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

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 𝑖𝜙_𝑡+Δ𝜙−|𝑥|²𝜙+𝜙^4/𝑁𝜙=0,𝑥∈ℝ^𝑁𝜙(),𝑡≥0,(1)0,𝑥=𝜙₀(𝑥),(2) 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 𝑖𝜓_𝑡+Δ𝜓+𝜓^4/𝑁𝜓=0,𝑥∈ℝ^𝑁𝜓(),𝑡≥0,(3)0,𝑥=𝜓₀(𝑥).(4) 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 −Δ𝑢+𝑢−|𝑢|^4/𝑁𝑢=0,𝑢∈𝐻¹ℝ^𝑁.(5) Weinstein [9] proved that the solutions of the Cauchy problem (3)-(4) are globally defined if ‖𝜓₀‖_𝐿2<‖𝑄‖_𝐿². On the other hand, for any 𝑙≥‖𝑄‖_𝐿², there exist blow-up solutions with ‖𝜓₀‖_𝐿2=𝑙. 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 ‖𝜓₀‖_𝐿2=‖𝑄‖_𝐿² (see [13, 15, 16]). By using the pseudoconformal invariance of (3), Weinstein [15] constructed the explicit blow-up solution with critical mass (‖𝜓₀‖_𝐿2=‖𝑄‖_𝐿²) for (3) in the form (𝑎+𝑏𝑡)^−(𝑁/2)𝑄𝑥𝑒𝑎+𝑏𝑡^{(𝑖𝑏|𝑥|2)/4(𝑎+𝑏𝑡)}𝑒^{𝑖(𝑐+𝑑𝑡)/(𝑎+𝑏𝑡)},(6) 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: lim_𝑡→𝑇𝜆(𝑡)^𝑁/2𝜓𝑡,𝜆(𝑡)𝑥+𝑦(𝑡)=𝑄(𝑥),(7) 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 𝜓()=𝜔𝑡,𝑥𝑇−𝑡^𝑁/2𝑒^{𝑖𝜃+(𝑖|𝑥−𝑥1|2/4(−𝑇+𝑡))−(𝑖𝜔2/(−𝑇+𝑡))}𝜔×𝑄𝑇−𝑡𝑥−𝑥₁−(𝑇−𝑡)𝑥₀.(8)

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 ‖𝜙₀‖_𝐿2<‖𝑄‖_𝐿². In other words, ‖𝜙₀‖_𝐿2≥‖𝑄‖_𝐿² 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 ‖𝜙₀‖_𝐿2=‖𝑄‖_𝐿² 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 ‖𝜙₀‖_𝐿2=‖𝑄‖_𝐿². Then there is 𝑦₀∈ℝ^𝑁 such that 𝜙(𝑡,𝑥)⟶‖𝑄‖²_𝐿2𝛿_𝑦0(9) in the sense of distribution as 𝑡→𝑇.

Theorem 3. There exists 𝐶>0 such that ∇𝜙(𝑡)_𝐿2≥𝐶[).𝑇−𝑡,∀𝑡∈0,𝑇(10)

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), ‖⋅‖_𝐿^𝑞_(ℝ^𝑁₎, and ∫_ℝ𝑁⋅𝑑𝑥 are denoted by 𝐿^𝑞, ‖⋅‖_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 Σ∶=𝑢∈𝐻¹,|𝑥|𝑢∈𝐿².(11) The inner product of the space Σ is defined as ⟨𝑢,𝑣⟩∶=∇𝑢∇𝑣+𝑢𝑣+|𝑥|²𝑢𝑣𝑑𝑥.(12) The norm of Σ is denoted by ‖⋅‖_Σ. Moreover, we define an energy functional ℰ on Σ by ℰ(𝑢)∶=|∇𝑢|²+|𝑥|²|𝑢|²−1|𝑢|1+2/𝑁^2+4/𝑁𝑑𝑥.(13)

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 lim_𝑡→𝑇‖𝜙(𝑡)‖_Σ=∞ (blowup). Moreover, for any 𝑡∈[0,𝑇), it holds the conservation laws of mass 𝜙(𝑡)_𝐿2=𝜙_0𝐿2(14) and the energy ℰ𝜙(𝑡)=ℰ𝜙₀.(15)

2.2. Variational Characterization of the Ground State

Consider the equation −Δ𝑢+𝜔𝑢−|𝑢|^4/𝑁𝑢=0,𝑢∈𝐻¹ℝ^𝑁.(16) For (16), we set some notations such as 𝒳_𝜔 (the solution set), 𝒢_𝜔 (the ground solution set), and 𝒢 as follows: 𝒳_𝜔=𝑢∈𝐻¹;𝑢≠0,−Δ𝑢+𝜔𝑢−|𝑢|^4/𝑁,𝒢𝑢=0_𝜔=𝑢∈𝒳_𝜔;𝑆(𝑢)≤𝑆(𝑣),∀𝑣∈𝒳𝜔,𝒢=_𝜔∈𝑅+𝒢_𝜔,(17) 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: |∇𝑢|²+𝜔|𝑢|²𝑑𝑥=|𝑢|^2+(4/𝑁)(|𝑑𝑥,𝑁−2)|∇𝑢²+𝑁𝜔|𝑢|²=𝑁𝑑𝑥|𝑢|1+2/𝑁^2+(4/𝑁)𝑑𝑥Pohozaev'sidentity.(18) The above two equalities imply ℋ(𝑢)=0,∀𝑢∈𝒳,(19) where ℋ(𝑢)∶=|∇𝑢|²−1|𝑢|2/𝑁+1^2+(4/𝑁)𝑑𝑥.(20)

Naturally, we get 𝑢∈𝒢_𝜔⟺𝑢∈𝒳_𝜔,‖𝑢‖_𝐿²≤‖𝑣‖_𝐿²,∀𝑣∈𝒳_𝜔.(21)

According to Cazenave [17], the set 𝒢_𝜔 can be described as 𝒢_𝜔=𝑒^𝑖𝜃𝜑_𝜔⋅−𝑦;𝜃∈𝑅,𝑦∈ℝ^𝑁,(22) 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 −Δ𝑢+𝑢−|𝑢|^4/𝑁𝑢=0,𝑢∈𝐻¹,𝑢(𝑥)𝑥,𝑢(𝑥)=𝑢(||)>0.(23) Noticing the fact that 𝑄(𝑥)=𝜑_𝜔|_𝜔=1, it is easy to check that 𝜑_𝜔=𝜔^𝑁/4𝑄𝜔^1/2𝑥∈𝒢_𝜔,𝜑_𝜔𝐿2=‖𝑄‖_𝐿².(24) It follows from (21), (22), and (24) that 𝑢∈𝒢_𝜔⟺𝑢∈𝒳_𝜔,‖𝑢‖_𝐿²=‖𝑄‖_𝐿²,𝒢(25)_𝜔=𝑒^𝑖𝜃𝜑_𝜔⋅−𝑦;𝜃∈𝑅,𝑦∈ℝ^𝑁,=𝑒^𝑖𝜃𝜔^𝑁/4𝑄𝜔^1/2⋅−𝑦;𝜃∈𝑅,𝑦∈ℝ^𝑁.(26)

With functional ℋ defined by (20), we now introduce the following constrained minimization problem ℐ‖𝑄‖_𝐿²≡infℋ𝑓∣𝑓∈𝐻¹,𝑓_𝐿2=‖𝑄‖_𝐿².(27)

Now, we claim that ().𝑢∈𝒢⟺𝑢isasolutiontotheminimizationproblem27(28)

In fact, (𝑁/(𝑁+2))‖𝑄‖_𝐿^4/𝑁₂ is the minimum of the functional (see Kwong [11] or Weinstein [9]) 𝐼𝜓=∇𝜓²_𝐿2𝜓_𝐿^4/𝑁₂𝜓_𝐿^2+4/𝑁_2+4/𝑁,𝜓∈𝐻¹,(29) which derives the Gagliardo-Nirenberg inequality 𝜓_𝐿^2+4/𝑁_2+4/𝑁≤𝑁+2𝑁𝜓_𝐿2‖𝑄‖_𝐿^24/𝑁∇𝜓²_𝐿2.(30) The inequality (30) implies the following lemma on the functional ℋ.

Lemma 6 (see Weinstein [9]). For any 𝑓∈𝐻¹, one has 𝑓1−^𝐿2‖𝑄‖_𝐿^24/𝑁∇𝑓²_𝐿2≤ℋ𝑓.(31)

Lemma 6 implies that ℋ𝑓≥0,if𝑓_𝐿2≤‖𝑄‖_𝐿².(32) It follows from (19), (27), and (32) that ℐ‖𝑄‖_𝐿²=0.(33) Hence, from (19) and (25), it holds that ().𝑢∈𝒢⟹𝑢isasolutiontotheminimizationproblem27(34)

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 ().𝑢∈𝒢⟸𝑢isasolutiontotheminimizationproblem27(35) 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)𝑢∈⋃_𝜔∈ℝ+𝒢_𝜔,(ii)𝑢 is a solution to the minimizing problem 𝑚𝑖𝑛{ℋ(𝑢),‖𝑢‖_𝐿²=‖𝑄‖_𝐿²}, (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 ℰ𝜙₀≤|𝑥|²𝜙₀²𝑑𝑥;(36) then 𝜙(𝑡) blows up in a finite time.

Consider the constrained minimization problem 𝐼(𝛼)≡minℋ𝑓∣𝑓∈𝐻¹,𝑓_𝐿2=𝛼.(37)

For 𝐼(𝛼), we cite a lemma in [15].

Lemma 9 (see Weinstein [15]). (a) Consider  𝐼(𝛼)=0 or 𝐼(𝛼)=−∞.
(b) Let 𝛼<‖𝑄‖_𝐿² 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 𝑓∈𝐿¹_loc, ‖∇𝑓‖_𝐿2≤𝐶, 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), 𝜇𝐵(0,1)∩𝑇_𝑦𝜀𝑔>2>𝛿.(38)

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 𝑓_𝑗⋅+𝑦_𝑗⇀𝑓≠0𝑤𝑒𝑎𝑘𝑙𝑦𝑖𝑛𝑊^1,𝑝.(39)

Lemma 12. Let 𝜃 be a real-valued function on ℝ^𝑁 and 𝑣∈𝐻¹(ℝ^𝑁) with ‖𝑣‖_𝐿²≤‖𝑄‖_𝐿². Then ℑ𝑣(𝑥)∇𝜃(𝑥)𝑑𝑥≤2ℋ(𝑣)|𝑣(𝑥)|²|∇𝜃(𝑥)|²𝑑𝑥^1/2.(40)

Proof. It follows from (30) and ‖𝑣‖_𝐿²≤‖𝑄‖_𝐿² that ℋ𝑒^𝑖𝛼𝜃𝑣≥0(41) for all real numbers 𝛼. On the other hand, it has ℋ𝑒^𝑖𝛼𝜃𝑣=𝛼²|𝑣|²|∇𝜃|²𝑑𝑥−𝛼ℑ(𝑣∇𝑣)∇𝜃𝑑𝑥+ℋ(𝑣).(42) 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 |𝑥|²𝜙(𝑡,𝑥)²𝑑𝑥≤𝑐₀.(43)

Proof. Setting 𝐽(𝑡)=∫|𝑥|²|𝜙(𝑡,𝑥)|²𝑑𝑥, we have 𝐽^′(𝑡)=2ℑ𝐽𝜙∇𝜙𝑑𝑥,_′′(𝑡)=4ℰ𝜙−4𝐽(𝑡).(44) It follows that 𝐽(𝑡)=(𝐽(0)−ℰ(0))cos𝑡+𝐽^′(0)sin𝑡+ℰ(0),(45) which implies the conclusion.

Lemma 14 (see [16, page 433]). Let 𝑢_𝑛∈𝐻¹, 𝑐₀>0, and 𝑅₀>0, for arbitrary 𝑛, satisfy ℋ𝑢_𝑛≤𝑐₀,𝑢_𝑛𝐿2≤‖𝑄‖_𝐿²,∇𝑢_𝑛𝐿2⟶∞,_|𝑥|>𝑟0𝑢_𝑛²(𝑛),(H)𝑑𝑥≤𝜀 where 𝜀(𝑛)>0 depends only on 𝑛. Then, it holds that _{|𝑥|>4𝑟0}∇𝑢_𝑛²𝑑𝑥≤𝐴,(46) 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 𝜙_0𝐿2=‖𝑄‖_𝐿².(47) From (15) and (19), the corresponding energy is ℰ𝜙₀=1−|𝑐|^4/𝑁|𝑐|²𝜆²|∇𝑄|²|𝑥|𝑑𝑥+²𝜙₀²𝑑𝑥=|𝑥|²𝜙₀²𝑑𝑥.(48) 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 𝜆(𝑡)=‖∇𝑄‖_𝐿²/‖∇𝜙(𝑡)‖_𝐿2 and (𝑆_𝜆𝜙)(𝑥,𝑡)=𝜆^𝑁/2𝜙(𝜆𝑥,𝑡). If 𝜙_0𝐿2=‖𝑄‖_𝐿²,(49) it holds that 𝑆_𝜆(𝑡)𝜙⋅+𝑦(𝑡),𝑡𝑒^{𝑖𝛾(𝑡)}⟶𝑄(⋅)in𝐻¹,as𝑡⟶𝑇(50) with 𝑦(𝑡)∈ℝ^𝑁 and 𝛾(𝑡)∈ℝ.

Proof. Let 𝑡_𝑘→𝑇. We choose 𝜆_𝑘=𝜆(𝑡_𝑘) to satisfy ∇𝑆_𝜆𝑘𝜙⋅+𝑦_𝑘,𝑡_𝑘𝐿2=𝜆_𝑘∇𝜙⋅+𝑦_𝑘,𝑡_𝑘𝐿2=‖∇𝑄‖_𝐿².(51) Setting 𝜙_𝑘≡𝑆_𝜆𝑘𝜙(⋅+𝑦_𝑘,𝑡_𝑘), noticing that ‖𝜙(𝑡_𝑘)‖_𝐿2 tends to ∞ as 𝑡_𝑘→𝑇, 𝜆_𝑘→0, and 𝜙_𝑘𝐿2=𝜙𝑡_𝑘𝐿2=𝜙_0𝐿2,(52) we know that 𝜙_𝑘 is uniformly bounded in 𝐻¹ and there is a weakly convergent subsequence 𝜙_𝑘𝑗 such that 𝜙_𝑘𝑗⇀𝜙in𝐿²,𝜙_𝑘𝑗⇀𝜙in𝐻¹.(53)
We note that ℋ𝜙_𝑘𝑗=𝜆²_𝑘𝑗ℋ𝜙𝑡_𝑘𝑗≤𝜆²_𝑘𝑗ℰ𝜙₀⟶0,𝑗⟶∞.(54) Since we have assumed ‖𝜙₀‖_𝐿2=‖𝑄‖_𝐿², 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 𝜙_𝑘𝑗 and a family 𝑦_𝑗 such that 𝜙_𝑘𝑗(⋅−𝑦_𝑘) has a strong limit in 𝐻¹. To see this, we need to make use of the concentration-compactness lemma (Lions [24]) which means that 𝜙_𝑘𝑗 has one of three properties: vanishing, dichotomy, and compactness.
Vanishing. For every 𝑀<∞, one has lim_𝑗→∞sup_{𝑦∈ℝ𝑁𝑦𝑗+𝐵𝑟(𝑀)}𝜙_𝑘𝑗(𝑥)²𝑑𝑥=0.(55)
Dichotomy. There exist a constant 𝛼∈(0,‖𝑄‖_𝐿²) and sequences 𝜓¹_𝑗 and 𝜓²_𝑗, bounded in 𝐻¹, such that, for all 𝜀>0, there exists 𝑗₀>0 such that for 𝑗>𝑗₀𝜓¹_𝑗𝐿2−𝛼≤𝜀,𝜓²_𝑗𝐿2−‖𝑄‖_𝐿²𝜙−𝛼≤𝜀,_𝑘𝑗−𝜓¹_𝑗−𝜓²_𝑗𝐻1𝜙≤𝜀,_𝑘𝑗−𝜓¹_𝑗−𝜓²_𝑗𝐿𝑝≤𝜀for2≤𝑝<2𝑁,𝑁−2distancesupp𝜓¹_𝑗,supp𝜓²_𝑗⟶∞.(56)
Compactness. There exists 𝑦_𝑗 in ℝ^𝑁. For any 𝜀>0, we can find 𝑀<∞ such that _{𝑦𝑗+𝐵𝑟(𝑀)}𝜙_𝑘𝑗²𝑑𝑥≥‖𝑄‖²_𝐿2−𝜀.(57)
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 𝜙_𝑘²_𝐿2≤𝐶₁,𝜙_𝑘𝐿^2+4/𝑁_2+4/𝑁≥𝐶₂>0.(58) By the boundness of ‖𝜙_𝑘‖_𝐻1 and the Sobolev inequality, there exist 𝛾>2+4/𝑁 and 𝐶₃>0 such that 𝜙_𝑘^𝛾_𝐿𝛾≤𝐶₃.(59) Now, we show the existence of positive constants 𝜀 and 𝛿 such that 𝜇𝜙_𝑘>𝜀≥𝛿>0.(60) Indeed, from (58) and (59), for sufficiently small 𝜀>0, we get 𝑐₂≤𝜙_𝑘^2+4/𝑁𝑑𝑥=_{𝜙𝑘<𝜀}𝜙_𝑘^2+4/𝑁𝑑𝑥+_{𝜙𝜀<𝑘<(1/𝜀)}𝜙_𝑘^2+4/𝑁𝑑𝑥+_{𝜙𝑘>(1/𝜀)}𝜙_𝑘^2+4/𝑁≤𝐶𝑑𝑥₂4𝐶_{1𝜙𝑘<𝜀}𝜙_𝑘²+𝑑𝑥_{𝜙𝜀<𝑘<(1/𝜀)}𝜙_𝑘^2+4/𝑁𝐶𝑑𝑥+²4𝐶_{3𝜙𝑘>(1/𝜀)}𝜙_𝑘^𝛾≤𝐶𝑑𝑥₂4𝐶₁𝜙_𝑘²_𝐿2+_{𝜙𝜀<𝑘<(1/𝜀)}𝜙_𝑘^2+4/𝑁+𝐶𝑑𝑥₂4𝐶₃𝜙_𝑘^𝛾_𝐿𝛾≤𝐶𝑑𝑥₂2+𝜇𝜙_𝑘1>𝜀𝜀^2+4/𝑁.(61) 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 𝜇{|𝑥|≤1}∩𝜙_𝑘⋅+𝑦_𝑘>𝜀2>𝛿.(62) Thus, _|𝑥|≤1𝜙_𝑘𝑗⋅+𝑦_𝑗²𝜀𝑑𝑥≥2²𝛿,(63) 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 0<𝜐<𝜇𝜃<𝜓¹_𝑗,(64) where 𝜃 and 𝜐 are two constants and 𝜓¹_𝑗 is bounded in 𝐻¹. Hence, by Lemma 11, there are a subsequence 𝜓¹_𝑗𝑟 and a sequence 𝑦_𝑟 such that 𝜓¹_𝑗𝑟⋅+𝑦_𝑟⇀𝜓≠0in𝐻¹.(65)
Using (56) gives rise to 0=𝐼‖𝑄‖_𝐿²≥liminf_𝑟→∞ℋ𝜓¹_𝑗𝑟+liminf_𝑟→∞ℋ𝜓²_𝑗𝑟=liminf_𝑟→∞ℋ𝜓¹_𝑗𝑟.(66) On the other hand, the fact ‖𝜓¹_𝑗𝑟‖_𝐿2<‖𝑄‖_𝐿² implies with Lemma 6 that liminf_𝑟→∞ℋ𝜓¹_𝑗𝑟≥0.(67) Thus, for any fixed 𝑛^∗, it has 0=𝐼‖𝑄‖_𝐿²≥liminf_𝑟→∞ℋ𝜓¹_𝑗𝑟=sup_𝑛inf_𝑟≥𝑛ℋ𝜓¹_𝑗𝑟≥inf_{𝑟≥𝑛∗}ℋ𝜓¹_𝑗𝑟.(68) We can then extract a minimizing subsequence, which we rename it by 𝜓¹_𝑗𝑟; that is, lim_𝑟→∞ℋ(𝜓¹_𝑗𝑟)=0. Using Lemma 9 yields 𝜓¹_𝑗𝑟⟶0,(69) which is impossible from (65).
Occurrence of Compactness. It follows from the previous arguments that compactness occurs. By (57), we get ‖𝑄‖²_𝐿2−𝜀≤_{𝑦𝑗+𝐵(𝑀)}𝜙_𝑘𝑗²𝑑𝑥≤𝜙_𝑘𝑗²𝑑𝑥≤‖𝑄‖²_𝐿2.(70) For 𝜙_𝑘𝑗(⋅+𝑦_𝑗) being bounded in 𝐻¹(ℝ^𝑁), there exist 𝜙∈𝐻¹(ℝ^𝑁) and a subsequence, which we again label it by 𝜙_𝑘𝑗, such that 𝜙_𝑘𝑗⋅+𝑦_𝑗⇀𝜙in𝐻¹.(71)
Given 𝑀>0, the embedding 𝐻¹(ℝ^𝑁)↪𝐿²({|𝑥|≤𝑟}) is compact and _{|𝑥|≤𝑟}𝜙²𝑑𝑥=lim_{𝑗→∞𝑥𝑚+𝐵(𝑟)}𝜙_𝑘𝑗²𝑑𝑥.(72)
Making use of (70) derives _ℝ𝑁𝜙²𝑑𝑥≥‖𝑄‖²_𝐿2−𝜀(73) for any 𝜀>0. Hence, it holds that _ℝ𝑁𝜙²𝑑𝑥=‖𝑄‖²_𝐿2.(74)
It follows that 𝜙_𝑘𝑗⋅+𝑦_𝑗⟶𝜙in𝐿²,(75) which implies with the Gagliardo-Nirenberg inequality (30) that 𝜙_𝑘𝑗⋅+𝑦_𝑗⟶𝜙in𝐿^2+4/𝑁.(76)
To show 𝜙_𝑘𝑗→𝜙 in 𝐻¹, we only need to show that ‖∇𝜙‖_𝐿2=‖∇𝑄‖_𝐿².
From (51) and (54), we know that 0=lim_𝑡→𝑇ℋ𝜙_𝜙𝑘𝑗=‖∇𝑄‖_𝐿²−12/𝑁+1lim_𝑡→𝑇𝜙_𝑘𝑗^4/𝑁+2=‖‖𝑑𝑥∇𝑄_𝐿²−12/𝑁+1lim_𝑡→𝑇𝜙^4/𝑁+2𝑑𝑥.(77) Hence, ‖∇𝜙‖_𝐿2<‖∇𝑄‖_𝐿² 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 ‖∇|𝜙|‖_𝐿2≤‖∇𝜙‖_𝐿2, we infer that |𝜙| is also a solution to problem (27). Thus it is a nonnegative solution of (16). It follows from ‖𝜙‖_𝐿2=‖𝑄‖_𝐿², ‖∇𝜙‖_𝐿2=‖∇𝑄‖_𝐿², and Proposition 7 that 𝜙=𝑄⋅+𝑦_𝑗𝑒^𝑖𝛾(78) 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 𝜆^𝑁(𝑡)𝜙(𝑡,𝜆(𝑡)(𝑥+𝑥(𝑡)))²⟶|𝑄(𝑥)|²in𝐿¹𝜙((𝑡as𝑡⟶𝑇,(79)𝑡,𝑥+𝑥))²⟶‖𝑄‖²_𝐿2𝛿_𝑥=0as𝑡⟶𝑇.(80) Using Lemma 13 derives that limsup_𝑡→𝑇|𝑥(𝑡≤√)|𝑐₀‖𝑄‖_𝐿².(81) Hence we have a positive constant 𝑟₀ such that ∀𝑡∈[0,𝑇),|𝑥(𝑡)|≤𝑟₀.(82)_𝐵(0,𝑟)𝜙(𝑡,𝑥)²=𝑥𝑑𝑥_𝐵(0,𝑟)𝜙(𝑡,𝑥)²(𝑥−𝑥(𝑡))𝑑𝑥+_𝐵(0,𝑟)𝜙(𝑡,𝑥)²𝑥(𝑡)=𝑑𝑥_{𝐵(−𝑥(𝑡),𝑟)}𝜙𝑡,𝑦+𝑥(𝑡)²+𝑦𝑑𝑦_{𝐵(−𝑥(𝑡),𝑟)}𝜙𝑡,𝑦+𝑥(𝑡)²𝑥(𝑡)𝑑𝑦.(83)
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 _𝐵(0,𝑟)𝜙(𝑡,𝑥)²𝑥𝑑𝑥−|𝑄(𝑥)|²𝑥(𝑡)𝑑𝑥=0.(84) On the other hand, Lemma 13 implies that _|𝑥|>𝑟𝜙(𝑡,𝑥)²𝑐𝑥𝑑𝑥≤⁰𝑟.(85) Thus lim_𝑡→𝑇𝜙(𝑡,𝑥)²𝑥𝑑𝑥−|𝑄(𝑥)|²𝑑𝑥𝑥(𝑡)=0.(86) By Lemma 12, we obtain 𝑑𝜙(𝑑𝑡𝑡,𝑥)²=𝑥𝑑𝑥2ℑ𝜙(𝑡,𝑥)∇𝜙(𝑡,𝑥)𝑑𝑥=2ℑ^𝑁_𝑗=1𝜙(𝑡,𝑥)∇𝜙(𝑡,𝑥)⋅∇𝜃_𝑗(𝑥)𝑑𝑥≤2^𝑁_𝑗=12ℋ𝜙(𝑡)𝜙(𝑡,𝑥)²∇𝜃_𝑗(𝑥)²𝑑𝑥^1/2≤𝐶,(87) where 𝜃_𝑗(𝑥)=𝑥_𝑗. Hence there exists 𝑥₁∈ℝ^𝑁 such that lim_𝑡→𝑇𝜙(𝑡,𝑥)²𝑥𝑑𝑥=−|𝑄(𝑥)|²𝑑𝑥𝑥₁.(88) Combining (86) with (88), we know that 𝑥(𝑡)→−𝑥₁ as 𝑡→𝑇 and we have |𝑢(𝑡,𝑥)|⟶‖𝑄‖²_𝐿2𝛿_𝑥=𝑥1.(89)

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 lim_𝑡→𝑇𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥=0.(90)

Proof. Let us define a positive function ℎ(𝑥)∈𝐶¹(ℝ^𝑁) such that ℎ(𝑥)𝑥=|𝑥||𝑥|=|𝑥|=ℎ(||)=0,<1,>0,1<<2,₂4,|𝑥|>2,(91) and ℎ_𝐴(𝑥)=𝐴²ℎ(𝑥/𝐴) for 𝐴>0 and it is valid that ∇ℎ_𝐴(𝑥)²≤𝐶ℎ_𝐴(𝑥),∀𝑥∈ℝ^𝑁.(92) Carrying out direct computation and using Hölder's inequality, we have 𝑑𝜙(𝑑𝑡𝑡,𝑥)²ℎ_𝐴𝑥−𝑦₀=𝑑𝑥2ℑ^𝑁_𝑗=1𝜙(𝑡,𝑥)∇𝜙(𝑡,𝑥)⋅∇ℎ_𝐴𝑥−𝑦₀𝑑𝑥≤𝐶_{|𝑥−𝑦0|≥𝐴}∇𝜙(𝑡,𝑥)²𝑑𝑥^1/2×𝜙(𝑡,𝑥)²∇ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2≤𝐶_{|𝑥−𝑦0|≥𝐴}∇𝜙(𝑡,𝑥)²𝑑𝑥^1/2×𝜙(𝑡,𝑥)²ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2,(93) which implies 𝑑𝜙(𝑑𝑡𝑡,𝑥)²ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2≤𝐶_{|𝑥−𝑦0|≥𝐴}∇𝜙(𝑡,𝑥)²𝑑𝑥^1/2.(94) Integrating on both sides gives rise to sup_{[)𝑡∈0,𝑇}𝜙(𝑡,𝑥)²ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2≤𝜙₀(𝑥)²ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2+𝐶^𝑇_{0|𝑥−𝑦0|≥𝐴}∇𝜙(𝑠,𝑥)²𝑑𝑥^1/2𝑑𝑠.(95) From the fact 𝜙₀∈Σ, we have sup_{[)𝑡∈0,𝑇}𝜙(𝑡,𝑥)²ℎ_𝐴𝑥−𝑦₀𝑑𝑥^1/2≤𝜀(𝐴)+𝐶^𝑇_{0|𝑥−𝑦0|≥𝐴}∇𝜙(𝑠,𝑥)²𝑑𝑥^1/2𝑑𝑠.(96)
By the virtue of Lemma 14 and Proposition 16, there exist 𝐴₁ and 𝐶₂>0 such that ∇𝜙(_{|𝑥−𝑦0|≥𝐴1}𝑠,𝑥)²𝑑𝑥^1/2𝑑𝑠≤𝐶₂,∀𝑠∈[0,𝑇).(97)
Using the dominated convergence theorem, we infer that lim_𝐴→∞^𝑇_{0|𝑥−𝑦0|≥𝐴}∇𝜙(𝑠,𝑥)²𝑑𝑥^1/2𝑑𝑠=0.(98) Thus, it holds that lim_𝐴→∞sup_{[)𝑡∈0,𝑇|𝑥−𝑦0|≥𝐴}𝜙(𝑡,𝑥)²𝑥−𝑦₀²𝑑𝑥=0,(99) which implies that there is 𝑎_𝜀>0 such that, for ∀𝑡∈[0,𝑇), _{|𝑥−𝑦0|≥𝑎𝜀}𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝜀𝑑𝑥≤2.(100) The identity ‖𝜙(𝑡)‖_𝐿2=‖𝜙₀‖_𝐿2=‖𝑄‖_𝐿² shows that _{|𝑥−𝑦0|≤𝑏𝜀}𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥≤𝑏²_𝜀‖𝑄‖_𝐿²≤𝜀2,for𝑏²_𝜀=𝜀2‖𝑄‖_𝐿².(101) In addition, we have _{𝑏𝜀≤|𝑥−𝑦0|≤𝑎𝜀}𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥≤𝑎²_{𝜀𝑏𝜀≤|𝑥−𝑦0|≤𝑎𝜀}𝜙(𝑡,𝑥)²𝑑𝑥.(102) Using Theorem 2 yields lim_{𝑡→𝑇𝑏𝜀≤|𝑥−𝑦0|≤𝑎𝜀}𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥=0.(103) In conclusion, for all 𝜀>0, we have shown that lim_𝑡→𝑇𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝜀𝑑𝑥≤2+𝜀2≤𝜀.(104)

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

Proof of Theorem 3. Simple calculation yields 𝑑𝑑𝑡𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥=4ℑ𝑥−𝑦₀𝜙(𝑡,𝑥)∇𝜙(𝑡,𝑥).(105) Therefore, the inequality (40) in the case 𝜃(𝑥)=|𝑥−𝑦₀|² implies that 𝑑𝑑𝑡𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥^1/2≤𝐶.(106) Integrating from 𝑡 to 𝑇, by Proposition 16, we obtain 𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥^1/2≤𝐶(𝑇−𝑡).(107)
Combining the above inequality and the following inequality 𝜙(𝑡,𝑥)²𝑑𝑥²≤𝑥−𝑦₀²𝜙(𝑡,𝑥)²𝑑𝑥∇𝜙(𝑡,𝑥)²𝑑𝑥,(108) we get the result ∇𝜙(𝑡)_𝐿2≥‖𝑄‖_𝐿²𝐶().𝑇−𝑡(109)

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).