Mass concentration for nonlinear Schrödinger equation with partial confinement

J. Math. Anal. Appl. 481 (2020) 123484

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Mass concentration for nonlinear Schrödinger equation with partial confinement ✩ Jingjing Pan, Jian Zhang ∗ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 6 June 2019 Available online 9 September 2019 Submitted by X. Zhang

This paper studies the dynamical properties of blow-up solutions for nonlinear Schrödinger equation with partial confinement, which may model the Bose-Einstein condensate under a partial trap potential. By using the variational characteristic of the classic nonlinear scalar field equation and the Hamilton conservations, we first get the threshold for global existence and blow-up of the Cauchy problem on mass in two-dimensional space. Then, in terms of the refined compactness Lemma and the variational characteristic of the ground state of nonlinear scalar field equation, we get mass concentration properties of the blow-up solutions as well as limiting profile of the blow-up solutions with small super-critical mass. © 2019 Elsevier Inc. All rights reserved.

Keywords: Nonlinear Schrödinger equation Partial confinement Blow-up solution Mass concentration Variational characteristic

1. Introduction In this paper, we study the nonlinear Schrödinger equation with partial confinement in R2 : 

iϕt + Δϕ − x21 ϕ + |ϕ|2 ϕ = 0, t > 0, x ∈ R2 ,

(1.1)

ϕ(0, x) = ϕ0 (x). Here ϕ = ϕ(t, x) = ϕ(t, x1 , x2 ) : [0, T ) × R2 → C is a complex value wave function and 0 < T ≤ ∞, i = √ ∂2 ∂2 2 −1, Δ = ∂x 2 + ∂x2 is the Laplace operator on R . 1 2 In [1], Antonelli, Carles and Silva studied the scattering properties for the nonlinear Schrödinger equation with partial confinement in RN . In [3], Bellazzini, Boussaid, Jeanjean and Visciglia studied the soliton solutions for the nonlinear Schrödinger equations with partial confinement in R3 . It is known that nonlinear Schrödinger equation with a trap potential may model the Bose-Einstein condensate (BEC) as the Gross✩

This research is supported by Natural Science Foundation of China 11871138.

* Corresponding author. E-mail addresses: [email protected] (J. Pan), [email protected] (J. Zhang). https://doi.org/10.1016/j.jmaa.2019.123484 0022-247X/© 2019 Elsevier Inc. All rights reserved.

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

2

Pitaevskii (GP) equation [9,14,27]. In the physical experiment, BEC is observed in presence of a confined potential trap and its macroscopic behavior strongly depends on the shape of this trap potential [15,25]. When the trap potential is confined on partial directions in the space, [1,3] showed the same properties as the whole confinement in the space under some assumptions. Bao [2] obtained mathematical theory and numerical methods for BEC. In this paper, we are interested in the mass concentration of the blow-up solutions for the Cauchy problem (1.1). We recall some known results about blow-up solutions for the classical nonlinear Schrödinger equation without any potential iϕt + Δϕ + |ϕ|p−1 ϕ = 0, t ≥ 0, x ∈ RN , p > 1.

(1.2)

Ginibre and Velo [10] showed the local well-posedness for the Cauchy problem of (1.2) in H 1 (RN ). Glassey [11], Ogawa and Tsutsumi [22], Merle and Raphaël [19] showed the existence of blow-up solutions for the Cauchy problem of (1.2). Weinstein [29] and Zhang [31] obtained the sharp conditions of the blow-up and global existence for L2 -critical and L2 -supercritical nonlinearities respectively. Merle and Tsutsumi [20] showed a new dynamical phenomenon for the blow-up solutions that is the L2 -mass concentration. Tsutsumi [28] got the rate of L2 -mass concentration of blow-up solutions with critical power. Li [18] obtained the rate of L2 -mass concentration of blow-up solutions with small super-critical mass. Hmidi and Keranni [13] established a profile decomposition theory of bounded sequence in H 1 (RN ) and then proved the L2 -mass concentration of blow-up solutions for the L2 -critical nonlinearities. Guo [12] further discussed the mass concentration properties about L2 -supercritical and H˙ 1 -subcritical case. For the nonlinear Schrödinger equation with a harmonic potential iϕt + Δϕ − |x|2 ϕ + |ϕ|p−1 ϕ = 0, t ≥ 0, x ∈ RN , p > 1,

(1.3)

Oh [23] and Cazenave [6] established the local existence of the Cauchy problem in the natural energy space. Tsurumi and Wadati [26], Zhang [30,32] showed existence of the blow-up solutions. Zhang [32] obtained the sharp threshold for global existence and blow-up of the solutions. Li and Zhang [17] showed that the mass of the radial symmetric blow-up solutions has concentration phenomena at the blow-up time. Li and Meng [16] obtained concentration phenomena in the case of non-radial symmetric case by using refined compactness lemma. Zhu, Zhang and Li [33] obtained the limiting profile of blow-up solutions with critical mass, and further extended this result to small-critical mass case. For the Cauchy problem (1.1), according to Cazenve [7], one could establish the local well-posedness in the corresponding energy space (also see [1,3]). Then, we can show the existence of blow-up solutions. In terms of Zhang [30], we choose the ground state solution of the nonlinear Schrödinger equation without any potential to describe the blow-up solutions of the Cauchy problem (1.1). In light of [5,29,30], by using the variational characteristic of the classic nonlinear scalar equation and the Hamilton conservations, we get the sharp threshold for global existence and blow-up of the Cauchy problem on mass. Furthermore, we are interested in some dynamical properties of the blow-up solutions. We study the mass concentration properties of blow-up solutions as well as limiting profile of the blow-up solutions with small super-critical mass. This paper is organized as follows: in section 2, we present some preliminaries. In section 3, we prove the existence of blow-up solutions and give a threshold for global existence of the Cauchy problem (1.1). In section 4, we consider some concentration properties of blow-up solutions for the Cauchy problem (1.1). In section 5, we obtain the limiting profile of blow-up solution with small super-critical mass. 2. Preliminaries For the Cauchy problem (1.1), we set a natural energy space

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

⎧ ⎨ H :=

 ϕ ∈ H 1 (R2 ),

⎩

x21 |ϕ|2 dx < ∞

R2

3

⎫ ⎬ ⎭

,

(2.1)

 where H 1 (R2 ) = ϕ : ϕ ∈ L2 (R2 ) and ∇ϕ ∈ L2 (R2 ) . Thus H becomes a Hilbert space, continuously embedded in H 1 (R2 ), when endowed with the inner product  (ϕψ + ∇ϕ · ∇ψ + x21 ϕψ)dx,

< ϕ, ψ >H =

(2.2)

R2

whose associated norm denoted by · H . In addition, we denote · Lp , Lp (R2 ) and R2 ·dx by · p , Lp and ·dx respectively, and C > 0 will stand for a constant that may be different from line to line when it does not cause any confusion. Then, we shall state two important propositions for the Cauchy problem (1.1) according to [7]. Proposition 2.1. Assume ϕ0 ∈ H. Then there exists a unique solution ϕ(t, x) of the Cauchy problem (1.1) in C([0, T ); H) for some T ∈ (0, ∞] (maximal existence time). At the same time, we have the alternatives T = ∞ (global existence) or else T < ∞ and lim ϕ H = ∞ (blow up). Moreover for all t ∈ [0, T ), the t→T −

solution ϕ satisfies the following two conservation laws of the mass 



|ϕ0 |2 dx,

|ϕ(t)|2 dx =

(2.3)

and the energy  E(ϕ(t)) =

 |∇ϕ(t)|2 dx +

x21 |ϕ(t)|2 dx −

1 2

 |ϕ(t)|4 dx

(2.4)

= E(ϕ0 ). By a direct calculation (also see [7]) we have the following result. Proposition 2.2. Assume that ϕ0 ∈ H, and |x|2 |ϕ0 |2 dx < ∞. Let ϕ(t, x) ∈ C([0, T ); H) be a solution of 2 the Cauchy problem (1.1). Put J(t) := |x| |ϕ(t, x)|2 dx, then one has J  (t) = −4Im

 xϕ∇ϕdx

(2.5)

and   1 |∇ϕ|2 − x21 |ϕ|2 − |ϕ|4 dx 2  = 8E(ϕ0 ) − 16 x21 |ϕ|2 dx.

J  (t) = 8

Corollary 2.3. Let ϕ be a classical solution to the Cauchy problem (1.1) with ϕ0 ∈ H. Assume that (i) E(ϕ0 ) < 0; (ii) E(ϕ0 ) = 0, Im xϕ0 ∇ϕ¯0 dx > 0; (iii) E(ϕ0 ) > 0, Im xϕ0 ∇ϕ¯0 dx ≥ 4[J(0)E(0)]1/2 .

(2.6)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

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Then there exists a finite time T such that lim ∇ϕ(t) 2 = +∞.

t→T

3. Sharp threshold for global existence Consider the following nonlinear scalar field equation −ΔQ + Q = |Q|2 Q, Q ∈ H 1 (R2 ).

(3.1)

From [24,29], we have Lemma 3.1. Equation (3.1) possesses a unique positive radially symmetric solution Q(x), that is, there is a unique positive radially symmetric function Q(x) determined by (3.1). Moreover 12 Q2 dx is the minimum of the functional  I(u) =

   |∇ϕ|2 dx |ϕ|2 dx , ϕ ∈ H 1 (R2 ). |ϕ|4 dx

(3.2)

We also call Q is the ground state solution of (3.1). In addition, from equation (3.1), we have Pohozaev identity. Lemma 3.2. Let u ∈ H 1 (R2 ) be a solution of equation (3.1). Then one has  (2|∇u|2 − |u|4 )dx = 0, 



(3.3)

 2|u|2 − |u|4 dx = 0.

(3.4)

From Proposition 3.1 and Lemma 3.2, one has the sharp Gagliardo-Nirenberg inequality (also see [29]). Lemma 3.3. For any ψ ∈ H 1 (R2 ), we have 1 2



 |ψ| dx ≤



 |∇ψ| dx

4

|ψ| dx

2

2

−1 2

Q dx

,

(3.5)

where Q is the ground state solution of (3.1). Then we state an inequality as follows (also see [29]). Lemma 3.4. Assume that ψ ∈ H 1 (R2 ). Then we have 

 |ψ| dx ≤ 2

|∇ψ| dx 2

 12 

|x| |ψ| dx 2

2

 12 .

(3.6)

In the following Theorem, we show a blow-up result for the Cauchy problem (1.1). Theorem 3.5. Let Q(x) be the positive radially symmetric solution of the equation (3.1). Then for arbitrarily  > 0, there exist ϕ0 (x) ∈ H and |x|2 |ϕ0 |2 dx < ∞ such that

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

5



 |ϕ0 | dx = 2

Q2 dx + ,

(3.7)

and the solutions of the Cauchy problem (1.1) blow up in a finite time. Proof. For arbitrary λ > 0, μ > 0, we put Qλ,μ (x) = μλQ(λx). In terms of scaling arguments, we have that 

 2

2

Q2 dx,

[Qλ,μ (x)] dx = μ 

[Qλ,μ (x)]4 dx = μ4 λ2

Q4 dx,

(3.9)



 2

[∇Qλ,μ (x)] dx = μ2 λ2 

(3.8)



x21 [Qλ,μ (x)]2 dx = λ−2 μ2

|∇Q|2 dx,

(3.10)

x21 Q2 dx.

(3.11)

Q2 dx.

(3.12)



By Lemma 3.2, one has that  |∇Q|2 dx =

1 2



 Q4 dx =

Now we take that

 μ= λ> It is obvious that ϕ0 (x) ∈ H and



 2

2

Q dx + 

Q dx

 12 ,

   14 1 x21 Q2 dx , ϕ0 (x) = μλQ(λx). 

|x|2 |ϕ0 |2 dx < ∞ (see [21]). Moreover from (3.8) it follows that 

 |ϕ0 | dx = 2

Q2 dx + .

From (2.4), (3.8), (3.9), (3.10), (3.11), (3.12), one has that   1 |∇Q|2 + λ−4 x21 Q2 − μ2 Q4 dx 2    = μ2 λ2 (1 − μ2 )|∇Q|2 + λ−4 x21 Q2 dx

    2 2 2 2 −4 2 2 x1 Q dx < μ λ − |∇Q| dx Q dx + λ

E(ϕ0 ) = μ2 λ2

< 0. This and Corollary 2.3 complete the proof. 2 Now we give a threshold for global existence of the Cauchy problem (1.1). Theorem 3.6. Let Q(x) be the positive radially symmetric solution of equation (3.1). If ϕ0 satisfies ϕ0 ∈ H and   (3.13) |ϕ0 |2 dx < |Q|2 dx,

6

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

then the solution ϕ(t, x) of the Cauchy problem (1.1) exists globally in time. Proof. Let ϕ(t, x) ∈ C((0, T ); H) be a solution of the Cauchy problem (1.1). From (2.4) and (3.2) we get that     |ϕ|2 dx 2 2 2 1− dx ≤ E(ϕ0 ). |∇ϕ| + x |ϕ| (3.14) 1 |Q|2 dx Thus from (2.3) and (3.13) we imply that |∇ϕ|2 dx and x21 |ϕ|2 dx are bounded for t ∈ [0, T ) and any T < ∞. By Proposition 2.1, it yields that ϕ(t, x) globally exists in t ∈ [0, ∞). The proof is then completed. 2 Remark 3.7. Here we only give the proof of the case of |μ| > 1 in the Theorem 3.5. Indeed, for the case of |μ| = 1, it is also true. When |μ| = 1, ϕ0 = μλQ(λx) which is corresponding to the critical case, then, the proof of the existence of blow-up solutions needs more complex tools and methods (see [8]), we will prove it later. 4. Concentration properties of blow-up solutions In this section, we firstly recall the following refined compactness result which will play an important role in proving the mass concentration properties of the blow-up solutions for the Cauchy problem (1.1), and it can be proved by using the profile decomposition theory of bounded sequences in H 1 and GagliardoNirenberg inequality (3.5)(see [13]). 1 Lemma 4.1. Let {vn }∞ n=1 be a bounded sequence in H , such that

lim sup ∇vn L2 ≤ M,

lim sup vn L4 ≥ m > 0.

n→∞

(4.1)

n→∞

2 Then, there exists a sequence {xn }∞ n=1 ⊂ R such that up to a subsequence

vn (x + xn )  V (x)

in H 1

weakly

with 1 m2 Q 2 , V L2 ≥ √ 2M where Q is the ground state solution of (3.1). Theorem 4.2. Let ϕ ∈ H 1 . If ϕ is the solution of Cauchy problem (1.1) which blows up at finite time T > 0. Then let a(t) be a real-valued nonnegative function defined on [0, T ) satisfying a(t) ∇ϕ(t) 2 → ∞ as t → T . Then, there exists x(t) ∈ R2 such that 

 |ϕ(t, x)| dx ≥ 2

lim inf t→T

Q2 dx,

|x−x(t)|≤a(t)

where Q is the ground state solution of (3.1). Proof. Let {tn }∞ n=1 be a sequence such that tn → T , we set λ(t) =

∇Q L2 , ∇ϕ(t) L2

v(t, x) = λ(t)ϕ(t, λ(t)x),

(4.2)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

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and λn = λ(tn ), vn = v(tn , x). Then the sequence {vn } satisfies vn L2 = ϕn L2 = ϕ0 L2 ,

(4.3)

∇vn L2 = λn ∇ϕn L2 = ∇Q L2 . Furthermore, by conservations of the energy and blow-up criteria, we deduce that  1 |vn |4 dx |∇vn |2 dx − 2    1 2 2 4 |ϕn | dx = λn |∇ϕn | dx − 2    = λ2n E(ϕ0 ) − x21 |ϕn |2 dx 

H(vn ) :=

→ 0 as n → ∞, where in the last step we use the fact that λn → 0 as n → ∞ and according to Proposition 2.2, one has that d2 dt2

 |x|2 |ϕ(t, x)|2 dx ≤ 8E(ϕ0 ),

which implies that there exists a constant c0 > 0 such that  ∀ t ∈ [t0 , T ),

 x21 |ϕ(t, x)|2 dx ≤

|x|2 |ϕ(t, x)|2 dx ≤ c0 .

Hence, it follows that |vn |4 dx → 2 |∇Q|2 dx. 4 2 The family {vn }∞ n=1 satisfies the hypotheses of Lemma 4.1 with m = 2 ∇Q L2 and M = ∇Q L2 . Thus ∞ 2 1 2 there exists a family {xn }n=1 ⊂ R and a profile V ∈ H (R ) with V L2 ≥ Q L2

(4.4)

vn (x + xn ) = λn ϕ(tn , λn x + xn )  V weakly in H 1 .

(4.5)

such that

Then for every A > 0, 

 λ2n |ϕ(tn , λn x + xn )|2 dx ≥

lim inf n→∞

|x|≤A

|V |2 dx.

(4.6)

|x|≤A

n) Note that lim a(t = ∞ as n → ∞, thus there exists n1 > 0 such that for every n > n1 , Aλn < a(tn ). n→∞ λn This gives immediately:





 |ϕ(tn , x)|2 dx ≥ lim inf

lim inf n→∞

|ϕ(tn , x)|2 dx ≥

n→∞

|x−xn |≤a(tn )

|x−xn |≤Aλn

|V (x)|2 dx. |x|≤A

Then 

 lim inf sup

n→∞ y∈R2 |x−y|≤a(tn )

|ϕ(tn , x)| dx ≥

|V (x)|2 dx.

2

|x|≤A

(4.7)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

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Since the sequence {tn }∞ n=1 is arbitrary, we finally get that for every A, 

 lim inf sup

n→∞ y∈R2 |x−y|≤a(t)

|ϕ(t, x)|2 dx ≥

Observe that for every t ∈ [0, T ), the function y → infinity, there exists x(t) such that

|x−y|≤a(t)

|u(t, x)|2 dx is continuous and goes to 0 at



 sup y∈R2 |x−y|≤a(t)

This and (4.7) as claimed.

|V (x)|2 dx.

|ϕ(t, x)|2 dx =

|V (x)|2 dx.

(4.8)

|x−x(t)|≤a(t)

2

Corollary 4.3. The Theorem in the above gives the L2 -concentration and rate of L2 -concentration of blow-up R solutions for the Cauchy problem (1.1). Indeed, we can choose a(t) = ∇ϕ(t) 1−δ with 0 < δ < 1 and R > 0. 2

It is obvious that lim a(t) = 0 and a(t) satisfies the assumption in Theorem 4.2. According to Theorem 4.2, t→T

if ϕ is a blow-up solution of Cauchy problem (1.1) and T is blow-up time, then for every A > 0, there exists a function x(t) ∈ R2 such that 

 |ϕ(t, x)|2 dx ≥

lim inf t→T

|x−x(t)|≤A

|Q(x)|2 dx. R2

Meanwhile, it follows from the choice of a(t) that for any function a(t) ≤ 2

R , ∇ϕ(t)1−δ 2

(4.2) can be estab-

lished, which implies that the rate of L -concentration of blow-up solutions of the Cauchy problem (1.1) is R with 0 < δ < 1. ∇ϕ(t)1−δ 2

In the end, we use the Theorem 4.2 to prove that the blow-up solution |ϕ(t, x)|2 converges to a δ-function as t close to the blow-up time T at the point x = y1 , which implies that the point y1 concentrates all mass of the blow-up solutions of (1.1). Theorem 4.4. Let ϕ0 ∈ H. According to Remark 3.7, if the solution ϕ(t, x) of (1.1) blows up in finite time T and ϕ0 L2 = Q L2 . Then there exists y1 ∈ R2 such that |ϕ(t, x)|2 → Q(x) 2L2 δy1 , as t → T, which means that  lim

t→T

ρ(x)|ϕ(t, x)|2 dx = Q(x) 22 ρ(y1 ), for any ρ ∈ C ∞ ,

where Q(x) is the unique ground state solution of (3.1). Proof. According to Theorem 4.2, one has that for all A > 0,  |ϕ(t, x)|2 dx ≥ Q 2L2 .

lim inf t→T

|x−x(t)|
This and the conservation of mass ϕ(t) L2 = Q L2 = ϕ0 L2 imply that for all A > 0,

(4.9)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

9

 |ϕ(t, x)|2 dx = Q 2L2 .

lim inf t→T

|x−x(t)|
Hence we deduce that |ϕ(t, x + x(t))|2 → Q 2L2 δx=0 , as t → T.

(4.10)

In addition, for any real number η and for any real function θ defined on R2 , one has that  1 |ϕ|4 dx | ± iηϕ∇θ + ∇ϕ|2 dx − 2   ¯ = η 2 |∇θ|2 |ϕ|2 dx ∓ ηIm ϕ∇ϕ∇θdx

H(e±iηθ ϕ) =



(4.11)

+ H(ϕ). Furthermore, from the inequality (3.5), it follows that ±iηθ

H(e

 ϕ) ≥

|∇e

±iηθ

  ϕ 2 = 0, ϕ| dx 1 − Q 2 2

which implies that 

 |Im

ϕ∇ϕ∇θdx| ¯ ≤

 H(ϕ)

|∇θ|2 |ϕ|2 dx

 12 .

Using the estimate (4.12) and H(ϕ(t)) ≤ E(ϕ0 ), we get           d   2     ¯  dt |ϕ(t, x)| xj dx = 2  xj ϕΔϕdx      R2 R2         = 2  ϕ∇ϕ∇x ¯ j dx 2  R

 1/2  ≤ 2 H(ϕ) |ϕ|2 |∇xj |2 dx ≤ C, for any j = 1, 2. Now taking tn , tm ∈ [0, T ), we let lim tn = lim tm = T , then for any j = 1, 2, one has n→∞

m→∞

        |ϕ(tn , x)|2 xj dx − |ϕ(tm , x)|2 xj dx   2  2 R

R

≤ C|tn − tm | → 0, as n, m → ∞, which implies that  |ϕ(t, x)|2 xj dx exists for any j = 1, 2.

lim

t→T R2

(4.12)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

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Let y1 = Q −2 L2 lim



t→T

|ϕ(t, x)|2 xdx, then  y1 Q 2L2 = lim

|ϕ(t, x)|2 xdx.

t→T

(4.13)

On the other hand, it follows from Proposition 2.2 that for any t ∈ [0, T ), there exists a constant c0 that  |x|2 |ϕ(t, x)|2 dx ≤ c0 .

(4.14)

Then there exists a constant c1 that   2 2 c1 |x(t)| |ϕ(t, x)| dx ≤ |x + x(t)|2 |ϕ(t, x + x(t))|2 dx ≤ c0 ,

(4.15)

it yields  lim |x(t)| ≤

t→T

c0 /c1 Q 2

(4.16)

Therefore, for every t ∈ [0, T ), there exists a constant A0 such that ⎧ ⎪ ⎨  lim

t→T

 |ϕ(t, x)|2 xdx −

⎪ ⎩

⎫ ⎪ ⎬

 Q2 dx x(t)

⎪ ⎭

B(0,A)

= 0,

(4.17)

where A > A0 , |x(t)| ≤ A0 , B(0, A) = {x ∈ R2 ||x| ≤ A}. Indeed, 





|ϕ(t, x)| xdx =

|ϕ(t, x)| (x − x(t))dx +

2

B(0,A)

|ϕ(t, x)|2 x(t)dx

2

B(0,A)

B(0,A)





|ϕ(t, y + x(t))|2 ydy +

= B(−x(t),A)

|ϕ(t, y + x(t))|2 x(t)dy B(−x(t),A)





|ϕ(t, y + x(t))| ydy +

|ϕ(t, y + x(t))|2 ydy

2

= B(0,δ)

B(−x(t),A) B(0,δ)



|ϕ(t, y + x(t))|2 x(t)dy.

+ B(−x(t),A)

Due to A > A0 , then there exists δ > 0 such that B(0, δ) ⊂ B(−x(t), A). Combine (4.10) we have 

 |ϕ(t, x)|2 xdx − B(0,A)

|ϕ(t, y + x(t))|2 x(t)dy → 0, as t → T.

B(−x(t),A)

In addition, for any t ∈ [0, T ), 

 |ϕ(t, x)| xdx| ≤ 2

|x|≥A

|x|≥A

|x|2 |ϕ(t, x)|2 dx A

c0 ≤ → 0, as A → +∞. A

(4.18)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

11

Combine (4.17) and (4.18), we obtain 

 |ϕ(t, x)|2 xdx −

lim

t→T

  |Q(x)|2 x(t) = 0.

(4.19)

This, together with (4.13) implies that lim x(t) = y1 . Therefore t→T

 lim

t→T

 |ϕ(t, x)| xdx = y1 2

|Q(x)|2 dx.

(4.20)

It follows that |ϕ(t, x)|2 → Q 2L2 δy1 as t → T, which concludes the Theorem. 2 5. Decomposition of blowup solution with small super-critical mass In this section, using the variational characterization of (3.1) and scaling technique, we prove that if the initial data ϕ0 ∈ H 1 are close to the ground state Q(x), then the blow-up solution ϕ(t, x) remains close to Q(x) in H 1 . Theorem 5.1. Let ϕ0 ∈ H 1 . Assume that there exists a α1 > 0 such that for all 0 < α ≤ α1 , there exists δ(α ) with δ(α ) → 0 as α → 0 such that for all u ∈ H 1 ,  0 < α(ϕ0 ) =

 |ϕ0 | dx − 2

|Q|2 dx < α .

(5.1)

Furthermore, if ϕ(t, x) is the corresponding finite time blow-up solution of Cauchy problem (1.1) and 0 < T < ∞ is the blow-up time. Then there exist functions x(t) ∈ R2 and θ(t) ∈ R such that when t → T , λ(t)ϕ(t, λ(t)x + x(t))eiθ(t) − Q(x) H 1 ≤ δ(α ), where λ(t) =

∇QL2 ∇ϕ(t)L2

(5.2)

, δ(α ) > 0, and δ(α ) → 0 as α → 0.

Proof. We first suppose that ϕ(t) L2 = Q L2 . Then we show that for any time sequence {tn }∞ n=1 such that tn → T as n → ∞ and for arbitrary ε > 0, the following inequality is true: λn ϕ(tn , λn x + xn )eiθn − Q(x) H 1 < ε, +∞ +∞ + 2 where parameters {λn }+∞ n=1 ⊂ R , {xn }n=1 ⊂ R and {θn }n=1 ⊂ R. Indeed, we have verified that

vn (x + xn )  V weakly in L2 with V L2 ≥ Q L2 in the proof of Theorem 4.2. Hence, applying the weakly lower semi-continuous of the L2 -norm that Q L2 ≤ V L2 ≤ lim inf vn L2 = lim inf ϕn L2 = ϕ0 L2 = Q L2 . n→∞

Thus

n→∞

(5.3)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

12

lim vn L2 = V L2 = Q L2 .

n→∞

Using the Brézis-Lieb Lemma [4], we have vn (x + xn ) → V (x) strongly in L2 as n → ∞. Applying the Gagliardo-Nirenberg inequality, there exists θn ∈ R such that vn (x + xn )eiθn − V 4L4 ≤ C vn (x + xn )eiθn − V 2L2 ∇(vn (x + xn )eiθn − V ) 2L2 . From ∇vn (x + xn ) 2L2 ≤ C, we get vn (x + xn )eiθn → V in L4 as n → ∞. Next, we shall show that vn (x + xn )eiθn converges to V strongly in H 1 . Observe that 0 = lim H(vn eiθn )   1 |vn |4 dx = |∇Q|2 dx − lim 2 n→∞   1 |V |4 dx. = |∇Q|2 dx − 2

(5.4)

Thus, we infer from the inequality (3.5) that  |∇Q|2 dx =

1 2

 |V |4 dx ≤

V 2L2 ∇V 2L2 = ∇V 2L2 . Q 2L2

(5.5)

On the other hand, we deduce from (4.3) that ∇V L2 ≤ lim ∇vn (x + xn ) L2 = ∇Q L2 . Hence, we have n→

Q H 1 = V H 1 and

vn (x + xn ) → V strongly in H 1 as n → ∞. This and (5.5) imply that  H(V ) =

1 |∇V | dx − 2 2

 |V |4 dx = 0.

Up to now, we have verified that V L2 = Q L2 , ∇V L2 = ∇Q L2 and H(V ) = 0. The variational characterization of the ground state implies that V (x) = eiθn Q(x + x0 ) for some θn ∈ R, x0 ∈ R2 , and vn = λn ϕ(tn , λn x + x0 ) → eiθn Q(x + x0 ) strongly in H 1 as n → ∞. By the definition of vn and (5.6), we can redefine the sequence xn and θn to ensure that (5.3) is true.

(5.6)

J. Pan, J. Zhang / J. Math. Anal. Appl. 481 (2020) 123484

13

Now, in the case 0 < α(ϕ0 ) = |ϕ0 |2 dx − |Q2 |dx < α , we need to prove that for arbitrary ε > 0 and any sequence {tn }∞ n=1 such that tn → T as n → ∞, we have λn ϕ(tn , λ(tn )x + x(t))eiθn − Q(x) H 1 ≤ δ(α ),

(5.7)

+∞ +∞ + 2 where parameters {λn }+∞ n=1 ⊂ R , {xn }n=1 ⊂ R and {θn }n=1 ⊂ R. From the assumption, there exists a sub∞ sequence {tn }∞ n=1 (still denoted {tn }n=1 ) such that lim ϕ(tn , x) 2 = Q 2 . From the above discussions, n→+∞

we can deduce that lim vn 2 = lim ϕ(tn ) 2 = Q 2 ,

n→+∞

n→+∞

(5.8)

∇vn 2 = ∇Q 2 ,

(5.9)

lim H(vn ) = 0.

(5.10)

and

n→+∞

For simplicity, we take another scaling transformation An = An 2 = Q 2 , and

Q2 vn 2 vn .

From (5.8)-(5.10), we see that

lim ∇An 2 = ∇Q 2 ,

n→+∞

lim H(An ) = 0.

n→+∞

2  +∞ Then, according to the above proof in the case ϕ 2 = Q 2 , there exists { xn }+∞ n=1 ⊂ R and {θn }n=1 ⊂ R such that 

An (tn , x + x n )eiθn → Q(x) strongly in H 1 .

(5.11)

We can get (5.7) by redefining the parameters in (5.11). The proof is then completed. 2 References [1] P. Antonelli, R. Carles, J.D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Comm. Math. Phys. 334 (1) (2015) 367–396. [2] W. Bao, Y. Cai, Mathematics theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models 6 (1) (2013) 1–135. [3] J. Bellazzini, N. Boussaid, L. Jeanjean, N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys. 353 (2017) 229–251. [4] H. Brézis, E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. [5] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci. 12 (10) (2002) 1513–1523. [6] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Mathematicos, vol. 26, 1996, Rio de Janeiro. [7] T. Cazenave, Semilinear Schrödinger Equations, Kluwer, New York, 2003. [8] S.L. Coz, Y. Martel, P. Raphaël, Minimal mass blow up solutions for a double power nonlinear Schrödinger equation, Rev. Mat. Iberoam. 32 (3) (2016) 795–833. [9] F. Dalfovo, S. Giorgini, Lev P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys. 71 (3) (1999) 463–512. [10] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979) 1–71. [11] R.T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (9) (1977) 1794–1797. [12] Q. Guo, A note on concentration for blowup solutions to supercritical Schrödinger equations, Proc. Amer. Math. Phys. 141 (2013) 4215–4227.

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