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

Authors: Xiaoguang Li; Chong Lai
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction and Main Results

In this paper, we deal with the Cauchy problem of the nonlinear Schrödinger equation with a harmonic potential(1)iϕt+Δϕ-|x|2ϕ+|ϕ|4/Nϕ=0,x∈ℝN,t≥0,(2)ϕ(0,x)=ϕ0(x),
where ϕ=ϕ(t,x): [0,T)×ℝN→ℂ is the wave function, N is the space dimension, and Δ denotes the Laplace operator on ℝN. 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(3)iψt+Δψ+|ψ|4/Nψ=0,x∈ℝN,t≥0,(4)ψ(0,x)=ψ0(x).
For Cauchy problem (3)-(4), Ginibre and Velo [7] established the local existence in H1(ℝN). 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
(5)-Δu+u-|u|4/Nu=0,u∈H1(ℝN).
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
(6)(a+bt)-(N/2)Q(xa+bt)e(ib|x|2)/4(a+bt)ei(c+dt)/(a+bt),
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:
(7)limt→Tλ(t)N/2ψ(t,λ(t)(x+y(t)))=Q(x),
where T is the blow-up time and y(t)∈ℝN 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∈ℝN, and x1∈ℝN such that
(8)ψ(t,x)=(ωT-t)N/2eiθ+(i|x-x1|2/4(-T+t))-(iω2/(-T+t))×Q(ωT-t((x-x1)-(T-t)x0)).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.Letϕ(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)
Ifϕ(t) blows up at a finite time Tϕ, then Tϕ≤π/2.(2)
Ifϕ(t) blows up at Tϕ<π/2, ψ(t) blows up at time Tψ<∞.(3)
Conversely,ψ(t) blows up at time Tψ<∞; then ϕ(t) blows up at Tϕ<π/2.(4)
Ifϕ(t) blows up at Tϕ=π/2, ψ(t) exists globally (Tψ=∞).Moreover, Carles studied the qualitative properties of minimal blow-up solutionsϕ(t) with Tϕ<π/2 (see [18, 20]). As for the minimal blow-up solutions with Tϕ=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 Tϕ=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 dataϕ0 with ∥ϕ0∥L2=∥Q∥L2 for which the solution of the Cauchy problem (1)-(2) blows up in a finite time.Theorem 2.
Letϕ(t) be a blow-up solution of (1) with ∥ϕ0∥L2=∥Q∥L2. Then there is y0∈ℝN such that
(9)ϕ(t,x)⟶∥Q∥L22δy0
in the sense of distribution as t→T.Theorem 3.
There existsC>0 such that
(10)∥∇ϕ(t)∥L2≥CT-t,∀t∈[0,T).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 eitQ(x). The minimal blow-up solutions with T=π/2 were sensible as pointed in [18].In this paper,Lq(ℝN), ∥·∥Lq(ℝN), and ∫ℝN‍·dx are denoted by Lq, ∥·∥Lq, and ∫‍·dx, respectively. The various positive constants are also denoted by C.This paper proceeds as follows. In Section2, 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
(11)Σ:={u∈H1,|x|u∈L2}.
The inner product of the space Σ is defined as
(12)〈u,v〉:=∫‍∇u∇v¯+uv¯+|x|2uv¯dx.
The norm of Σ is denoted by ∥·∥Σ. Moreover, we define an energy functional ℰ on Σ by
(13)ℰ(u):=∫‍|∇u|2+|x|2|u|2-11+2/N|u|2+4/Ndx.From Cazenave [17], we have the local well-posedness for the Cauchy problem of (1) follows.Proposition 5.
For anyϕ0∈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 limt→T∥ϕ(t)∥Σ=∞ (blowup). Moreover, for any t∈[0,T), it holds the conservation laws of mass
(14)∥ϕ(t)∥L2=∥ϕ0∥L2
and the energy
(15)ℰ(ϕ(t))=ℰ(ϕ0).


### 2.2. Variational Characterization of the Ground State

Consider the equation(16)-Δu+ωu-|u|4/Nu=0,u∈H1(ℝN).
For (16), we set some notations such as 𝒳ω (the solution set), 𝒢ω (the ground solution set), and 𝒢 as follows:
(17)𝒳ω={u∈H1;u≠0,-Δu+ωu-|u|4/Nu=0},𝒢ω={u∈𝒳ω;S(u)≤S(v),∀v∈𝒳ω},𝒢=⋃ω∈R+‍𝒢ω,
where S(u)=∫‍(1/2)|∇u|2+(ω/2)|u|2-(1/4/N+2)|u|2+(4/N)dx.For anyu∈𝒳ω, the following two identities hold true:
(18)∫‍|∇u|2+ω|u|2dx=∫|u|2+(4/N)‍dx,∫‍(N-2)|∇u|2+Nω|u|2dx=∫‍N1+2/N|u|2+(4/N)dx(Pohozaev’sidentity).
The above two equalities imply
(19)ℋ(u)=0,∀u∈𝒳,
where
(20)ℋ(u):=∫|∇u|2‍-12/N+1|u|2+(4/N)dx.Naturally, we get(21)u∈𝒢ω⟺{u∈𝒳ω,∥u∥L2≤∥v∥L2,∀v∈𝒳ω.According to Cazenave [17], the set 𝒢ω can be described as
(22)𝒢ω=⋃‍{eiθφω(·-y);θ∈R,y∈ℝN},
where φω 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
(23)-Δu+u-|u|4/Nu=0,u∈H1,u(x)=u(|x|),u(x)>0.
Noticing the fact that Q(x)=φω|ω=1, it is easy to check that
(24)φω=ωN/4Q(ω1/2x)∈𝒢ω,∥φω∥L2=∥Q∥L2.
It follows from (21), (22), and (24) that
(25)u∈𝒢ω⟺{u∈𝒳ω,∥u∥L2=∥Q∥L2,(26)𝒢ω=⋃‍{eiθφω(·-y);θ∈R,y∈ℝN},=⋃‍{eiθωN/4Q(ω1/2(·-y));θ∈R,y∈ℝN}.With functionalℋ defined by (20), we now introduce the following constrained minimization problem
(27)ℐ(∥Q∥L2)≡inf{ℋ(f)∣f∈H1,∥f∥L2=∥Q∥L2}.Now, we claim that(28)u∈𝒢⟺uisasolutiontotheminimizationproblem(27).In fact,(N/(N+2))∥Q∥L24/N is the minimum of the functional (see Kwong [11] or Weinstein [9])
(29)I(ψ)=∥∇ψ∥L22∥ψ∥L24/N∥ψ∥L2+4/N2+4/N,ψ∈H1,
which derives the Gagliardo-Nirenberg inequality
(30)∥ψ∥L2+4/N2+4/N≤N+2N(∥ψ∥L2∥Q∥L2)4/N∥∇ψ∥L22.
The inequality (30) implies the following lemma on the functional ℋ.Lemma 6 (see Weinstein [9]).
For anyf∈H1, one has
(31)[1-(∥f∥L2∥Q∥L2)4/N]∥∇f∥L22≤ℋ(f).Lemma6 implies that
(32)ℋ(f)≥0,if∥f∥L2≤∥Q∥L2.
It follows from (19), (27), and (32) that
(33)ℐ(∥Q∥L2)=0.
Hence, from (19) and (25), it holds that
(34)u∈𝒢⟹uisasolutiontotheminimizationproblem(27).On the other hand, ifu is a minimizer of the variational problem of (27), it solves the Euler-Lagrange equation (16). So u∈𝒳ω for some ω>0, and by (27) and (25), we know u∈𝒢ω⊂𝒢. This implies that
(35)u∈𝒢⟸uisasolutiontotheminimizationproblem(27).
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+𝒢ω,(ii)
uis a solution to the minimizing problemmin{ℋ(u),∥u∥L2=∥Q∥L2},(iii)
u=eiθωN/4Q(ω1/2(x-x0)), for someθ∈ℝ,ω∈ℝ+,
andx0∈ℝN.


### 2.3. Lemmas

Lemma 8 (see Zhang [6]).
Letϕ0≠0, the initial datum of Cauchy problem (1)-(2), satisfy
(36)ℰ(ϕ0)≤∫‍|x|2|ϕ0|2dx;
then ϕ(t) blows up in a finite time.Consider the constrained minimization problem(37)I(α)≡min{ℋ(f)∣f∈H1,∥f∥L2=α}.ForI(α), we cite a lemma in [15].Lemma 9 (see Weinstein [15]).
(a) ConsiderI(α)=0 or I(α)=-∞.
(b) Letα<∥Q∥L2 and un be a minimizing sequence; then it holds that I(α)=0 and un⇀0 weakly in H1.Now, we recall some lemmas on the compactness.Lemma 10 (see Brezis and Lieb [22]).
Letf∈Lloc1, ∥∇f∥L2≤C, and μ(|f|>ε)≥δ>0. Then there exists a shift Tyf(x)=f(x+y) such that, for some constant α=α(C,δ,ε),
(38)μ(B(0,1)∩[Tyg>ε2])>δ.Lemma 11 (see Lieb [23]).
Letfj be a uniformly bounded sequence of functions in W1,p with 1<p<∞. Assume further that there are positive constant C and η satisfying μ(|fj|>η)≥C. Then there exists a sequence yj∈ℝN such that
(39)fj(·+yj)⇀f≠0weaklyinW1,p.Lemma 12.
Letθ be a real-valued function on ℝN and v∈H1(ℝN) with ∥v∥L2≤∥Q∥L2. Then
(40)|ℑ∫v¯(x)∇θ(x)dx|≤(2ℋ(v)∫|v(x)|2|∇θ(x)|2dx)1/2.Proof.
It follows from (30) and ∥v∥L2≤∥Q∥L2 that
(41)ℋ(eiαθv)≥0
for all real numbers α. On the other hand, it has
(42)ℋ(eiαθv)=α2∫|v|2|∇θ|2dx-α∫ℑ(v∇v¯)∇θdx+ℋ(v).
Thus the discriminant of the equation in α must be negative or null and the desired inequality follows.Lemma 13.
There is a constantc0 such that
(43)∫‍|x|2|ϕ(t,x)|2dx≤c0.Proof.
SettingJ(t)=∫‍|x|2|ϕ(t,x)|2dx, we have
(44)J′(t)=2ℑ∫ϕ¯∇ϕdx,J′′(t)=4ℰ(ϕ)-4J(t).
It follows that
(45)J(t)=(J(0)-ℰ(0))cost+J′(0)sint+ℰ(0),
which implies the conclusion.Lemma 14 (see [16, page 433]).
Letun∈H1, c0>0, and R0>0, for arbitrary n, satisfy
(H)ℋ(un)≤c0,∥un∥L2≤∥Q∥L2,∥∇un∥L2⟶∞,∫|x|>r0‍|un|2dx≤ε(n),
where ε(n)>0 depends only on n. Then, it holds that
(46)∫|x|>4r0‍|∇un|2dx≤A,
with A=A(r0,c0>0).


## 2.1. Local Wellposedness

The energy space of (1) was defined as
(11)Σ:={u∈H1,|x|u∈L2}.
The inner product of the space Σ is defined as
(12)〈u,v〉:=∫‍∇u∇v¯+uv¯+|x|2uv¯dx.
The norm of Σ is denoted by ∥·∥Σ. Moreover, we define an energy functional ℰ on Σ by
(13)ℰ(u):=∫‍|∇u|2+|x|2|u|2-11+2/N|u|2+4/Ndx.From Cazenave [17], we have the local well-posedness for the Cauchy problem of (1) follows.Proposition 5.
For anyϕ0∈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 limt→T∥ϕ(t)∥Σ=∞ (blowup). Moreover, for any t∈[0,T), it holds the conservation laws of mass
(14)∥ϕ(t)∥L2=∥ϕ0∥L2
and the energy
(15)ℰ(ϕ(t))=ℰ(ϕ0).


## 2.2. Variational Characterization of the Ground State

Consider the equation(16)-Δu+ωu-|u|4/Nu=0,u∈H1(ℝN).
For (16), we set some notations such as 𝒳ω (the solution set), 𝒢ω (the ground solution set), and 𝒢 as follows:
(17)𝒳ω={u∈H1;u≠0,-Δu+ωu-|u|4/Nu=0},𝒢ω={u∈𝒳ω;S(u)≤S(v),∀v∈𝒳ω},𝒢=⋃ω∈R+‍𝒢ω,
where S(u)=∫‍(1/2)|∇u|2+(ω/2)|u|2-(1/4/N+2)|u|2+(4/N)dx.For anyu∈𝒳ω, the following two identities hold true:
(18)∫‍|∇u|2+ω|u|2dx=∫|u|2+(4/N)‍dx,∫‍(N-2)|∇u|2+Nω|u|2dx=∫‍N1+2/N|u|2+(4/N)dx(Pohozaev’sidentity).
The above two equalities imply
(19)ℋ(u)=0,∀u∈𝒳,
where
(20)ℋ(u):=∫|∇u|2‍-12/N+1|u|2+(4/N)dx.Naturally, we get(21)u∈𝒢ω⟺{u∈𝒳ω,∥u∥L2≤∥v∥L2,∀v∈𝒳ω.According to Cazenave [17], the set 𝒢ω can be described as
(22)𝒢ω=⋃‍{eiθφω(·-y);θ∈R,y∈ℝN},
where φω 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
(23)-Δu+u-|u|4/Nu=0,u∈H1,u(x)=u(|x|),u(x)>0.
Noticing the fact that Q(x)=φω|ω=1, it is easy to check that
(24)φω=ωN/4Q(ω1/2x)∈𝒢ω,∥φω∥L2=∥Q∥L2.
It follows from (21), (22), and (24) that
(25)u∈𝒢ω⟺{u∈𝒳ω,∥u∥L2=∥Q∥L2,(26)𝒢ω=⋃‍{eiθφω(·-y);θ∈R,y∈ℝN},=⋃‍{eiθωN/4Q(ω1/2(·-y));θ∈R,y∈ℝN}.With functionalℋ defined by (20), we now introduce the following constrained minimization problem
(27)ℐ(∥Q∥L2)≡inf{ℋ(f)∣f∈H1,∥f∥L2=∥Q∥L2}.Now, we claim that(28)u∈𝒢⟺uisasolutiontotheminimizationproblem(27).In fact,(N/(N+2))∥Q∥L24/N is the minimum of the functional (see Kwong [11] or Weinstein [9])
(29)I(ψ)=∥∇ψ∥L22∥ψ∥L24/N∥ψ∥L2+4/N2+4/N,ψ∈H1,
which derives the Gagliardo-Nirenberg inequality
(30)∥ψ∥L2+4/N2+4/N≤N+2N(∥ψ∥L2∥Q∥L2)4/N∥∇ψ∥L22.
The inequality (30) implies the following lemma on the functional ℋ.Lemma 6 (see Weinstein [9]).
For anyf∈H1, one has
(31)[1-(∥f∥L2∥Q∥L2)4/N]∥∇f∥L22≤ℋ(f).Lemma6 implies that
(32)ℋ(f)≥0,if∥f∥L2≤∥Q∥L2.
It follows from (19), (27), and (32) that
(33)ℐ(∥Q∥L2)=0.
Hence, from (19) and (25), it holds that
(34)u∈𝒢⟹uisasolutiontotheminimizationproblem(27).On the other hand, ifu is a minimizer of the variational problem of (27), it solves the Euler-Lagrange equation (16). So u∈𝒳ω for some ω>0, and by (27) and (25), we know u∈𝒢ω⊂𝒢. This implies that
(35)u∈𝒢⟸uisasolutiontotheminimizationproblem(27).
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+𝒢ω,(ii)
uis a solution to the minimizing problemmin{ℋ(u),∥u∥L2=∥Q∥L2},(iii)
u=eiθωN/4Q(ω1/2(x-x0)), for someθ∈ℝ,ω∈ℝ+,
andx0∈ℝN.


## 2.3. Lemmas

Lemma 8 (see Zhang [6]).
Letϕ0≠0, the initial datum of Cauchy problem (1)-(2), satisfy
(36)ℰ(ϕ0)≤∫‍|x|2|ϕ0|2dx;
then ϕ(t) blows up in a finite time.Consider the constrained minimization problem(37)I(α)≡min{ℋ(f)∣f∈H1,∥f∥L2=α}.ForI(α), we cite a lemma in [15].Lemma 9 (see Weinstein [15]).
(a) ConsiderI(α)=0 or I(α)=-∞.
(b) Letα<∥Q∥L2 and un be a minimizing sequence; then it holds that I(α)=0 and un⇀0 weakly in H1.Now, we recall some lemmas on the compactness.Lemma 10 (see Brezis and Lieb [22]).
Letf∈Lloc1, ∥∇f∥L2≤C, and μ(|f|>ε)≥δ>0. Then there exists a shift Tyf(x)=f(x+y) such that, for some constant α=α(C,δ,ε),
(38)μ(B(0,1)∩[Tyg>ε2])>δ.Lemma 11 (see Lieb [23]).
Letfj be a uniformly bounded sequence of functions in W1,p with 1<p<∞. Assume further that there are positive constant C and η satisfying μ(|fj|>η)≥C. Then there exists a sequence yj∈ℝN such that
(39)fj(·+yj)⇀f≠0weaklyinW1,p.Lemma 12.
Letθ be a real-valued function on ℝN and v∈H1(ℝN) with ∥v∥L2≤∥Q∥L2. Then
(40)|ℑ∫v¯(x)∇θ(x)dx|≤(2ℋ(v)∫|v(x)|2|∇θ(x)|2dx)1/2.Proof.
It follows from (30) and ∥v∥L2≤∥Q∥L2 that
(41)ℋ(eiαθv)≥0
for all real numbers α. On the other hand, it has
(42)ℋ(eiαθv)=α2∫|v|2|∇θ|2dx-α∫ℑ(v∇v¯)∇θdx+ℋ(v).
Thus the discriminant of the equation in α must be negative or null and the desired inequality follows.Lemma 13.
There is a constantc0 such that
(43)∫‍|x|2|ϕ(t,x)|2dx≤c0.Proof.
SettingJ(t)=∫‍|x|2|ϕ(t,x)|2dx, we have
(44)J′(t)=2ℑ∫ϕ¯∇ϕdx,J′′(t)=4ℰ(ϕ)-4J(t).
It follows that
(45)J(t)=(J(0)-ℰ(0))cost+J′(0)sint+ℰ(0),
which implies the conclusion.Lemma 14 (see [16, page 433]).
Letun∈H1, c0>0, and R0>0, for arbitrary n, satisfy
(H)ℋ(un)≤c0,∥un∥L2≤∥Q∥L2,∥∇un∥L2⟶∞,∫|x|>r0‍|un|2dx≤ε(n),
where ε(n)>0 depends only on n. Then, it holds that
(46)∫|x|>4r0‍|∇un|2dx≤A,
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 Theorem1.
Settingϕ0=ϕ(c,λ)=cλN/2Q(λx) with λ being arbitrary positive real number and c being complex number satisfying |c|=1, then
(47)∥ϕ0∥L2=∥Q∥L2.
From (15) and (19), the corresponding energy is
(48)ℰ(ϕ0)=(1-|c|4/N)|c|2λ2∫‍|∇Q|2dx+∫‍|x|2|ϕ0|2dx=∫‍|x|2|ϕ0|2dx.
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 (Theorem2).Proposition 15.
Letϕ(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 (Sλϕ)(x,t)=λN/2ϕ(λx,t). If
(49)∥ϕ0∥L2=∥Q∥L2,
it holds that
(50)Sλ(t)ϕ(·+y(t),t)eiγ(t)⟶Q(·)inH1,ast⟶T
with y(t)∈ℝN and γ(t)∈ℝ.Proof.
Lettk→T. We choose λk=λ(tk) to satisfy
(51)∥∇Sλkϕ(·+yk,tk)∥L2=λk∥∇ϕ(·+yk,tk)∥L2=∥∇Q∥L2.
Setting ϕk≡Sλkϕ(·+yk,tk), noticing that ∥ϕ(tk)∥L2 tends to ∞ as tk→T, λk→0, and
(52)∥ϕk∥L2=∥ϕ(tk)∥L2=∥ϕ0∥L2,
we know that ϕk is uniformly bounded in H1 and there is a weakly convergent subsequence ϕkj such that
(53)ϕkj⇀ϕinL2,ϕkj⇀ϕinH1.
We note that(54)ℋ(ϕkj)=λkj2ℋ(ϕ(tkj))≤λkj2ℰ(ϕ0)⟶0,j⟶∞.
Since we have assumed ∥ϕ0∥L2=∥Q∥L2, by (52), (54), and (31), we know that ϕk is a minimizing sequence for the variational problem (27).
Next, we will prove that the minimizing sequenceϕk has a subsequence ϕkj and a family yj such that ϕkj(·-yk) has a strong limit in H1. To see this, we need to make use of the concentration-compactness lemma (Lions [24]) which means that ϕkj has one of three properties: vanishing, dichotomy, and compactness.
Vanishing. For every M<∞, one has
(55)limj→∞supy∈ℝN∫yj+Br(M)‍|ϕkj(x)|2dx=0.
Dichotomy. There exist a constant α∈(0,∥Q∥L2) and sequences ψj1 and ψj2, bounded in H1, such that, for all ε>0, there exists j0>0 such that for j>j0(56)|∥ψj1∥L2-α|≤ε,|∥ψj2∥L2-(∥Q∥L2-α)|≤ε,∥ϕkj-ψj1-ψj2∥H1≤ε,∥ϕkj-ψj1-ψj2∥Lp≤εfor2≤p<2NN-2,distance(suppψj1,suppψj2)⟶∞.
Compactness. There exists yj in ℝN. For any ɛ>0, we can find M<∞ such that
(57)∫yj+Br(M)‍|ϕkj|2dx≥∥Q∥L22-ε.
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
(58)∥ϕk∥L22≤C1,∥ϕk∥L2+4/N2+4/N≥C2>0.
By the boundness of ∥ϕk∥H1 and the Sobolev inequality, there exist γ>2+4/N and C3>0 such that
(59)∥ϕk∥Lγγ≤C3.
Now, we show the existence of positive constants ε and δ such that
(60)μ(|ϕk|>ε)≥δ>0.
Indeed, from (58) and (59), for sufficiently small ε>0, we get
(61)c2≤∫|ϕk|2+4/Ndx=∫{|ϕk|<ε}|ϕk|2+4/Ndx+∫{ε<|ϕk|<(1/ε)}|ϕk|2+4/Ndx+∫{|ϕk|>(1/ε)}|ϕk|2+4/Ndx≤C24C1∫{|ϕk|<ε}|ϕk|2dx+∫{ε<|ϕk|<(1/ε)}|ϕk|2+4/Ndx+C24C3∫{|ϕk|>(1/ε)}|ϕk|γdx≤C24C1∥ϕk∥L22+∫{ε<|ϕk|<(1/ε)}|ϕk|2+4/Ndx+C24C3∥ϕk∥Lγγdx≤C22+μ(|ϕk|>ε)(1ε)2+4/N.
Thus we know that (60) with δ=(C2/2)ε2+4/N is valid. From (60) and Lemma 10, there exist α and yk satisfying
(62)μ({|x|≤1}∩{|ϕk(·+yk)|}>ε2)>δ.
Thus,
(63)∫|x|≤1‍|ϕkj(·+yj)|2dx≥(ε2)2δ,
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
(64)0<υ<μ{θ<|ψj1|},
where θ and υ are two constants and ψj1 is bounded in H1. Hence, by Lemma 11, there are a subsequence ψjr1 and a sequence yr such that
(65)ψjr1(·+yr)⇀ψ≠0inH1.
Using (56) gives rise to
(66)0=I(∥Q∥L2)≥liminfr→∞ℋ(ψjr1)+liminfr→∞ℋ(ψjr2)=liminfr→∞ℋ(ψjr1).
On the other hand, the fact ∥ψjr1∥L2<∥Q∥L2 implies with Lemma 6 that
(67)liminfr→∞ℋ(ψjr1)≥0.
Thus, for any fixed n*, it has
(68)0=I(∥Q∥L2)≥liminfr→∞ℋ(ψjr1)=supninfr≥nℋ(ψjr1)≥infr≥n*ℋ(ψjr1).
We can then extract a minimizing subsequence, which we rename it by ψjr1; that is, limr→∞ℋ(ψjr1)=0. Using Lemma 9 yields
(69)ψjr1⟶0,
which is impossible from (65).
Occurrence of Compactness. It follows from the previous arguments that compactness occurs. By (57), we get
(70)∥Q∥L22-ε≤∫yj+B(M)‍|ϕkj|2dx≤∫‍|ϕkj|2dx≤∥Q∥L22.
For ϕkj(·+yj) being bounded in H1(ℝN), there exist ϕ∈H1(ℝN) and a subsequence, which we again label it by ϕkj, such that
(71)ϕkj(·+yj)⇀ϕinH1.
GivenM>0, the embedding H1(ℝN)↪L2({|x|≤r}) is compact and
(72)∫|x|≤r‍|ϕ|2dx=limj→∞∫xm+B(r)‍|ϕkj|2dx.
Making use of (70) derives
(73)∫ℝN‍|ϕ|2dx≥∥Q∥L22-ε
for any ε>0. Hence, it holds that
(74)∫ℝN‍|ϕ|2dx=∥Q∥L22.
It follows that(75)ϕkj(·+yj)⟶ϕinL2,
which implies with the Gagliardo-Nirenberg inequality (30) that
(76)ϕkj(·+yj)⟶ϕinL2+4/N.
To showϕkj→ϕ in H1, we only need to show that ∥∇ϕ∥L2=∥∇Q∥L2.
From (51) and (54), we know that
(77)0=limt→Tℋ(ϕϕkj)=∥∇Q∥L2-12/N+1limt→T∫‍|ϕkj|4/N+2dx=∥∇Q∥L2-12/N+1limt→T∫‍|ϕ|4/N+2dx.
Hence, ∥∇ϕ∥L2<∥∇Q∥L2 derives ℰ(ϕ)<0. This contradicts Lemma 6 and the fact ϕ≠0.
Sinceϕ 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
(78)ϕ=Q(·+yj)eiγ
for some y∈ℝN and γ∈ℝ. By redefining the sequence γj, we can set γ=0.Proof of Theorem2.
It follows from Proposition15 that
(79)λN(t)|ϕ(t,λ(t)(x+x(t)))|2⟶|Q(x)|2inL1ast⟶T,(80)|ϕ(t,x+x(t))|2⟶∥Q∥L22δx=0ast⟶T.
Using Lemma 13 derives that
(81)limsupt→T|x(t)|≤c0∥Q∥L2.
Hence we have a positive constant r0 such that
(82)∀t∈[0,T),|x(t)|≤r0.(83)∫B(0,r)‍|ϕ(t,x)|2xdx=∫B(0,r)‍|ϕ(t,x)|2(x-x(t))dx+∫B(0,r)‍|ϕ(t,x)|2x(t)dx=∫B(-x(t),r)‍|ϕ(t,y+x(t))|2ydy+∫B(-x(t),r)‍|ϕ(t,y+x(t))|2x(t)dy.
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
(84)∫B(0,r)‍|ϕ(t,x)|2xdx-∫‍|Q(x)|2x(t)dx=0.
On the other hand, Lemma 13 implies that
(85)∫|x|>r‍|ϕ(t,x)|2xdx≤c0r.
Thus
(86)limt→T{∫‍|ϕ(t,x)|2xdx-∫‍|Q(x)|2dxx(t)}=0.
By Lemma 12, we obtain
(87)ddt|∫‍|ϕ(t,x)|2xdx|=|2ℑ∫‍ϕ¯(t,x)∇ϕ(t,x)dx|=2ℑ∑j=1N‍∫‍|ϕ¯(t,x)∇ϕ(t,x)·∇θj(x)dx|≤2∑j=1N‍(2ℋ(ϕ(t))∫‍|ϕ(t,x)|2|∇θj(x)|2dx)1/2≤C,
where θj(x)=xj. Hence there exists x1∈ℝN such that
(88)limt→T∫‍|ϕ(t,x)|2xdx=-(∫‍|Q(x)|2dx)x1.
Combining (86) with (88), we know that x(t)→-x1 as t→T and we have
(89)|u(t,x)|⟶∥Q∥L22δx=x1.


## 4. Blow-Up Rate

To establish the lower bound of the blow-up rate, we use the following proposition.Proposition 16.
Lettingy0 be the blow-up point determined in Theorem 2, it has
(90)limt→T∫‍|x-y0|2|ϕ(t,x)|2dx=0.Proof.
Let us define a positive functionh(x)∈C1(ℝN) such that
(91)h(x)=h(|x|)={=0,|x|<1,>0,1<|x|<2,=|x|24,|x|>2,
and hA(x)=A2h(x/A) for A>0 and it is valid that
(92)|∇hA(x)|2≤ChA(x),∀x∈ℝN.
Carrying out direct computation and using Hölder's inequality, we have
(93)|ddt∫‍|ϕ(t,x)|2hA(x-y0)dx|=|2ℑ∑j=1N‍∫‍ϕ¯(t,x)∇ϕ(t,x)·∇hA(x-y0)dx|≤C(∫|x-y0|≥A‍|∇ϕ(t,x)|2dx)1/2×(∫‍|ϕ(t,x)|2∇hA(x-y0)dx)1/2≤C(∫|x-y0|≥A‍|∇ϕ(t,x)|2dx)1/2×(∫‍|ϕ(t,x)|2hA(x-y0)dx)1/2,
which implies
(94)|ddt(∫‍|ϕ(t,x)|2hA(x-y0)dx)1/2|≤C(∫|x-y0|≥A‍|∇ϕ(t,x)|2dx)1/2.
Integrating on both sides gives rise to
(95)supt∈[0,T)(∫‍|ϕ(t,x)|2hA(x-y0)dx)1/2≤(∫‍|ϕ0(x)|2hA(x-y0)dx)1/2+C∫0T‍(∫|x-y0|≥A‍|∇ϕ(s,x)|2dx)1/2ds.
From the fact ϕ0∈Σ, we have
(96)supt∈[0,T)(∫‍|ϕ(t,x)|2hA(x-y0)dx)1/2≤ε(A)+C∫0T‍(∫|x-y0|≥A‍|∇ϕ(s,x)|2dx)1/2ds.
By the virtue of Lemma14 and Proposition 16, there exist A1 and C2>0 such that
(97)∫|x-y0|≥A1‍(|∇ϕ(s,x)|2dx)1/2ds≤C2,∀s∈[0,T).
Using the dominated convergence theorem, we infer that(98)limA→∞∫0T‍(∫|x-y0|≥A‍|∇ϕ(s,x)|2dx)1/2ds=0.
Thus, it holds that
(99)limA→∞supt∈[0,T)(∫|x-y0|≥A‍|ϕ(t,x)|2|x-y0|2dx)=0,
which implies that there is aε>0 such that, for ∀t∈[0,T),
(100)∫|x-y0|≥aε‍|x-y0|2|ϕ(t,x)|2dx≤ε2.
The identity ∥ϕ(t)∥L2=∥ϕ0∥L2=∥Q∥L2 shows that
(101)∫|x-y0|≤bε‍|x-y0|2|ϕ(t,x)|2dx≤bε2∥Q∥L2≤ε2,forbε2=ε2∥Q∥L2.
In addition, we have
(102)∫bε≤|x-y0|≤aε‍|x-y0|2|ϕ(t,x)|2dx≤aε2∫bε≤|x-y0|≤aε‍|ϕ(t,x)|2dx.
Using Theorem 2 yields
(103)limt→T∫bɛ≤|x-y0|≤aɛ‍|x-y0|2|ϕ(t,x)|2dx=0.
In conclusion, for all ε>0, we have shown that
(104)limt→T∫‍|x-y0|2|ϕ(t,x)|2dx≤ε2+ε2≤ε.Now, we establish the lower bound of the blow-up rate.Proof of Theorem3.
Simple calculation yields(105)ddt∫‍|x-y0|2|ϕ(t,x)|2dx=4ℑ∫‍(x-y0)ϕ(t,x)∇ϕ¯(t,x).
Therefore, the inequality (40) in the case θ(x)=|x-y0|2 implies that
(106)|ddt(∫‍|x-y0|2|ϕ(t,x)|2dx)1/2|≤C.
Integrating from t to T, by Proposition 16, we obtain
(107)|(∫‍|x-y0|2|ϕ(t,x)|2dx)1/2|≤C(T-t).
Combining the above inequality and the following inequality(108)(∫‍|ϕ(t,x)|2dx)2≤(∫‍|x-y0|2|ϕ(t,x)|2dx)(∫|∇ϕ(t,x)|‍2dx),
we get the result
(109)∥∇ϕ(t)∥L2≥∥Q∥L2C(T-t).



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*Source: 813417-2013-12-31.xml*