Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

Gong, Min; Jian, Hui

doi:https://doi.org/10.1155/2023/4316819

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2023 | Article ID 4316819 | https://doi.org/10.1155/2023/4316819

Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

Min Gong¹and Hui Jian¹

Academic Editor: Igor Freire

Received23 Apr 2023

Revised03 Oct 2023

Accepted04 Oct 2023

Published31 Oct 2023

Abstract

This article is concerned with the initial-value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass in the -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criterion in the -critical and -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the -minimal blow-up solutions in the -critical case. The main ingredients of the proofs are the variational characterisation of the ground state, a suitably refined compactness lemma, and scaling techniques. Our conclusions extend and compensate for some previous results.

1. Introduction

In this paper, we consider the initial-value problem of the following Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement:where is a complex valued function, and is a given function in , , and (), , , is the Riesz potential defined by the following:with and is the Gamma function.

Nonlinear Schrödinger equations of Hartree-type have a broad physical background. They often appear as models of quantum semiconductor devices [1]. When , Equation (1), known as Schrödinger–Hartree equation with complete harmonic confinement, can be used to characterise Bose–Einstein condensation (BEC) in a gas with very weak two-body interactions, which was found in or atomic systems [2]. When , Equation (1) is called a nonlinear Hartree equation with partial confinement, arising also as a typical model to describe the BEC [3]. When removing the harmonic confinement in Equation (1), for , , and , Equation (1) is used to describe electrons trapped in their own holes, which is similar to the Hartree–Fock theory of single component plasma to some extent [4].

When , Equation (1) with complete harmonic confinement has been well-studied. In the special case and , Huang et al. [5] applied the Hamiltonian invariants and the Gagliardo–Nirenberg inequality of convolution type and scaling technique to investigate the sharp threshold of global existence and showed the stability of standing waves in the mass-critical case . Wang [6] proved the existence of blow-up solutions and studied the strong instability of standing waves by variational methods in the mass-supercritical case . It’s worth mentioning that, Feng [7] derived the sharp threshold for global existence and finite time blow-up on mass for and in Equation (1), by using the variational characterisation of the ground state solution to a nonlinear Schrödinger–Hartree equation without potential (see Equation (14)). Moreover, in the general -supercritical case with , Feng [7] obtained blow-up versus global well-posedness criteria in both the -critical and -supercritical cases by constructing some cross-invariant manifolds and variational problems and studied the stability and instability of standing waves. If the nonlinearity is replaced by , there exist extensive literatures on the Cauchy problem of nonlinear Schrödinger equation with complete harmonic potential, see, e.g., [8–10]. In particular, Shu and Zhang [9] and Zhang [10] derived the sharp criterion of global existence to Equation (1) in the -critical and -supercritical cases by variational methods and constructing different cross-constrained variational problems and so-called invariant sets.

When , the main difference between nonlinear Schrödinger-type equation with partial harmonic confinement and complete confinement is that the embedding from natural energy space (see Section 2) to () is lack of compactness, resulting the main difficulty on the study of blow-up dynamics and stability of standing waves to the corresponding Cauchy problem. Due to the fact, the existence of stable standing waves, global and blow-up dynamics, and sharp criterion of global existence to the nonlinear Schrödinger-type equations with partial confinement have attracted considerable interest. A lot of studies have been made in these directions to Equation (1) with power type nonlinearity , see [11–16] for example and the references therein. More precisely, in the -supercritical case with and , Bellazzini et al. [11] applied the concentration compactness principle to overcome the lack of compactness and obtained the existence and stability of normalised standing waves. Ardila and Carles [12] studied the criteria of blow-up and scattering below the ground state in the focusing -supercritical case. Zhang [13] and Pan and Zhang [14] studied the sharp threshold for finite time blow-up and global existence in the mass-critical case by making full use of the ground state to a classical nonlinear elliptic equation without harmonic confinement and Hamilton conservation, as well as scaling arguments. It is worth noting that by exploiting the refined compactness lemma proposed by Hmidi and Keraani [17] and the variational characterisation of the ground state and scaling techniques, Pan and Zhang [14] investigated the mass concentration properties and limiting profile of the blow-up solutions possessing small super-critical mass for the 2D -critical Schrödinger equation with . More recently, when and , Wang and Zhang [15] derived the sharp condition for global existence and blow-up to the solutions by constructing cross-constrained variational problems and invariant manifolds of the evolution flow. Liu et al. [16] studied the existence and stability of normalised standing waves for Equation (1) with anisotropic partial confinement and inhomogeneous nonlinearity by making use of profile decomposition theory and concentration compactness principle. Besides, based on the ideas of Bellazzini et al. [11], the existence and orbital stability of standing waves for Equation (1) with were obtained by Xiao et al. [18]. As far as we know, the research of the sharp threshold of global existence and mass concentration phenomenon to the blow-up solutions of nonlinear Schrödinger-type equation with Hartree nonlinearity and partial confinement are still open, which is greatly pursued in physics. This is the main motivation for us to study the Cauchy problem (1).

In the absence of harmonic confinement in Equation (1), the corresponding equation is also known as the Choquard equation, which has also been extensively studied, see for instance [8, 19–23]. In particular, by constructing invariant sets and using variational methods, Chen and Guo [19] obtained the existence of blow-up solutions for some suitable initial data and showed strong instability of standing waves in the case and . Miao et al. [20] studied the mass concentration properties of blow-up solutions as well as the dynamics of blow-up solutions with minimal mass for Equation (1) in the -critical case with and . When , Genev and Venkov [21] gave a sharp sufficient condition of global existence to Equation (1). Furthermore, they proved the existence of blow-up solutions and considered the blow-up dynamics to the solutions in the -critical setting, i.e., with . Notice that Feng and Yuan [22] not only considered the local and global well-posedness and finite time blow-up to the corresponding initial-value problem (1) in the general case with , but also took into account the concentration phenomenon of blow-up solutions and the blow-up dynamics of blow-up solutions possessing minimal mass in the case , by establishing a new refined compactness lemma with respect to the nonlocal nonlinearity .

To the best of our knowledge, there are few papers dealing with the global well-posedness and blow-up dynamics to the Cauchy problem (1) in the presence of anisotropic partial/whole harmonic confinement. Inspired by the literatures aforementioned, the purposes of this present article are devoted to investigate the sharp criterion of global existence and mass-concentration phenomenon of blow-up solutions as well as the dynamical properties of minimal mass blow-up solutions of Equation (1) with anisotropic partial/whole confinement. The main difficulties come from the presence of anisotropic harmonic confinement and the nonlocal nonlinearity , resulting in the loss of compactness and pseudo-conformal transformation. Motivated by Feng [7], Zhang [13], Pan and Zhang [14], and Zhang [24], we utilise the ground state to the nonlinear Schrödinger–Hartree Equation (14), which is without any confined potential, to study the blow-up phenomenon and overcome the difficulties. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass for the Schrödinger–Hartree equation with anisotropic partial/whole harmonic confinement in the -critical case. This result extends and compensates the work of Feng [7], which only considered the case with complete confinement for and in Equation (1). Then, in the -critical and -supercritical cases, by constructing some new cross-invariant manifolds of the evolution flow and some variational problems associated with Equation (1), we derive blow-up versus global well-posedness criterion for Equation (1). In the present case, the constructed cross-invariant sets and variational problems are in light of Shu and Zhang [9], which differ from those of Feng [7], and some new criterion of global existence is derived, generalising and complementing the corresponding result of. Finally, based on the ideas of Pan and Zhang [14], Hmidi and Keraani [17], and Feng and Yuan [22], we research the mass concentration phenomenon of blow-up solutions and the dynamics of the -minimal blow-up solutions, including the precise mass-concentration and blow-up rate of the minimal mass blow-up solutions. The main ingredients of the proofs are the variational characterisation of the ground state to Equation (14), a refined compactness lemma established by Feng and Yuan [22], and scaling techniques. Our conclusions about the mass concentration phenomenon of blow-up solutions and the dynamics of the -minimal blow-up solutions extend the results of Feng and Yuan [22], in which the case without any potential was considered, to the Schrödinger–Hartree equation with anisotropic partial/whole confinement.

The rest of this paper is organised as follows: in Section 2, some notations and preliminaries are given. Section 3 considers the sharp threshold for global existence and finite time blow-up of Equation (1) in both the -critical and -supercritical cases. The last section focuses on the mass concentration phenomenon of blow-up solutions and the dynamics of the -minimal blow-up solutions.

2. Notations and Preliminaries

Throughout this paper, we use to represent and denote , and use to stand for positive constants, which may vary from line to line. Without loss of generality, we assume in this and subsequent sections.

For Equation (1), we equip the natural energy spacewith the inner productand the corresponding norm is given by the following:The energy function associated with Equation (1) is defined as follows:

Let us now state the local well-posedness of Equation (1) in energy space according to Feng [7] and Feng and Yuan [22].

Proposition 1. Let and . Then there exists such that Equation (1) admits a unique solution Let be the maximal time interval such that the solution is well-defined. If , then (blow-up). Furthermore, depends continuously on initial data and for any , the following conservation laws of mass and energy hold,

Then we introduce some vital lemmas.

Lemma 2 (see [25]). Let and , be constants such thatAssume that and . Then

By inequality Equation (10), we can obtain the following generalised Gagliardo–Nirenberg inequalityFollowing Weinstein [26], Feng and Yuan [22] derived the best constant in the inequality Equation (11) by discussing the existence of the minimiser of the functional

Lemma 3 (see [22]). It follows that the best constant in the generalised Gagliardo-Nirenberg inequality Equation (11) iswhere is the ground state of the elliptic equationIn particular, in the -critical case, .

It is known that the ground state is of great importance in studying global existence and blow-up dynamics to the initial-value problem of the nonlinear Schrödinger equation; in the following lemma, we recall some existence results and properties of the ground state solution to Equation (14).

Lemma 4 (see [27]). Let and It follows that Equation (14) admits a ground state solution in Every ground state of Equation (14) is in , and there exist and a monotone real function such that for every , . Moreover, the following Pohoaev identity holds,From Equations (15) and (16), one has

In order to study the blow-up phenomenon of Equation (1), we also need the following lemma obtained in Weinstein [26].

Lemma 5 (see [26]). Let , then we have that

Following the idea of Glassey [28] (see also Feng [7]), we will adopt the convexity method to study the existence of blow-up solutions. More precisely, we need to consider the varianceand show that there exists time such that . With some formal computations (which can be rigorously proved by Cazenave [8]), we have the following virial identities:

Proposition 6. Let and assume that is a solution of problem (1) in with and . Then the function belongs to . Furthermore, the function belongs to , then we obtain thatandfor all In particular, when we have

Using Lemma 5 and Proposition 6, we can easily get the following sufficient conditions on the existence of blow-up solutions.

Corollary 7. Assume that and . Let and , and satisfy one of the following conditions:Case (1):;Case (2): and ;Case (3): and .Then the corresponding solution of Equation (1) blows up in finite time.

3. Sharp Threshold for Global Existence and Blow-up

3.1. The -Critical Case

The aim of this subsection is mainly to consider the global existence and blow-up of the solutions to Equation (1) in the -critical case, i.e., . The ground state mass gives a sufficient condition on the global existence of the solution to Equation (1).

Theorem 8. Let and be the positive radially symmetric ground state solution of Equation (14). If and satisfies then the Cauchy problem (1) has a global solution in . Furthermore, we have for any

Proof. Let be the corresponding solution of Equation (1) in with initial value . By Equation (8), Equation (6), Lemma 3, and Equation (7), we obtainFrom Equations (25) and (23), we have for all , where is arbitrary and , there exists such thatThen, according to Proposition 1, exists globally in time. Moreover, we haveandIt follows from Equations (27) and (28) that Equation (24) holds true.

Remark 9. (i)When and in Equation (1), Feng [7] proved that the solution of Equation (1) exists globally (see Theorem 3.2 by Feng [7]). Theorem 8 can be viewed as the complement of the corresponding result of Feng [7] for Equation (1) with whole harmonic confinement.(ii)We give an explicit bound to the global solution of Equation (1) in (see Equation (24)).By using the variational characterisation of the ground state solution to Equation (14), some scaling arguments and energy conservation, we can get the existence result of blow-up solutions to Equation (1).

Theorem 10. Let be the positive radially symmetric solution of Equation (14), . Then for any , there exist and such that and the solution of the Cauchy problem (1) blows up in finite time.

Proof. For any , we take . Based on some scaling arguments, one has thatNow we setthen we have and . In fact, since , by utilising the exponential decay of ground state solution (see [27]):we conclude that and so and . Thus, we also deduce that . Moreover, it follows from Equation (30) thatFrom Equations (6), (8), (17), and (31)–(33), we getThus, it follows from Corollary 7 that the solution of Equation (1) blows up in finite time.

Remark 11. (i)When and in Equation (1), Feng [7] proved the existence of blow-up solutions (see Theorem 3.2 by Feng [7]). When considering Equation (1) in the presence of anisotropic partial/complete harmonic confinement, we derive the corresponding blow-up result by scaling approach, which differs from the method of Feng [7].(ii)Theorems 8 and 10 declare that provides a sharp threshold for global existence and blow-up to Equation (1) in terms of the initial data, which is called minimal mass for the blow-up solutions.

3.2. The -Supercritical Case

For and , define the following functionals:Then, we define the setand consider the following two constrained minimisation problems:

Proposition 12. If , then .

Proof. First, we prove . According to Lemma 3, there exists such that is a solution of Equation (14). By multiplying both sides of Equation (14) by and integrating over , we getIt follows from Equation (44) that Moreover, by taking the inner product of Equation (14) with , we have the following Pohoaev identityThen multiplying both sides of Equation (44) by , we haveFrom Equations (45) and (46), one has thatwhich implies . Thus, there exists such that and .
SetBy some simple computations, we obtainandNote that . Thus for every . Moreover,Thus, there exists such that . Therefore, when , we have and which implies .
Next, we prove Let , from , we get . Since , we haveIt follows from , Equation (52) and that for all . Thus, by Equation (43), we obtain . In the following, we will divide the proof into two cases: the -supercritical case and the -critical case.
We first consider the -supercritical case . In this case, it follows from Equation (10) thatwhich together with impliesThus, one has thatSince , we deduce from Equation (52) and (55) thatwhich implies for .
Now we deal with the -critical case . Suppose by contradiction that , then we derive from Equation (43) that there exists a sequence such that and as . Since , one can derive from Equation (52) thatOn the other hand, it follows from and Equation (11) thatHowever, when is sufficiently large, from Equation (57), one has thatIt is obvious that Equation (59) contradicts Equation (58). Thus, for . Therefore, we have for .

Now we defineThen, we have the following conclusion.

Proposition 13. Let , then .

Proof. From Equations (38) and (39), we obtainTherefore, . This, together with Proposition 12, implies that the proposition holds true.

To study the sharp threshold of global existence for Equation (1) in the -supercritical case, we introduce some new cross-constrained invariant sets as follows:

Proposition 14. Define Then , , , are invariant sets of Equation (1), that is, if , , or then the solution of the Equation (1) also satisfies , , or for any

Proof. We first prove that is an invariant set of Equation (1). Let and be the corresponding solution of Equation (1). From Equations (7) and (8), one has thatThus implies that for any
Now we show for If otherwise, by the continuity of on , there exists such that . By Equation (63), we have . It is clear that Equations (42) and (60) imply This is contradictory to for . Thus for all .
Then we show for all . On the contrary, from the continuity, there exists such that . Because we have shown and , it follows that . Thus, Equations (43) and (60) imply . This contradicts to for all . Therefore for all . From the above we have proved for any .
Similar to the proof above, we can also prove that , , are invariant manifolds.

In the below, we will use the cross-constrained variational approach to investigate the sharp condition of global existence for Equation (1).

Theorem 15. If , then the solution of the Cauchy problem (1) globally exists.

Proof. For , we have for by Proposition 14. For , one has and . It follows from Equations (38) and (40) thatFirst, we deal with the -critical case . In this case, we infer from Equation (64) thatDenote , then one hasIt follows from that there exists such that . Combining Equation (38) with Equation (40), we deduce thatwhich, together with Equation (64), yieldsNow we see , which only has two possibilities. One is . In this case, noting that , we infer from Equations (43) and (60) thatThus,That is,It follows thatBy Equation (65), we obtainFor , the other possible case is . In the present case, we deduce from the inequality Equation (68) thatSince and Equation (74), one hasIt follows from Equation (75) thatThus, according to Proposition 1, we obtain that the solution is global in time.
When , by Equation (64), we also haveBy Proposition 1, the solution of Equation (1) exists globally. Thus, the solution of Equation (1) with initial data exists globally on .
Now we consider . In view of Proposition 14, this gives immediately that the solution of Equation (1) satisfies that for . That is, and for . By Equations (38) and (39), we getThus, the solution of of Equation (1) exists globally. This completes the proof.

Theorem 16. Let . If and , then the solution of Cauchy problem (1) blows up in finite time.

Proof. For , we know from Proposition 14 that the solution of Equation (1) satisfies: for . For , it follows from Equations (22) and (40) thatThus for , satisfies that , . For , we take . ThusSince , , then there exists such that , and when , . For since , has the following two cases:(i) for ;(ii)There exists such that .For the case (i), we have and . It follows from Equations (43) and (60) thatFurthermore, one hasTaking into account that and , we infer from Equations (82) and (83) thatFor the case (ii), we have and . Thus, Equations (42) and (60) yield thatIt follows from Equations (82) and (83) thatSince , , from Equations (84) and (86), we obtainFrom , and Equation (87), one can estimate as follows:Then, by the convexity method introduced by Glassey [28], there must exist time such that . Then from Proposition 1 or Lemma 5, we haveThus, the proof is completed.

Remark 17. When and in Equation (1), Feng [7] derived the sharp threshold for global existence and blow-up to the solutions of Equation (1) (see Theorem 3.10 and Theorem 3.11 by Feng [7]). Our results in Theorems 15 and 16 extend and compensate for the ones of Feng [7] for Equation (1) with anisotropic partial/whole harmonic confinement by constructing some new cross-invariant sets and minimisation problems.

Remark 18. It is obvious thatIn this sense, Theorem 16 implies that Theorem 15 is sharp when .

By the above corollary, we immediately have

Corollary 19. Let and satisfy and . Then the solution of Equation (1) blows up in finite time if and only if .

By Theorem 15, we can get another sufficient condition of the global existence of Equation (1).

Corollary 20. If and , then the corresponding solution of Equation (1) exists globally.

Proof. Since , we have . Thus, we only need to prove . If otherwise, there exists with , such that . From Equations (40), (59), and , we haveOn the other hand,Therefore, we have , which gives a contradiction. Thus one has . It follows from Theorem 15 that the corollary holds true.

4. Mass Concentration and Dynamics of the -Minimal Blow-up Solutions

In this section, we are devoted to the dynamical properties of blow-up solutions to Equation (1) with partial/whole harmonic confinement. We first study the mass concentration phenomenon and then the dynamics of the -minimal blow-up solutions, including the precise mass-concentration and blow-up rate to the blow-up solutions with minimal mass.

In order to study the dynamical properties of the blow-up solutions of Equation (1), we recall the refined compactness lemma established by Feng and Yuan [22].

Lemma 21. Let , be a bounded sequence in and satisfyThen, there exists such that, up to a subsequence,with .
Using the refined compactness lemma, we can establish the following concentration property to the blow-up solutions of Equation (1).

Theorem 22. (-concentration) Assume and . Let be a solution of Equation (1) that blows up in finite time , and be a real-valued nonnegative function on such that as . Then there exists a function for such thatwhere is the ground state solution of Equation (14).

Proof. SetLet be an arbitrary time sequence such that as , and denote and . By Equations (7), (8), and (96), we obtainFor , we define the functionalFrom Equations (97), (6), and (96), one has thatwhich yields, in particular,Take and . Then by Lemma 21, there exist and such that, up to a subsequence,with . From Equation (101), it follows thatThen, from Equation (102) and the weakly lower semicontinuous of the -norm, it ensures that for any ,Sincethen there exists such that for any , we obtain that . It follows from and Equation (103) thatwhich implies thatDue to the arbitrariness of the sequence , from , we get thatFor every , one can easily see that the function is continuous on and . Therefore, for every , there exists a function such thatThus, it follows from Equations (107) and (108) that (101) holds true.

Remark 23. According to Theorem 22, we know that the blow-up solutions of Equation (1) must have a lower -bound, i.e., , which on the contrary, indicates that Theorem 8 holds true.

By Theorem 22, we can immediately obtain the conclusion below.

Corollary 24. Let be a solution of Equation (1) that blows up in finite time . Then for all , there exists for such thatwhere is the ground state solution of Equation (14) and

Theorem 25. Assume that and . Let and be the corresponding solution of problem Equation (1) that blows up in finite time with . Then(i)(Location of -concentration point) there exists such thatwhere is the ground state solution of Equation (14).(ii)(Blow-up rate) There exists a positive constant such that

Proof. (i)According to Equation (7) and , for , we haveOn the other hand, from Theorem 22 and Corollary 24, for all , one has thatIt is distinct that Equations (112) and (113) deducewhich implies thatNext, we will prove that there exists such thatIn fact, for any real-valued function defined on and any real number , from Equations (11) and (7), one can estimateTherefore, for any , we infer from Equation (8) thatwhich implies thatThen, choosing for in Equation (119), using Equations (1), (119), and (8), we deriveTaking any two sequences , such that . Therefore, for all , we deduce from the inequality Equation (120) thatwhich implies thatIn other words,Set , then and we obtainOn the other hand, we infer from Equation (22) thatThus, for any , there exists a constant such thatHence, we deduce thatFrom Equation (115), it follows thatFrom Equation (128), one can estimateCombined Equation (127) with Equation (129), we havewhere . Thus, for any , one hasTherefore, for any , there exists a large enough such thatThen, using Equations (132) and (115), we infer that for any It follows from Equations (124) and (133) thatTherefore, there exists (see Equation (124)) such thatThus, we know that Equation (110) holds true.(ii)Taking is a nonnegative radial function such thatFor , we define that and with define by Equation (124) (see also Equation (135)). From Equation (119), for every , we derivewhich implies thatBy integrating on both sides, one has thatIt is clear that Equation (110) implieswhere . Thus, by letting in Equation (139), we deduce thatFix and let , we obtainIt follows from the uncertainty principle and the above inequality thatwhich means thatTherefore, the whole proof is completed.

Remark 26. (i)For Equation (1) without harmonic confinement, Feng and Yuan [22] derived the similar mass concentration properties of blow-up solutions and dynamical properties of the -minimal blow-up solutions in the -critical case (see Theorems 1.4 and 1.5 by Feng and Yuan [22]). Theorems 22 and 25 in our present paper extend the corresponding conclusions of Feng and Yuan [22] to the Schrödinger–Hartree equation with anisotropic partial/whole harmonic confinement.(ii)As we know, the characterisation of the blow-up solutions with minimal mass depends strongly on the uniqueness of the ground state of Equation (14). However, in the general case and , the uniqueness of the ground state of Equation (14) is still open, so we cannot obtain the limiting profile of the minimal mass blow-up solutions to initial-value problem Equation (1) at the moment, except for some special cases discussed by Miao et al. [20] and Genev and Venkov [21].

Data Availability

No underlying data were collected or produced in this study.

Disclosure

A preprint has previously been published (Min Gong, Hui Jian, Sharp threshold of global existence and mass concentration for the Schrödinger–Hartree equation with anisotropic harmonic confinement, 2022), and a reference to the preprint has been included in the reference list, see [29]. The present manuscript is an improved version.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is partially supported by the Jiangxi Provincial Natural Science Foundation (grant no. 20212BAB211006) and the National Natural Science Foundation of China (grant no. 11761032).

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Copyright © 2023 Min Gong and Hui Jian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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