that from (4.26) and considering the assumptions (3.1) and (3.2) wecan write

�ag�uk��2L2�0�T L2�� ≤ �a�L��

∫ T

0

∫�a�x��g�uk��2dxdt

≤ �a�L��c22

∫ T

0

∫�a�x��uk�2dxdt

≤ �a�L��c22c

−11

∫ T

0

∫�a�x�g�uk�ukdxdt −→ 0�

that is,

ag�uk� −→ 0 strongly in L2�0� T L2�� (4.30)

Our objective is to pass to the limit in (4.29). Initially, we recall the followingembedded chain:

L2�0� T H10 �� ↪→ L2�0� T L2�� ≡ �L2�Q��′ ↪→ L2�0� T H−1�� ↪→ �′�Q��

(4.31)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


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 thefirst equation of (4.29) by �� and integrating over Q, we arrive at∫ T

0

∫�u′k�x� t��t��x�− i�uk�x� t��t��x�+ a�x�g�uk�x� t��t��x�dxdt = 0�

(4.33)

Integrating by parts and taking the real part it follows that

− Re∫ T

0

∫�uk�x� t��

′�t��x�dxdt − Re∫ T

0

∫�iuk�x� t��t��x�dxdt

+ Re∫ T

0

∫�a�x�g�uk�x� t��t��x�dxdt = 0� (4.34)

Taking k → � in (4.34) and taking (4.30) and (4.32) into account, we concludethat

−∫ T

0�u�t�� ′�t��L2��dt −

∫ T

0�iu�t�� t��L2��dt = 0 (4.35)

From the density of � � � ∈ D�0� T� and � ∈ �� in ��Q� it results that

−∫ T

0�u�t�� ′�t��L2��dt −

∫ T

0�iu�t�� t��L2��dt = 0� (4.36)

for all � ∈ ��Q�� namely,

u′ − i�u = 0 in �′�Q�� (4.37)

where u ∈ L��0� T L2�� and, from (4.28),

u�x� t� = 0 a�e� em �× �0� T�� (4.38)

Let us consider, now, the ball centered at the origin and radius 2R, B2R, andlet us define � = �\B2R ⊂ � keeping in mind that � = �\BR and �′ = �\�. Thus,from (4.37) and (4.38) we deduce{

u′ − i�u = 0 in �′��× �0� T��

u�x� t� = 0 in ��\�′�× �0� T��(4.39)

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�0�T L2��′�� and vk =uk

ck� (4.40)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


we deduce

�vk�L2�0�T L2��′�� = 1� (4.41)

Dividing (4.29) by ck we obtainv′k − i�vk + a�x�g�uk�

ck= 0 in �× �0��

vk�x� t� = 0 on ��× �0��

vk�0� = v0k �= u0kck

in ��

(4.42)

We observe that∫ T

0 �uk�2L2��′�dt∫ T

0


= c2kc2k

∫ T

0 �uk�2L2��′�dt∫ T

0


= c2k�vk�2L2�0�T L2��′��∫ T

0


= �vk�2L2�0�T L2��′��∫ T

0

∫�a�x�g�uk�

ukc2kdxdt

≤ c1�vk�2L2�0�T L2��′��∫ T

0

∫�a�x��vk�2dxdt

� (4.43)

Thus, taking (4.22) and (4.43) into account, we have

limk→�

�vk�2L2�0�T L2��′��∫ T

0


= +�� (4.44)

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

limk→�

∫ T

0

∫�a�x��vk�2dxdt = 0 (4.45)

From the fact that a�x� ≥ a0 > 0 em �, we conclude that∫ T

0

∫��vk�2dxdt ≤ a−1

0

∫ T

0


≤ a−10

∫ T

0

∫�a�x��vk�2dxdt −→ 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∫ T

0Ek�t�dt ≤

∫ T

0

∫�′�uk�2dxdt + a−1

0 c−11

∫ T

0

∫�a�x�g�uk�ukdxdt� (4.47)

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

2TEk�T� ≤ 2∫ T

0Ek�t�dt ≤

∫ T

0

∫�′�uk�2dxdt + a−1

0 c−11

∫ T

0

∫�a�x�g�uk�ukdxdt�

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Consequently,

Ek�T� ≤ C

[∫ T

0

∫�′�uk�2dxdt +

∫ T

0


]� (4.48)

Making use of the identity of the energy established in (4.19), we arrive at

Ek�T�− Ek�0� = −∫ T

0

∫�a�x�g�uk�ukdxdt�

so that,

Ek�0� =∫ T

0

∫�a�x�g�uk�ukdxdt + Ek�T�� (4.49)

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

Ek�0� ≤ C

[∫ T

0

∫�′�uk�2dxdt +

∫ T

0


]� (4.50)

Dividing (4.50) by c2k and having in mind that the energy is a nonincreasingfunction on the parameter t, we have, for all t ≥ 0, that

Ek�t�

c2k≤ Ek�0�

c2k≤ C

[∫ T

0


c2k+ 1

]� (4.51)

and from (4.23) and (4.51) we guarantee the existence of a constanteM > 0 such that

�v0k�2L2�� =�u0

k�2L��

c2k= 2Ek�0�

c2k≤ M� (4.52)

for all k ∈ �, and, therefore,

v0k� is bounded in L2�� (4.53)

Since vk is a regular solution to problem (4.42), we have that vk satisfies theintegral equation

vk�t� = S�t�v0k −∫ t

0S�t − s�

a

ckg�uk�s��ds� (4.54)

where S�t� is the semigroup generated by −i�. Employing the local smoothing effectdue to Burq, Gérard and Tzvetkov (see [9], Prop. 2.7, p. 302), we deduce, for allT > 0, and for all � ∈ C�

0 ��n�, that

��2wk�L2�0�T H1�� ≤ C1��fk�L2�0�T L2�� (4.55)

where

wk�t� =∫ t

0S�t − s�fk�s�ds and fk�s� =

a

ckg�uk�s��

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


In addition,

��S�t�v0k�L2�0�T H

12D ��

≤ C2��v0k�L2�� (4.56)

where H12D �� is the domain of the operator ��+ I�

14 . Considering � ∈ ��

0 ��n� such

that � = 1 in �′ and 0 ≤ � ≤ 1 in �n, we obtain, taking (4.55) and (4.56) intoaccount, that

��2wk�L2�0�T H1��′�� ≤ ��2wk�L2�0�T H1�� ≤ C1��fk�L2�0�T L2�� ≤ C3� (4.57)

since, from (4.23), is valid that

∫ T

0

∫�

�a�x�g�uk��2c2k

dxdt ≤ �a�L��

∫ T

0

∫�

a�x��g�uk��2c2k

dxdt

≤ �a�L��c22

∫ T

0

∫�

a�x��uk�2c2k

dxdt

≤ �a�L��c22c

−11

∫ T

0

∫�a�x�g�uk�ukdxdt∫ T

0

∫�′ �uk�2dxdt

−→ 0�

that is,

ag�uk�

ck−→ 0 strongly in L2�0� T L2��

and since � = 1 in �′ we deduce that

�wk�L2�0�T H1��′�� ≤ C3�

Furthermore, H1��′� ↪→ H12 ��′�, then

�wk�L2�0�T H12 ��′��

≤ C4� (4.58)

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

�S�t�v0k�L2�0�T H12 ��′��

≤ C5� (4.59)

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

vk� is bounded in L2�0� T H12 ��′�� (4.60)

In what follows, we shall estimate the term v′k. Since Ek�t� ≤ Ek�0� for all t ≥ 0,we have that

��vk�t��D�A��′ =��uk�t��D�A��′

ck≤ C6

�uk�t��L2��

ck≤ C6

�uk�0��L2��

ck�

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


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

�vk� is bounded in L��0� T� �D�A��′��

and since D�A� = H10 �� ∩H2�� and

�w��D�A��′ = sup�∈D�A�

��w�� ≥ sup

�∈H10 ��

′�∩H2��′�

��w�� = �w��H1

0 ��′�∩H2��′��′ �

we infer

�vk� is bounded in L��0� T� �H10 ��

′� ∩H2��′��′��

Therefore, from (4.42), it results that

v′k� is bounded in L2�0� T� �H10 ��

′� ∩H2��′��′�� (4.61)

once {ag�uk�

ck

}is bounded in L2�0� T L2��′��

Now, making use of the embedded chain

H12 ��′�

c↪→ L2��′� ↪→ �H1

0 ��′� ∩H2��′��′�

it follows from the boundness (4.60) and (4.61) and employing Aubin-Lionstheorem, that there exists a subsequence of vk�, still denote by the same form suchthat,

vk −→ v strongly in L2�0� T� L2��′�� (4.62)

Again, considering the fact that the energy is a nonincreasing function on theparameter 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

v�t� ={v�t� in �′ = �\��0 in ��

(4.64)

Thus, taking k → � in (4.42) we conclude that{v′ − i�v = 0 in �′��× �0� T��

v�x� t� = 0 in �× �0� T��(4.65)

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

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


is a contradiction, since from (4.41)

�vk�L2�0�T 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), weobtain the desired inequality, namely,∫ T

0E�t�dt ≤ C

∫ T

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 theassumption (3.1) we have that, for all T > 0, there exists C = C�T� > 0 such that∫ T

0E�t�dt ≤ C

∫ T

0

∫�a�x�g�u�udxdt

≤ C∫ T

0

∫�a�x��u�2dxdt� (4.66)

Since the energy is a nonincreasing function on the parameter t, we obtain, forall T ≥ T0, T0 > 0, and t ∈ �0� T0� that

E�T� ≤ E�T0� ≤ E�t� �⇒ T0E�T� ≤∫ T0

0E�t�dt�

On the other hand, from (4.66), we can write

T0E�T� ≤∫ T0

0E�t�dt ≤ C�T0�

∫ T0

0

∫�a�x��u�2dxdt

≤ C�T0�∫ T

0

∫�a�x��u�2dxdt�

thus,

E�T� ≤ C∫ T

0

∫�a�x��u�2dxdt� (4.67)

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

E�0� = E�T�+∫ T

0

∫�a�x�g�u�udxdt ≥ E�T�+ c1

∫ T

0

∫�a�x��u�2dxdt�

that is,

E�T�− E�0� ≤ −c1

∫ T

0

∫�a�x��u�2dxdt� (4.68)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


From (4.67) it results that

−E�T� ≥ −C∫ T

0

∫�a�x��u�2dxdt� (4.69)

and, consequently, multiplying (4.68) by C and taking (4.69) into consideration weconclude, for all T ≥ T0, that

C�E�T�− E�0�� ≤ c1

(−C

∫ T

0

∫�a�x��u�2dxdt

)≤ −c1E�T��

namely,

E�T� ≤ C

C + c1E�0� = 1

1+ CE�0��

where C = Cc1> 0.

Repeating the procedure for nT , n ∈ �, we deduce

E�nT� ≤ 1

�1+ C�nE�0��

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

E�t� ≤ E�t − r� = E�nT0� ≤1

�1+ C�nE�0� = 1

�1+ C�t−rT0

E�0��

Setting C0 = erT0

ln�1+C� and �0 = ln�1+C�

T0> 0 we obtain

E�t� ≤ C0e−�0tE�0� ∀t ≥ T0�

which proves the exponential decay for regular solutions to problem (3.16). Fromstandard arguments of density the exponential decay holds for weak solutionsas well.

5. Schrödinger Equation on Noncompact Manifolds

In what follows we shall omit some variables in order to make easier the notationand we will denote the Laplace-Beltrami operator by � . We shall study followingdamped problem: {

iut + �u+ ia�x�g�u� = 0 in �× �0��

u�0� = u0 in ��(5.70)

where ��n� g� a noncompact, nontrapping n-dimensional Rimannian manifold,n≥ 2, simply connected, orientable and without boundary endowed by aRiemannian metric g�·� ·� = �·� ·�. In addition, assume that g is complete of class C�.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


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

A Riemannian manifold � is nontrapping when, roughly speaking, there is nogeodesic completely contained in a compact subset of �. Remember that trappedgeodesics breaks the local smoothing effect (see [9], [14]) which plays an importantrole in the proof (see Figure 6).

The assumptions on the function g � � → � are the same announced inAssumption 3.1.

Assumption 5.1. Hypotheses on the function a � � → � �

i) a�x� ∈ L�� is nonnegative function;ii) a�x� ≥ 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] (seeSection 6, p. 945), namely, �∗ ⊃ �\V , where V = ∪k

i=1Vi ⊂ � is an open set withboundary �V = �1V ∪ �2V regular, such that �1V intercepts �� transversally,meas�V� ≥ meas��− � and meas��2V� ≥ ��− �, 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 complex-valued functions so that in order that the spaces L2��, s well as, H1��, becomereal Hilbert spaces, we shall define

�w� v�L2�� = Re∫�wv d��

5.1. Existence and Uniqueness of Solutions: Noncompact Manifolds

Problem (5.70) can be rewritten as{ut − i�u+ a�x�g�u� = 0 in �× �0��

u�0� = u0 in ��(5.71)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


The energy associated with problem (5.70) is, again, defined as:

E�t� = 12

∫��u�x� t��2d�� (5.72)

Defining

A � D�A� ⊂ L2�� −→ L2��

u �−→ Au = −i�u�

and,

B � D�B� ⊂ L2�� −→ L2��

u �−→ Bu = ag�u��

we can show, exactly as considered previously in Section 3.1 that A is maximalmonotone and B is monotone, hemicontinuous and take bounded sets in boundedsets. It is important to observe that in this case we have D�A� = w ∈ H1�� w ∈L2��. Then, according to Barbu ([4], Corol. 1.1, p. 39) it results that A+ B �D�A+ B� ⊂ L2�� → L2�� is maximal monotone. Consequently, for all initialdata u0 ∈ D�A+ B� = D�A� 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 weaksolution to problem (5.70).

5.2. Stability Result: Noncompact Manifolds

In order to establish the exponential decay rate desired to problem (5.70), westrongly need two results, namely:

1) A unique continuation principle for the linear and homogeneous Schrödingerequation.

2) A local smoothing effect for the linear and nonhomogeneous Schrödingerequation.

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 traditionalHolgren’s uniqueness theorem. However, the second result, posed in a generalsetting, will be assumed, that is, we shall impose:

Assumption 5.2. Let u0 ∈ L2��, F ∈ L1�0� T L2��, for all T > 0. Then thesolution u of problem {

iut + �u = F in �× �0��

u�0� = u0 in ��(5.73)

belongs to the class u ∈ L2�0� T H12loc��. In addition, for all � ∈ C�

0 ��×�� suchthat supp�� ⊂ �× �0� T�, one has

��u�L2�0�T H

12 ��

≤ C�

(�u0�L2�� + �F�L1�0�T L2��

)� (5.74)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Remark 6.

• When � = �n endowed with the Euclidean topology, Hypothesis 5.2 is nolonger 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 theRiemannian manifold ��n� g� is nontrapping, the result assumed in theHypothesis 5.2 remains valid. This is a direct consequence on the argumentspresented in Burq [8], combined with those ones established in Burq, Gérard,and Tzvetkov in [9]. In this direction we would like to thank professorNicolas Burq for fruitful discussion regarding this issue.

Before considering the stability result, let us recall the identity of the energywhich will play an important role in the proof, namely,

E�t2�− E�t1� = −∫ t2

t1

∫�a�x�g�u�x� t��u�x� t� d�dt� (5.75)

for all t2 > t1 ≥ 0. The above identity proved initially for regular solutions remainsvalid 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 toAssumption 5.1. Assume that problem

iut + �u = 0 in �∗ × �0� T��

u�x� t� = 0 in ��∗ \�� ∪�∗ × �0� T��

u�0� = u0 ∈ L2��∗��(5.76)

admits a solution u in the class u ∈ L��0� T L2��∗�� T > 0. Our objective is toprove that u = 0 in �∗ × �0� T� by exploiting the unique continuation principle dueto Triggini and Xu [21]. For this purpose, let us consider the following problem

ivt + �v = 0 in �∗ × �0� T��

v�x� t� = 0 in ��∗ × �0� T��

v�0� = u0 ∈ L2��∗��(5.77)

Problem (5.77) admits an unique solution v belonging to the class

v ∈ C��0� T� L2��∗��

However, u is also a solution to problem (5.77), thus u = v almost everywhere.Consequently

u ∈ C��0� T� L2��∗��

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


and therefore, u�x� 0� = u0�x� = 0 almost everywhere in ��∗\�� ∪�∗, that is,

u ∈ C��0� T� H� and u0 ∈ H�

where H = w ∈ L2��∗� w = 0 q�s� in ��∗\�� ∪�∗�. Setting H1∗ ��

∗� = w ∈H1

0 ��∗� w = 0 q�s� in ��∗\�� ∪�∗� and V = H1

∗ ��∗� ∩H2��∗�, it results that V

possesses a continuous and dense embedding in H . Since u0 ∈ H , there exists asequence u0

m� ⊂ V� such that

u0m −→ u0 in H�

For each m ∈ �� let us consider the problem:iu′

m + �um = 0 in �∗ × �0� T� �

um�x� t� = 0 in ��∗\�� ∪�∗ × �0� T��

um�0� = u0m ∈ V�

(5.78)

According to the semigroup theory, for each m ∈ �, problem (5.78) admits aunique solutions um in the class

um ∈ C��0� T� V� ∩ C1��0� T� H�� (5.79)

where V = D�−i��, where we are considering −i� � D�−i�� ⊂ H → H .Consequently

um ∈ H1�0� T L2��∗�� ∩ L2�0� T H1��∗�� (5.80)

In addition, employing Lummer-Philips theorem, −i� is the infinitesimalgenerator of a contraction semigroup S�t�, and, therefore,

�um�t�− un�t��L2��∗� = �S�t�u0m − S�t�u0

n�L2��∗� ≤ �u0m − u0

n�L2��∗��

which proves that um� is a Cauchy sequence in C��0� T� H�. Thus, there exists w ∈C��0� T� H�� such that

um −→ w strongly in C��0� T� H��

So, it results

w�0� = limm→� um�0� = lim

m→� u0m = u0 in H�

Then, taking m → � in (5.78), we deduce that w is a weak solution to problemiwt + �w = 0 in �∗ × �0� T�

w�x� t� = 0 in ��∗\�� ∪�∗ × �0� T�

w�0� = u0 ∈ H�

(5.81)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


and, by uniqueness of solution, we conclude that w = u, that is,

um −→ u in C��0� T� H�� (5.82)

The next step is to employ the unique continuation principle due to Triggianiand 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 constructedin a compact Riemannian manifold with boundary. In the present paper �∗ is justa compact set of �. However, in spite of this, the construction is analogous andconsequently it will be omitted.

So, we are in a position to employ the unique continuation property abovementioned, since, for all m ∈ �, we have

um ∈ H1�0� T L2��∗�� ∩ L2�0� T H1��∗��

is a solution of (5.78) which satisfies:

• um = 0 in ��∗\�� ∪�∗ × �0� T� where, as previously mentioned, �∗ is anopen subset of � which contains �\V and V = ∪k

i=1Vi.• Since ��\V� ⊂ �∗, for each i = 1� � � � � k, there exist open, bounded andconnected 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 inCavalcanti 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 uniquecontinuation 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 thesolution of (5.76).

From the above we can announce the following theorem:

Theorem 5.1. Let � ⊂⊂ �∗ ⊂ �, two open, bounded and connected subsets of � withsmooth boundaries �� ∗ and such that � and �∗ are compact sets. Let u be asolution to problem

iut + �u = 0 in �∗ × �0� T��

u�x� t� = 0 in ��∗ \��× �0� T��

u�0� = u0 ∈ L2��∗��

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 eachcomponent Vi of V so that we have to put damping in a subset inside � withmeasure arbitrarily small according to the Figure 2.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Following similar arguments as in [12], we can avoid put damping in radiallysymmetric disjoint regions as well (see Figure 3):

There exist some Riemannian manifolds �M� g� such that they admit a stronglyconvex function f , namely, there exists a positive constant c such that:

Hess�f��X�X� ≥ c g�X�X�� ∀X ∈ TM�

This happens, in general, when one has sectional curvature Kg non positive. In thiscase 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 possibleto 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

0E�t�dt ≤ C

∫ T

0

∫�a�x�g�u�u d�dt�

where C is a positive constant, since proceeding analogously as we have donepreviously we obtain the exponential decay of the energy. It is sufficient to workwith regular solutions to problem (5.70) since the decay for weak solutions isrecovered 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

2∫ T

0E�t�dt =

∫ T

0

∫��u�x� t��2 d�dt +

∫ T

0

∫�\�

�u�x� t��2 d�dt

≤∫ T

0

∫��u�x� t��2 d�dt + a−1

0

∫ T

0

∫�\�

a�x��u�x� t��2 d�dt

≤∫ T

0

∫��u�x� t��2 d�dt + a−1

0 c−11

∫ T

0

∫�a�x�g�u�u d�dt� (5.83)

Therefore, it remains to estimate∫ T

0

∫��u�x� t��2 d�dt in terms of the damping

term. For this purpose we shall announce the following lemma whose proof isidentical 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 suchthat �u0�L2�� ≤ L� L > 0. Then, for all T > 0, there exists a constant C = C�T� > 0,such that ∫ T

0

∫��u�x� t��2 d�dt ≤ C

∫ T

0

∫�a�x�g�u�u d�dt� (5.84)

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Proof. Is verbatim the same of Lemma 4.1. The main ingredients are theAssumption 5.2 and Theorem 5.1 and the rest of the proof is exactly the same as inLemma 4.1. �

We observe that from (5.83) and applying Lemma (5.1) we have proved thedesired inequality: ∫ T

0E�t�dt ≤ C

∫ T

0


where C is a positive constant.

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

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 Euclideanmetric, where the geodesics are straight lines. A more refined one is when �n isendowed with a metric g such that ��n� g� is nontrapping.

2) Let � be an arbitrary simply connected Riemannian manifold endowed by acomplete Riemannian metric g∗. We define in �×� the following metric:

�X� Y � �= xy + erg∗�X∗� Y ∗� X = �x� X∗�� Y = �y� Y ∗� ∈ T�r�p��×��

The Riemannian metric � � � is clearly complete. We shall prove that �×�endowed with the metric � � � defined above is nontrapping. Indeed, let c�t� =�r�t�� u�t�� be a geodesic in �×�. We shall prove that the function r�t� does notpossess maximum, which implies that there are no closed geodesics in �×�. Let usassume that there exists t0 maximum of r�t�. Let �U� �u1� � � � � un�� a local coordinatesystem in a neighborhood of u�t0� and let �g∗ik� be the local representation of g∗ inU . In the local coordinate system

��× U� �Id� u1� � � � � un��

the Riemannian metric � � � possesses the following form:

g00 = 1�

gi0 = 0 for i ≥ 1�

gik = erg∗ik for i� k ≥ 1�

The Christoffel symbols of the differential equation associated with r�t� are

�00k = 0 for k ≥ 0�

�0ik = −er

2g∗ik for i� k ≥ 1�

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


The differential equation corresponding to r�t� is

r�t�+ ∑i�j≥1

�0ij u

i�t�uj�t� = 0�

or, still,

r�t�− er

2

∑i�j≥1

g∗ij ui�t�uj�t� = 0�

where r = drdt.

Thus, we conclude that

r�t� = er

2g∗�u�t�� u�t��

Since r�t0� = 0, it results that u�t0� = 0. Consequently r�t0� > 0, which impliesthat t0 is not a maximum point of r�t�, 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 realvariable,

��0� = lims↓0

��s� ≥ 0� ��s� > 0 for s > 0� such that s → s��s� is monotone increasing.

(6.85)

Let us define the continuous function g�z� by

g�z� = ��z��z� z ∈ ��

and assume that g�0� = 0� so that g�s� is increasing on �0� s0�. Then g�z� = ��z��zsatisfies 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 suchthat ��0� = +�, since we are in a different scenario, namely, we are consideringunbounded sets. As a consequence we need that c1�z�2 ≤ g�z�z ≤ c1�z�2, 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, g�z� = 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��

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Thus,

g�z� = �z�rz+ Cz� with �z� ∈ �0� 1��

If one considers �z� ≥ 1, we define

g�z� = �C + 1�z�

Employing Lemma 6.1, we guarantee that g�z� satisfies Assumption 3.1(ii) and(iii). We observe that g is continuous. In addition, if �z� ∈ �0� 1�, we deduce,

�g�z�� ≤ �z�r+1 + C�z� ≤ �C + 1��z��

and,

�g�z�� = ��z�r + C��z� ≥ C�z��

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

��s� = s2e− 1

s2 + Cs� with s ∈ �0� 1��

Then g�0� = 0 and

g�z� = �z�e− 1�z�2 z+ Cz� with �z� ∈ �0� 1��

For �z� ≥ 1, we define

g�z� = �C + 1�z�

Applying Lemma 6.1, we guarantee that g�z� satisfies Assumption 3.1(ii) and (iii).We note that g is continuous and g�0� = 0� Furthermore, if �z� ∈ �0� 1�, we infer,

�g�z�� ≤ �z�2e− 1�z�2 + C�z� ≤ �e−1 + C��z��

and,

�g�z�� ≥ ��z�2e− 1�z�2 + C��z� ≥ C�z��

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 discussionsduring the preparation of this manuscript and the anonymous referee for carefulreading the manuscript and for his(her) comments and suggestions which resultedin the present version of it.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


Funding

Research of César Augusto Bortot partially supported by the CNPq Grant141122/2011-0. Research of Marcelo M. Cavalcanti partially supported by theCNPq Grant 300631/2003-0.

References

[1] Aloui, L., Khenissi, M. (2007). Stabilization of Schrödinger equation in exteriordomains. ESAIM Control Optim. Calc. Var. 13:570–579.

[2] Aloui, L., Khenissi, M., Vodev, G. (2013). Smoothing effect for the regularizedSchrödinger equation with non-controlled orbits. Comm. Partial DifferentialEquations 38:265–275.

[3] Aloui, L. (2002). Stabilisation Neumann pour l’ équation des ondes dans undomaine extérieur. [Neumann stabilization for the wave equation in an exteriordomain] J. Math. Pures Appl. 81:1113–1134 (in French).

[4] Barbu V. (1976). Nonlinear Semigroups and Differential Equations in BanachSpaces. Bucuresti, Romania: Editura Academiei.

[5] Brézis H. (1973). Operateurs Maximaux Monotones et Semigroups de Contractionsdans les Spaces de Hilbert [Monotone Operators and Maximum Semigroups ofContractions in Hilbert Spaces]. Amsterdam: North Holland Publishing Co.

[6] Bortot, C.A., Cavalcanti, M.M., Corrêa, W.J., Domingos Cavalcanti, V.N.(2013). Uniform decay rate estimates for Schrödinger and plate equations withnonlinear locally distributed damping. J. Differential Equations 254:3729–3764.

[7] Burq, N. (2004). Smoothing effect for Schrödinger boundary value problems.Duke Math. J. 123:403–427.

[8] Burq, N. (2001). Semi-classical estimates for the resolvent in nontrappinggeometries. International Mathematics Research Notices 5:221–241.

[9] Burq, N., Gérard, P., Tzvetkov, N. (2004). On nonlinear Schrödinger equationsin exterior domains. [Equations de Schrödinger non linéaires dans des domainesextérieurs.] Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 21:295–318.

[10] Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Natali, F. (2009).Exponential stability for the 2-D defocusing Schrödinger equation with locallydistributed damping. Differential Integral Equations 22:617–636.

[11] Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A., Natali, F. (2010).Qualitative aspects for the cubic nonlinear Schrödinger equations with localizeddamping: Exponential and polynomial stabilization. J. Differential Equations248:2955–2971.

[12] Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Soriano, J.A.(2010). Asymptotic stability of the wave equation on compact manifolds andlocally distributed damping: A sharp result. Arch. Rational Mech. Anal. 197:925–964.

[13] Counstantin, P., Saut, J.C. (1998). Local smoothing properties of dispersiveequation. Journal of the American Mathematical Society 1:413–439.

[14] Doi, S.I. (1996). Smoothing effects of Schrödinger evolution groups onRiemannian manifolds. Duke Math. J. 82:679–706.

[15] Greene, R.E., Wu, H. (1979). Function Theory on Manifolds which Possess a Pole.Lecture Notes in Mathematics, Vol. 699. Berlin: Springer.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014


[16] Hebey, E. (1996). Sobolev Space on Riemannian Manifolds. Berlin: Springer.[17] Lasiecka, I., Triggiani, R. (1999). Inverse/Observability estimates for second-

order hiperbolic equations with variable coefficients. Journal. Math. Anal. Apl.235:13–57.

[18] Lasiecka, I., Triggiani, R. (2006). Well-posedness and sharp uniform decayrates at the L2��-level of the Schrödinger equation with nonlinear boundarydissipation. Journal. Evol. Eq. 6:485–537.

[19] Ralston, J.V. (1969). Solutions of the wave equation with localized energy.Comm. Pure Appl. Math. 22:807–823.

[20] Taylor, E.M. (2010). Partial Differential Equations I. Basic Theory. 2nd ed.New York: Springer.

[21] Triggiani, R., Xu, X. (2007). Pointwise Calerman estimates, Global uniqueness,observability, and stabilization for de Schrödinger equations on Riemannianmanifolds at the H1��−level. Contemporary Mathematics 426:339–404.

[22] Thorbergsson, G. (1978). Closed geodesics on non-compact Riemannianmanifolds. Mathematische Zeitschrift 159:249–258.

[23] Tsutsumi, Y. (1984). Local energy decay of solutions to the free Schrödingerequation in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31:97–108.

[24] Schoen, R., Yau, S. (2010). Lectures on Differential Geometry. Somerville, MA:International Press.

Dow

nloa

ded

by [

Yor

k U

nive

rsity

Lib

rari

es]

at 1

5:46

12

Aug

ust 2

014

Documents

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