Energy Scattering for Schrödinger Equation with Exponential Nonlinearity in Two Dimensions

Abstract

When the spatial dimensions 𝑛=2, the initial data 𝑢₀∈𝐻¹, and the Hamiltonian 𝐻(𝑢₀)≤1, we prove that the scattering operator is well defined in the whole energy space 𝐻¹(ℝ²) for nonlinear Schrödinger equation with exponential nonlinearity (𝑒^𝜆|𝑢|2−1)𝑢, where 0<𝜆<4𝜋.

1. Introduction

We consider the Cauchy problem for the following nonlinear Schrödinger equation: 𝑖𝑢_𝑡(𝑢),𝑓(𝑢)𝑒+▵𝑢=𝑓∶=^𝜆|𝑢|2−1𝑢, (1)(2) in two spatial dimensions with initial data 𝑢₀∈𝐻¹ and 0<𝜆<4𝜋. Solutions of the above problem satisfy the conservation of mass and Hamiltonian: 𝑀(𝑢;𝑡)∶=_ℝ2|𝑢|²𝑑𝑥=𝑀𝑢₀,𝐻(𝑢;𝑡)∶=_ℝ2|∇𝑢|²+𝐹(𝑢)𝑑𝑥=𝐻𝑢₀, (3) where 𝐹(𝑢)=1𝜆𝑒^𝜆|𝑢|2−𝜆|𝑢|²−1. (4)

Nakamura and Ozawa [1] showed the existence and uniqueness of the scattering operator of (1) with (2). Then, Wang [2] proved the smoothness of this scattering operator. However, both of these results are based on the assumption of small initial data 𝑢₀. In this paper, we remove this assumption and show that for arbitrary initial data 𝑢₀∈𝐻¹(ℝ²) and 𝐻(𝑢₀)≤1, the scattering operator is always well defined.

Wang et al. [3] proved the energy scattering theory of (1) with 𝑓(𝑢)=(𝑒^𝜆|𝑢|2−1−𝜆|𝑢|²−(𝜆²/2)|𝑢|⁴)𝑢, where 𝜆∈ℝ and the spatial dimension 𝑛=1. Ibrahim et al. [4] showed the existence and asymptotic completeness of the wave operators for (1) with 𝑓(𝑢)=(𝑒^𝜆|𝑢|2−1−𝜆|𝑢|²)𝑢 when the spatial dimensions 𝑛=2, 𝜆=4𝜋, and 𝐻(𝑢₀)≤1. Under the same assumptions as [4], Colliander et al. [5] proved the global well-posedness of (1) with (2).

Theorem 1. Assume that 𝑢₀∈𝐻¹(ℝ²), 𝐻(𝑢₀)≤1, and 𝜆=4𝜋. Then problem (1) with (2) has a unique global solution 𝑢 in the class 𝐶(ℝ,𝐻¹(ℝ²)).

Remark 2. In fact, by the proof in [5], the global well-posedness of (1) with (2) is also true for 0<𝜆≤4𝜋.

In this paper, we further study the scattering of this problem. Note that 𝑓(𝑢)=(𝑒^𝜆|𝑢|2−1)𝑢=∑^∞_𝑘=1(𝜆^𝑘/𝑘!)|𝑢|^2𝑘𝑢. Nakanishi [6] proved the existence of the scattering operators in the whole energy space 𝐻¹(ℝ²) for (1) with 𝑓(𝑢)=|𝑢|^𝑝𝑢 when 𝑝>2. Then, Killip et al. [7] and Dodson [8] proved the existence of the scattering operators in 𝐿²(ℝ²) for (1) with 𝑓(𝑢)=|𝑢|²𝑢. Inspired by these two works, we use the concentration compactness method, which was introduced by Kenig and Merle in [9], to prove the existence of the scattering operators for (1) with (2).

For convenience, we write (1) and (2) together; that is, 𝑖𝑢_𝑡+▵𝑢=𝑓(𝑢)∶=𝑒^𝜆|𝑢|2−1𝑢,𝑢(0,𝑥)=𝑢₀, (5) where 𝑢₀∈𝐻¹(ℝ²) and 0<𝜆<4𝜋. Our main result is as follows.

Theorem 3. Assume that the initial data 𝑢₀∈𝐻¹(ℝ²), 𝐻(𝑢₀)≤1, and 0<𝜆<4𝜋. Let 𝑢 be a global solution of (5). Then ‖𝑢‖_𝐿⁴_{𝑡,𝑥(ℝ×ℝ}²₎<∞. (6)

In Section 2, Lemma 9 will show us that Theorem 3 implies the following scattering result.

Theorem 4. Assume that the initial data 𝑢₀∈𝐻¹(ℝ²), 𝐻(𝑢₀)≤1, and 0<𝜆<4𝜋. Then the solution of (5) is scattering in the energy space 𝐻¹(ℝ²).

We will prove Theorem 3 by contradiction in Section 5. In Section 2, we give some nonlinear estimates. In Section 3, we prove the stability of solutions. In Section 4, we give a new profile decomposition for 𝐻¹ sequence which will be used to prove concentration compactness.

Now, we introduce some notations: 𝐺(𝑢)∶=𝑢𝑓(𝑢)−𝐹(𝑢)=𝑒^𝜆|𝑢|2|𝑢|²−1𝜆𝑒^𝜆|𝑢|2=−1_∞𝑘=1𝑘𝜆^𝑘|𝑢|^2𝑘+2()!,()()().𝑘+1𝐸=𝐸𝑢;𝑡∶=𝑀𝑢;𝑡+𝐻𝑢;𝑡 (7)

We define ‖𝑢‖_𝐻^𝑠_𝑞(ℝ²₎∶=(𝐼−▵)^𝑠/2𝑢_{𝐿𝑞(ℝ2)},‖𝑢‖_̇𝐻^𝑠_𝑞(ℝ²₎∶=(−▵)^𝑠/2𝑢_{𝐿𝑞(ℝ2)}. (8)

For Banach space 𝑋=𝐻^𝑠_𝑞(ℝ²), 𝐻̇^𝑠_𝑞(ℝ²), or 𝐿^𝑞(ℝ²), we denote ‖𝑢‖_𝐿^𝑝_(ℝ;𝑋)∶=_ℝ‖𝑢(𝑡)‖^𝑝_𝑋𝑑𝑡^1/𝑝. (9)

When 𝑞=𝑟, 𝐿^𝑞_𝑡𝐿^𝑟_𝑥 is abbreviated to  𝐿^𝑞_𝑡,𝑥. When 𝑞 or 𝑟 is infinity or when the domain ℝ×ℝ² is replaced by 𝐼×ℝ², we make the usual modifications. Specially, we denote 𝑆(𝑢)∶=‖𝑢‖⁴_{𝐿4𝑡,𝑥ℝ×ℝ2}. (10)

For 𝑡₀∈ℝ, we split 𝑆(𝑢)=𝑆_≤𝑡0(𝑢)+𝑆_≥𝑡0(𝑢), where 𝑆_≤𝑡0(𝑢)∶=^𝑡0_−∞ℝ2|𝑢(𝑡,𝑥)|⁴𝑆𝑑𝑥𝑑𝑡,_≥𝑡0(𝑢)∶=_𝑡^+∞_0ℝ2|𝑢(𝑡,𝑥)|⁴𝑑𝑥𝑑𝑡. (11)

For any two Banach spaces 𝑋 and 𝑌, ‖⋅‖_𝑋∩𝑌∶=max{‖⋅‖_𝑋,‖⋅‖_𝑌}. 𝐶 denotes positive constant. If 𝐶 depends upon some parameters, such as 𝜆, we will indicate this with 𝐶(𝜆).

Remark 5. Note that 0<𝜆<4𝜋 in Theorem 3; we only need to prove the result for 0<𝜆<4(1−4𝜀)²𝜋, 𝜀∈(0,1/8). Hence, we always suppose that 0<𝜆<4(1−4𝜀)²𝜋 in the context.
Moreover, we always suppose that the initial data 𝑢₀ of (5) satisfies 𝑢₀∈𝐻¹(ℝ²) and 𝐻(𝑢₀)≤1.

2. Nonlinear Estimates

In order to estimate (2), we need the following Trudinger-type inequality.

Lemma 6 (see [10]). Let 𝜆∈[0,4𝜋). Then for all 𝑢∈𝐻¹(ℝ²) satisfying ‖∇𝑢‖_𝐿²_(ℝ²₎≤1, one has _ℝ2𝑒^𝜆|𝑢|2−1𝑑𝑥≤𝐶(𝜆)‖𝑢‖²_𝐿2ℝ2. (12)

Note that for for all 𝛼≥1, 𝑒^𝜆|𝑢|2−1^𝛼≤𝑒^{𝜆𝛼|𝑢|2}−1. (13)

By Lemma 6 and Hölder inequality, for 𝜆∈(0,4𝜋) and for all 𝛽≥0, we have _ℝ2𝑒^𝜆|𝑢|2−1|𝑢|^𝛽𝑑𝑥≤𝑒^𝜆|𝑢|2−1_{𝐿𝑥1/(1−𝜀)(ℝ2)}‖𝑢‖^𝛽_{𝐿𝑥𝛽/𝜀ℝ2}≤‖𝑢‖²_𝐿2ℝ2‖𝑢‖^𝛽_𝐻1ℝ2≤𝐶𝜆,𝛽‖𝑢‖²_𝐿2ℝ2, (14) and thus _ℝ2𝑒^𝜆|𝑢|2−𝜆|𝑢|²−1𝑑𝑥≤𝜆_ℝ2𝑒^𝜆|𝑢|2−1|𝑢|²(𝜆𝑢‖𝑑𝑥≤𝐶)‖²_𝐿2ℝ2. (15)

Lemma 7 (Strichartz estimates). For 𝑠=0 or 1, 12≤𝑟,𝑝<∞,𝛾𝑝+1𝑝=12,1𝑝+1𝑝^′=1, (16) (the pairs (𝛾(𝑝),𝑝) were called admissible pairs) we have 𝑒^{𝑖(𝑡−𝑡0)▵}𝑢𝑡_{0𝐿𝛾(𝑝)(ℝ;𝐻𝑠𝑝)}≤𝐶𝑢𝑡_{0𝐻𝑠(ℝ2)},_𝑡𝑡0𝑒^{𝑖(𝑡−𝜏)▵}𝑓(𝑢(𝜏))𝑑𝜏_{𝐿𝛾(𝑝)(ℝ;𝐻𝑠𝑝)}≤𝐶𝑓(𝑢)_{𝐿′𝛾(𝑟)(ℝ;𝐻𝑠𝑟′)}. (17)

Lemma 8 (see [3, Proposition 2.3]). Let 1<𝑟<𝑝<∞ be fixed indices. Then for any 𝑞∈[𝑝,∞), ‖𝑢‖_𝐿^𝑞_(ℝ²₎≤𝐶𝑝,𝑟𝑞^{1/𝑟′+𝑝/𝑟𝑞}‖𝑢‖_𝐿^𝑝/𝑞_𝑝ℝ2‖𝑢‖_̇𝐻^{1−𝑝/𝑞}_{𝑟2/𝑟ℝ2}. (18)

As shown in [6, 11], to obtain the scattering result, it suffices to show that any finite energy solution has a finite global space-time norm. In fact, if Theorem 3 is true, we have the following theorem.

Lemma 9 (Theorem 3 implies Theorem 4). Let 𝑢 be a global solution of (5), 𝐻(𝑢)≤1, and ‖𝑢‖_𝐿⁴_{𝑡,𝑥(ℝ×ℝ}²₎<∞. Then, for all admissible pairs, we have ‖𝑢‖_𝐿^𝛾(𝑝)_(ℝ;𝐻¹_𝑝)<∞. (19)
Moreover, there exist 𝑢_±∈𝐻¹ such that lim_𝑡→±∞𝑢−𝑒^𝑖𝑡▵𝑢_{±𝐻1(ℝ2)}=0. (20)

Proof. Defining 𝑋(𝐼)=𝐿^{2/(1−2𝜀)}(𝐼;𝐻¹_1/𝜀), 𝑌(𝐼)=𝐿⁴(𝐼;𝐻¹₄), by Strichartz estimates, (14) and (15), ‖𝑢‖_{𝑋(𝐼)∩𝑌(𝐼)}≤𝐶‖𝑢(𝑆)‖_𝐻¹_(ℝ²₎+𝐶𝜆|𝑢|²𝑢_{𝐿4/3(𝐼;𝐻14/3)}+𝐶𝑒^𝜆|𝑢|2−𝜆|𝑢|²−1𝑢_{𝐿2/(1+2𝜀)(𝐼;𝐻11/(1−𝜀))}≤𝐶(𝐸)+𝐶‖𝑢‖²_{𝐿4𝑡,𝑥𝐼×ℝ2}‖𝑢‖_𝑌(𝐼)+𝐶‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥𝐼×ℝ2}‖𝑢‖_𝑋(𝐼)×𝑒^𝜆|𝑢|2−𝜆|𝑢|²−1𝑢^−8𝜀_{𝐿∞𝑡𝐿𝑥1/(1−4𝜀)(𝐼×ℝ2)}+𝐶‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥𝐼×ℝ2}‖𝑢‖_𝑋(𝐼)×𝑒^𝜆|𝑢|2−1|𝑢|^2−8𝜀_{𝐿∞𝑡𝐿𝑥1/(1−4𝜀)(𝐼×ℝ2)}≤𝐶(𝐸)+𝐶(𝐸)‖𝑢‖²_{𝐿4𝑡,𝑥𝐼×ℝ2}+‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥𝐼×ℝ2}‖𝑢‖_{𝑋(𝐼)∩𝑌(𝐼)}. (21)
Using the same way as in Bourgain [12], one can split ℝ into finitely many pairwise disjoint intervals: ℝ=^𝐽_𝑗=1𝐼_𝑗,‖𝑢‖_𝐿⁴_{𝑡,𝑥(𝐼𝑗×ℝ}²₎≤𝜂,𝐶(𝐸)𝜂²+𝜂^8𝜀≤12. (22)
By (21), ‖𝑢‖_𝐿^{2/(1−2𝜀)}_{(𝐼𝑗;𝐻}¹_{1/𝜀)∩𝐿}⁴_{(𝐼𝑗;𝐻}¹₄₎≤𝐶(𝐸). (23)
Since ‖𝑢‖_𝐿^∞_{(𝐼𝑗;𝐻}¹₎≤𝐶(𝐸) and 𝜀∈(0,1/8) can be chosen small arbitrarily, by interpolation, ‖𝑢‖_𝐿^𝛾(𝑝)_{(𝐼𝑗;𝐻}¹_𝑝)≤𝐶(𝐸), (24) for all admissible pairs and 𝑗=1,2,…,𝐽. The desired result (19) follows.
Let 𝑢_±=𝑢₀−𝑖₀^±∞𝑒^{−𝑖𝜏▵}𝑓(𝑢(𝜏))𝑑𝜏. (25)
By (19) and (21), 𝑢_{±𝐻1(ℝ2)}≲1+‖𝑢‖³_{𝑌()0,±∞}+‖𝑢‖_𝑋()()^1+8𝜀_{0,±∞∩𝑌0,±∞}≲1. (26)
Thus, 𝑢_± were well defined and belong to 𝐻¹. Since 𝑢−𝑒^𝑖𝑡▵𝑢_{±𝐻1(ℝ2)}=_𝑡^±∞𝑒^{𝑖(𝑡−𝜏)▵}𝑓(𝑢(𝜏))𝑑𝜏_𝐻1(ℝ2)≲‖𝑢‖³_{𝑌()𝑡,±∞}+‖𝑢‖_𝑋()()^1+8𝜀_{𝑡,±∞∩𝑌𝑡,±∞},‖𝑢‖_{𝑋(0,±∞)∩𝑌(0,±∞)}≲1, (27) we must have lim_𝑡→±∞𝑢−𝑒^𝑖𝑡▵𝑢_{±𝐻1(ℝ2)}≲lim_𝑡→±∞‖𝑢‖³_{𝑌()𝑡,±∞}+‖𝑢‖_𝑋()()^1+8𝜀_{𝑡,±∞∩𝑌𝑡,±∞}=0. (28) (20) was proved.

3. Stability

Lemma 10 (stability). For any 𝐴>0 and 𝜎>0, there exists 𝛿>0 with the following property: suppose that 𝑢∶[0,∞)×ℝ²→ℂ satisfies ‖∇𝑢‖_𝐿²_𝑥(ℝ²₎≤1 for all 𝑡∈[0,∞), ‖𝑢‖_𝐿⁴_{𝑡,𝑥([0,∞)×ℝ}²₎≤𝐴 and approximately solves (5) in the sense that ^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑖𝑢_𝑡+▵𝑢−𝑓(𝑢)(𝜏)𝑑𝜏_{𝐿4𝑡,𝑥∩𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀([0,∞)×ℝ2)}≤𝛿. (29)
Then for any initial data 𝑣(0)∈𝐻¹(ℝ²) satisfying 𝐻(𝑣(0))≤1 and ‖𝑢(0)−𝑣(0)‖_𝐿²_𝑥(ℝ²₎≤𝛿, there is a unique global solution 𝑣 to (5) satisfying ‖𝑢−𝑣‖_𝐿⁴_{𝑡,𝑥([0,∞)×ℝ}²₎≤𝜎.

Proof. Denote 𝑣=𝑢+𝑤, then 𝑖𝑤_𝑡+▵𝑤=𝑓(𝑢+𝑤)−𝑓(𝑢)−𝑖𝑢_𝑡+▵𝑢−𝑓(𝑢) (30) and ‖𝑤(0)‖_𝐿²_(ℝ²₎≤𝛿. Let 𝑋=𝐿^∞_𝑡𝐿²_𝑥∩𝐿⁴_𝑡,𝑥∩𝐿_𝑡^{2/(1−2𝜀)}𝐿_𝑥^1/𝜀. By the similar estimates as (21), we have ‖𝑤‖_𝑋≤𝛿+^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓(𝑢+𝑤)−𝑓(𝑢)(𝜏)𝑑𝜏_𝑋|+𝛿≤2𝛿+𝐶|𝑢+𝑤²(𝑢+𝑤)−|𝑢|²𝑢_{𝐿4/3𝑡,𝑥}+𝐶𝑒^{𝜆|𝑢+𝑤|2}−𝜆|𝑢+𝑤|²)−𝑒−1(𝑢+𝑤_𝜆|𝑢|2−𝜆|𝑢|²−1𝑢_{𝐿𝑡2/(1+2𝜀)𝐿𝑥1/(1−𝜀)}≤2𝛿+𝐶|𝑢+𝑤|²|𝑤|+|𝑢|²|𝑤|_{𝐿4/3𝑡,𝑥}+𝐶𝑒^{𝜆|𝑢+𝑤|2}−𝜆|𝑢+𝑤|²|𝑤|+𝑒−1_𝜆|𝑢|2−𝜆|𝑢|²−1|𝑤|_{𝐿𝑡2/(1+2𝜀)𝐿𝑥1/(1−𝜀)}+𝐶𝑒^{𝜆|𝑢+𝑤|2}−1|𝑢+𝑤|²|𝑤|+𝑒^𝜆|𝑢|2−1|𝑢|²|𝑤|_{𝐿𝑡2/(1+2𝜀)𝐿𝑥1/(1−𝜀)}≤2𝛿+𝐶‖𝑢‖²_{𝐿4𝑡,𝑥}+‖𝑢+𝑤‖²_{𝐿4𝑡,𝑥}‖𝑤‖_𝐿⁴_𝑡,𝑥+𝐶‖𝑢‖_𝐿^8𝜀_4𝑡,𝑥+‖𝑢+𝑤‖_𝐿^8𝜀_4𝑡,𝑥‖𝑤‖_𝐿𝑡^{2/(1−2𝜀)}_𝐿𝑥^1/𝜀≤2𝛿+𝐶‖𝑢‖²_{𝐿4𝑡,𝑥}+‖𝑢‖_𝐿^8𝜀_4𝑡,𝑥‖𝑤‖_𝑋+‖𝑤‖_𝑋^1+8𝜀+‖𝑤‖³_𝑋. (31)
Then we subdivide the time interval [0,∞) into finite subintervals 𝐼_𝑗=[𝑡_𝑗,𝑡_𝑗+1], 𝑗=1,…,𝐽, 𝐽=𝐽(𝐴) such that 𝐶‖𝑢‖²_{𝐿4𝑡,𝑥𝐼𝑗×ℝ2}+‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥𝐼𝑗×ℝ2}<14 (32) for each 𝑗. Let 𝛿=𝛿(𝐴,𝐽) be small such that 𝐶8^2𝐽𝛿≪1,8^2𝐽𝛿≪𝜎. (33)
Then by (31) on 𝐼₁×ℝ², we have ‖𝑤‖_{𝑋(𝐼1×ℝ}²₎≤8𝛿 and 𝑤𝑡_{2𝐿2𝑥(ℝ2)}≤𝑒^{𝑖(𝑡2−𝑡1)▵}𝑤𝑡_{1𝐿2𝑥(ℝ2)}+^𝑡2_𝑡1𝑒^{𝑖(𝑡2−𝜏)▵}𝑓(𝑢+𝑤)−𝑓(𝑢)(𝜏)𝑑𝜏_{𝐿2𝑥(ℝ2)}‖𝑢‖+𝛿≤2𝛿+𝛿+𝐶²_{𝐿4𝑡,𝑥𝐼1×ℝ2}+‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥𝐼1×ℝ2}‖𝑤‖_{𝑋(𝐼1×ℝ}²₎+‖𝑤‖_𝑋𝐼^1+8𝜀_1×ℝ2+‖𝑤‖³_{𝑋𝐼1×ℝ2}≤2𝛿+4⋅8𝛿≤8²𝛿. (34)
Using the same analysis as above, we can get ‖𝑤‖_{𝑋(𝐼2×ℝ}²₎≤8³𝛿. Iterating this for 𝐼₂,𝐼₃,…,𝐼_𝐽, we obtain ‖𝑤‖_{𝑋([0,∞)×ℝ}²₎≤8^2𝐽𝛿≤𝜎; the desired result was obtained.

4. Linear Profile Decomposition

In this section, we will give the linear profile decomposition for Schrödinger equation in 𝐻¹(ℝ²). First, we give some definitions and lemmas.

Definition 11 (symmetry group, [13]). For any phase 𝜃∈ℝ/2𝜋ℤ, position 𝑥₀∈ℝ², frequency 𝜉₀∈ℝ², and scaling parameter 𝜆>0, we define the unitary transformation 𝑔_{𝜃,𝜉0,𝑥0,𝜆}∶𝐿²_𝑥(ℝ²)→𝐿²_𝑥(ℝ²) by the formula 𝑔_{𝜃,𝜉0,𝑥0,𝜆}𝑓(𝑥)1∶=𝜆𝑒^𝑖𝜃𝑒^{𝑖𝑥⋅𝜉0}𝑓𝑥−𝑥₀𝜆. (35)
We let 𝐺 be the collection of such transformations; this is a group with identity 𝑔_0,0,0,1, inverse 𝑔⁻¹_{𝜃,𝜉0,𝑥0,𝜆}=𝑔_{−𝜃−𝑥0𝜉0,−𝜆𝜉0,−𝑥0/𝜆,1/𝜆}, and group law 𝑔_{𝜃,𝜉0,𝑥0,𝜆}𝑔_{𝜃′,𝜉′0,𝑥′0,𝜆′}=𝑔_{𝜃+𝜃′−𝑥0𝜉′0/𝜆,𝜉0+𝜉′0/𝜆,𝑥0+𝜆𝑥′0,𝜆𝜆′}. (36)
If 𝑢∶𝐼×ℝ²→ℂ is a function, we define 𝑇_{𝑔00𝜃,𝜉,𝑥,𝜆}𝑢∶𝜆²𝐼×ℝ²→ℂ, where 𝜆²𝐼∶={𝜆²𝑡∶𝑡∈𝐼} by the formula 𝑇_{𝑔00𝜃,𝜉,𝑥,𝜆}𝑢()1𝑡,𝑥∶=𝜆𝑒^𝑖𝜃𝑒^{𝑖𝑥⋅𝜉0}𝑒^{−𝑖𝑡|𝜉0|2}𝑢𝑡𝜆²,𝑥−𝑥₀−2𝜉₀𝑡𝜆, (37) or equivalently 𝑇_{𝑔00𝜃,𝜉,𝑥,𝜆}𝑢(𝑡)=𝑔_{𝜃−𝑡|𝜉0|}²_{,𝜉0,𝑥0+2𝜉0𝑡,𝜆}𝑢𝑡𝜆². (38)

If 𝑔∈𝐺, we can easily prove that 𝑀(𝑇_𝑔𝑢)=𝑀(𝑢) and 𝑆(𝑇_𝑔𝑢)=𝑆(𝑢).

Definition 12 (enlarged group, [13]). For any phase 𝜃∈ℝ/2𝜋ℤ, position 𝑥₀∈ℝ², frequency 𝜉₀∈ℝ², scaling parameter 𝜆>0, and time 𝑡₀, we define the unitary transformation 𝑔_{𝜃,𝜉0,𝑥0,𝜆,𝑡0}∶𝐿²_𝑥(ℝ²)→𝐿²_𝑥(ℝ²) by the formula 𝑔_{𝜃,𝜉0,𝑥0,𝜆,𝑡0}=𝑔_{𝜃,𝜉0,𝑥0,𝜆}𝑒^𝑖𝑡0▵, (39) or in other words 𝑔_{𝜃,𝜉0,𝑥0,𝜆,𝑡0}𝑓(𝑥)1∶=𝜆𝑒^𝑖𝜃𝑒^{𝑖𝑥⋅𝜉0}𝑒^𝑖𝑡0▵𝑓𝑥−𝑥₀𝜆. (40)
Let 𝐺^′ be the collection of such transformations. We also let 𝐺^′ act on global space-time function 𝑢∶ℝ×ℝ²→ℂ by defining 𝑇_{𝑔000𝜃,𝜉,𝑥,𝜆,𝑡}𝑢()1𝑡,𝑥∶=𝜆𝑒^𝑖𝜃𝑒^{𝑖𝑥⋅𝜉0}𝑒^{−𝑖𝑡|𝜉0|2}𝑒^𝑖𝑡0▵𝑢𝑡𝜆²,𝑥−𝑥₀−2𝜉₀𝑡𝜆, (41) or equivalently 𝑇_{𝑔000𝜃,𝜉,𝑥,𝜆,𝑡}𝑢(𝑡)=𝑔_{𝜃−𝑡|𝜉0|}²_{,𝜉0,𝑥0+2𝜉0𝑡,𝜆,𝑡0}𝑢𝑡𝜆². (42)

Lemma 13 (linear profiles for 𝐿² sequence, [14]). Let 𝑢_𝑛 be a bounded sequence in 𝐿²_𝑥(ℝ²). Then (after passing to a subsequence if necessary) there exists a family 𝜙^(𝑗), 𝑗=1,2,… of functions in 𝐿²_𝑥(ℝ²) and group elements 𝑔_𝑛^(𝑗)∈𝐺^′ for 𝑗,𝑛=1,2,… such that one has the decomposition 𝑢_𝑛=^𝑙_𝑗=1𝑔_𝑛^(𝑗)𝜙^(𝑗)+𝑤_𝑛^(𝑙) (43) for all 𝑙=1,2,…; here, 𝑤_𝑛^(𝑙)∈𝐿²_𝑥(ℝ²) is such that its linear evolution has asymptotically vanishing scattering size: lim_𝑙→∞limsup_𝑛→∞𝑆𝑒^𝑖𝑡▵𝑤_𝑛^(𝑙)=0. (44)
Moreover, for any 𝑗≠𝑗^′, 𝜆^(𝑗′)_𝑛𝜆_𝑛^(𝑗)+𝜆_𝑛^(𝑗)𝜆^(𝑗′)_𝑛+𝜆_𝑛^(𝑗)𝜆^(𝑗′)_𝑛𝜉_𝑛^(𝑗)−𝜉^(𝑗′)_𝑛²+𝑥_𝑛^(𝑗)−𝑥^(𝑗′)_𝑛²𝜆_𝑛^(𝑗)𝜆^(𝑗′)_𝑛+𝑡_𝑛^(𝑗)𝜆_𝑛^(𝑗)2−𝑡^(𝑗′)_𝑛𝜆^(𝑗′)_𝑛²𝜆_𝑛^(𝑗)𝜆^(𝑗′)_𝑛⟶∞. (45)
Furthermore, for any 𝑙≥1, one has the mass decoupling property lim_𝑛→∞𝑀𝑢_𝑛−^𝑙_𝑗=1𝑀𝜙^(𝑗)−𝑀𝑤_𝑛^(𝑙)=0. (46) For any 𝑗≤𝑙, we have 𝑔_𝑛^(𝑗)−1𝑤_𝑛^(𝑙)⟶0weaklyin𝐿²_𝑥ℝ². (47)

Remark 14. If the orthogonal condition (45) holds, then (see [14]) lim_𝑛→∞𝑔_𝑛^(𝑗)𝜙^(𝑗),𝑔^(𝑗′)_𝑛𝜙^(𝑗′)_𝐿2(ℝ2)=0,𝑗≠𝑗^′,lim_𝑛→∞𝑔_𝑛^(𝑗)𝜙^(𝑗),𝑤_𝑛^(𝑙)_𝐿2(ℝ2)=0. (48)
Moreover, if 𝑣^(𝑗),𝑣^(𝑗′)∈𝐿⁴_𝑡,𝑥(ℝ×ℝ²), then (see [14, 15]), for any 0<𝜃<1, lim_𝑛→∞𝑇_𝑔𝑛^(𝑗)𝑣^{(𝑗)1−𝜃}𝑇_{𝑔′)𝑛(𝑗}𝑣^{(𝑗′)𝜃}_{𝐿4𝑡,𝑥(ℝ×ℝ2)}=0. (49) If 𝑣⁽¹⁾,…,𝑣^(𝑙)∈𝐿⁴_𝑡,𝑥(ℝ×ℝ²), then (see [16, Lemma 5.5]) lim_𝑛→∞𝑆^𝑙_𝑗=1𝑔_𝑛^(𝑗)𝑣^(𝑗)≤^𝑙_𝑗=1𝑆𝑣^(𝑗). (50)

Remark 15. As each linear profile 𝜙^(𝑗) in Lemma 13 is constructed in the sense that 𝑒^{𝑖𝑡𝑛(𝑗)▵}𝑒^{𝑖𝜉𝑛(𝑗)⋅𝑥}𝜆_𝑛^(𝑗)𝑢_𝑛𝜆_𝑛^(𝑗)𝑥𝑥+𝑥_𝑛^(𝑗)⟶𝜙^(𝑗) (51) weakly in 𝐿²_𝑥(ℝ²) (see [14]), after passing to a subsequence in 𝑛, rearrangement, translation, and refining 𝜙^(𝑗) accordingly, we may assume that the parameters satisfy the following properties:(i)𝑡_𝑛^(𝑗)→±∞ as 𝑛→∞, or 𝑡_𝑛^(𝑗)≡0 for all 𝑛,𝑗;(ii)𝜆_𝑛^(𝑗)→0 or ∞ as 𝑛→∞, or 𝜆_𝑛^(𝑗)≡1 for all 𝑛,𝑗;(iii)|𝜉_𝑛^(𝑗)|→∞ as 𝑛→∞, or 𝜉_𝑛^(𝑗)≡𝜉^(𝑗) with |𝜉^(𝑗)|<∞;(iv)when 𝜆_𝑛^(𝑗)≡1, 𝜉_𝑛^(𝑗)≡𝜉^(𝑗) and |𝜉^(𝑗)|<∞, we can let 𝜉^(𝑗)≡0.

Our main result in this section is the following lemma.

Lemma 16 (linear profiles for 𝐻¹ sequence). Let 𝑢_𝑛 be a bounded sequence in 𝐻¹(ℝ²). Then up to a subsequence, for any 𝐽≥1, there exists a sequence 𝜙^𝐽_𝛼 in 𝐻¹(ℝ²) and a sequence of group elements 𝑔_𝑛𝛼=𝑔_{𝜃𝑛𝛼,𝜉𝑛𝛼,𝑥𝑛𝛼,𝜆𝑛𝛼,𝑡𝑛𝛼}∈𝐺^′ such that 𝑢_𝑛=^𝐽_𝛼=1𝑔_𝑛𝛼𝜙^𝐽_𝛼+𝑅(𝑛,𝐽). (52)
Here, for each 𝛼, 𝜆_𝑛𝛼 and 𝜉_𝑛𝛼 must satisfy "𝜆_𝑛𝛼≡1and𝜉_𝑛𝛼≡0,"or"𝜆_𝑛𝛼⟶∞." (53)𝑅(𝑛,𝐽)∈𝐻¹(ℝ²) is such that lim_𝐽→∞limsup_𝑛→∞𝑆𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)=0. (54)
Moreover, for any 𝛼≠𝛼^′, one has the same orthogonal conditions as (45). For any 𝐽≥1, one has the following decoupling properties: lim_𝑛→∞𝑢_𝑛²_𝐿2ℝ2−^𝐽_𝛼=1𝜙^𝐽_𝛼²_𝐿2ℝ2−‖𝑅(𝑛,𝐽)‖²_𝐿2ℝ2=0,lim_𝑛→∞𝑢_𝑛²_̇𝐻1ℝ2−^𝐽_𝛼=1𝑔_𝑛𝛼𝜙^𝐽_𝛼²_̇𝐻1ℝ2−‖𝑅(𝑛,𝐽)‖²_̇𝐻1ℝ2=0,lim_𝐽→∞limsup_𝑛→∞𝐻𝑢_𝑛−^𝐽_𝛼=1𝐻𝑔_𝑛𝛼𝜙^𝐽_𝛼−𝐻(𝑅(𝑛,𝐽))=0. (55)(56)(57)

Proof. Let □_𝑘=ℱ⁻¹𝜒_𝑘ℱ,𝜒_𝑘=12^𝑘−1<𝜉≤2^𝑘,0else. (58)
Then, we have 𝑢_𝑛=^+∞_𝑘=−∞□_𝑘𝑢_𝑛∶=_{|𝑘|≤𝑁}□_𝑘𝑢_𝑛+𝑅_𝑁,𝑢_𝑛²_𝐿2ℝ2=_{|𝑘|≤𝑁}□_𝑘𝑢_𝑛²_𝐿2ℝ2+𝑅_𝑁²_𝐿2ℝ2,𝑢_𝑛²_̇𝐻1ℝ2=_{|𝑘|≤𝑁}□_𝑘𝑢_𝑛²_̇𝐻1ℝ2+𝑅_𝑁²_̇𝐻1ℝ2,lim_𝑁→∞limsup_𝑛→∞𝑅_{𝑁𝐿2(ℝ2)}=0. (59)
By Lemma 13, after passing to a subsequence if necessary, we can obtain □_𝑘𝑢_𝑛=^𝑙𝑘_𝑗=1𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗)+𝑤^(𝑙𝑘)_𝑛𝑘 (60) with the stated properties (i)–(iv) in Remark 15 and (43)–(47). Denote Λ_1,0=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘≡1,𝜉^(𝑗)_𝑛𝑘,Λ≡0_1,∞=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘≡1,𝜉^(𝑗)_𝑛𝑘,Λ⟶∞₀=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘,Λ⟶0_∞,0=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘⟶∞,𝜉^(𝑗)_𝑛𝑘≡𝜉_𝑘^(𝑗),𝜉_𝑘^(𝑗)<2^𝑘−1,Λ_∞,1=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘⟶∞,𝜉^(𝑗)_𝑛𝑘≡𝜉_𝑘^(𝑗),𝜉_𝑘^(𝑗)∈2^𝑘−1,2^𝑘,Λ_∞,∞=𝑘,𝑗∣|𝑘|≤𝑁,1≤𝑗≤𝑙_𝑘,𝜆^(𝑗)_𝑛𝑘⟶∞,𝜉^(𝑗)_𝑛𝑘⟶∞or𝜉^(𝑗)_𝑛𝑘≡𝜉_𝑘^(𝑗),𝜉_𝑘^(𝑗)>2^𝑘. (61)
Step 1. We prove that 𝑢_𝑛=_{𝑘,𝑗∈Λ1,0∪Λ∞,1}𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗)+𝑅 (62) with 𝜙_𝑘^(𝑗)∈𝐻¹ and for each fixed 𝑁, lim_𝑛→∞𝑢_𝑛²_𝐿2ℝ2−_{𝑘,𝑗∈Λ1,0∪Λ∞,1}𝜙_𝑘^(𝑗)2_𝐿2ℝ2−‖𝑅‖²_𝐿2ℝ2=0,lim_𝑛→∞𝑢_𝑛²_̇𝐻1ℝ2−_{𝑘,𝑗∈Λ1,0∪Λ∞,1}𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗)2_̇𝐻1ℝ2−‖𝑅‖²_̇𝐻1ℝ2=0,lim_𝑁→∞lim_{𝑙𝑘→∞}limsup_𝑛→∞𝑆𝑒^𝑖𝑡▵𝑅=0, (63)(64) where 𝑅=𝑅_𝑁+𝑅_𝑤,𝑅_𝑤=_{|𝑘|≤𝑁}𝑤^(𝑙𝑘)_𝑛𝑘. (65)
By (44) and lim_𝑁→∞limsup_𝑛→∞‖𝑅_𝑁‖_𝐿2(ℝ2)=0, (64) holds obviously. For (62), we prove it by induction. For every 𝑘, suppose that □_𝑘𝑢_𝑛=𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾+𝑤⁽¹⁾_𝑛𝑘. (66)
Case 1. If (𝑘,1)∈Λ_1,∞∪Λ₀∪Λ_∞,0∪Λ_∞,∞, we have 𝜙_𝑘⁽¹⁾=0.
In fact, by (66), 𝜙_𝑘⁽¹⁾=𝑔⁽¹⁾_𝑛𝑘⁻¹□_𝑘𝑢_𝑛−𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘. (67)
Thus, 𝜙_𝑘⁽¹⁾²_𝐿2ℝ2=𝑔⁽¹⁾_𝑛𝑘⁻¹□_𝑘𝑢_𝑛−𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘,𝜙_𝑘⁽¹⁾_𝐿2(ℝ2)=𝑢_𝑛,□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾_𝐿2(ℝ2)−𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘,𝜙_𝑘⁽¹⁾_𝐿2(ℝ2). (68)
Using (47), 𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘,𝜙_𝑘⁽¹⁾_𝐿2(ℝ2)⟶0as𝑛⟶∞. (69)
By direct calculation, □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾=ℱ⁻¹𝜒_𝑘(𝜉)𝜆⁽¹⁾_𝑛𝑘𝑒^{𝑖𝜃(1)𝑛𝑘}𝑒^{−𝑖𝑥(1)𝑛𝑘(𝜉+𝜉(1)𝑛𝑘)}𝑒^{−𝑖𝑡(1)𝑛𝑘(𝜆(1)𝑛𝑘)2|𝜉+𝜉(1)𝑛𝑘|2}×𝜙_𝑘⁽¹⁾𝜆⁽¹⁾_𝑛𝑘𝜉+𝜉⁽¹⁾_𝑛𝑘=1𝜆⁽¹⁾_𝑛𝑘𝑒^{𝑖𝜃(1)𝑛𝑘}𝑒^{𝑖𝑥⋅𝜉(1)𝑛𝑘}ℱ⁻¹𝜒_𝑘𝜉𝜆⁽¹⁾_𝑛𝑘+𝜉⁽¹⁾_𝑛𝑘𝑒^{−𝑖𝑡(1)𝑛𝑘|𝜉|2}𝜙_𝑘⁽¹⁾×𝑥−𝑥⁽¹⁾_𝑛𝑘𝜆⁽¹⁾_𝑛𝑘. (70)
Let 𝑛→∞. When (𝑘,1)∈Λ_1,∞, □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤_{2𝑘−1≤|𝜉+𝜉(1)𝑛𝑘|≤2𝑘}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉⟶0. (71) When (𝑘,1)∈Λ₀, □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤_{𝜆(1)𝑛𝑘2𝑘−1≤|𝜉+𝜆(1)𝑛𝑘𝜉(1)𝑛𝑘|≤𝜆(1)𝑛𝑘2𝑘}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉⟶0. (72) When (𝑘,1)∈Λ_∞,0, □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤_{|𝜉|≥𝜆(1)𝑛𝑘(2𝑘−1−|𝜉𝑘(1)|)}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉⟶0. (73) When (𝑘,1)∈Λ_∞,∞, □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤_{|𝜉|≥𝜆(1)𝑛𝑘(|𝜉(1)𝑛𝑘|−2𝑘)}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉⟶0. (74)
By (68)–(74), ‖𝜙_𝑘⁽¹⁾‖²_𝐿2(ℝ2)=0 and thus 𝜙_𝑘⁽¹⁾=0.
Case 2. If (𝑘,1)∈Λ_1,0∪Λ_∞,1, we can prove 𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾−□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾_𝐿2(ℝ2)⟶0as𝑛⟶∞. (75)
By absorbing the error into 𝑤⁽¹⁾_𝑛𝑘, we can suppose 𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾=□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾. Since □_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾∈𝐻¹ for each fixed 𝑛, we must have 𝜙_𝑘⁽¹⁾∈𝐻¹.
Now, we begin to prove (75). Let 𝜒_𝐴𝑘⁽¹⁾ be the characteristic function of the set 𝐴_𝑘⁽¹⁾ and 𝑃_𝐴𝑘⁽¹⁾=ℱ⁻¹𝜒_𝐴𝑘⁽¹⁾ℱ, and then 𝑔⁽¹⁾_𝑛𝑘𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾+𝑃_𝐴𝑘⁽¹⁾𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘𝐴=𝑃_𝑘⁽¹⁾𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾+𝑤⁽¹⁾_𝑛𝑘𝐴=𝑃_𝑘⁽¹⁾□_𝑘𝑢_𝑛, (76) where 𝑃𝐴_𝑘⁽¹⁾=ℱ⁻¹𝜒_𝐴𝑘⁽¹⁾𝜆⁽¹⁾_𝑛𝑘𝜉−𝜉⁽¹⁾_𝑛𝑘ℱ. (77)
Note that 𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾,𝑃_𝐴𝑘⁽¹⁾𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_{𝑛𝑘𝐿2(ℝ2)}=𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾,𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_{𝑛𝑘𝐿2(ℝ2)}⟶0. (78) We have lim_𝑛→∞𝑃𝐴_𝑘⁽¹⁾□_𝑘𝑢_𝑛²_𝐿2ℝ2−𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾²_𝐿2ℝ2−𝑃_𝐴𝑘⁽¹⁾𝑔⁽¹⁾_𝑛𝑘⁻¹𝑤⁽¹⁾_𝑛𝑘²_𝐿2ℝ2=0. (79)
When (𝑘,1)∈Λ_1,0, we have 𝑃_𝐴𝑘⁽¹⁾𝐴=𝑃_𝑘⁽¹⁾. Choosing 𝐴_𝑘⁽¹⁾={𝜉||𝜉|≤2^𝑘−1or|𝜉|>2^𝑘}, then by (79), 𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾=0, the desired result follows.
When (𝑘,1)∈Λ_∞,1 and |𝜉_𝑘⁽¹⁾|∈(2^𝑘−1,2^𝑘), we have 𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾−□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤_{|𝜉+𝜆(1)𝑛𝑘𝜉𝑘(1)|≤𝜆(1)𝑛𝑘2𝑘−1}ℱ𝜙_𝑘⁽¹⁾²+𝑑𝜉_{|𝜉+𝜆(1)𝑛𝑘𝜉𝑘(1)|≥𝜆(1)𝑛𝑘2𝑘}ℱ𝜙_𝑘⁽¹⁾²≤𝑑𝜉_{|𝜉|≥𝜆(1)𝑛𝑘(|𝜉𝑘(1)|−2𝑘−1)}ℱ𝜙_𝑘⁽¹⁾²+𝑑𝜉_{|𝜉|≥𝜆(1)𝑛𝑘(2𝑘−|𝜉𝑘(1)|)}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉⟶0as𝑛⟶∞. (80)
When (𝑘,1)∈Λ_∞,1 and |𝜉_𝑘⁽¹⁾|=2^𝑘, we denote 𝜉=(𝜉₁,𝜉₂) and 𝜉_𝑘⁽¹⁾=(𝜉⁽¹⁾_𝑘1,𝜉⁽¹⁾_𝑘2). The line 𝜉₂=−(𝜉⁽¹⁾_𝑘1/𝜉⁽¹⁾_𝑘2)𝜉₁ (when 𝜉^(𝑗)_𝑘2=0, we use the line 𝜉₁=0 instead) separates the frequency space ℝ² into two half-planes. We let 𝐴_𝑘⁽¹⁾ to be the half-plane which contains the point 𝜉_𝑘⁽¹⁾, and then 𝑃𝐴_𝑘⁽¹⁾□_𝑘𝑢_𝑛=ℱ⁻¹𝜒_𝐴𝑘⁽¹⁾𝜆⁽¹⁾_𝑛𝑘𝜉−𝜉_𝑘⁽¹⁾𝜒_𝑘ℱ𝑢_𝑛=0. (81)
By (79), we have 𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾=0. Note that lim_𝑛→∞𝑔⁽¹⁾_𝑛𝑘𝐼−𝑃_𝐴𝑘⁽¹⁾𝜙_𝑘⁽¹⁾−□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘^(𝑗)2_𝐿2ℝ2=lim_𝑛→∞1−𝜒_𝐴𝑘⁽¹⁾(⋅)−𝜒_𝑘⋅𝜆⁽¹⁾_𝑛𝑘+𝜉_𝑘⁽¹⁾ℱ𝜙_𝑘⁽¹⁾²_𝐿2ℝ2≤lim_{𝑛→∞𝜉∈ℝ2⧵(𝐴𝑘(1)∪{2𝑘−1≤|𝜉/𝜆(1)𝑛𝑘+𝜉𝑘(1)|≤2𝑘})}ℱ𝜙_𝑘⁽¹⁾²𝑑𝜉=0. (82) (75) holds.
When (𝑘,1)∈Λ_∞,1 and |𝜉_𝑘⁽¹⁾|=2^𝑘−1, let 𝐴_𝑘⁽¹⁾ be the half-plane which does NOT contain the point 𝜉_𝑘⁽¹⁾; we can prove (75) similarly as above.
By the proof above, we get □_𝑘𝑢_𝑛=𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾+𝑤⁽¹⁾_𝑛𝑘,𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾=□_𝑘𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾,𝜙_𝑘⁽¹⁾∈𝐻¹ (83) and (𝑘,1)∈Λ_1,0∪Λ_∞,1. Denote 𝑢_𝑛⁽¹⁾=𝑢_𝑛−𝑔⁽¹⁾_𝑛𝑘𝜙_𝑘⁽¹⁾ and suppose □_𝑘𝑢_𝑛⁽¹⁾=𝑔⁽²⁾_𝑛𝑘𝜙_𝑘⁽²⁾+𝑤⁽²⁾_𝑛𝑘. (84)
Repeating the proof above, we can get 𝑔⁽²⁾_𝑛𝑘𝜙_𝑘⁽²⁾=□_𝑘𝑔⁽²⁾_𝑛𝑘𝜙_𝑘⁽²⁾, 𝜙_𝑘⁽²⁾∈𝐻¹, and (𝑘,2)∈Λ_1,0∪Λ_∞,1…; by induction, we obtain (62).
By the orthogonal condition (45), following the proof in [14], we can obtain that for fix 𝑘 and for all 𝑗≠𝑗^′, lim_𝑛→∞𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗),𝑔^(𝑗′)_𝑛𝑘𝜙^(𝑗′)_{𝑘̇𝐻1(ℝ2)}=lim_𝑛→∞□_𝑘𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗),□_𝑘𝑔^(𝑗′)_𝑛𝑘𝜙^(𝑗′)_{𝑘̇𝐻1(ℝ2)}=0,lim_𝑛→∞𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗),𝑤^(𝑙𝑘)_{𝑛𝑘̇𝐻1(ℝ2)}=lim_𝑛→∞□_𝑘𝑔^(𝑗)_𝑛𝑘𝜙_𝑘^(𝑗),□_𝑘𝑤^(𝑙𝑘)_{𝑛𝑘̇𝐻1(ℝ2)}=0, (85) (63) were proved.
Step 2. For arbitrary (𝑘₁,𝑗₁),(𝑘₂,𝑗₂)∈Λ_1,0∪Λ_∞,1, we define (𝑘₁,𝑗₁)∼(𝑘₂,𝑗₂) if the orthogonal condition (45) is NOT true for any subsequence; that is, limsup_𝑛→∞𝜆^(𝑗1)_𝑛𝑘1𝜆^(𝑗2)_𝑛𝑘2+𝜆^(𝑗2)_𝑛𝑘2𝜆^(𝑗1)_𝑛𝑘1+𝜆^(𝑗2)_𝑛𝑘2𝜆^(𝑗1)_𝑛𝑘1𝜉^(𝑗2)_𝑛𝑘2−𝜉^(𝑗1)_𝑛𝑘1²+𝑥^(𝑗2)_𝑛𝑘2−𝑥^(𝑗1)_𝑛𝑘1²𝜆^(𝑗2)_𝑛𝑘2𝜆^(𝑗1)_𝑛𝑘1+𝑡^(𝑗2)_𝑛𝑘2𝜆^(𝑗2)_𝑛𝑘2²−𝑡^(𝑗1)_𝑛𝑘1𝜆^(𝑗1)_𝑛𝑘1²𝜆^(𝑗2)_𝑛𝑘2𝜆^(𝑗1)_𝑛𝑘1<∞. (86)
By the definition above, if (𝑘₁,𝑗₁)∼(𝑘₂,𝑗₂), we have 𝜆^(𝑗1)_𝑛𝑘1∼𝜆^(𝑗2)_𝑛𝑘2,𝜆^(𝑗1)_𝑛𝑘1𝜉^(𝑗1)_𝑛𝑘1∼𝜆^(𝑗2)_𝑛𝑘2𝜉^(𝑗2)_𝑛𝑘2,𝑥^(𝑗1)_𝑛𝑘1𝜆^(𝑗1)_𝑛𝑘1∼𝑥^(𝑗2)_𝑛𝑘2𝜆^(𝑗2)_𝑛𝑘2,𝑡^(𝑗1)_𝑛𝑘1∼𝑡^(𝑗2)_𝑛𝑘2. (87)
Note that 𝑔_{𝜃,𝜉0,𝑥0,𝜆,𝑡0}𝑓(𝑥)1∶=𝜆𝑒^𝑖𝜃𝑒^{𝑖(𝑥/𝜆)⋅𝜆𝜉0}𝑒^𝑖𝑡0▵𝑓𝑥𝜆−𝑥₀𝜆. (88) By Remark 15, we can put these two profiles together as one profile. Then, by denoting (Λ_1,0∪Λ_∞,1)/∼ as {1,2,…,𝐽}, we can obtain the sequence 𝜙^𝐽_𝛼, 𝛼=1,2,…,𝐽; and (52)–(56) were proved.
Specially, since 𝜙_𝑘^(𝑗)∈𝐻¹ for each (𝑘,𝑗)∈Λ_1,0∪Λ_∞,1, we have _{𝑘,𝑗∈Λ1,0∪Λ∞,1}𝜙_𝑘^(𝑗)∈𝐻¹ (89) for fixed 𝑁 and 𝑙_𝑘, and hence 𝜙^𝐽_𝛼∈𝐻¹(ℝ²) for any fixed 𝐽 and 1≤𝛼≤𝐽.
Step 3. We prove (57) now. By (56), we only need to prove that for all 𝑚∈ℕ, 𝑚≥2, lim_𝐽→∞limsup_𝑛→∞𝑢_𝑛𝐿^2𝑚_2𝑚ℝ2−^𝐽_𝛼=1𝑔_𝑛𝛼𝜙^𝐽_𝛼𝐿^2𝑚_2𝑚ℝ2−‖𝑅(𝑛,𝐽)‖_𝐿^2𝑚_2𝑚ℝ2=0. (90)
As ‖𝑅(𝑛,𝐽)‖_𝐿^2𝑚_(ℝ²₎≲‖𝑅(𝑛,𝐽)‖_𝐿^1/(𝑚−1)_4ℝ2‖𝑅(𝑛,𝐽)‖_𝐻^{(𝑚−2)/(𝑚−1)}_1ℝ2 (91) and for 1/4<𝜃<1/2, ‖𝑅(𝑛,𝐽)‖_𝐿⁴_(ℝ²₎≤𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿∞𝑡𝐿4𝑥(ℝ×ℝ2)}≲𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐻𝜃4𝐿4𝑥(ℝ×ℝ2)}=𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^1−2𝜃_{4𝑡,𝑥ℝ×ℝ2}𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐻^2𝜃_{41/2𝐿4𝑥ℝ×ℝ2}=𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^1−2𝜃_{4𝑡,𝑥ℝ×ℝ2}𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^2𝜃_{4𝑡𝐻14ℝ×ℝ2}≲𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^1−2𝜃_{4𝑡,𝑥ℝ×ℝ2}. (92) By (54), we have lim_𝐽→∞limsup_𝑛→∞‖𝑅(𝑛,𝐽)‖_𝐿^2𝑚_2𝑚ℝ2=0. (93)
We separate the set 1≤𝛼≤𝐽 into two subsets: Λ₁=𝛼∣1≤𝛼≤𝐽,𝜆_𝑛𝛼≡1,𝜉_𝑛𝛼,Λ≡0_∞=𝛼∣1≤𝛼≤𝐽,𝜆_𝑛𝛼⟶∞. (94)
When 𝛼∈Λ_∞, lim_𝑛→∞𝑔_𝑛𝛼𝜙^𝐽_{𝛼𝐿2𝑚(ℝ2)}=lim_𝑛→∞𝜆_𝑛𝛼^−1+1/𝑚𝑒^{𝑖𝑡𝑛𝛼▵}𝜙^𝐽_{𝛼𝐿2𝑚(ℝ2)}≤lim_𝑛→∞𝜆_𝑛𝛼^−1+1/𝑚𝜙^𝐽_{𝛼𝐻1(ℝ2)}=0. (95)
Hence, in order to prove (90), one only needs to prove lim_{𝑛→∞𝛼∈Λ1}𝑔_𝑛𝛼𝜙^𝐽_𝛼𝐿^2𝑚_2𝑚ℝ2−_𝛼∈Λ1𝑔_𝑛𝛼𝜙^𝐽_𝛼𝐿^2𝑚_2𝑚ℝ2=0. (96)
If 𝛼∈Λ₁ and 𝑡_𝑛𝛼→∞, for a function 𝜙^𝐽_𝛼∈𝐻̇^1/2∩𝐿^4/3, we have 𝑔_𝑛𝛼𝜙^𝐽_{𝛼𝐿4(ℝ2)}≤𝑔_𝑛𝛼𝜙^𝐽_𝛼−𝑔_𝑛𝛼𝜙^𝐽_{𝛼𝐿4(ℝ2)}+𝑔_𝑛𝛼𝜙^𝐽_{𝛼𝐿4(ℝ2)}≤𝜙^𝐽_𝛼−𝜙^𝐽_{𝛼̇𝐻1/2(ℝ2)}+𝑡_𝑛𝛼^−1/2𝜙^𝐽_{𝛼𝐿4/3(ℝ2)}. (97)
By approximating 𝜙^𝐽_𝛼 by 𝜙^𝐽_𝛼∈𝐶^∞_𝑐 in 𝐻̇^1/2 and sending 𝑛→∞, we have ‖𝑔_𝑛𝛼𝜙^𝐽_𝛼‖_𝐿4(ℝ2)→0. Note that 𝑔_𝑛𝛼𝜙^𝐽_𝛼∈𝐻¹; we obtain ‖𝑔_𝑛𝛼𝜙^𝐽_𝛼‖_{𝐿2𝑚(ℝ2)}→0 for all 𝑚≥2.
If 𝛼∈Λ₁ and 𝑡_𝑛𝛼≡0, we have orthogonal condition |𝑥_𝑛𝛼−𝑥_𝑛𝛼′|→∞ for any 𝛼≠𝛼^′. Thus, lim_{𝑛→∞𝛼∈Λ1,𝑡𝑛𝛼≡0}𝑔_𝑛𝛼𝜙^𝐽_𝛼𝐿^2𝑚_2𝑚ℝ2−_{𝛼∈Λ1,𝑡𝑛𝛼≡0}𝑔_𝑛𝛼𝜙^𝐽_𝛼𝐿^2𝑚_2𝑚ℝ2=lim_{𝑛→∞𝛼∈Λ1,𝑡𝑛𝛼≡0}𝑒^{𝑖𝜃𝑛𝛼}𝜙^𝐽_𝛼⋅−𝑥_𝑛𝛼𝐿^2𝑚_2𝑚ℝ2−_{𝛼∈Λ1,𝑡𝑛𝛼≡0}𝜙^𝐽_𝛼⋅−𝑥_𝑛𝛼𝐿^2𝑚_2𝑚ℝ2=0. (98) (96) holds and then (57) was proved.

5. The Proof of Theorem 3

Let 𝑢 be a solution of (5), 𝐻(𝑢₀)≤1; by Strichartz estimate and (21), ‖𝑢‖_{𝐿4𝑡,𝑥∩𝐿𝑡}^{2/(1−2𝜀)}_𝐿𝑥^1/𝜀_(ℝ×ℝ2)≤𝐶𝑢_0𝐿2(ℝ2)+𝐶‖𝑢‖³_{𝐿4𝑡,𝑥ℝ×ℝ2}+𝐶(𝐸)‖𝑢‖_𝐿𝑡^{2/(1−2𝜀)}_𝐿𝑥^1/𝜀_(ℝ×ℝ2)‖𝑢‖_𝐿^8𝜀_{4𝑡,𝑥ℝ×ℝ2}≤𝐶𝑢_0𝐿2(ℝ2)+𝐶‖𝑢‖³_{𝐿4𝑡,𝑥∩𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀ℝ×ℝ2}+𝐶(𝐸)‖𝑢‖_𝐿^1+8𝜀_{4𝑡,𝑥∩𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀ℝ×ℝ2}. (99)

When ‖𝑢₀‖_𝐿2(ℝ2)≪1, by standard continuity argument, we have ‖𝑢‖_𝐿⁴_{𝑡,𝑥(ℝ×ℝ}²₎≤𝐶𝑢_0𝐿2(ℝ2)<∞. (100)

Hence, if 𝑀(𝑢)≪1, then ‖𝑢‖_𝐿⁴_{𝑡,𝑥(ℝ×ℝ}²₎<∞. In particular, we have scattering in both directions.

For any mass 𝑚≥0, we define 𝐴(𝑚)𝑆(𝑢)(5),𝑀(𝑢)(𝑢)}.∶=sup∶𝑢istheglobalsolutionof≤𝑚,𝐻≤1 (101)

Then 𝐴∶[0,+∞)→[0,+∞] is a monotone increasing function of 𝑚. As 𝐴 is left-continuous and finite for small 𝑚, there must exist a unique critical mass 𝑚₀∈(0+∞] such that 𝐴(𝑚) is finite for all 𝑚<𝑚₀ but infinite for all 𝑚≥𝑚₀.

To prove Theorem 3, one only needs to prove that the critical mass 𝑚₀ is infinite. We will prove that by contradiction.

Proposition 17. Suppose that the critical mass 𝑚₀ is finite. Let 𝑢_𝑛∶ℝ×ℝ²→ℂ for 𝑛=1,2,… be a sequence of solutions and let 𝑡_𝑛∈ℝ be a sequence of times such that 𝐻(𝑢_𝑛)≤1, limsup_𝑛→∞𝑀(𝑢_𝑛)=𝑚₀, and lim_𝑛→∞𝑆_≥𝑡𝑛𝑢_𝑛=lim_𝑛→∞𝑆_≤𝑡𝑛𝑢_𝑛=+∞. (102)
Then there exists a sequence of 𝑥_𝑛=𝑥_𝑛(𝑡_𝑛)∈ℝ²such that 𝑢_𝑛(𝑡_𝑛,𝑥+𝑥_𝑛) has a subsequence which converges strongly into 𝐿²_𝑥(ℝ²). Especially, the Hamiltonian of the limiting function is not greater than 1.

Proof. We can take 𝑡_𝑛=0 for all 𝑛 by translating 𝑢_𝑛 in time. Thus, lim_𝑛→∞𝑆_≥0𝑢_𝑛=lim_𝑛→∞𝑆_≤0𝑢_𝑛=+∞. (103)
By Lemma 16, up to a subsequence if necessary, we have 𝑢_𝑛(0)=_{𝛼∈Λ1∪Λ∞}𝑔_𝑛𝛼𝜙^𝐽_𝛼+𝑅(𝑛,𝐽), (104) where Λ₁ and Λ_∞ were defined by (94). Suppose that 𝑔_𝑛𝛼=ℎ_𝑛𝛼𝑒^{𝑖𝑡𝑛𝛼▵}, (105) where 𝑡_𝑛𝛼∈ℝ and ℎ_𝑛𝛼∈𝐺. By (55), _{𝛼∈Λ1∪Λ∞}𝑀𝜙^𝐽_𝛼≤limsup_𝑛→∞𝑀𝑢_𝑛(0)≤𝑚₀. (106)
Hence, sup_{𝛼∈Λ1∪Λ∞}𝑀𝜙^𝐽_𝛼≤𝑚₀. (107)
Suppose sup_{𝛼∈Λ1∪Λ∞}𝑀𝜙^𝐽_𝛼≤𝑚₀−𝜎. (108) For some 𝜎>0, we will prove that this leads to a contradiction: defining the nonlinear profile 𝑣^𝐽_𝛼∶ℝ×ℝ²→ℂ as follows.(i)When 𝛼∈Λ₁,(1)if 𝑡_𝑛𝛼≡0, we define 𝑣^𝐽_𝛼 to be the global solution of (5) with initial data 𝑣^𝐽_𝛼(0)=𝜙^𝐽_𝛼;(2)if 𝑡_𝑛𝛼→+∞, we define 𝑣^𝐽_𝛼 to be the global solution of (5) which scatters to 𝑒^𝑖𝑡▵𝜙^𝐽_𝛼 in 𝐻¹ when 𝑡→+∞;(3)if 𝑡_𝑛𝛼→−∞, we define 𝑣^𝐽_𝛼 to be the global solution of (5) which scatters to 𝑒^𝑖𝑡▵𝜙^𝐽_𝛼 in 𝐻¹ when 𝑡→−∞.(ii)When 𝛼∈Λ_∞,(1)if 𝑡_𝑛𝛼≡0, we define 𝑣^𝐽_𝛼 to be the global solution of 𝑖𝑢_𝑡+▵𝑢=|𝑢|²𝑢 with initial data 𝑣^𝐽_𝛼(0)=𝜙^𝐽_𝛼;(2)if 𝑡_𝑛𝛼→+∞, we define 𝑣^𝐽_𝛼 to be the global solution of 𝑖𝑢_𝑡+▵𝑢=|𝑢|²𝑢 which scatters to 𝑒^𝑖𝑡▵𝜙^𝐽_𝛼 in 𝐻¹ when 𝑡→+∞;(3)if 𝑡_𝑛𝛼→−∞, we define 𝑣^𝐽_𝛼 to be the global solution of 𝑖𝑢_𝑡+▵u=|𝑢|²𝑢 which scatters to 𝑒^𝑖𝑡▵𝜙^𝐽_𝛼 in 𝐻¹ when 𝑡→−∞.
For each 𝛼∈Λ₁, 𝑣^𝐽_𝛼 is well defined by the definition of 𝐴(𝑚), (100), and (108). For each 𝛼∈Λ_∞, 𝑣^𝐽_𝛼 is well defined by the scattering of cubic Schrödinger equation in 𝐿²(ℝ²) (see [7, 8]) and the same analysis as in Lemma 9.
Now, we define 𝑢_𝑛=_{𝛼∈Λ1∪Λ∞}𝑇_ℎ𝑛𝛼𝑣^𝐽_𝛼⋅+𝑡_𝑛𝛼+𝑒^𝑖𝑡▵𝑅(𝑛,𝐽) (109) for 𝑛,𝐽=1,2,…, and then we have the following two lemmas.
Lemma 18 lim_𝑛→∞𝑀𝑢_𝑛(0)−𝑢_𝑛(0)=0,lim_𝐽→∞limsup_𝑛→∞𝑢_𝑛²_̇𝐻1ℝ2≤1. (110)
Proof. Since lim_𝑛→∞𝑀𝑢_𝑛(0)−𝑢_𝑛(0)≤_{𝛼∈Λ1∪Λ∞}lim_𝑛→∞𝑀𝑇_ℎ𝑛𝛼𝑣^𝐽_𝛼⋅+𝑡_𝑛𝛼(0)−𝑔_𝑛𝛼𝜙^𝐽_𝛼=_{𝛼∈Λ1∪Λ∞}lim_𝑛→∞𝑀ℎ_𝑛𝛼𝑣^𝐽_𝛼𝑡_𝑛𝛼−ℎ_𝑛𝛼𝑒^{𝑖𝑡𝑛𝛼▵}𝜙^𝐽_𝛼=_{𝛼∈Λ1∪Λ∞}lim_𝑛→∞𝑀𝑣^𝐽_𝛼𝑡_𝑛𝛼−𝑒^{𝑖𝑡𝑛𝛼▵}𝜙^𝐽_𝛼=0, (111) we get the first equality.
Note that 𝑇_ℎ𝑛𝛼 maps the solutions of 𝑖𝑢_𝑡+▵𝑢=|𝑢|²𝑢 from one to another; using (57) and energy conservation, we have lim_𝐽→∞limsup_𝑛→∞𝑢_𝑛²_̇𝐻1ℝ2≤lim_𝐽→∞limsup_𝑛→∞^𝐽_𝛼=1𝑇_ℎ𝑛𝛼𝑣^𝐽_𝛼⋅+𝑡_𝑛𝛼²_̇𝐻1ℝ2+‖𝑅(𝑛,𝐽)‖²_̇𝐻1ℝ2≤lim_𝐽→∞limsup_𝑛→∞^𝐽_𝛼=1𝐻𝑔_𝑛𝛼𝜙^𝐽_𝛼+𝐻(𝑅(𝑛,𝐽))≤lim_𝐽→∞limsup_𝑛→∞𝐻𝑢_𝑛≤1. (112)
Lemma 19.  Iflim_𝐽→∞limsup_𝑛→∞𝑢_{𝑛𝐿4𝑡,𝑥(ℝ×ℝ2)}𝑣<∞,_{𝐽𝛼𝐿4𝑡,𝑥(ℝ×ℝ2)}<∞(∀𝛼), (113)thenlim_𝐽→∞limsup_𝑛→∞^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑖𝜕_𝑡+▵𝑢_𝑛−𝑓𝑢_𝑛(𝜏)𝑑𝜏_𝑋=0, (114)where 𝑋=𝐿⁴_𝑡,𝑥∩𝐿_𝑡^{2/(1−2𝜀)}𝐿_𝑥^1/𝜀(ℝ×ℝ²).
Proof. Denote 𝑣^𝐽_𝑛𝛼=𝑇_ℎ𝑛𝛼𝑣^𝐽_𝛼⋅+𝑡_𝑛𝛼. (115)
By the definition of 𝑢_𝑛, we have 𝑢_𝑛=_{𝛼∈Λ1∪Λ∞}𝑣^𝐽_𝑛𝛼+𝑒^𝑖𝑡▵𝑅(𝑛,𝐽),𝑖𝜕_𝑡+▵𝑢_𝑛=_𝛼∈Λ1𝑓𝑣^𝐽_𝑛𝛼+_𝛼∈Λ∞𝑣^𝐽_𝑛𝛼²𝑣^𝐽_𝑛𝛼. (116)
Thus, by triangle inequality, it suffices to show that lim_𝐽→∞limsup_𝑛→∞^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓𝑢_𝑛−𝑒^𝑖𝑡▵𝑅()𝑢𝑛,𝐽−𝑓_𝑛(𝜏)𝑑𝜏_𝑋=0,lim_𝑛→∞^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓_{𝛼∈Λ1∪Λ∞}𝑣^𝐽_𝑛𝛼−_{𝛼∈Λ1∪Λ∞}𝑓𝑣^𝐽_𝑛𝛼(𝜏)𝑑𝜏_𝑋=0,lim_𝑛→∞^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}_𝛼∈Λ∞𝑓𝑣^𝐽_𝑛𝛼−𝑣^𝐽_𝑛𝛼²𝑣^𝐽_𝑛𝛼(𝜏)𝑑𝜏_𝑋=0. (117)(118)(119)
By Lemma 18, let 𝐽 and 𝑛 be sufficiently large; we have 𝑢_𝑛²_̇𝐻1ℝ2≤1,𝑢1−4𝜀^𝑛−𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)²_̇𝐻1ℝ2≤1.1−4𝜀 (120)
Since we have supposed 0<𝜆<4(1−4𝜀)²𝜋, Lemma 6 can also be used here. Then by the same estimates as (31), ^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓𝑢_𝑛−𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)−𝑓𝑢_𝑛(𝜏)𝑑𝜏_𝑋≲𝑢_𝑛²_{𝐿4𝑡,𝑥ℝ×ℝ2}+𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)²_{𝐿4𝑡,𝑥ℝ×ℝ2}×𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿4𝑡,𝑥(ℝ×ℝ2)}+𝑢_𝑛𝐿^8𝜀_{4𝑡,𝑥ℝ×ℝ2}+𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^8𝜀_{4𝑡,𝑥ℝ×ℝ2}×𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀(ℝ×ℝ2)}. (121)
By (54), lim_𝐽→∞limsup_𝑛→∞𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿4𝑡,𝑥(ℝ×ℝ2)}=0. (122)
As 𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀(ℝ×ℝ2)}≲𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^{2𝜀/(1−2𝜀)}_{4𝑡,𝑥ℝ×ℝ2}𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_𝐿^{(1−4𝜀)/(1−2𝜀)}_{𝑡2/(1−𝜀)𝐿𝑥2/𝜀ℝ×ℝ2},𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿𝑡2/(1−𝜀)𝐿𝑥2/𝜀(ℝ×ℝ2)}≲‖𝑅(𝑛,𝐽)‖_𝐿²_𝑥(ℝ²₎<∞, (123) we have lim_𝐽→∞limsup_𝑛→∞𝑒^𝑖𝑡▵𝑅(𝑛,𝐽)_{𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀(ℝ×ℝ2)}=0. (124)
By lim_𝐽→∞limsup_𝑛→∞‖𝑢_𝑛‖_{𝐿4𝑡,𝑥(ℝ×ℝ2)}<∞, (117) was obtained.
Using Strichartz estimate, ^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓𝑣^𝐽_𝑛𝛼−𝑣^𝐽_𝑛𝛼²𝑣^𝐽_𝑛𝛼(𝜏)𝑑𝜏_𝑋≤^∞_𝑚=2𝑣^𝐽_𝑛𝛼^2𝑚𝑣^𝐽_𝑛𝛼𝑚!_{𝐿4/3𝑡,𝑥(ℝ×ℝ2)}≤^∞_𝑚=21𝑣𝑚!^{𝐽𝑛𝛼𝐿4𝑡,𝑥(ℝ×ℝ2)}𝑣^𝐽_𝑛𝛼𝐿^2𝑚_{4𝑚𝑡,𝑥ℝ×ℝ2}≤^∞_𝑚=21𝜆𝑚!^{𝑛𝛼−2(𝑚−1)}𝑣^𝐽_{𝛼𝐿4𝑡,𝑥(ℝ×ℝ2)}𝑣^𝐽_𝛼𝐿^2𝑚_{4𝑚𝑡,𝑥ℝ×ℝ2}. (125)
By Lemma 8, 𝑣^𝐽_{𝛼𝐿4𝑚𝑡,𝑥(ℝ×ℝ2)}≤𝐶𝑚^1/2+1/2𝑚𝑣^𝐽_𝛼𝐿^1/𝑚_{4𝑡,𝑥ℝ×ℝ2}𝑣^𝐽_𝛼𝐿^1−1/𝑚_{∞̇𝐻ℝ,1}. (126)
Note that ‖𝑣^𝐽_𝛼‖_{𝐿4𝑡,𝑥(ℝ×ℝ2)}<∞. Equation (119) was obtained.
To prove Lemma 19, it is only left to prove (118). Note that 𝑓^𝐽_𝛼=1𝑧_𝛼−^𝐽_𝛼=1𝑓𝑧_𝛼≲_{𝛼≠𝛼′}𝑧_𝛼𝑒^{𝜆|𝑧𝛼′|2}−1. (127) By (57), we have ^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓_{𝛼∈Λ1∪Λ∞}𝑣^𝐽_𝑛𝛼−_{𝛼∈Λ1∪Λ∞}𝑓𝑣^𝐽_𝑛𝛼(𝜏)𝑑𝜏_𝑋≲_{𝛼≠𝛼′}𝑣^𝐽_𝑛𝛼𝑣^𝐽_𝑛𝛼′²_{𝐿4/3𝑡,𝑥(ℝ×ℝ2)}+𝑣^𝐽_𝑛𝛼𝑒^{𝜆|𝑣𝐽′𝑛𝛼|2}−𝜆𝑣^𝐽_𝑛𝛼′²−1_{𝐿𝑡2/(1+2𝜀)𝐿𝑥1/(1−𝜀)(ℝ×ℝ2)}≲_{𝛼≠𝛼′}𝑣^𝐽_𝑛𝛼^1/2𝑣^𝐽_𝑛𝛼′^1/22_{𝐿4𝑡,𝑥ℝ×ℝ2}𝑣^𝐽_{𝑛𝛼′𝐿4𝑡,𝑥(ℝ×ℝ2)}+_{𝛼≠𝛼′}𝑣^𝐽_𝑛𝛼^{𝜀/(1+𝜀)}𝑣^𝐽_𝑛𝛼′^1/(1+𝜀)_𝐿^{4𝜀(1+𝜀)/(1−2𝜀)}_{4𝑡,𝑥ℝ×ℝ2}×𝑣^𝐽_𝑛𝛼^{(1−2𝜀−4𝜀2}_𝐿^{)/(1−2𝜀)22}_{)𝑡2(1−2𝜀−4𝜀)/(1−2𝜀−6𝜀𝐿}²²_{𝑥(1−2𝜀−4𝜀)/𝜀ℝ×ℝ2}×𝑣^𝐽_𝑛𝛼′^{−4𝜀/(1−2𝜀)}×𝑒^{𝜆|𝑣𝐽′𝑛𝛼|2}−𝜆𝑣^𝐽_𝑛𝛼′²−1_{𝐿∞𝑡𝐿𝑥(1−2𝜀)/(1−4𝜀)(ℝ×ℝ2)}. (128)
By (49), we immediately obtain (118).
If sup_{𝛼∈Λ1∪Λ∞}𝑀𝜙_𝛼≤𝑚₀−𝜎 (129) for some 𝜎>0, by the definition of 𝐴(𝑚) and (100), we have 𝐴(𝑚)≤𝐵𝑚∀𝑚∈0,𝑚₀−𝜎, (130) where 𝐵=𝐵(𝜎)∈(0,+∞). Then 𝑣^𝐽_𝛼 satisfies 𝑀𝑣^𝐽_𝛼=𝑀𝜙^𝐽_𝛼≤𝑚₀𝑆𝑣−𝜎,_𝐽𝛼≤𝐴𝑀𝜙^𝐽_𝛼≤𝐵𝑀𝜙^𝐽_𝛼. (131)(132)
By (50), (106), and (132), we have lim_𝐽→∞limsup_𝑛→∞𝑆𝑢_𝑛≤𝐵𝑚₀. (133)
Using Lemmas 18 and 19, we have 𝑀𝑢_𝑛(0)−𝑢_𝑛(0)≤𝛿,𝑆𝑢_𝑛≤2𝐵𝑚₀,_𝑡0𝑒^{𝑖(𝑡−𝜏)▵}𝑖𝜕_𝑡+▵𝑢_𝑛−𝑓𝑢_𝑛(𝜏)𝑑𝜏_{𝐿4𝑡,𝑥∩𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀(ℝ×ℝ2)}≤𝛿, (134) for 𝛿>0 sufficiently small, 𝐽=𝐽(𝛿), and 𝑛=𝑛(𝐽,𝛿) sufficiently large. By Lemma 10, we obtain that 𝑆(𝑢_𝑛)≤3𝐵𝑚₀ which contradicts (103). Thus, (129) fails for all 𝜎>0, and then sup_{𝛼∈Λ1∪Λ∞}𝑀𝜙_𝛼=𝑚₀. (135)
Comparing this with (106), we have 𝑢_𝑛(0)=ℎ_𝑛𝑒^{𝑖𝑡𝑛▵}𝜙+𝑅_𝑛 (136) with 𝑡_𝑛 converging to ±∞ or 𝑡_𝑛≡0, ℎ_𝑛∈𝐺, 𝑀(𝜙)=𝑚₀, and 𝑀𝑅_𝑛⟶0,𝑆𝑒^𝑖𝑡▵𝑅_𝑛⟶0as𝑛⟶∞. (137)
Specially, the parameters 𝜆_𝑛, 𝜉_𝑛 of ℎ_𝑛 must satisfy "𝜆_𝑛≡1and𝜉_𝑛≡0,"or"𝜆_𝑛⟶∞." (138)
If 𝜆_𝑛→∞, similar to the former case, we can define the approximate solution 𝑢_𝑛=𝑇_ℎ𝑛𝑣⋅+𝑡_𝑛+𝑒^𝑖𝑡▵𝑅_𝑛. (139)
By the scattering of cubic Schrödinger equation, we have 𝑆(𝑣)<∞ and lim_𝑛→∞𝑆(𝑢_𝑛)<∞. By Lemmas 18, 19, and 10, for 𝑛 sufficiently large, we obtain 𝑆(𝑢_𝑛)<∞ which contradicts (103).
If 𝜆_𝑛≡1, 𝜉_𝑛≡0, and 𝑡_𝑛→+∞, by Strichartz estimate and monotone convergence, we have lim_𝑛→∞𝑆_≥0𝑒^𝑖𝑡▵𝑒^{𝑖𝑡𝑛▵}𝜙=0. (140)
Thus, lim_𝑛→∞𝑆_≥0𝑒^𝑖𝑡▵ℎ_𝑛𝑒^{𝑖𝑡𝑛▵}𝜙=lim_𝑛→∞𝑆_≥0𝑒^𝑖𝑡▵𝑒^{𝑖𝑡𝑛▵}𝜙=0. (141)
Since lim_𝑛→∞𝑆(𝑒^𝑖𝑡▵𝑅_𝑛)=0, we can see from (136) that lim_𝑛→∞𝑆_≥0𝑒^𝑖𝑡▵𝑢_𝑛(0)=0. (142)
Hence, lim_𝑛→∞^𝑡₀𝑒^{𝑖(𝑡−𝜏)▵}𝑓𝑒^𝑖𝑡▵𝑢_𝑛(0)(𝜏)𝑑𝜏_{𝐿4𝑡,𝑥∩𝐿𝑡2/(1−2𝜀)𝐿𝑥1/𝜀([0,∞)×ℝ2)}=0. (143)
By Lemma 10 (with 𝑒^𝑖𝑡▵𝑢_𝑛(0) as the approximate solution and 𝑢_𝑛(0) as the initial data), we have lim_𝑛→∞𝑆_≥0𝑢_𝑛=0, (144) which contradicts one of the estimates in (103).
If 𝜆_𝑛≡1, 𝜉_𝑛≡0, and 𝑡_𝑛→−∞, the argument is the same and we can obtain a contradiction by using the other half of (103).
Now, the only case left is 𝜆_𝑛≡1, 𝜉_𝑛≡0, and 𝑡_𝑛≡0. In this case, we have 𝑀𝑢_𝑛(0)−ℎ_𝑛𝜙=𝑀𝑅_𝑛⟶0as𝑛⟶∞. (145)
Thus, (ℎ_𝑛)⁻¹𝑢_𝑛(0)=𝑒^𝑖𝜃𝑛𝑢_𝑛(0,𝑥+𝑥_𝑛) converges to 𝜙 in 𝐿²_𝑥(ℝ²). Since 𝐻(𝜙)=𝐻(𝑔_𝑛𝜙)≤1, after passing to a subsequence if necessary and rotating 𝜙, the desired result follows.

Let {𝑢_𝑛} be the sequence given in Proposition 17 and satisfy 𝑀(𝑢_𝑛)≤𝑚₀ and suppose that 𝑢_𝑛(0,𝑥+𝑥_𝑛) converges to 𝑢₀ strongly in 𝐿²_𝑥(ℝ²); then 𝑀(𝑢₀)≤𝑚₀ and 𝐻(𝑢₀)≤1. Let 𝑢 be the global solution with initial data 𝑢(0)=𝑢₀; by Lemma 10, we must have 𝑆_≥0(𝑢)=𝑆_≤0(𝑢)=+∞. (146)

By the definition of 𝑚₀, 𝑀(𝑢₀)≥𝑚₀ and hence 𝑀(𝑢₀)=𝑚₀.

Since 𝑢 is locally in 𝐿⁴_𝑡,𝑥, for all 𝑡_𝑛∈ℝ, we have 𝑆_≥𝑡𝑛(𝑢)=𝑆_≤𝑡𝑛(𝑢)=+∞. (147)

Using Proposition 17 for {𝑢(𝑡_𝑛)}, we have that 𝑢(𝑡_𝑛,𝑥+𝑥(𝑡_𝑛)) converges into 𝐿²_𝑥(ℝ²). By Ascoli-Arzela Theorem, we have the following.

Proposition 18. Suppose that the critical mass 𝑚₀ is finite. Then there exists a global solution 𝑢 with mass 𝑚₀, and for every 𝜂>0 there exists 0<𝐶(𝜂)<∞ such that _{|𝑥−𝑥(𝑡)|≥𝐶(𝜂)}|𝑢(𝑡,𝑥)|²𝑑𝑥+_{|𝜉−𝜉(𝑡)|≥𝐶(𝜂)}𝑢(𝑡,𝜉)²𝑑𝜉≤𝜂 (148) for all 𝑡∈ℝ, where the functions 𝑥,𝜉∶ℝ→ℝ².

Proposition 19. The solution described in Proposition 18 does not exist.

Once we proved Proposition 19, we can say that 𝑚₀=∞ and thus Theorem 3 is true. In order to prove Proposition 19, we need the following two lemmas.

Lemma 20 (see [6, Lemma 5.2]). Let u be a global solution of (5). Then one has _ℝ×ℝ2⟨𝑡⟩²𝐺(𝑢)⟨𝑡⟩³+|𝑥|³𝑑𝑥𝑑𝑡≤𝐶(𝐸), (149) where ⟨𝑡⟩=1+|𝑡|².

Lemma 21 (see [6, Lemma 6.2]). Let u be a global solution of (5). Let 𝐵 be a compact subset of ℝ². Then for any 𝑅>0 and 𝑇>0, one has _𝐵(𝑅)|𝑢(𝑇,𝑥)|²𝑑𝑥≥_𝐵|𝑢(0,𝑥)|²𝐶(𝐸)𝑇𝑑𝑥−𝑅, (150) where 𝐵(𝑅)∶={𝑥∈ℝ²∣∃𝑦∈𝐵𝑠.𝑡.|𝑥−𝑦|<𝑅}.

Proof of Proposition 19. By Lemma 21, choosing 𝜂 sufficiently small, _{|𝑥−𝑥(0)|≤𝐶(𝜂)+𝑅|𝑡|}|𝑢(𝑡,𝑥)|²𝑑𝑥≥_{|𝑥−𝑥(0)|≤𝐶(𝜂)}|𝑢(0,𝑥)|²𝐶(𝐸)𝑑𝑥−𝑅≥𝑚₀𝐶(𝐸)−𝜂−𝑅. (151)
By Proposition 18, _{|𝑥−𝑥(𝑡)|≤𝐶(𝜂)}|𝑢(𝑡,𝑥)|²𝑑𝑥≥𝑚₀−𝜂. (152)
For a fixed large number 𝑅, we must have |𝑥(𝑡)−𝑥(0)|≤2𝐶(𝜂)+𝑅|𝑡|. By Lemma 20 and Hölder inequality, ∞>_ℝ×ℝ2⟨𝑡⟩²|𝑢|⁴⟨𝑡⟩³+|𝑥|³≳𝑑𝑥𝑑𝑡_{ℝ|𝑥−𝑥(𝑡)|≤𝐶(𝜂)}⟨𝑡⟩²|𝑢|⁴⟨𝑡⟩³+|𝑥|³≳𝑑𝑥𝑑𝑡_{ℝ|𝑥−𝑥(𝑡)|≤𝐶(𝜂)}|𝑢|⁴⟨𝑡⟩≳+1𝑑𝑥𝑑𝑡_{ℝ|𝑥−𝑥(𝑡)|≤𝐶(𝜂)}|𝑢|²⟨𝑡⟩≳𝑚+1𝑑𝑥𝑑𝑡₀−𝜂_ℝ1⟨𝑡⟩+1𝑑𝑥𝑑𝑡=∞. (153)
This is a contradiction. Proposition 19 was obtained.

Acknowledgment

This work is supported by China Scholarship Council and NNSF of China (no. 11271023).