This paper is concerned with the blow-up solutions of the critical Gross-Pitaevskii equation, which models the Bose-Einstein condensate. The existence and qualitative properties of the minimal blow-up solutions are obtained.

1. Introduction and Main Results

In this paper, we deal with the Cauchy problem of the nonlinear Schrödinger equation with a harmonic potential

()

where ϕ = ϕ(t, x): [0, T) × ℝ^N → ℂ is the wave function, N is the space dimension, and Δ denotes the Laplace operator on ℝ^N. Equation (1) is also called Gross-Pitaevskii equation (see [1, 2]), which models the Bose-Einstein condensate (see [3, 4]). The harmonic potential |x|² describes a magnetic field. With the nonlinear term |ϕ|^4/Nϕ being replaced by |ϕ|^p−1ϕ, it is well known that the exponent p = 1 + 4/N is the minimal value for the existence of blow-up solutions (see e.g., [5, 6]). Hence (1) is called critical Gross-Pitaevskii equation.

Let us recall the classical nonlinear Schrödinger equation

()

For Cauchy problem (3)-(4), Ginibre and Velo [7] established the local existence in H¹(ℝ^N). Glassey [8], Weinstein [9], and Zhang [10] proved that, for some initial data, the solutions of the Cauchy problem (3)-(4) blow up in finite time.

For the Cauchy problem (3)-(4), it is well known that there exists a minimum of L² norm for the initial data of blow-up solutions (see [9]). More precisely, let Q(x) be the ground state, which is the unique, positive, radially symmetric solution (see [11]) of the semilinear elliptic equation

()

Weinstein [9] proved that the solutions of the Cauchy problem (3)-(4) are globally defined if

. On the other hand, for any

, there exist blow-up solutions with

. Since then, much progress has been made on the blow-up rate and profile of the blow-up solutions of the Cauchy problem (3)-(4) (see [12–15]). In particular, based on the pseudoconformal invariance of (3) and the variational characterization of the ground, elaborate and interesting conclusions were established on the existence and profile of the minimal blow-up solution, which is the blow-up solution ψ(t, x) such that

(see [13, 15, 16]). By using the pseudoconformal invariance of (3), Weinstein [15] constructed the explicit blow-up solution with critical mass (

) for (3) in the form

()

where a, b, c, d ∈ R, ad − bc = 1, and ab < 0. Moreover, Weinstein proved that, for any minimal blow-up solution ψ(t), the following holds:

()

where T is the blow-up time and y(t) ∈ ℝ^N and λ(t) ∈ ℝ are some suitable functions.

Merle [13, 16] proved that ψ(t, x) is a minimal blow-up solution of (3) if and only if there exist θ ∈ ℝ, ω > 0, x₀ ∈ ℝ^N, and x₁ ∈ ℝ^N such that

()

For the Cauchy problem (1)-(2), local well-posedness in energy space was established in Cazenave [17]. Moreover, from the result of Carles [18] and Zhang [6, 19], it is known that ϕ(t) is globally defined if . In other words, if ϕ(t) blows up in finite time.

Let ϕ(t) and ψ(t) be the solutions of the Cauchy problems (1)-(2) and (3)-(4), respectively. Under the condition of ϕ₀(x) = ψ₀(x), Carles [18] established a formula, which reflects the relation between ϕ(t) and ψ(t). According to the formula, Carles [18] established the following statements.

(1)
If ϕ(t) blows up at a finite time T_ϕ, then T_ϕ ≤ π/2.
(2)
If ϕ(t) blows up at T_ϕ < π/2, ψ(t) blows up at time T_ψ < ∞.
(3)
Conversely, ψ(t) blows up at time T_ψ < ∞; then ϕ(t) blows up at T_ϕ < π/2.
(4)
If ϕ(t) blows up at T_ϕ = π/2, ψ(t) exists globally (T_ψ = ∞).

Moreover, Carles studied the qualitative properties of minimal blow-up solutions ϕ(t) with T_ϕ < π/2 (see [18, 20]). As for the minimal blow-up solutions with T_ϕ = π/2, though the existence was established by the formula in [5], there is no further information on the qualitative properties obtained by the formula. Up to our knowledge, there is no result about the qualitative properties of the minimal blow-up solutions ϕ(t) of (1) with T_ϕ = π/2.

The purpose of the present paper is to investigate the qualitative properties of the minimal blow-up solutions without any limit to the blow-up time. The formula presented in [18] is not used to carry out the objective. We follow the ideas of Merle [13, 16], as well as Weinstein [15], in which the profile and uniqueness of the minimal blow-up solutions for (3) were investigated. However, in contrast to (3), (1) loses the invariance of pseudoconformal invariance, which is very important in the arguments of [13, 15, 16]. Therefore, some appropriate modifications will be made in the argument of this work to reach our goal. In particular, we note that some techniques developed by Pang et al. [21] are adopted in this paper.

We state our main results.

Theorem 1. There exist initial data ϕ₀ with for which the solution of the Cauchy problem (1)-(2) blows up in a finite time.

Theorem 2. Let ϕ(t) be a blow-up solution of (1) with . Then there is y₀ ∈ ℝ^N such that

()

in the sense of distribution as t → T.

Theorem 3. There exists C > 0 such that

()

Remark 4. For any blow-up solutions of (1), we know that T ≤ π/2 (T is a blow-up time). When T < π/2, the formula presented in [18] is valid. For the minimal blow-up solutions with T < π/2, the conclusion of the above theorems can be found in [18]. However, there exist minimal blow-up solutions with T = π/2. For example, if the initial ϕ₀(x) = ψ₀(x) = Q(x), with Q(x) being the solution of problem (5), the solution ϕ(t) of (1) will blow up at T = π/2, while the corresponding solution of (3) is a solitary wave e^itQ(x). The minimal blow-up solutions with T = π/2 were sensible as pointed in [18].

In this paper, L^q(ℝ^N), , and are denoted by L^q, , and ∫ · dx, respectively. The various positive constants are also denoted by C.

This paper proceeds as follows. In Section 2, we establish some preliminaries. In Section 3, we give the proof of the existence and profile of the minimal blow-up solutions of (1) (Theorems 1 and 2). In Section 4, we derive the argument of the lower bound of the blow-up rate of the minimal blow-up solutions of (1) (Theorem 3).

2. Preliminaries

2.1. Local Wellposedness

The energy space of (1) was defined as

()

The inner product of the space Σ is defined as

()

The norm of Σ is denoted by ∥·∥_Σ. Moreover, we define an energy functional ℰ on Σ by

()

From Cazenave [17], we have the local well-posedness for the Cauchy problem of (1) follows.

Proposition 5. For any ϕ₀ ∈ Σ, there exist T > 0 and a unique solution ϕ(t, x) of the Cauchy problem (1)-(2) in C([0, T); Σ) such that either T = ∞ (global existence) or T < ∞ and (blowup). Moreover, for any t ∈ [0, T), it holds the conservation laws of mass

()

and the energy

()

2.2. Variational Characterization of the Ground State

Consider the equation

()

For (16), we set some notations such as 𝒳_ω (the solution set), 𝒢_ω (the ground solution set), and 𝒢 as follows:

()

where S(u) = ∫ (1/2) | ∇u|² + (ω/2) | u|² − (1/4/N + 2) | u|^2+(4/N)dx.

For any u ∈ 𝒳_ω, the following two identities hold true:

()

The above two equalities imply

()

where

()

Naturally, we get

()

According to Cazenave [17], the set 𝒢_ω can be described as

()

where φ_ω is a positive, spherically symmetric, decreasing, and real valued function.

It is of importance that Kwong [11] proved the uniqueness for the solution Q(x) of the problem

()

Noticing the fact that Q(x) = φ_ω |_ω=1, it is easy to check that

()

It follows from (21), (22), and (24) that

()

With functional ℋ defined by (20), we now introduce the following constrained minimization problem

()

Now, we claim that

()

In fact,

is the minimum of the functional (see Kwong [11] or Weinstein [9])

()

which derives the Gagliardo-Nirenberg inequality

()

The inequality (30) implies the following lemma on the functional ℋ.

Lemma 6 (see Weinstein [9].)For any f ∈ H¹, one has

()

Lemma 6 implies that

()

It follows from (19), (27), and (32) that

()

Hence, from (19) and (25), it holds that

()

On the other hand, if u is a minimizer of the variational problem of (27), it solves the Euler-Lagrange equation (16). So u ∈ 𝒳_ω for some ω > 0, and by (27) and (25), we know u ∈ 𝒢_ω ⊂ 𝒢. This implies that

()

Hence (28) holds true.

Putting together (22), (25), and (28), we summarize the variational characterization.

Proposition 7. Each of the following three statements is equivalent:

(i)
,
(ii)
u is a solution to the minimizing problem ,
(iii)
u = e^iθω^N/4Q(ω^1/2(x − x₀)), for some θ ∈ ℝ, ω ∈ ℝ⁺, and x₀ ∈ ℝ^N.

2.3. Lemmas

Lemma 8 (see Zhang [6].)Let ϕ₀ ≠ 0, the initial datum of Cauchy problem (1)-(2), satisfy

()

then ϕ(t) blows up in a finite time.

Consider the constrained minimization problem

()

For I(α), we cite a lemma in [15].

Lemma 9 (see Weinstein [15].) (a) Consider I(α) = 0 or I(α) = −∞.

(b) Let and u_n be a minimizing sequence; then it holds that I(α) = 0 and u_n⇀0 weakly in H¹.

Now, we recall some lemmas on the compactness.

Lemma 10 (see Brezis and Lieb [22].)Let , , and μ(|f| > ε) ≥ δ > 0. Then there exists a shift T_yf(x) = f(x + y) such that, for some constant α = α(C, δ, ε),

()

Lemma 11 (see Lieb [23].)Let f_j be a uniformly bounded sequence of functions in W^1,p with 1 < p < ∞. Assume further that there are positive constant C and η satisfying μ(|f_j| > η) ≥ C. Then there exists a sequence y_j ∈ ℝ^N such that

()

Lemma 12. Let θ be a real-valued function on ℝ^N and v ∈ H¹(ℝ^N) with . Then

()

Proof. It follows from (30) and that

()

for all real numbers α. On the other hand, it has

()

Thus the discriminant of the equation in α must be negative or null and the desired inequality follows.

Lemma 13. There is a constant c₀ such that

()

Proof. Setting J(t) = ∫ |x|²|ϕ(t,x)|²dx, we have

()

It follows that

()

which implies the conclusion.

Lemma 14 (see [16], page 433.)Let u_n ∈ H¹, c₀ > 0, and R₀ > 0, for arbitrary n, satisfy

where ε(n) > 0 depends only on n. Then, it holds that

()

with A = A(r₀, c₀ > 0).

3. Profile of the Minimal Blow-Up Solution

Now we prove the existence of the minimal blow-up solutions.

Proof of Theorem 1. Setting ϕ₀ = ϕ(c, λ) = cλ^N/2Q(λx) with λ being arbitrary positive real number and c being complex number satisfying |c| = 1, then

()

From (15) and (19), the corresponding energy is

()

Thus Lemma 8 infers that ϕ(t, x) blows up in a finite time.

Employing the concentration compactness lemma, we can prove the following proposition which is crucial to the study of the blow-up profile (Theorem 2).

Proposition 15. Let ϕ(t) ∈ C([0, T), Σ) be a blow-up solution of the Cauchy problem (1)-(2) and T is the blow-up time. Set and (S_λϕ)(x, t) = λ^N/2ϕ(λx, t). If

()

it holds that

()

with y(t) ∈ ℝ^N and γ(t) ∈ ℝ.

Proof. Let t_k → T. We choose λ_k = λ(t_k) to satisfy

()

Setting

, noticing that

tends to ∞ as t_k → T, λ_k → 0, and

()

we know that ϕ_k is uniformly bounded in H¹ and there is a weakly convergent subsequence

such that

()

We note that

()

Since we have assumed

, by (52), (54), and (31), we know that ϕ_k is a minimizing sequence for the variational problem (27).

Next, we will prove that the minimizing sequence ϕ_k has a subsequence and a family y_j such that has a strong limit in H¹. To see this, we need to make use of the concentration-compactness lemma (Lions [24]) which means that has one of three properties: vanishing, dichotomy, and compactness.

Vanishing. For every M < ∞, one has

()

Dichotomy. There exist a constant and sequences and , bounded in H¹, such that, for all ε > 0, there exists j₀ > 0 such that for j > j₀

()

Compactness. There exists y_j in ℝ^N. For any ɛ > 0, we can find M < ∞ such that

()

Now, we exclude the cases of vanishing and dichotomy.

Exclusion of Vanishing. By (52), (51), and (54) there are C₁ > 0 and C₂ > 0 such that

()

By the boundness of

and the Sobolev inequality, there exist γ > 2 + 4/N and C₃ > 0 such that

()

Now, we show the existence of positive constants ε and δ such that

()

Indeed, from (58) and (59), for sufficiently small ε > 0, we get

()

Thus we know that (60) with δ = (C₂/2)ε^2+4/N is valid. From (60) and Lemma 10, there exist α and y_k satisfying

()

Thus,

()

which excludes the occurrence of vanishing.

Exclusion of Dichotomy. Suppose by contradiction that dichotomy occurs. Then, by the same argument as that in the case of vanishing we can get

()

where θ and υ are two constants and

is bounded in H¹. Hence, by Lemma 11, there are a subsequence

and a sequence y_r such that

()

Using (56) gives rise to

()

On the other hand, the fact

implies with Lemma 6 that

()

Thus, for any fixed n^*, it has

()

We can then extract a minimizing subsequence, which we rename it by

; that is,

. Using Lemma 9 yields

()

which is impossible from (65).

Occurrence of Compactness. It follows from the previous arguments that compactness occurs. By (57), we get

()

For

being bounded in H¹(ℝ^N), there exist ϕ ∈ H¹(ℝ^N) and a subsequence, which we again label it by

, such that

()

Given M > 0, the embedding H¹(ℝ^N)↪L²({|x| ≤ r}) is compact and

()

Making use of (70) derives

()

for any ε > 0. Hence, it holds that

()

It follows that

()

which implies with the Gagliardo-Nirenberg inequality (30) that

()

To show in H¹, we only need to show that .

From (51) and (54), we know that

()

Hence,

derives ℰ(ϕ) < 0. This contradicts Lemma 6 and the fact ϕ ≠ 0.

Since ϕ solves the minimizing problem (27), it satisfies the Euler-Lagrange equation (16). Noticing the fact , we infer that |ϕ| is also a solution to problem (27). Thus it is a nonnegative solution of (16). It follows from , , and Proposition 7 that

()

for some y ∈ ℝ^N and γ ∈ ℝ. By redefining the sequence γ_j, we can set γ = 0.

Proof of Theorem 2. It follows from Proposition 15 that

()

Using Lemma 13 derives that

()

Hence we have a positive constant r₀ such that

()

From (82), for arbitrary r > r₀, there is a δ > 0 such that B(0, δ) ⊂ B(−x(t), r). The formula (80) implies that

()

On the other hand, Lemma 13 implies that

()

Thus

()

By Lemma 12, we obtain

()

where θ_j(x) = x_j. Hence there exists x₁ ∈ ℝ^N such that

()

Combining (86) with (88), we know that x(t)→−x₁ as t → T and we have

()

4. Blow-Up Rate

To establish the lower bound of the blow-up rate, we use the following proposition.

Proposition 16. Letting y₀ be the blow-up point determined in Theorem 2, it has

()

Proof. Let us define a positive function h(x) ∈ C¹(ℝ^N) such that

()

and h_A(x) = A²h(x/A) for A > 0 and it is valid that

()

Carrying out direct computation and using Hölder′s inequality, we have

()

which implies

()

Integrating on both sides gives rise to

()

From the fact ϕ₀ ∈ Σ, we have

()

By the virtue of Lemma 14 and Proposition 16, there exist A₁ and C₂ > 0 such that

()

Using the dominated convergence theorem, we infer that

()

Thus, it holds that

()

which implies that there is a_ε > 0 such that, for ∀t ∈ [0, T),

()

The identity

shows that

()

In addition, we have

()

Using Theorem 2 yields

()

In conclusion, for all ε > 0, we have shown that

()

Now, we establish the lower bound of the blow-up rate.

Proof of Theorem 3. Simple calculation yields

()

Therefore, the inequality (40) in the case

implies that

()

Integrating from t to T, by Proposition 16, we obtain

()

Combining the above inequality and the following inequality

()

we get the result

()

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 10771151), the Key Project of Chinese Ministry of Education (Grant no. 211162), and Sichuan Province Science Foundation for Youths (no. 2012JQ0011).

References

1 Gross E. P., Hydrodynamics of a super uid condensate, Journal of Mathematical Physics. (1963) 4, 195–207.
10.1063/1.1703944
CAS Web of Science® Google Scholar
2 Pitaevskii L. P., Vortex lines in an imparfect Bose gas, Soviet Physics JETP. (1961) 13, 451–454.
Web of Science® Google Scholar
3 Bradley C. C., Sackett C. A., and Hulet R. G., Bose-Einstein condensation of Lithium, observation of limted condensate number, Physical Review Letters. (1997) 78, 985–989.
10.1103/PhysRevLett.78.985
CAS Web of Science® Google Scholar
4 Huepe C., Métens S., Dewel G., Borckmans P., and Brachet M. E., Decay rates in attractive Bose-Einstein condensate, Physical Review Letters. (1997) 82, 1616–1619.
10.1103/PhysRevLett.82.1616
Web of Science® Google Scholar
5 Carles R., Remarks on nonlinear Schrödinger equations with harmonic potential, Annales Henri Poincaré. (2002) 3, no. 4, 757–772, https://doi.org/10.1007/s00023-002-8635-4, MR1933369, ZBL1021.81013.
10.1007/s00023-002-8635-4
Web of Science® Google Scholar
6 Zhang J., Stability of attractive Bose-Einstein condensates, Journal of Statistical Physics. (2000) 101, no. 3-4, 731–746, https://doi.org/10.1023/A:1026437923987, MR1804895, ZBL0989.82024.
10.1023/A:1026437923987
Web of Science® Google Scholar
7 Ginibre J. and Velo G., On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, Journal of Functional Analysis. (1979) 32, no. 1, 33–71, https://doi.org/10.1016/0022-1236(79)90077-6, MR533219, ZBL0396.35029.
10.1016/0022-1236(79)90077-6
Web of Science® Google Scholar
8 Glassey R. T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Journal of Mathematical Physics. (1977) 18, no. 9, 1794–1797, MR0460850, https://doi.org/10.1063/1.523491, ZBL0372.35009.
10.1063/1.523491
Web of Science® Google Scholar
9 Weinstein M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Communications in Mathematical Physics. (1983) 87, no. 4, 567–576, MR691044.
10.1007/BF01208265
Web of Science® Google Scholar
10 Zhang J., Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Analysis: Theory, Methods & Applications. (2002) 48, 191–207, https://doi.org/10.1016/S0362-546X(00)00180-2, MR1870752, ZBL1038.35131.
10.1016/S0362-546X(00)00180-2
Web of Science® Google Scholar
11 Kwong M. K., Uniqueness of positive solutions of ∆u − u + u^p = 0 in ℝ^N, Archive for Rational Mechanics and Analysis. (1989) 105, no. 3, 243–266, https://doi.org/10.1007/BF00251502, MR969899, ZBL0676.35032.
10.1007/BF00251502
Web of Science® Google Scholar
12 Hmidi T. and Keraani S., Remarks on the blowup for the L²-critical nonlinear Schrödinger equations, SIAM Journal on Mathematical Analysis. (2006) 38, no. 4, 1035–1047, https://doi.org/10.1137/050624054, MR2274472.
10.1137/050624054
Web of Science® Google Scholar
13 Merle F., Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Mathematical Journal. (1993) 69, no. 2, 427–454, https://doi.org/10.1215/S0012-7094-93-06919-0, MR1203233, ZBL0808.35141.
10.1215/S0012-7094-93-06919-0
Web of Science® Google Scholar
14 Merle F. and Raphael P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Communications in Mathematical Physics. (2005) 253, no. 3, 675–704, https://doi.org/10.1007/s00220-004-1198-0, MR2116733, ZBL1062.35137.
10.1007/s00220-004-1198-0
Web of Science® Google Scholar
15 Weinstein M. I., On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Communications in Partial Differential Equations. (1986) 11, no. 5, 545–565, https://doi.org/10.1080/03605308608820435, MR829596, ZBL0596.35022.
10.1080/03605308608820435
Web of Science® Google Scholar
16 Merle F., On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Communications on Pure and Applied Mathematics. (1992) 45, no. 2, 203–254, https://doi.org/10.1002/cpa.3160450204, MR1139066, ZBL0767.35084.
10.1002/cpa.3160450204
Web of Science® Google Scholar
17 Cazenave T., An Introduction to Nonlinear Schrodinger Equations, 1993, 26, IM-UFRJ, Rio de Janeiro, Brazil, Textos de Metodos Mathmatics.
Google Scholar
18 Carles R., Critical nonlinear Schrödinger equations with and without harmonic potential, Mathematical Models & Methods in Applied Sciences. (2002) 12, no. 10, 1513–1523, https://doi.org/10.1142/S0218202502002215, MR1933935, ZBL1029.35208.
10.1142/S0218202502002215
Web of Science® Google Scholar
19 Zhang J., Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Communications in Partial Differential Equations. (2005) 30, no. 10–12, 1429–1443, https://doi.org/10.1080/03605300500299539, MR2182299.
10.1080/03605300500299539
Web of Science® Google Scholar
20 Carles R., Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM Journal on Mathematical Analysis. (2003) 35, no. 4, 823–843 , https://doi.org/10.1137/S0036141002416936, MR2049023, ZBL1054.35090.
10.1137/S0036141002416936
Web of Science® Google Scholar
21 Pang P. Y. H., Tang H., and Wang Y., Blow-up solutions of inhomogeneous nonlinear Schrödinger equations, Calculus of Variations and Partial Differential Equations. (2006) 26, no. 2, 137–169, https://doi.org/10.1007/s00526-005-0362-5, MR2222242, ZBL1091.35094.
10.1007/s00526-005-0362-5
Web of Science® Google Scholar
22 Brezis H. and Lieb E. H., Minimum action solutions of some vector field equations, Communications in Mathematical Physics. (1984) 96, no. 1, 97–113, MR765961, https://doi.org/10.1007/BF01217349, ZBL0579.35025.
10.1007/BF01217349
Web of Science® Google Scholar
23 Lieb E. H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Inventiones Mathematicae. (1983) 74, no. 3, 441–448, https://doi.org/10.1007/BF01394245, MR724014, ZBL0538.35058.
10.1007/BF01394245
Web of Science® Google Scholar
24 Lions P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Annales de l′Institut Henri Poincaré. (1984) 1, no. 2, 109–145, MR778970, ZBL0541.49009.
10.1016/s0294-1449(16)30428-0
Web of Science® Google Scholar

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Remarks on the Blow-Up Solutions for the Critical Gross-Pitaevskii Equation

Abstract

1. Introduction and Main Results

2. Preliminaries

2.1. Local Wellposedness

2.2. Variational Characterization of the Ground State

2.3. Lemmas

3. Profile of the Minimal Blow-Up Solution

4. Blow-Up Rate

Conflict of Interests

Acknowledgments

References

References

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Remarks on the Blow-Up Solutions for the Critical Gross-Pitaevskii Equation

Abstract

1. Introduction and Main Results

2. Preliminaries

2.1. Local Wellposedness

2.2. Variational Characterization of the Ground State

2.3. Lemmas

3. Profile of the Minimal Blow-Up Solution

4. Blow-Up Rate

Conflict of Interests

Acknowledgments

References

References

Related

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