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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 813417, 9 pages
http://dx.doi.org/10.1155/2013/813417
Research Article

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

1Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu 610066, China
2School of Finance, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 22 August 2013; Accepted 29 November 2013

Academic Editor: Benchawan Wiwatanapataphee

Copyright © 2013 Xiaoguang Li and Chong Lai. 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.

Abstract

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 : is the wave function, is the space dimension, and denotes the Laplace operator on . Equation (1) is also called Gross-Pitaevskii equation (see [1, 2]), which models the Bose-Einstein condensate (see [3, 4]). The harmonic potential describes a magnetic field. With the nonlinear term being replaced by , it is well known that the exponent 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 . 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 norm for the initial data of blow-up solutions (see [9]). More precisely, let 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 [1215]). 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 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 , , and . Moreover, Weinstein proved that, for any minimal blow-up solution , the following holds: where is the blow-up time and and are some suitable functions.

Merle [13, 16] proved that is a minimal blow-up solution of (3) if and only if there exist , ,   , and 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 is globally defined if . In other words, if blows up in finite time.

Let and be the solutions of the Cauchy problems (1)-(2) and (3)-(4), respectively. Under the condition of , Carles [18] established a formula, which reflects the relation between and . According to the formula, Carles [18] established the following statements.(1)If blows up at a finite time , then .(2)If blows up at , blows up at time .(3)Conversely, blows up at time ; then blows up at .(4)If blows up at , exists globally ( ).

Moreover, Carles studied the qualitative properties of minimal blow-up solutions with (see [18, 20]). As for the minimal blow-up solutions with , 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 of (1) with .

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 with for which the solution of the Cauchy problem (1)-(2) blows up in a finite time.

Theorem 2. Let be a blow-up solution of (1) with . Then there is such that in the sense of distribution as .

Theorem 3. There exists such that

Remark 4. For any blow-up solutions of (1), we know that ( is a blow-up time). When , the formula presented in [18] is valid. For the minimal blow-up solutions with , the conclusion of the above theorems can be found in [18]. However, there exist minimal blow-up solutions with . For example, if the initial , with being the solution of problem (5), the solution of (1) will blow up at , while the corresponding solution of (3) is a solitary wave . The minimal blow-up solutions with were sensible as pointed in [18].

In this paper, , , and are denoted by , , and , respectively. The various positive constants are also denoted by .

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 is defined as The norm of is denoted by . Moreover, we define an energy functional on by

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

Proposition 5. For any , there exist and a unique solution of the Cauchy problem (1)-(2) in such that either (global existence) or and (blowup). Moreover, for any , 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 (the solution set), (the ground solution set), and as follows: where .

For any , the following two identities hold true: The above two equalities imply where

Naturally, we get

According to Cazenave [17], the set can be described as where is a positive, spherically symmetric, decreasing, and real valued function.

It is of importance that Kwong [11] proved the uniqueness for the solution of the problem Noticing the fact that , it is easy to check that It follows from (21), (22), and (24) that

With functional 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 .

Lemma 6 (see Weinstein [9]). For any , 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 is a minimizer of the variational problem of (27), it solves the Euler-Lagrange equation (16). So for some , and by (27) and (25), we know . 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) is a solution to the minimizing problem , (iii) , for some , , and .

2.3. Lemmas

Lemma 8 (see Zhang [6]). Let , the initial datum of Cauchy problem (1)-(2), satisfy then blows up in a finite time.

Consider the constrained minimization problem

For , we cite a lemma in [15].

Lemma 9 (see Weinstein [15]). (a) Consider   or .
(b) Let and be a minimizing sequence; then it holds that and weakly in .

Now, we recall some lemmas on the compactness.

Lemma 10 (see Brezis and Lieb [22]). Let , , and . Then there exists a shift such that, for some constant ,

Lemma 11 (see Lieb [23]). Let be a uniformly bounded sequence of functions in with . Assume further that there are positive constant and satisfying . Then there exists a sequence such that

Lemma 12. Let be a real-valued function on and with . Then

Proof. It follows from (30) and that for all real numbers . On the other hand, it has Thus the discriminant of the equation in must be negative or null and the desired inequality follows.

Lemma 13. There is a constant such that

Proof. Setting , we have It follows that which implies the conclusion.

Lemma 14 (see [16, page 433]). Let , , and , for arbitrary , satisfy where depends only on . Then, it holds that with .

3. Profile of the Minimal Blow-Up Solution

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

Proof of Theorem 1. Setting with being arbitrary positive real number and being complex number satisfying , then From (15) and (19), the corresponding energy is Thus Lemma 8 infers that 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 be a blow-up solution of the Cauchy problem (1)-(2) and is the blow-up time. Set and . If it holds that with and .

Proof. Let . We choose to satisfy Setting , noticing that tends to as , , and we know that is uniformly bounded in and there is a weakly convergent subsequence such that
We note that Since we have assumed , by (52), (54), and (31), we know that is a minimizing sequence for the variational problem (27).
Next, we will prove that the minimizing sequence has a subsequence and a family such that has a strong limit in . 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 , one has
Dichotomy. There exist a constant and sequences and , bounded in , such that, for all , there exists such that for
Compactness. There exists in . For any , we can find such that
Now, we exclude the cases of vanishing and dichotomy.
Exclusion of Vanishing. By (52), (51), and (54) there are and such that By the boundness of and the Sobolev inequality, there exist and such that Now, we show the existence of positive constants and such that Indeed, from (58) and (59), for sufficiently small , we get Thus we know that (60) with is valid. From (60) and Lemma 10, there exist and 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 and are two constants and is bounded in . Hence, by Lemma 11, there are a subsequence and a sequence such that
Using (56) gives rise to On the other hand, the fact implies with Lemma 6 that Thus, for any fixed , 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 , there exist and a subsequence, which we again label it by , such that
Given , the embedding is compact and
Making use of (70) derives for any . Hence, it holds that
It follows that which implies with the Gagliardo-Nirenberg inequality (30) that
To show in , we only need to show that .
From (51) and (54), we know that Hence, derives . This contradicts Lemma 6 and the fact .
Since solves the minimizing problem (27), it satisfies the Euler-Lagrange equation (16). Noticing the fact , we infer that is also a solution to problem (27). Thus it is a nonnegative solution of (16). It follows from , , and Proposition 7 that for some and . By redefining the sequence , we can set .

Proof of Theorem 2. It follows from Proposition 15 that Using Lemma 13 derives that Hence we have a positive constant such that
From (82), for arbitrary , there is a such that . The formula (80) implies that On the other hand, Lemma 13 implies that Thus By Lemma 12, we obtain where . Hence there exists such that Combining (86) with (88), we know that as 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 be the blow-up point determined in Theorem 2, it has

Proof. Let us define a positive function such that and for 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 , we have
By the virtue of Lemma 14 and Proposition 16, there exist and such that
Using the dominated convergence theorem, we infer that Thus, it holds that which implies that there is such that, for , The identity shows that In addition, we have Using Theorem 2 yields In conclusion, for all , 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 to , 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).

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