Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential

Applied Mathematics and Computation 208 (2009) 389–396

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Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential Jian Zhang a, Xiaoguang Li a,*,1, Yonghong Wu b a b

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

a r t i c l e

i n f o

Keywords: Blow-up rate Nonlinear Schrödinger equation Harmonic potential Attractive Bose–Einstein condensates

a b s t r a c t We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a harmonic potential, which models the attractive Bose–Einstein condensate. We establish the sharp lower and upper bounds of blow-up rate as t ! T (blow-up time), which improve the result of [Q. Liu, Y. Zhou, J. Zhang, Upper and lower bound of the blow-up rate for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput. 172 (2006) 1121–1132]. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction This paper is concerned with the Cauchy problem of the following attractive nonlinear Schrödinger equation with a harmonic potential:

1 1 i/t þ D/  x2 jxj2 / þ j/j4=N / ¼ 0; 2 2 /ð0; xÞ ¼ uðxÞ;

x 2 RN ;

t P 0;

ð1:1Þ ð1:2Þ

pffiffiffiffiffiffiffi where x > 0, N is the space dimension, / ¼ /ðt; xÞ: ½0; TÞ  RN ! C and 0 < T 6 1, i ¼ 1, D is the Laplace operator on RN . p1 If we replace the nonlinear term by j/j /, it is well known that the exponent p ¼ pc ¼ 1 þ 4=N in dimension N is the critical values for nonexistence of global solutions (see e.g. [2,18]). Eq. (1.1) models the Bose–Einstein condensate with attractive inter-particle interactions under magnetic trap, which is also called as Gross–Pitaevski equation (see [1,16]). The harmonic potential jxj2 models a magnetic field whose role is to confine the movement of particles, x is the trap frequency. For Eq. (1.1), Oh [13] established the local well-posedness of solutions of the Cauchy problem in the corresponding energy field. Cazenave [4], Tsurumi and Wadati [15], Zhang [18,19], Carles [3] established the existence of blow-up solutions. Liu et al. [8] studied the blow-up rate of Eq. (1.1). Under some conditions, for the blow-up solution /ðtÞ of Eq. (1.1), Liu et al. [8], established the following lower bound

C kr/ðtÞkL2 P pffiffiffiffiffiffiffiffiffiffiffi ; T t

* Corresponding author. E-mail address: [email protected] (X. Li). 1 This work was supported by National Natural Science Foundation of PR China (10771151). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.12.046

ð1:3Þ

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J. Zhang et al. / Applied Mathematics and Computation 208 (2009) 389–396

and upper bound

kr/ðtÞkL2 6 C

 1 ln j ln sin xðT  tÞ þ Cj 2 þ C; sin xðT  tÞ

ð1:4Þ

where T is blow-up time. In this paper, by applying Merle and Raphaël’s idea [9–12] and the transform provided by Carles [3], with some technical treatment, we can establish an upper and a lower bound of blow-up rate for Eq. (1.1). From the results of this paper, it is obvious that the blow-up solution of Eq. (1.1) /ðtÞ, if has small super critical mass initial data and negative energy, behaves like near the blow-up time T

kr/ðtÞkL2  C

 1 ln j ln sin xðT  tÞj 2 : sin xðT  tÞ

We recall the Cauchy problem of the classical nonlinear Schrödinger equation

1 iwt þ Dw þ jwj4=N w ¼ 0; 2 wð0; xÞ ¼ uðxÞ:

t P 0; x 2 RN ;

ð1:5Þ ð1:6Þ

In recent years, some profound results on the blow-up properties of Eq. (1.5) were obtained ([10–12,14]). In particular, Merle and Raphaël [9,11,12] proved a sharp upper and lower bound of the blow-up rate, which is according with the numerical computations [7]. More precisely, Merle and Raphaël’s results suggest that the blow-up solution wðtÞ, if has small super critR R R ical mass initial data ( Q 2 dx < juj2 dx < Q 2 dx þ a with a small enough) and negative energy, behaves like near the blow-up time s

krwðtÞkL2  C

 1 ln j lnðs  tÞj 2 : st

In this paper, by applying Merle and Raphaël’s idea [9–12] and the transform provided by Carles [3], with some technical treatment, we can establish an upper and a lower bound of blow-up rate for Eq. (1.1). From the results of this paper, it is obvious that the blow-up solution of Eq. (1.1) /ðtÞ, if has small super critical mass initial data and negative energy, behaves like near the blow-up time T

kr/ðtÞkL2  C

 1 ln j ln sin xðT  tÞj 2 : sin xðT  tÞ

We conclude this section with several notations. We denote Lq ðRN Þ and k  kLq ðRN Þ by Lq and k  kLq , respectively. The various positive constants will be simply denoted by C. 2. Preparation We define the energy space for the Cauchy problem (1.1) and (1.2) in the course of nature by

H :¼ fu 2 H1 ðRN Þ; jxju 2 L2 ðRN Þg; with the inner product

hu; v i :¼

Z

rurv þ uv þ jxj2 uv :

The norm of H is denoted by k  kH . Moreover, we define a energy functional E on H by

EðuÞ :¼

Z

1 1 1 jruj2 þ x2 jxj2 juj2  juj2þ4=N : 2 2 1 þ 2=N

According to the Sobolev embedding theorem, the functional E is well defined. From Oh [13], we note that the local wellposedness for the cauchy problem to (1.1) holds in H. Proposition 2.1. For any u 2 H, there exist T > 0 and a unique solution /ðt; xÞ of the Cauchy problem (1.1) and (1.2) in Cð½0; TÞ; HÞ such that either T ¼ 1 (global existence), or T < 1 and lim k/ðtÞkH ¼ 1 (blow-up). Furthermore, for all t 2 ½0; TÞ, /ðtÞ t!T satisfies the following two conservation laws of mass and energy: (i) Conversation of mass

k/ðtÞkL2 ¼ kukL2 ;

ð2:1Þ

(ii) Conversation of energy

Eð/ðtÞÞ ¼ EðuÞ:

ð2:2Þ

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It is well known that the same local well-posedness for the Cauchy problem (1.5) and (1.6) holds in H. In the same way, If we define the energy functional E on H by

E ðwÞ :¼

Z

4 1 1 jwj2þN dx; jrwj2  2 1 þ N2

8w 2 H;

E is also well defined. In addition, let w be a solution of Cauchy problem (1.5) and (1.6) in Cð½0; sÞ; H for some (maximal time), then w satisfies the following two conservation laws for t 2 ½0; sÞ.

s 2 ½0; þ1Þ

(i) Conservation of mass

kwðtÞkL2 ¼ kukL2 ;

ð2:3Þ

(ii) Conversation of energy

E ðwðtÞÞ ¼ E ðuÞ:

ð2:4Þ

Consider the following field equation

1  Du þ u  juj4=N u ¼ 0; 2

u 2 H1 ðRN Þ:

ð2:5Þ

From Kwong [6], we have the following proposition Proposition 2.2. There exists a unique ground solution Q ðxÞ to (2.5), which is a positive and spherically symmetric function with exponentially decay at infinity. The ground state Q of Eq. (2.5) plays an important role in the formation of singularities for the solutions of the Cauchy problem (1.1) and (1.2). Precisely, for the Cauchy problem (1.1) and (1.2), Zhang [18] and Carles [2] proved the following results: if kukL2 < kQ kL2 , the corresponding solution / is globally defined; if kukL2 P kQ kL2 , for some initial data u, the corresponding solution / blow-up in finite time. That is to say, kQ kL2 is a sharp lower bound of the L2 norm of the blow-up solutions. From now on, we restrict ourselves to considering small super critical mass initial data, that is:



u 2 Ba ¼ u 2 H1 with

Z

Q 2 dx <

Z

juj2 dx <

Z

 Q 2 dx þ a ;

for some parameter a > 0 small enough. Next, in terms of Merle and Raphaël’s arguments in [9–11], we state the following property, which has been proved in [11] for dimension N = 1. Spectral property Let N P 1 consider the two real Schrödinger operators

Ł1 ¼ D þ

  4 2 4 þ 1 Q N1 y  rQ ; N N

and the real valued quadratic form for

Ł2 ¼ D þ

2 N4 1 Q y  rQ; N

e ¼ e1 þ ie2 2 H1 :

Hðe; eÞ ¼ ðŁ1 e1 ; e1 Þ þ ðŁ2 e2 ; e1 Þ: Then there exists a universal d > 0, such that 8e 2 H1 ; if ðe1 ; Q Þ ¼ ðe1 ; Q 1 Þ ¼ ðe1 ; yQ Þ ¼ ðe2 ; Q 2 Þ ¼ ðe2 ; rQ Þ ¼ 0, then

Z  Z Hðe; eÞ P d jrej2 dx þ jej2 ejyj dx ;

where Q 1 ¼ N2 Q þ y  rQ , Q 2 ¼ N2 Q 1 þ y  rQ 1 and Q is the ground state of (2.5). Remark 1. This property has been proved in [11] for dimension N ¼ 1. This property accords with the numerical computation for dimensions N ¼ 2; 3; 4 [5]. By using the spectral property, Merle and Raphaël [9–12] obtained the upper and lower bounds of the blowup solutions for the Cauchy problem (1.5) and (1.6), as follows: Lemma 2.1 [12]. Let N ¼ 1 or N P 2 assuming spectral property holds true. There exist universal constants a > 0, universal constant M1 > 0 such that the following holds true. Let u 2 Ba and assume that the corresponding solution wðtÞ to (1.5) blows up in finite time 0 < s < þ1. Then one has the following lower bound on the blow-up rate for t close to T:

krwðtÞkL2 P M 1

 1 ln j lnðs  tÞj 2 : st

ð2:6Þ

Lemma 2.2 ([9,10]). Let N ¼ 1 or N P 2 assuming spectral property holds true. There exist universal constants a > 0 M 2 > 0 such that the following is true. Let u 2 H such that

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J. Zhang et al. / Applied Mathematics and Computation 208 (2009) 389–396

 2  R 1 I ruu dx E ðuÞ 6 : 2 kukL2 

u 2 Ba ;

Then the corresponding solution wðtÞ to (1.5) blow up in finite time and there holds for t ! s

krwðtÞkL2 6 M 2

 1 ln j lnðs  tÞj 2 : st

ð2:7Þ

Remark 2. Merle and Raphaël [9] proved that if

u 2 Ba ; E ðuÞ <

 2  R 1 I ruu dx ; 2 kukL2

then the corresponding solution wðtÞ to (1.5) blows up in finite time Merle and Raphaël [10] proved that if

u 2 H \ Ba ; E ðuÞ ¼

s > 0, and (2.7) holds.

 2  R 1 I ruu dx ; kukL2 2

then the corresponding solution wðtÞ to (1.5) blows up in finite time

s > 0, and (2.7) holds.

The following lemma gives results on the blow-up rate of the blow-up solutions of (1.5) and (1.6), which has initial datum

u such that

u 2 Ba ; E ðuÞ >

 2  R 1 I ruu dx : kukL2 2

Lemma 2.3 (Raphaël [14]). Let N ¼ 1 or N P 2 assuming spectral property holds true. There exist universal constants a > 0, M3 > 0 and M 4 > 0 such that the following is true. Let u 2 H such that

 2  R 1 I ruu dx E ðuÞ > : kukL2 2 

u 2 Ba ;

and assume that the corresponding solution wðtÞ to (1.5) blows up in finite time 0 < s < þ1, then there holds for t to T either

krwðtÞkL2 6 M 3

 1 ln j lnðs  tÞj 2 ; st

ð2:8Þ

or

krwðtÞkL2 P

M 4 pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðs  tÞ EG ðuÞ

ð2:9Þ

 12 0 R I ruu dx  1@ A : where EG ðuÞ ¼ E ðuÞ  2 kuk 2 L

Carles [3] gave the transform which shows the relationship between the solutions of (1.1) and (1.2) and that of (1.5) and (1.6), as follows: Lemma 2.4 [3] (1) Assume that w is a solution of the Cauchy problem (1.5) and (1.6) in Cð½0; sÞ; HÞ where

/ðt; xÞ ¼

1

x 2 tan xt

N

ðcos xtÞ 2

ei 2 x

w

 tan xt

x

;

x cos xt

 ;

s > 0. Let ð2:10Þ

xs then /ðt; xÞ 2 Cð½0; arctan x  Þ; HÞ  is a solution of the Cauchy problem (1.1) and (1.2). In particular, if wðtÞ 2 Cð½0; þ1Þ; HÞ (global solution), /ðtÞ 2 C 0; 2px ; H is a local solution of Cauchy problem (1.1) and (1.2). 

(2) Assume that / is a solution of the Cauchy problem (1.1) and (1.2) in Cð½0; s0 Þ; HÞ where s0 2 0; 2px . Let

wðt; xÞ ¼

1

2

ix2

N

ð1 þ ðxtÞ2 Þ 4

e

x2 t 1þx2 t2

/

arctan xt

x

;

!

x 1

ð1 þ ðxtÞ2 Þ2

;

ð2:11Þ

     0 then wðt; xÞ 2 C 0; tanxxs ; H is a solution of the Cauchy problem (1.5) and (1.6). In particular, if / 2 C 0; 2px ; H , wðt; xÞ 2 Cð½0; þ1Þ; HÞ is a global solution of the Cauchy problem (1.5) and (1.6).

J. Zhang et al. / Applied Mathematics and Computation 208 (2009) 389–396

393

Remark 3 (1) The transform of Lemma 2.4 is based on the fact that /ð0; xÞ ¼ wð0; xÞ ¼ u. Moreover, the solution obtained by Lemma 2.4 is usually a local solution. The maximal existence time of the local solution for the Cauchy problem relies on the initial data u. (2) If the maximal existence time of (1.5) and (1.6) is t0 ¼ 1, then t 00 , the maximal existence time of (1.1) and (1.2), has the following alternative: t 00 ¼ 1 or else t00 ¼ 2px. Based on Lemma 2.4, we get following two lemmas directly.   Lemma 2.5. Assume that / is a blow-up solution of the Cauchy problem (1.1) and (1.2) in Cð½0; TÞ; HÞ where T 2 0; 2px (maximal existence time). Let wðt; xÞ is defined by (2.11), then (1) wðt; xÞ 2 Cð½0; tanxxT Þ; HÞ is a blow-up solution of the Cauchy problem (1.5) and (1.6), where ½0; tanxxT Þ is the maximal existence time. (2)

/ðt; xÞ ¼

1

x 2 tan xt

i 2 x N e

ðcos xtÞ 2

w

 tan xt

x

;

 x ; cos xt

t 2 ½0; TÞ:

Lemma 2.6. Assume that w is a blow-up solution of the Cauchy problem (1.5) and (1.6) in Cð½0; sÞ; HÞ where imal existence time). Let /ðt; xÞ is defined by (2.10), then

ð2:5Þ

s 2 ð0; þ1Þ (max-

 xsÞ; HÞ is a blow-up solution of the Cauchy problem (1.1) and (1.2), where 0; arctan xT is the maximal (1) /ðt; xÞ 2 Cð½0; arctan x x existence time. (2)

wðt; xÞ ¼

1

2

ix2

N

ð1 þ ðxtÞ2 Þ 4

e

x2 t 1þx2 t2

/

arctan xt

x

;

!

x 1

ð1 þ ðxtÞ2 Þ2

:

ð2:6Þ

3. Blow-up rate In this section, we will give our main results. Theorem 3.1. Let N ¼ 1 or N P 2 assuming spectral property holds true. There exist constants a > 0, C 1 > 0 such that the following holds true. Let u 2 Ba and assume that the corresponding solution /ðtÞ to (1.1) blows up in finite time 0 < T < 2px. Then one has the following lower bound on the blow-up rate for t close to T:

kr/ðtÞkL2 P C 1



1 ln j ln sin xðT  tÞj 2 : sin xðT  tÞ

ð3:1Þ

Remark 4. Carles [2] points out that if the solution of (1.1) and (1.2) collapses at time T, then T 6 2px. Moreover, Carles [3] proved that if EðuÞ < 12 x2 kxukL2 , then the solution of (1.1) and (1.2) collapses at time T < 2px. Theorem 3.2. Let N ¼ 1 or N P 2 assuming spectral property holds true. There exists a > 0 and a constant C 2 > 0 such that the following is true. Let u 2 H such that

u 2 Ba ; E ðuÞ 6

 2  R 1 I ruu dx : 2 kukL2

Then the corresponding solution /ðtÞ to (1.1) blow up in finite time 0 < T < 2px and there holds for t ! T

kr/ðtÞkL2 6 C 2

 1 ln j ln sin xðT  tÞj 2 : sin xðT  tÞ

Remark 5. Liu et al. [8] proved that if

u 2 Ba ; E ðuÞ <

 2  R 1 I ruu dx ; 2 kukL2

then /ðtÞ the corresponding solution of (1.1) and (1.2) blows up in finite time T > 0, and (1.4) holds.

ð3:2Þ

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J. Zhang et al. / Applied Mathematics and Computation 208 (2009) 389–396

Remark 6. From Theorems 3.1 and 3.2, it is obvious that if all the assumptions of Theorem 3.2 are satisfied, then

kr/ðtÞkL2  C

 1 ln j ln sin xðT  tÞj 2 : sin xðT  tÞ

Theorem 3.3. Let N ¼ 1 or N P 2 assuming spectral property holds true. There exists a > 0 and constants C 3 > 0 and C 4 > 0 such that the following is true. Let u 2 H such that

u 2 Ba ; E ðuÞ >

 2  R 1 jI ruu dxj ; kukL2 2

and assume that the corresponding solution /ðtÞ to (1.1) blows up in finite time 0 < T < 2px. Then there holds for t to T either

kr/ðtÞkL2 6 C 3

 1 ln j lnðT  tÞj 2 ; sin xðT  tÞ

ð3:3Þ

or

kr/ðtÞkL2 P

C 4

pffiffiffiffiffiffiffiffiffiffiffiffiffi ; sin xðT  tÞ EG ðuÞ

ð3:4Þ

 12 0 R I ruu dx A : where EG ðuÞ ¼ E ðuÞ  12 @ kuk 2 

L

Proof of Theorem 3.1. By Lemma 2.5, let wðt; xÞ be defined by (2.11), then wðt; xÞ 2 Cð½0; tanxxT Þ; HÞ is a blow-up solution of Cauchy problem (1.5) and (1.6) where ½0; tanxxT Þ is a maximal existence time, and (2.10) holds true. That is

/ðt; xÞ ¼

1

x 2 tan xt

ei 2 x N

ðcos xtÞ 2

w

 tan xt

x

;

 x ; cos xt

t 2 ½0; TÞ:

ð3:5Þ

  Then for t 2 ½0; TÞ; T 2 0; 2px ; we have

      1  ixx sinðxtÞw tan xt ;  þ rw tan xt ;   2 cos xt  x x L            1 tan x t tan x t rw  P ;  ;   2  xxw 2 cos xt  x x L L         1  tan xt x  tan xt   rw ;   ;  P xw  2:   cos xt x cos x t x 2 L L

kr/ðtÞkL2 ¼

ð3:11Þ

Then we claim that there is a constant C > 0, such that

 

xxw



tan xt

x

  ;   2 6 C;

t 2 ½0; TÞ:

ð3:12Þ

L

In fact, consider

Jðt  Þ 

Z

jxj2 jwðt ; xÞj2 dx ¼ kxwðt Þk22 ;

By Weinstein [17], we have that

Jðt  Þ ¼ Jð0Þ þ J 0 ð0Þt  þ

Z

t

d2 dt

  tan xT : t  2 0;

x

Jðt Þ ¼ 4E ðuÞ, E ðuÞ is a constant. By an analytical identity

J 00 ðsÞðt   sÞ ds ¼ Jð0Þ þ J 0 ð0Þt  þ 2E ðuÞðt Þ2 :

ð3:13Þ

0

Let t ¼ tanxxt, t 2 ½0; TÞ, 0 < T < 2px, we have

   2 2   xw tan xt ;   ¼ Jð0Þ þ J 0 ð0Þ tan xt þ 2E ðuÞ tan xt :  

x

2

x

This equation implies (3.12). By t < T < 2px, there is a constant C > 0 such that

    x  tan xt  6 C; t 2 ½0; TÞ: ;  xw 2  cos xt x L

x

x

cos xt

ð3:14Þ

6 cosxxT 6 C

ð3:15Þ

J. Zhang et al. / Applied Mathematics and Computation 208 (2009) 389–396

395

By (3.11) and (3.15), Lemma 2.1, we have

  !1 ln ln tanxxT  tanxxt 2 1 tan xT tan xt cos xt x  x  1 x cos xT ln j ln sin xðT  tÞ  ln x  ln cos xT  ln cos xtj 2 ¼ cos xt sin xðT  tÞ  1 x cos xT ln j ln sin xðT  tÞ  ln x  ln cos xT  ln cos xtj 2 P : sin xðT  tÞ

kr/ðtÞkL2 þ C P

ð3:16Þ

It follows that

kr/ðtÞkL2 P C

 1 ln j ln sin xðT  tÞj 2 ; sin xðT  tÞ

as t ! T:

ð3:17Þ

Proof of Theorem 3.2. By Lemma 2.2 there exist universal constant a > 0 and M1 > 0 such that the following is true. Let u 2 H such that

u 2 Ba ; E ðuÞ <

 2  R 1 jI ruu dxj : kukL2 2

Let wðtÞ be a solution for the Cauchy problem (1.5) and (1.6), then wðtÞ blows up in finite time 0 < s < þ1 and (2.7) holds for t close to s. That is

krwðtÞkL2 6 M 2

 1 ln j lnðs  tÞj 2 : st

ð3:8Þ

 xsÞ; H is a blow-up solution of the Cauchy problem By Lemma 2.6, let /ðt; xÞ be defined by (2.10), then /ðt; xÞ 2 Cð 0; arctan x xsÞ is the maximal existence time. We denote arctan xs by T, then 0 < T < p . And we have (1.1) and (1.2), where ½0; arctan x x 2x

      1  ixx sinðxtÞw tan xt ;  þ rw tan xt ;   2  cos xt x x L            1 tan x t tan x t rw  6 ;  ;   2 þ xxw 2 : cos xt  x x

kr/ðtÞkL2 ¼

L

ð3:18Þ

L

By (3.18) and (3.12) and Lemma 2.2, we have

  !1 ln j ln tanxxT  tanxxt j 2 1 kr/ðtÞkL2 6 þC tan xT tan xt cos xt x  x  1 x cos xT ln j ln sin xðT  tÞ  ln x  ln cos xT  ln cos xtj 2 ¼ þC cos xt sin xðT  tÞ  1 x ln j ln sin xðT  tÞ  ln x  ln cos xT  ln cos xtj 2 6 þ C: cos xt sin xðT  tÞ

ð3:19Þ

It follows that

kr/ðtÞkL2 6 C

 1 ln j ln sin xðT  tÞj 2 ; sin xðT  tÞ

as t ! T:

Proof of Theorem 3.3. By the Lemma 2.3 and the assumptions on initial datum u of Theorem 3.3, corresponding solution wðtÞ to (1.5) blow up in finite time. Moreover, either (2.8) or (2.9) holds true. When (2.8) holds, the same argument as that of Theorem 3.2 yields (3.3). When (2.9) holds true, similar argument as that of Theorem 3.1 yields (3.4). References [1] C.C. Bradley, C.A. Sackett, R.G. Hulet, Bose–Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett. 78 (1997) 985– 989. [2] R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. Henri Poincare 3 (2002) 757–772. [3] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Mod. Meth. Appl. Sci. 12 (2002) 1513–1523. [4] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de mètodos màthmatics, 26, IM-UFRJ, Rio de Janeiro, 1993. [5] G. Fibich, F. Merle, P. Raphaël, Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation, Physica D 220 (2006) 1–13. [6] M.K. Kwong, Uniqueness of positive solutions of Du  u þ up ¼ 0 in RN , Arch. Rational Mech. Anal. 105 (1989) 243–266. [7] M.J. Landam, G.C. Papanicolao, C. Suelm, P.L. Sulem, Rate of blowup for solutions of nonlinear Schrödinger equation at critical dimension, Phys. Rev. A 38 (1988) 3837–3843.

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