Final State Problem for the Dirac-Klein-Gordon Equations in Two Space Dimensions

Abstract

We study the fi…nal state problem for the Dirac-Klein-Gordon equations (DKG) in two space dimensions. We prove that if the nonresonance mass condition is satisfi…ed, then the wave operator for DKG is well defi…ned from a neighborhood at the origin in lower order weighted Sobolev space to some Sobolev space.

1. Introduction

We study the final state problem for the Dirac-Klein-Gordon equations (DKG) in two space dimensions: 𝜕_𝑡𝜕+𝛼⋅∇+𝑖𝑀𝛽𝜓=𝜙𝛽𝜓,_2𝑡−Δ+𝑚²𝜙=𝜓^∗()𝛽𝜓,𝑡,𝑥∈ℝ×ℝ², (DKG) where (𝜓,𝜙) is a ℂ²×ℝ-valued unknown function of (𝑡,𝑥),  𝜓=(𝜓₁,𝜓₂)^𝑡 stands a spinor field and 𝜙 denotes a scalar field, 𝑀,𝑚>0 denote masses of the spinor field and the scalar field, respectively, and 𝜓^∗ denotes a transposed conjugate to 𝜓. The operators 𝛼⋅∇ and Δ are defined by 𝛼⋅∇=∑²_𝑗=1𝛼_𝑗𝜕_𝑥𝑗 and Δ=∑²_𝑗=1𝜕²_𝑥𝑗, respectively. Here, 𝛼_𝑗  (𝑗=1,2) and 𝛽 are Dirac matrices, that is, 2 × 2 self-adjoint matrices with constant elements such that 𝛼²_𝑗=𝛽²=𝐼,𝛼_𝑗𝛽+𝛽𝛼_𝑗𝛼=𝑂,for𝑗=1,2,_𝑗𝛼_𝑘+𝛼_𝑘𝛼_𝑗=𝑂,for𝑗,𝑘=1,2,𝑗≠𝑘. (1)

Our aim in the present paper is to show existence of the wave operator for the DKG system (DKG) under the nonresonance mass condition 𝑚≠2𝑀 in two space dimensions.

First, we recall some well-posedness results for (DKG). Many local well-posedness results in low-order Sobolev spaces have been obtained for these ten years (for recent information see, e.g., [1, 2] and references therein). Global well-posedness results in 2d case were also obtained (see, e.g., [3]). Moreover, very recently, unconditional uniqueness in 2d case was discussed in [4, 5]. On the other hand, there are few results about scattering for (DKG) in 2d case.

In [6, 7], the asymptotic behavior of solutions for DKG system was studied in 3d case by reducing it to a nonlinear Klein-Gordon system (KG). Denote 𝒟_±≡𝜕_𝑡±(𝛼⋅∇+𝑖𝑀𝛽). In view of the properties (1), we have 𝒟₋𝒟₊=𝜕²_𝑡−𝛼⋅∇+𝑖𝑀𝛽𝛼⋅∇+𝑖𝑀𝛽=𝜕²_𝑡+⟨∇⟩²_𝑀, (2) where ⟨∇⟩_𝑀≡𝑀²−Δ. Hence, multiplying both sides of the Dirac part by 𝒟₋, we obtain 𝜕²_𝑡+⟨∇⟩²_𝑀𝜓=𝒟₋=𝒟𝜙𝛽𝜓₋𝜙𝛽𝜓−𝑖𝑀𝜙𝐼𝜓+𝜙𝛽𝒟₊𝜓=𝒟₋𝜙𝛽−𝑖𝑀𝜙𝐼+𝜆𝜙²𝐼𝜓, (3) where we have used the fact that 𝜓 is the solution of the DKG system. Thus, the solution of the DKG system satisfies the following KG one: 𝜕²_𝑡+⟨∇⟩²_𝑀𝜓=𝒟₋𝜙𝛽−𝑖𝑀𝜙𝐼+𝜆𝜙²𝐼𝜕𝜓,_2𝑡+⟨∇⟩²_𝑚𝜙=𝜓^∗𝛽𝜓. (4)

If we want to obtain a priori estimates to the local solution for the DKG system, we can use estimates to solutions for the above KG one. Moreover, in the present two-dimensional case, the initial value problem for nonlinear KG systems including (4) was studied in [8] (see also [9]). In [8], Sunagawa proved existence of a unique global asymptotically free solution under the nonresonance mass conditions, if the initial data are sufficiently small, smooth and decay fast at infinity. However, asymptotic behavior of solutions for DKG is not clear because (DKG) is not equivalent to (4) in general. In this paper, we will consider the DKG system itself without reducing it into (4) such as in [10]. Though the initial value problem for DKG was treated in [11], the final value problem which will be discussed in this paper is more delicate because of the derivative loss difficulties.

In [10], the wave operator for the DKG system has been obtained in a three-dimensional case. They dealt with the DKG system itself. Nevertheless, from a point of time decay property for the free solutions of the DKG system, two dimensional-case is critical, that is, borderline case between the long range scattering and the short range one. Therefore, their argument cannot be applicable to the two-dimensional case. To overcome the lack of time decay property, we will use the algebraic normal form transformation developed in paper [8] and the decomposition of the Klein-Gordon operator, that is, 𝜕²_𝑡+⟨∇⟩²_𝑀=𝒟₊𝒟₋. (5)

By this combination, we will find a suitable second approximate solution to 𝜓 (given by (42)). We note that the implicit null structure for (DKG) was discovered in [12], and it was used to prove local well-posedness in low regular setting in [2]. On the other hand, in this paper, by explicit null structure, wave operator for (DKG) will be constructed.

Next, we recall the problem of existence of the wave operator for (DKG). We define the free-Dirac-and Klein-Gordon evolution groups as follows: 𝒱_𝐷(𝑡)≡𝐼cos𝑡⟨∇⟩_𝑀−𝛼⋅∇+𝑖𝑀𝛽⟨∇⟩_𝑀⁻¹sin𝑡⟨∇⟩_𝑀,𝒱_𝐾(𝑡)≡cos⟨∇⟩_𝑚𝑡sin⟨∇⟩_𝑚𝑡−sin⟨∇⟩_𝑚𝑡cos⟨∇⟩_𝑚𝑡. (6) For given final data (𝜓⁺,(⟨∇⟩_𝑚𝜙⁺₁,𝜙⁺₂))∈(𝐗)⁴ with some Banach spaces 𝐗 defined explicitly later, we put 𝜓₀(𝑡)≡𝒱_𝐷(𝑡)𝜓⁺,𝜙₀(𝑡)⟨∇⟩_𝑚⁻¹𝜕_𝑡𝜙₀(𝑡)≡𝒱_𝐾(𝑡)𝜙⁺₁⟨∇⟩_𝑚⁻¹𝜙⁺₂. (7) We will look for a unique time local solution of (DKG) which satisfies the final state conditions as follows: lim_𝑡→∞𝜓(𝑡)−𝜓₀(𝑡)_𝐗=0,lim_𝑡→∞⟨∇⟩_𝑚^1/2𝜙(𝑡)⟨∇⟩_𝑚^−1/2𝜕_𝑡𝜙(𝑡)−⟨∇⟩_𝑚^1/2𝜙₀(𝑡)⟨∇⟩_𝑚^−1/2𝜕_𝑡𝜙₀(𝑡)_𝐗=0, (8)(9) where 𝐗 is also a suitable Banach space. If there exist 𝑇>0 and a unique solution (𝜓,⟨∇⟩_𝑚^1/2𝜙,⟨∇⟩_𝑚^−1/2𝜕_𝑡𝜙)∈(𝐂([𝑇,∞);𝐗))⁴ for (DKG) satisfying (8)-(9), then the wave operator 𝒲⁺ for (DKG) is defined by the mapping as follows: 𝒲⁺∶(𝐗)²×⟨∇⟩⁻¹⟶𝐗𝐗×𝐗₂×⟨∇⟩^−1/2𝐗×⟨∇⟩^1/2𝐗,𝜓(𝑡),𝜙(𝑡),𝜕_𝑡𝜙(𝑡)=𝒲⁺𝜓⁺,𝜙⁺₁,𝜙⁺₂,for𝑡∈[𝑇,∞), (10) where ⟨∇⟩^−𝑠𝐗≡{𝜙;‖⟨∇⟩^𝑠𝜙‖_𝐗<∞}.

2. Several Notations and Main Results

We introduce several notations to state our main results. For 𝑚,𝑘∈ℝ, and 1≤𝑝≤∞, we introduce the weighted Sobolev space as follows: 𝐻_𝑝^𝑚,𝑘=𝜙;𝜙_𝐻𝑝^𝑚,𝑘≡⟨𝑥⟩^𝑘⟨∇⟩^𝑚𝜙_𝐿𝑝<∞, (11)

where ⟨𝑥⟩=(1+|𝑥|²)^1/2, ⟨∇⟩=(1−Δ)^1/2. We also write for simplicity 𝐻^𝑚,𝑘=𝐻₂^𝑚,𝑘, 𝐻^𝑚=𝐻₂^𝑚,0, and  𝐻^𝑚_𝑝=𝐻_𝑝^𝑚,0, and so we usually omit the index 0 and 𝑝=2 if it does not cause a confusion.

We now state our main results in this paper. We introduce the function space as follows: 𝐷_𝑞≡𝐻^4−4/𝑞_{𝑞/𝑞−1}∩𝐻^5/2,1∩𝐻²₁. (12)

Theorem 1. Let 𝑚,𝑀>0,  𝑚≠2𝑀,  4<𝑞≤∞ and (𝜓⁺,(⟨∇⟩𝜙⁺₁,𝜙⁺₂))∈(𝐷_𝑞)⁴. If the norm 𝜌≡‖(𝜓⁺,(⟨∇⟩𝜙⁺₁,𝜙⁺₂))‖_𝐻21 is sufficiently small, then there exist a positive constant 𝑇>0 and a unique solution 𝜓(𝑡),⟨∇⟩_𝑚^1/2𝜙(𝑡)⟨∇⟩_𝑚^−1/2𝜕_𝑡𝜙(𝑡)∈𝐶[𝑇,∞);𝐻^1/24, (13) for the system (DKG). Moreover, there exists a positive constant 𝐶>0 such that the following estimate 𝜓(𝑡)−𝜓₀(𝑡)_𝐻1/2+𝜙(𝑡)⟨∇⟩_𝑚⁻¹𝜕_𝑡𝜙(𝑡)−𝜙₀(𝑡)⟨∇⟩_𝑚⁻¹𝜕_𝑡𝜙₀(𝑡)_𝐻1≤𝐶𝑡^−𝜇 (14)
is true for all 𝑡≥𝑇, where 1/2<𝜇<1−2/𝑞 and (𝜓₀,𝜙₀) is given by (7).

By Theorem 1, we can get existence of the wave operator for (DKG) as follows.

Corollary 2. Let 𝑚,𝑀>0,  𝑚≠2𝑀, and 4≤𝑞<∞. Then the wave operator 𝒲⁺ for (DKG) is well defined from a neighborhood at the origin in the space (𝐷_𝑞)²×(⟨∇⟩⁻¹𝐷_𝑞×𝐷_𝑞) to the space (𝐻^1/2)²×(𝐻¹×𝐿²).

The rest of this paper is organized as follows. In Section 3, we state some basic estimates for free solutions of the DKG system and we introduce “null forms” and state their properties. In Section 4, we decompose two harmful terms by the algebraic normal form transformation and we find a second approximation for 𝜓 through the decomposition of the Klein-Gordon operator by the Dirac one. In Section 5, following paper [10], we will also change the transformed DKG system into another form in order to apply the Strichartz type estimates to the Dirac part. In Section 6, we will prove Theorem 1 by an iteration scheme based on paper [13].

3. Elementary Estimates and Null Forms

Through the paper, we write 𝐴≃𝐵 if there exist some positive constants 𝐶₁,𝐶₂>0 such that 𝐶₁𝐵≤𝐴≤𝐶₂𝐵, and we also write 𝐴≲𝐵 if there exists a positive constant 𝐶>0 such that 𝐴≤𝐶𝐵.

We introduce the free evolution groups as follows: 𝒰_±,𝑚(𝑡)≡𝑒^{±𝑖𝑡⟨∇⟩𝑚}=ℱ⁻¹𝑒^{±𝑖𝑡⟨𝜉⟩𝑚}ℱ. (15) Then, we have the following decomposition: 𝒱_𝐷(𝑡)=_±𝒰_±,𝑀(𝑡)𝒜^𝐷_±, (16) where 𝒜^𝐷_±≡12𝐼±𝑖⟨∇⟩_𝑀⁻¹𝛼⋅∇+𝑖𝑀𝛽 (17) is 0th order matrix operator. We note that for any ℂ²-valued function 𝜓, the following equivalency is valid: 𝒜^𝐷_±𝜓_{𝐻𝑝𝑚,𝑘}≃𝜓_𝐻𝑝^𝑚,𝑘. (18)

Now, we state 𝐿^𝑝−𝐿^𝑞 time decay estimates through the free evolution groups 𝒰_±,𝑚(𝑡) obtained in paper [14].

Lemma 3. Let 𝑚≠0 and 2≤𝑝≤∞. Then the estimate 𝒰_±,𝑚(𝑡)𝜙_𝐿𝑝≲𝑡^2/𝑝−1𝜙_𝐻𝑞^{2(1−2/𝑝)} (19)
is true for any 𝑡>0, where 𝑞 is a conjugate exponent of 𝑝: 1/𝑝+1/𝑞=1.

By the lemma, we can easily get 𝐿^𝑝−𝐿^𝑞 time decay estimates to free solutions for the DKG system.

Corollary 4. Under the same assumption of Lemma 3 and 𝑀>0, the following estimates 𝒱_𝐷(𝑡)𝜓⁺_𝐿𝑝≲𝑡^2/𝑝−1𝜓⁺_𝐻𝑞^{2(1−2/𝑝)},𝒱_𝐾(𝑡)𝜙⁺₁⟨∇⟩_𝑚⁻¹𝜙⁺_2𝐿𝑝≲𝑡^2/𝑝−1𝜙⁺₁⟨∇⟩_𝑚⁻¹𝜙⁺_{2𝐻𝑞2(1−2/𝑝)} (20) are valid for any 𝑡>0, where 𝑞 is a conjugate exponent of 𝑝: 1/𝑝+1/𝑞=1.

Remark 5. Let 𝜅∈ℝ,  𝑀,𝑚≠0, and 2≤𝑝<∞. Then the following estimates 𝒱_𝐷(𝑡)𝜓⁺_𝐻𝜅𝑝≲𝑡^2/𝑝−1𝜓⁺_{𝐻𝜅+2−4/𝑝,1},𝒱_𝐾(𝑡)𝜙⁺₁⟨∇⟩_𝑚⁻¹𝜙⁺_{2𝐻𝜅𝑝}≲𝑡^2/𝑝−1𝜙⁺₁⟨∇⟩_𝑚⁻¹𝜙⁺_{2𝐻𝜅+2−4/𝑝,1} (21)
hold for any 𝑡>0.

Next, we introduce the Strichartz estimates, which enable us to treat the problem in lower order Sobolev spaces. Denote the space-time norm 𝜙_{𝐿𝑟𝑡(𝐼;𝐿}^𝑞_𝑥)≡𝜙(𝑡)_𝐿^𝑞_{𝑥𝐿𝑟𝑡(𝐼)}, (22) where 𝐼 is a bounded or unbounded time interval. We define the integral operator as follows: 𝒢_±,𝑚𝑔(𝑡)≡^𝑡_𝑇𝒰_±,𝑚(𝑡−𝜏)⟨∇⟩_𝑚⁻¹𝑔(𝜏)𝑑𝜏 (23) for any 𝑇∈𝐼, where 𝑚>0. By the duality argument of [15] along with Lemma 3, we have the following (see also [10, 13]).

Lemma 6. Let 2≤𝑞<∞ and 2/𝑟=1−(2/𝑞). Then for any time interval 𝐼, the following estimates are true: 𝒢_±,𝑚𝑔_{𝐿𝑟𝑡(𝐼;𝐿}^𝑞_𝑥)≲𝑔_{𝐿𝑟′𝑡(𝐼;𝐻𝑞′2𝛾−1)},𝒢_±,𝑚𝑔_{𝐿∞𝑡(𝐼;𝐿2𝑥)}≲𝑔_{𝐿𝑟′𝑡(𝐼;𝐻𝑞′𝛾−1)},𝒢_±,𝑚𝑔_{𝐿𝑟𝑡(𝐼;𝐿}^𝑞_𝑥)≲𝑔_{𝐿1𝑡(𝐼;𝐻𝛾−1)}, (24)
where 𝑟^′=𝑟/(𝑟−1),  𝑞^′=𝑞/(𝑞−1) and 𝛾=1−(2/𝑞).

Next, we introduce the Leibniz rule for fractional derivatives.

Lemma 7. Let 𝜅>0,  1<𝑝,𝑞₁,𝑞₂<∞,  1<𝑟₁,𝑟₂≤∞, and 1/𝑝=1/𝑞₁+1/𝑟₁=1/𝑞₂+1/𝑟₂. Then the following estimate holds: ‖𝑢𝑣‖_𝐻^𝜅_𝑝≲‖𝑢‖_𝐻^𝜅_𝑞1‖𝑣‖_𝐿^𝑟₁+‖𝑣‖_𝐻^𝜅_𝑞2‖𝑢‖_𝐿^𝑟₂. (25)

For the proof of (25) see, for example, [16].

We introduce the operator 𝒵=(𝒵₁,𝒵₂), where 𝒵_𝑘≡𝑥_𝑘𝜕_𝑡+𝑡𝜕_𝑘 for 𝑘=1,2. Let 𝒵^𝛼=𝒵^𝛼1₁𝒵^𝛼2₂ for a multi-index 𝛼=(𝛼₁,𝛼₂)∈(ℕ∪{0})². We can see the commutation relations (see [6, 17]) as follows: 𝒟₊,𝒵_𝑘−12𝛼_𝑘=𝛼_𝑘𝒟₊,𝜕²_𝑡−Δ+𝑚²,𝒵_𝑘=0, (26)

for 𝑘=1,2, where [𝐴,𝐵]≡𝐴𝐵−𝐵𝐴.

We introduce the quadratic null forms as follows: 𝒬₀𝑓,𝑔≡𝜕_𝑡𝑓𝜕_𝑡𝑔−⋅,𝒬∇𝑓∇𝑔_𝑗,𝑘𝑓,𝑔≡𝜕_𝑗𝑓𝜕_𝑘𝑔−𝜕_𝑘𝑓𝜕_𝑗𝑔, (27) for 0≤𝑗<𝑘≤2, where 𝜕≡(𝜕₀,∇)≡(𝑖𝜕_𝑡,𝜕₁,𝜕₂). In particular, 𝒬_𝑗,𝑘 is called a strong null form and has an additional time decay property through the operator 𝒵_𝑘, obtained in [18] (see also [8, 13, 19], etc.).

Lemma 8. Let 𝑗,𝑘=1,2. Then, for any smooth function 𝑓,𝑔, the identities 𝒬_0,𝑗𝑓,𝑔=𝑡⁻¹𝜕₀𝑓𝒵_𝑗𝑔−𝑡⁻¹𝒵_𝑗𝑓𝜕₀𝑔,𝒬_𝑗,𝑘𝑓,𝑔=𝑡⁻²𝒵_𝑗𝑔𝒵_𝑘𝑓−𝑡⁻²𝒵_𝑗𝑓𝒵_𝑘𝑔+𝑡⁻¹𝜕_𝑗𝑓𝒵_𝑘𝑔−𝑡⁻¹𝜕_𝑗𝑔𝒵_𝑘𝑓+𝑡⁻¹𝒵_𝑗𝑓𝜕_𝑘𝑔−𝑡⁻¹𝒵_𝑗𝑔𝜕_𝑘𝑓 (28) are valid for any 𝑡∈ℝ⧵{0}.

4. Decomposition of Critical Terms

We study a structure of some harmful terms of  (DKG). By the difference of  (DKG) and the free DKG system, it follows that 𝒟₊𝜓−𝜓₀=𝜙−𝜙₀𝛽𝜓+𝜙₀𝛽𝜓−𝜓₀+𝜙₀𝛽𝜓₀,□+𝑚²𝜙−𝜙₀=𝜓−𝜓₀^∗𝛽𝜓+𝜓^∗₀𝛽𝜓−𝜓₀+𝜓^∗₀𝛽𝜓₀, (29) where □=𝜕²_𝑡−Δ. The last two terms 𝜙₀𝛽𝜓₀ and  𝜓^∗₀𝛽𝜓₀ are critical, both of which have the worst time decay property. Especially, since 𝜙₀𝛽𝜓₀,𝜓^∗₀𝛽𝜓₀=𝑂𝑡⁻¹in𝐿²as𝑡⟶+∞ (30)

(see Corollary 4), the 𝐿²-norm of these terms is not integrable with respect to time 𝑡 over [1,∞). Therefore, it can not be expected that usual perturbation technique is applicable to (29). To overcome this lack of time decay property, we will decompose them into an image of a Klein-Gordon operator and a remainder term following paper [8], based on papers [19–21].

Let (𝑣₁,𝑣₂) be a solution for the following homogeneous KG system with masses 𝑀₁,𝑀₂>0, □+𝑀²_𝑗𝑣_𝑗=0,(𝑡,𝑥)∈ℝ×ℝ²,for𝑗=1,2. (31) By the masses 𝑀₁,  𝑀₂, we introduce the symmetric matrix as follows: ℳ=ℳ𝑀₁,𝑀₂=𝑀²₁+𝑀²₂2𝑀₁𝑀₂2𝑀₁𝑀₂𝑀²₁+𝑀²₂. (32) We have the following.

Lemma 9 (see [8]). Let 𝑚>0 with det(𝑚²𝐼−ℳ)≠0. Then the quadratic term 𝑣₁𝑣₂ can be decomposed as 𝑣₁𝑣₂=1det𝑚²𝑚𝐼−ℳ□+²𝑓−4ℛ, (33) where 𝑓=𝑓𝑣₁,𝑣₂≡−𝑀²₁−𝑀²₂+𝑚²𝑣₁𝑣₂−2𝒬₀𝑣₁,𝑣₂,ℛ=ℛ𝑣₁,𝑣₂≡²_𝑚=1𝒬_0,𝑚𝜕_𝑡𝑣₁,𝜕_𝑚𝑣₂+²_𝑚=1𝒬_0,𝑚𝜕_𝑡𝑣₂,𝜕_𝑚𝑣₁−𝒬_1,2𝜕₁𝑣₁,𝜕₂𝑣₂−𝒬_2,1𝜕₂𝑣₁,𝜕₁𝑣₂. (34)

Under the nonresonance mass condition 𝑚,𝑀>0, and 𝑚≠2𝑀, we can apply Lemma 9 to the critical terms 𝜙₀𝛽𝜓₀ and  𝜓^∗₀𝛽𝜓₀. Before doing so, we prepare for several notations. We put 1ℳ≡𝑚²(2𝑀+𝑚)(𝑚−2𝑀) (35) which is well defined if 𝑚,𝑀>0 and 𝑚≠2𝑀. For a real-valued function 𝜙 and a ℂ²-valued function 𝜓=(𝜓₁,𝜓₂)^𝑡, we define ℂ²-valued functions of bilinear form: 𝑓_𝐷=𝑓_𝐷𝜙,𝜓≡𝑓𝜙,𝜓₁,𝑓𝜙,𝜓₂^𝑡,ℛ_𝐷=ℛ_𝐷𝜙,𝜓≡ℛ𝜙,𝜓₁,ℛ𝜙,𝜓₂^𝑡,𝒬^𝐷₀=𝒬^𝐷₀𝜙,𝜓≡𝒬₀𝜙,𝜓₁,𝒬₀𝜙,𝜓₂^𝑡, (36) Moreover, for 𝐂²-valued functions 𝜑=(𝜑₁,𝜑₂)^𝑡,  𝜓=(𝜓₁,𝜓₂)^𝑡, we put the following bilinear forms: 𝑓_𝐾=𝑓_𝐾𝜑^𝑡,𝜓≡²_𝑗=1𝑓𝜑_𝑗,𝜓_𝑗,ℛ_𝐾=ℛ_𝐾𝜑^𝑡,𝜓≡²_𝑗=1ℛ𝜑_𝑗,𝜓_𝑗,𝒬^𝐾₀=𝒬^𝐾₀𝜑^𝑡,𝜓≡²_𝑗=1𝒬₀𝜑_𝑗,𝜓_𝑗. (37) We have the following.

Corollary 10. Let 𝑚,𝑀>0,  𝑚≠2𝑀, and (𝜓₀,𝜙₀) be a free solution for the Dirac-Klein-Gordon equations. Then the quadratic terms 𝜙₀𝛽𝜓₀,  𝜓^∗₀𝛽𝜓₀ can be expressed as 𝜙₀𝛽𝜓₀=ℳ□+𝑀²𝑓_𝐷𝜙₀,𝛽𝜓₀−4ℛ_𝐷𝜙₀,𝛽𝜓₀,𝜓^∗₀𝛽𝜓₀=ℳ□+𝑚²𝑓_𝐾𝜓^∗₀,𝛽𝜓₀−4ℛ_𝐾𝜓^∗₀,𝛽𝜓₀. (38)

Proof. We consider the Dirac part of (38). Multiplying by 𝒟₋ both hand sides of 𝒟₊𝜓₀=0, we get 𝒟₋𝒟₊𝜓₀=□+𝑀²𝜓₀=0, (39) which implies that 𝜓₀=(𝜓_0,1,𝜓_0,2)^𝑡 is also a solution of the free KG equation. Note that by the condition 𝑚,𝑀>0 and 𝑚≠2𝑀, we can apply Lemma 9 with 𝑚=𝑀,  𝑣₁=𝜙₀, and 𝑣₂=𝜓_0,𝑘 to get, for 𝑘=1,2, 𝜙₀𝜓_0,𝑘=ℳ□+𝑀²𝑓𝜙₀,𝜓_0,𝑘−4ℛ𝜙₀,𝜓_0,𝑘. (40) Thus, by a simple calculation, we obtain (38). Next, note that from equality (39), we see that 𝜓₀ satisfies the free KG equation. Thus in the same manner as the proof of the Dirac part, we can prove the KG part, which completes the proof of the corollary.

Next, we will change the DKG equations into another form without critical nonlinearities. We introduce a new unknown function (Ψ,Φ) as follows: Ψ≡𝜓−𝜓₀−𝑓_𝐷≡𝜓−𝑓_𝐷,Φ≡𝜙−𝜙₀−𝑓_𝐾≡𝜙−𝑓_𝐾, (41) where (𝜓₀,𝜙₀) is defined by (7) and 𝑓_𝐷=𝑓_𝐷𝜙₀,𝜓₀≡ℳ𝒟₋𝑓_𝐷𝜙₀,𝛽𝜓₀=ℳ𝑓_𝐷𝒟₋𝜙₀,𝛽𝜓₀−𝑖𝑀𝑓_𝐷𝜙₀,𝜓₀,𝑓_𝐾=𝑓_𝐾𝜓₀≡ℳ𝑓_𝐾𝜓^∗₀,𝛽𝜓₀ (42)(43)

are the second approximate solution to (𝜓,𝜙), where we have used the identities 𝛼_𝑗𝛽+𝛽𝛼_𝑗=𝑂,  𝛽²=𝐼 and 𝒟₊𝜓₀=0 to obtain the third equality in (42).

Here, we remember that by the anticommutation relations (1) of the Dirac matrices, we can decompose the KG operator as follows: □+𝑀²=𝒟₊𝒟₋. (44) By combining Corollary 10 and this decomposition, we can rewrite (DKG) as follows.

Lemma 11. Let 𝑚,𝑀>0 and 𝑚≠2𝑀. Then (𝜓,𝜙) satisfies (DKG) if and only if the new variable (Ψ,Φ) defined by (41) is a solution of 𝒟₊Ψ=𝐹,□+𝑚²()Φ=𝐺,𝑡,𝑥∈ℝ×ℝ², (45) where 𝜓≡𝐹=𝐹𝜙,𝜙𝛽𝜓+𝜙𝛽𝜓₀+𝜙₀𝛽𝜓−4ℳℛ_𝐷𝜙₀,𝛽𝜓₀,𝜓≡𝜓𝐺=𝐺_∗𝛽𝜓+𝜓^∗𝛽𝜓₀+𝜓^∗₀𝛽𝜓−4ℳℛ_𝐾𝜓^∗₀,𝛽𝜓₀, (46)
and ℳ,  ℛ_𝐷 and ℛ_𝐾 are defined by (35), (36), and (37), respectively.

This lemma enables us to treat the Dirac-Klein-Gordon equations (DKG) as well as the reduced KG system (4) in two space dimensions.

Proof. From (29), we see that (𝜓,𝜙) is a solution of (DKG) if and only if the new variable (𝜓,𝜙) satisfies the following DKG equations: 𝒟₊𝜓=𝜙𝛽𝜓+𝜙𝛽𝜓₀+𝜙₀𝛽𝜓+𝜙₀𝛽𝜓₀,□+𝑚²𝜙=𝜓^∗𝛽𝜓+𝜓^∗𝛽𝜓₀+𝜓^∗₀𝛽𝜓+𝜓^∗₀𝛽𝜓₀. (47) We consider the Dirac part of (47) only, since it is easier to handle the KG part. Note that by the assumption 𝑚,𝑀>0 and 𝑚≠2𝑀, we can apply Corollary 10 to 𝜙₀𝛽𝜓₀. Thus, we have 𝜙₀𝛽𝜓₀=ℳ□+𝑀²𝑓_𝐷𝜙₀,𝛽𝜓₀−4ℛ_𝐷𝜙₀,𝛽𝜓₀. (48) Moreover, by the decomposition (44), we can transform the first term of the right hand side of (48) as follows: 𝜆ℳ□+𝑀²𝑓_𝐷𝜙₀,𝛽𝜓₀=ℳ𝒟₊𝒟₋𝑓_𝐷𝜙₀,𝛽𝜓₀=𝒟₊𝑓_𝐷, (49) where we have used the definition of 𝑓_𝐷 given by (42). Inserting (48) and (49) into the Dirac part of (47), we obtain the Dirac part of (45), which completes the proof of the lemma.

Remark 12. The null structure of (DKG) was characterized in [12] by using Fourier space. On the other hand, we note that in the above argument, Fourier space does not appear at all.

5. Reduction to Some First Order System

To construct a solution for the final value problem of the DKG system, we will use the Strichartz type estimates (Lemma 6). However, it seems difficult to apply these estimates to the Dirac part for (45) due to a derivative loss difficulty. To gain first order differentiability properties of nonlinear term, we use the matrix operators ℬ^𝐷_±≡12𝐼1∓𝑖⟨∇⟩_𝑀⁻¹𝜕_𝑡𝑖=∓2⟨∇⟩_𝑀⁻¹ℒ^𝐷_∓,ℒ^𝐷_±≡𝜕_𝑡∓𝑖⟨∇⟩_𝑚𝐼, (50) though we do not necessarily need the operator ℬ in dealing with the initial value problem for the DKG system (see [11]). We will construct the desired solution (𝜓,𝜙) for the DKG system by the iteration scheme. Let {(𝜓^𝑙,𝜙^𝑙)}_𝑙≥0 be a sequence such that 𝒟₊𝜓^𝑙+1=𝜙^𝑙𝛽𝜓^𝑙,□+𝑚²𝜙^𝑙+1=𝜓^𝑙∗𝛽𝜓^𝑙,𝜓𝑙≥0,₀,𝜙⁰=𝜓₀,𝜙₀, (51) under the final conditions lim_𝑡→∞𝜓^𝑙(𝑡)−𝜓₀(𝑡)_𝐻1/2=0,lim_𝑡→∞𝜙^𝑙(𝑡)⟨∇⟩_𝑚⁻¹𝜕_𝑡𝜙^𝑙(𝑡)−𝜙₀(𝑡)⟨∇⟩_𝑚⁻¹𝜕_𝑡𝜙₀(𝑡)_𝐻1=0, (52)(53)

for 𝑙≥0, where (𝜓₀,𝜙₀) is given by (7). It suffices to prove that the sequence {𝜓^𝑙,(⟨∇⟩_𝑚^1/2𝜙^𝑙,⟨∇⟩_𝑚^−1/2𝜕_𝑡𝜙^𝑙)}_𝑙≥0 is a Cauchy one in the Banach space (𝐶([𝑇,∞);𝐻^1/2))⁴ for some 𝑇>0.

As the previous section, we introduce the new sequence {(Ψ^𝑙,Φ^𝑙)} as follows: Ψ^𝑙≡𝜓^𝑙−𝜓₀−𝑓_𝐷≡𝜓^𝑙−𝑓_𝐷,Φ^𝑙≡𝜙^𝑙−𝜙₀−𝑓_𝐾≡𝜙^𝑙−𝑓_𝐾. (54) By Lemma 11, the sequence {(𝜓^𝑙,𝜙^𝑙)} is a solution of (51) if and only if the new one {(Ψ^𝑙,Φ^𝑙)} satisfies the transformed DKG equations as follows: 𝒟₊Ψ^𝑙+1=𝐹^𝑙,□+𝑚²Φ^𝑙+1=𝐺^𝑙,𝑙≥1,Ψ⁰,Φ⁰=−𝑓_𝐷,𝑓_𝐾, (55)(56) where 𝐹^𝑙≡𝐹𝜙^𝑙,𝜓^𝑙,𝐺^𝑙≡𝐺𝜓^𝑙, (57)

for 𝑙≥0 (𝑓_𝐷,  𝑓_𝐾, and  𝐹 and 𝐺 are defined by (42)-(43) and (46), resp.).

By the decomposition of the Klein-Gordon operator by the Dirac operator, we have ℒ^𝐷_±ℬ^𝐷_±𝑖=∓2⟨∇⟩_𝑀⁻¹𝐼𝜕²_𝑡+⟨∇⟩²_𝑀𝑖=∓2⟨∇⟩_𝑀⁻¹𝒟₋𝒟₊. (58) Thus, from the Dirac part for (55), we can deduce the following: ℒ^𝐷_±ℬ^𝐷_±Ψ^𝑙+1𝑖=∓2⟨∇⟩_𝑀⁻¹𝒟₋𝒟₊Ψ^𝑙+1=⟨∇⟩_𝑀⁻¹𝐹^𝑙_±, (59) for 𝑙≥0, where 𝐹^𝑙_±≡∓(𝑖/2)𝒟₋𝐹^𝑙. Therefore, from Dirac part of (55), we have ℒ^𝐷_±ℬ^𝐷_±Ψ^𝑙+1=⟨∇⟩_𝑀⁻¹𝐹^𝑙_±ℬ,𝑙≥0,_𝐷±Ψ⁰=−ℬ^𝐷_±𝑓_𝐷. (60)

Remark 13. By properties (1) of the Dirac matrices, we can transform 𝐹^𝑙_± into another form without any derivatives of 𝜓 or the free solution 𝜓₀ (see (78)-(79), precisely). This fact enables us to use the Strichartz estimates for (60).

Next we will also transform the KG part of (55) as in [10, 13]. We also use the operator (1-component version of the Dirac part) as follows: ℬ^𝐾_±≡121∓𝑖⟨∇⟩_𝑚⁻¹𝜕_𝑡,ℒ^𝐾_±≡𝜕_𝑡∓𝑖⟨∇⟩_𝑚. (61) We can see that the sequence {Φ^𝑙} is a solution of the KG part for (55) if and only if the sequence {ℬ^𝐾_±Φ^𝑙} satisfies ℒ^𝐾_±ℬ^𝐾_±Φ^𝑙+1=⟨∇⟩_𝑚⁻¹𝐺^𝑙_±ℬ,for𝑙≥0,_𝐾±Φ⁰=−ℬ^𝐾_±𝑓_𝐾, (62)

where 𝐺^𝑙_±≡𝐺^𝑙_±(𝜓^𝑙)≡∓(𝑖/2)𝐺^𝑙.

Therefore, by (60) and (62), we get ℒ^𝐷_±ℬ^𝐷_±Ψ^𝑙+1=⟨∇⟩_𝑀⁻¹𝐹^𝑙_±,ℒ^𝐾_±ℬ^𝐾_±Φ^𝑙+1=⟨∇⟩_𝑚⁻¹𝐺^𝑙_±,for𝑙≥0,ℬ^𝐷_±Ψ⁰,ℬ^𝐾_±Φ⁰=−ℬ^𝐷_±𝑓_𝐷,ℬ^𝐾_±𝑓_𝐾. (63)(64)

Remark 14. The identity ∑_±ℬ^∗_±=𝐼 holds, which enables us to reconstruct a solution (Ψ,Φ) for (45) from (ℬ^𝐷_±Ψ,ℬ^𝐾_±Φ).

Inserting the identities 𝜓^𝑙=_±ℬ^𝐷_±Ψ^𝑙+𝑓_𝐷,𝜙^𝑙=_±ℬ^𝐾_±Φ^𝑙+𝑓_𝐾, (65)

into the nonlinearities 𝐹^𝑙_±,  𝐺^𝑙_±, we can express (63) by the new variable (ℬ^𝐷_±Ψ^𝑙,ℬ^𝐾_±Φ^𝑙) only without (𝜙^𝑙,𝜓^𝑙).

At the end of this section, we will lead the integral equations associated with (63). We introduce a new unknown function sequence {𝑣^𝑙} whose components are defined by 𝑣^𝑙≡ℬ^𝐷₊Ψ^𝑙,ℬ^𝐷₋Ψ^𝑙,⟨∇⟩_𝑚^1/2ℬ^𝐾₊Φ^𝑙,⟨∇⟩_𝑚^1/2ℬ^𝐾₋Φ^𝑙𝑡, (66) a nonlinear term 𝒩=𝒩𝑣^𝑙≡⟨∇⟩_𝑀⁻¹𝐹^𝑙₊,⟨∇⟩_𝑀⁻¹𝐹^𝑙₋,⟨∇⟩_𝑚^−1/2𝐺^𝑙₊,⟨∇⟩_𝑚^−1/2𝐺^𝑙₋^𝑡 (67) for 𝑙≥0, and a matrix-operator ℒ ≡diag(ℒ^𝐷₊,ℒ^𝐷₋,ℒ^𝐾₊,ℒ^𝐾₋). Then by using these notations, (63) can be simplified as ℒ𝑣^𝑙+1=𝒩𝑣^𝑙for𝑙≥0. (68) To lead the integral equations for (68), we need to study the asymptotic behavior of the new variable 𝑣^𝑙. We can obtain the following.

Lemma 15. Let (𝜓⁺,(⟨∇⟩𝜙⁺₁,𝜙⁺₂))∈(𝐻^5/2,1)⁴. The function (𝜓^𝑙,𝜙^𝑙) defined by (51) satisfies (52)-(53) for any 𝑙≥0 if and only if the new function 𝑣^𝑙 satisfies (68) and lim_𝑡→∞𝑣^𝑙_𝐻1/2=0,for𝑙≥0. (69)

The proof of the lemma will be given in Appendix.

We introduce a matrix evolution operator as follows: 𝒰(𝑡)≡diag𝒰_+,𝑀(𝑡),𝒰_−,𝑀(𝑡),𝒰_+,𝑚(𝑡),𝒰_−,𝑚(𝑡). (70) From Lemma 15, we can lead the integral equations associated with (68) as follows: 𝑣^𝑙+1(𝑡)=−^∞_𝑡𝒰(𝑡−𝑠)𝒩𝑣^𝑙𝑑𝑠. (71)

6. Proof of Theorem 1

In this section, we give a proof of Theorem 1. Note that the identities 𝜕_𝑡Ψ^𝑙=𝑖⟨∇⟩_𝑀𝑣^𝑙₁−𝑣^𝑙₂,𝜕_𝑡Φ^𝑙=𝑖⟨∇⟩_𝑚^1/2𝑣^𝑙₃−𝑣^𝑙₄ (72)

hold; the nonlinearity 𝒩(𝑣^𝑙) can be expressed in terms of the space derivatives of 𝑣^𝑙 (so excluding the time derivatives).

For 𝑇>1, where 𝑇 is sufficiently large, we introduce the following function space: 𝐗_𝑇=𝑣∈𝐶[𝑇,∞);𝐻^1/26;‖𝑣‖_𝐗𝑇<∞, (73) with the norm ‖𝑣‖_𝐗𝑇≡sup_{[)𝑡∈𝑇,∞}𝑡^𝜇‖𝑣‖_𝐿⁴_{𝑡(𝐼;𝐿}⁴₎+‖𝑣‖_𝐿^∞_{𝑡(𝐼;𝐻}^1/2₎, (74) where 1/2<𝜇<1−2/𝑞,  4<𝑞≤∞, and 𝐼=[𝑡,∞). We define 𝐴≡𝐶𝜓⁺,⟨∇⟩𝜙⁺₁,𝜙⁺_2𝐻^4−4/𝑞_{𝑞/(𝑞−1)∩𝐻5/2,1}. (75)

In order to obtain the theorem, we will show that the sequence {𝑣^𝑙} is a Cauchy one in a closed ball 𝐗_𝑇,𝐴 for appropriate 𝑇 and 𝜌, where 𝐗_𝑇,𝐴≡{𝑣∈𝐗_𝑇;‖𝑣‖_𝐗𝑇≤𝐴}.

Hereafter, we will use the notation 𝐿^𝑟_𝑡𝑋=𝐿^𝑟_𝑡(𝐼;𝑋),  𝒟=𝒟₋ and ℬΨ=ℬ^𝐷_±Ψ,ℬΦ=ℬ^𝐾_±Φ, (76) for simplicity if it does not cause a confusion.

Proof. We will prove that 𝑣^𝑙∈𝐗_𝑇,𝐴 for any 𝑙≥0 by induction. In the case of  𝑙=0, it is easy to see that 𝑣⁰∈𝐗_𝑇,𝐴 for some 𝑇 and 𝜌. We omit the details. For 𝑙≥1, we assume that 𝑣^𝑘∈𝐗_𝑇,𝐴 for 0≤𝑘≤𝑙. We will show that 𝑣^𝑙+1∈𝐗_𝑇,𝐴 for some 𝑇 and 𝜌.
First, by the identities 𝒟₊𝜓₀=0 and 𝒟₊𝜓^𝑙=𝜆𝜙^𝑙−1𝛽𝜓^𝑙−1 for 𝑙≥1, we get, for 𝑙≥1, 𝒟₋𝜙^𝑙𝛽𝜓^𝑙=𝒟₋𝜙^𝑙𝛽𝜓^𝑙−𝑖𝑀𝜙^𝑙𝐼𝜓^𝑙+𝜆𝜙^𝑙𝜙^𝑙−1𝐼𝜓^𝑙−1+𝜆𝜙^𝑙𝜙^𝑙−1𝐼𝜓₀+𝜆𝜙^𝑙𝜙₀𝐼𝜓^𝑙−1+𝜆𝜙^𝑙𝜙₀𝐼𝜓₀,𝒟₋ℛ_𝐷𝜙₀,𝛽𝜓₀=ℛ_𝐷𝒟₋𝜙₀,𝛽𝜓₀−𝑖𝑀ℛ_𝐷𝜙₀,𝛽𝜓₀. (77) From these identities, we can express 𝐹^𝑙_± as follows: 𝐹^𝑙_±𝑖=∓2³_𝑗=1𝐹^𝑙_𝑗+“remainder”for𝑙≥1, (78) where 𝐹^𝑙₁≡𝒟₋𝜙^𝑙𝛽𝜓^𝑙,𝐹^𝑙₂≡𝒟₋𝜙₀𝛽𝜓^𝑙+𝒟₋𝜙^𝑙𝛽𝜓₀,𝐹^𝑙₃≡4𝑖ℳℛ_𝐷𝒟₋𝜙₀,𝛽𝜓₀. (79)(80) Here, we note that “remainder” (given by (78)) can be handled in the same manner as 𝐹^𝑙_𝑗(𝑗=1,2,or3). Thus, we will omit the estimate of them. We also decompose 𝐺^𝑙_± as 𝐺^𝑙_±=∓(𝑖/2)∑³_𝑗=1𝐺^𝑙_𝑗, where 𝐺^𝑙₁=𝜓^𝑙∗𝛽𝜓^𝑙,𝐺^𝑙₂=𝜓^𝑙∗𝛽𝜓₀+𝜓^∗₀𝛽𝜓^𝑙,𝐺^𝑙₃=4𝑖ℳℛ_𝐾𝜓^∗₀,𝛽𝜓₀. (81) Taking 𝐿⁴_𝑡𝐿⁴_𝑥-norm and 𝐿^∞_𝑡𝐻^1/2-norm of (71) and applying Lemma 6 with (𝑞,𝑟,𝛾)=(4,4,1/2) and (2,∞,0), we have 𝑣^𝑙+1_{𝐿4𝑡𝐿4𝑥}+𝑣^𝑙+1_{𝐿∞𝑡𝐻1/2}≲𝐹^𝑙_{1𝐿𝑡4/3𝐿𝑥4/3}+𝐺^𝑙_{1𝐿𝑡4/3𝐻1/24/3}+_𝑗=2,3𝐹^𝑙_{𝑗𝐿1𝑡𝐻−1/2}+𝐺^𝑙_{𝑗𝐿1𝑡𝐿2𝑥}. (82)
Moreover, we remember that (𝜙^𝑙,𝜓^𝑙) is expressed as (65).
Now, we will estimate 𝐹^𝑙₁. By the Hölder inequality, we have 𝒟ℬΦ^𝑙ℬΨ^𝑙_{𝐿𝑡4/3𝐿𝑥4/3}≲ℬΦ^𝑙(𝑠)_𝐻1ℬΨ^𝑙(𝑠)_{𝐿4𝑥𝐿𝑡4/3}≲𝐴𝑠^−𝜇ℬΨ^𝑙(𝑠)_{𝐿4𝑥𝐿𝑡4/3}≤𝐴ℬΨ^𝑙_{𝐿4𝑡𝐿4𝑥}𝑠^−𝜇_{𝐿2𝑡(𝐼)}≲𝐴²𝑡^1/2−2𝜇, (83) for any 𝑡≥𝑇 since 𝑣^𝑘∈𝐗_𝑇,𝐴 for 0≤𝑘≤𝑙. By the Hölder inequality and Remark 5 with 𝑝=8, we obtain 𝑓_𝐷(𝑠)_𝐿4≲𝜙₀(𝑠)_𝐻28𝜓₀(𝑠)_𝐻18≲𝐴²𝑠^−3/2, (84) for any 𝑠≥𝑡. In the same manner as the proof of the estimate (83), we also obtain 𝒟ℬΦ^𝑙𝑓_{𝐷𝐿𝑡4/3𝐿𝑥4/3}≲𝐴³𝑡^{−3/4−𝜇}, (85) for all 𝑡≥𝑇, due to 𝑣^𝑘∈𝐗_𝑇,𝐴 for 0≤𝑘≤𝑙 and (84). By the Hölder inequality and Remark 5 with 𝑝=8/3,8, we obtain 𝒟𝑓_𝐾(𝑠)_𝐿2≲𝜓₀(𝑠)_𝐻28/3𝜓₀(𝑠)_𝐻18≲𝑠⁻¹𝜓⁺_𝐻^5/2_8/5𝜓⁺_𝐻^5/2_8/7≲𝐴²𝑠⁻¹ (86) for any 𝑠≥𝑡, where we have used properties (1) of 𝛼,𝛽, and 𝒟₊𝜓₀=0. Thus, in the same manner as the proof of the estimate (83), we obtain 𝒟𝑓_𝐾ℬΨ^𝑙_{𝐿𝑡4/3𝐿𝑥4/3}≲𝐴³𝑡^{−1/2−𝜇}, (87) for all 𝑡≥𝑇 due to 𝑣^𝑙∈𝐗_𝑇,𝐴 and (86). By the Hölder inequality and estimates (84) and (86), we get 𝒟𝑓_𝐾𝑓_{𝐷𝐿𝑡4/3𝐿𝑥4/3}≲𝒟𝑓_𝐾(𝑠)_𝐿2𝑥𝑓_𝐷(𝑠)_{𝐿4𝑥𝐿𝑡4/3(𝐼)}≲𝐴⁴𝑡^−7/4, (88) for all 𝑡≥𝑇. Thus by combining (83), (85), and (87)-(88), we obtain 𝐹^𝑙_{1𝐿𝑡4/3𝐿𝑥4/3}≲𝐴²𝑡^1/2−2𝜇, (89) for 𝑡≥𝑇≥1 since 𝜇<1. Next, we consider 𝐹^𝑙₂. We have 𝐹^𝑙_{2𝐿1𝑡𝐻−1/2}≤𝒟𝜙^𝑙𝜓_{0𝐿1𝑡𝐿2𝑥}+𝒟𝜙₀𝜓^𝑙_{𝐿1𝑡𝐿2𝑥}. (90) By Corollary 4 with 𝑝=∞, we have 𝒟ℬΦ^𝑙𝜓_{0𝐿1𝑡𝐿2𝑥}≲ℬΦ^𝑙(𝑠)_𝐻1𝜓₀(𝑠)_{𝐿∞𝑥𝐿1𝑡}≲𝜌𝐴𝑡^−𝜇, (91) for all 𝑡≥𝑇 since 𝑣^𝑘∈𝐗_𝑇,𝐴 for 0≤𝑘≤𝑙. In the same manner as the estimate (91), we get 𝒟𝑓_𝐾𝜓_{0𝐿1𝑡𝐿2𝑥}≲𝒟𝑓_𝐾(𝑠)_𝐿2𝑥𝜓₀(𝑠)_{𝐿∞𝑥𝐿1𝑡}≤𝜌𝐴²𝑡⁻¹, (92) for any 𝑡≥𝑇, where we have used the estimate (86). Moreover, we also have 𝒟𝜙₀ℬΨ^𝑙_{𝐿1𝑡𝐿2𝑥}≲𝜙₀(𝑠)_𝐻1∞ℬΨ^𝑙(𝑠)_{𝐿2𝑥𝐿1𝑡}≤𝐴𝜌𝑡^−𝜇, (93) for all 𝑡≥𝑇 since 𝑣^𝑙∈𝐗_𝑇,𝐴. In the same proof as the estimate (84), by the Hölder inequality and Remark 5 with 𝑝=4, we get 𝑓_𝐷(𝑠)_𝐿2≲𝜙₀(𝑠)_𝐻24𝜓₀(𝑠)_𝐻14≲𝐴²𝑠⁻¹, (94) for any 𝑠≥𝑡. By estimate (94) and Corollary 4 with 𝑝=∞, we have 𝒟𝜙₀𝑓_{𝐷𝐿1𝑡𝐿2𝑥}≲𝜙₀(𝑠)_𝐻1∞𝑓_𝐷(𝑠)_{𝐿2𝑥𝐿1𝑡}≲𝜌𝐴²𝑡⁻¹, (95) for all 𝑡≥𝑇. Therefore, by combining estimates (90)–(93) and (95), we obtain 𝐹^𝑙_{2𝐿1𝑡𝐻−1/2}≲𝜌𝐴𝑡^−𝜇, (96) for any 𝑡≥𝑇≥1 since 𝜇<1. Next, we consider 𝐹^𝑙₃. By the definition of ℛ_𝐷, we have ℛ_𝐷𝒟𝜙₀,𝜓_{0𝐿1𝑡𝐻−1/2}≲_𝑗=1,2ℛ𝒟𝜙₀,𝜓_{0,𝑗𝐿1𝑡𝐿2𝑥}, (97) where we put 𝜓₀=(𝜓_0,1,𝜓_0,2)^𝑡. By Lemma 8, we can express ℛ as ℛ𝒟𝜙₀,𝜓_0,𝑗≡𝑠⁻¹𝑍₁+𝑠⁻²𝑍₂, (98) for 𝑠∈ℝ⧵{0}, where 𝑍₁≡𝜕₀𝜕_𝑡𝒟𝜙₀𝒵₁𝜕₁𝜓_0,𝑗−𝒵₁𝜕_𝑡𝒟𝜙₀𝜕₀𝜕₁𝜓_0,𝑗𝑍+similar,₂≡−𝒵₁𝜕₂𝜓_0,𝑗𝒵₂𝜕₁𝒟𝜙₀+𝒵₁𝜕₁𝒟𝜙₀𝒵₂𝜕₂𝜓_0,𝑗+similar. (99) By applying the Hölder inequality, we have 𝑠⁻¹𝑍_{1𝐿1𝑡𝐿2𝑥}≲^∞_𝑡𝑠⁻¹𝜙_0𝐻3𝑞𝒵𝜓_{0𝐻12𝑞/(𝑞−2)}+𝜓_0𝐻2𝑞𝒵𝜙_{0𝐻22𝑞/(𝑞−2)}𝑑𝑠. (100) By Corollary 4 with 𝑝=𝑞, we get 𝜙₀(𝑠)_𝐻3𝑞≲𝑠^−1+2/𝑞⟨∇⟩𝜙⁺₁,𝜙⁺_2𝐻^4−4/𝑞_{𝑞/(𝑞−1)}≲𝐴𝑠^−1+2/𝑞,𝜓₀(𝑠)_𝐻2𝑞≲𝑠^−1+2/𝑞𝜓⁺_𝐻^4−4/𝑞_{𝑞/(𝑞−1)}≲𝐴𝑠^−1+2/𝑞, (101) for any 𝑠≥𝑡. On the other hand, note that the commutation relations (26) hold. By applying the Sobolev inequality and the charge and energy conservation laws, we obtain 𝒵𝜓_{0𝐻12𝑞/(𝑞−2)}≲𝒵𝜓_{0𝐻1+2/𝑞}≲𝒵𝜓_0𝐻3/2≲𝒵𝜓₀(0)_𝐻3/2≲𝐴,𝒵𝜙_{0𝐻22𝑞/(𝑞−2)}≲𝒵𝜙_{0𝐻2+2/𝑞}≲𝒵𝜙_0𝐻5/2≲𝒵𝜙₀(0)_𝐻5/2≲𝐴, (102) since 𝑞>4. Thus, by combining (100)–(102), we get 𝑠⁻¹𝑍_{1𝐿1𝑡𝐿2𝑥}≲𝐴²𝑡^−1+2/𝑞, (103) for any 𝑡≥𝑇. By the Hölder inequality, we have 𝑠⁻²𝑍_{2𝐿1𝑡𝐿2𝑥}≲^∞_𝑡𝑠⁻²𝒵𝜓₀(𝑠)_𝐻14𝒵𝜙₀(𝑠)_𝐻24𝑑𝑠≲𝐴²𝑡⁻¹, (104) since in the same manner as the proof of estimates (102), we obtain 𝒵𝜓₀(𝑠)_𝐻14+𝒵𝜙₀(𝑠)_𝐻24≲𝐴, (105) for any 𝑠≥𝑡. Therefore, combining (97)-(98), (103), and (104), we have 𝐹^𝑙_{3𝐿1𝑡𝐻−1/2}≲𝐴²𝑡^−1+2/𝑞, (106)
for all 𝑡≥𝑇≥1 since 𝑞>4.
Next, we will estimate 𝐺^l₁. By the Leibniz formula (25) with 𝜅=1/2,  𝑝=4/3,  𝑞₁=𝑞₂=2, and 𝑟₁=𝑟₂=4 and the Hölder inequality, we obtain ℬΨ^𝑙∗ℬΨ^𝑙_{𝐿𝑡4/3𝐻1/24/3}≲ℬΨ^𝑙(𝑠)_𝐻1/2ℬΨ^𝑙(𝑠)_{𝐿4𝑥𝐿𝑡4/3}≲𝐴𝑠^−𝜇ℬΨ^𝑙(𝑠)_{𝐿4𝑥𝐿𝑡4/3}≲𝐴𝑠^−𝜇_{𝐿2𝑡(𝐼)}ℬΨ^𝑙_{𝐿4𝑡𝐿4𝑥}≲𝐴²𝑡^1/2−2𝜇, (107) for any 𝑡≥𝑇 since 𝑣^𝑙∈𝐗_𝑇,𝐴. By the fractional Leibniz rule (25) again and Remark 5 with 𝑝=4, we have 𝑓_𝐷(𝑠)_𝐻1/2≲𝜙₀(𝑠)_𝐻4^5/2𝜓₀(𝑠)_𝐻4^3/2≲𝐴²𝑠^−3/2, (108) for any 𝑠≥𝑡. In the same manner as the proof of the estimate (107), we obtain ℬΨ^𝑙∗𝑓_{𝐷𝐿𝑡4/3𝐻1/24/3}≲ℬΨ^𝑙(𝑠)_𝐻1/2𝑓_𝐷(𝑠)_{𝐻1/2𝐿𝑡4/3}≲𝐴³𝑡^{−3/4−𝜇}, (109) for any 𝑡≥𝑇 due to 𝑣^𝑙∈𝐗_𝑇,𝐴 and (108). In the same manner as the proof of the estimate (109), we get 𝑓_𝐷^∗𝑓_{𝐷𝐿𝑡4/3𝐻1/24/3}≲𝐴⁴𝑡^−7/4, (110) for all 𝑡≥𝑇. Thus, by combining the estimates (107) and (109)-(110), we obtain 𝐺^𝑙_{1𝐿𝑡4/3𝐻1/24/3}≲𝐴²𝑡^1/2−2𝜇, (111) for 𝑡≥𝑇≥1 since 𝜇<1. In the same manner as the proof of the estimates (96) and (106), we obtain 𝐺^𝑙_{2𝐿1𝑡𝐿2𝑥}≲𝜌𝐴𝑡^−𝜇,𝐺^𝑙_{3𝐿1𝑡𝐿2𝑥}≲𝐴²𝑡^−1+2/𝑞, (112) for any 𝑡≥𝑇. Finally, by combining (82), (89), (96), (106), and (111)-(112), we obtain 𝑣^𝑙+1_𝐗𝑇≲𝐴𝐴𝑇^1/2−𝜇+𝜌+𝐴𝑇^{−1+𝜇+2/𝑞}, (113) for 𝑇≥1. By the estimate (113) and 1/2<𝜇<1−2/𝑞, there exist a large 𝑇>0 and a small 𝜌>0 such that 𝑣^𝑙+1∈𝐗_𝑇,𝐴. In the same manner as the proof of (113), we can prove the estimate 𝑣^𝑙+1−𝑣^𝑙_𝐗𝑇≤12𝑣^𝑙−𝑣^𝑙−1_𝐗𝑇, (114) for 𝑙≥1 if 𝑇>1 is sufficiently large and 𝜌>0 is sufficiently small, which implies that {𝑣^𝑙}_𝑙≥0 is a Cauchy sequencein 𝐗_𝑇,𝐴. Theorem 1 is proved.

Appendix

In this section, we give a proof of Lemma 15. First, we prepare the following.

Lemma 16 (see [10]). Let 𝜅∈ℝ and let 𝜓⁺=𝜓⁺(𝑥) be a ℂ²-valued given function. Then, for any 𝐂²-valued function 𝜓=𝜓(𝑡,𝑥), the equivalency 𝜓(𝑡)−𝒱_𝐷(𝑡)𝜓⁺_𝐻𝜅≃_±𝒜^𝐷_±𝜓(𝑡)−𝒰_±,𝑀(𝑡)𝒜^𝐷_±𝜓⁺_𝐻𝜅 (A.1)
holds for all 𝑡∈ℝ.

For the proof of the lemma, see [10].

By the lemma and a decay property of 𝑓_𝐷 given by (42), we also have the following.

Corollary 17. Let (𝜓⁺,(⟨∇⟩𝜙⁺₁,𝜙⁺₂))∈(𝐻^5/2,1)⁴. The final state condition (8) with 𝐗=𝐻^1/2 holds if and only if the identity lim_𝑡→∞±𝒜^𝐷_±Ψ(𝑡)_𝐻𝜅=0 (A.2)
is valid, where Ψ is defined by (41).

Before proving the corollary, we remember some properties of the operators 𝒜^𝐷_± given by (17) (see [10] in detail). We note that the identity 𝛼⋅∇+𝑖𝑀𝛽²=−⟨∇⟩²_𝑀𝐼 (A.3) holds due to properties (1) of Dirac matrices. Hence, by a direct calculation, we get the following identities: 𝒜^𝐷_±𝒜^𝐷_∓=𝑂,_±𝒜^𝐷_±=𝐼,𝒜^𝐷_±²=𝒜^𝐷_±. (A.4) We put 𝐵=‖(𝜓⁺,(⟨∇⟩𝜙⁺₁,𝜙⁺₂))‖_𝐻5/2,1.

Proof. By Lemma 16, we see that (8) with 𝐗=𝐻^1/2 is equivalent to lim_𝑡→∞±𝒜^𝐷_±𝜓(𝑡)−𝒰_±,𝑀(𝑡)𝒜^𝐷_±𝜓⁺_𝐻1/2=0. (A.5) By decomposition (16) and identities (A.4), we have 𝒜^𝐷_±Ψ(𝑡)_𝐻1/2=𝒜^𝐷_±𝜓(𝑡)−𝒰_±,𝑀(𝑡)𝒜^𝐷_±𝜓⁺−𝒜^𝐷_±𝑓_𝐷𝐻1/2. (A.6) By estimate (18), the fractional Leibniz rule (25) with 𝑝=2 and 𝑞_𝑖=𝑟_𝑖=4  (𝑖=1,2), and Remark 5 with 𝑝=4, we get 𝒜^𝐷_±𝑓_𝐷𝐻1/2≲𝜙_0𝐻45/2𝜓_0𝐻43/2≲𝑡⁻¹𝐵², (A.7)
for all 𝑡>0, which completes the proof of the corollary.

Next we will prove Lemma 15.

Proof of Lemma 15. First we prove the Dirac part. By Corollary 17, we see that (52) is equivalent to lim_𝑡→∞±𝒜^𝐷_±Ψ^𝑙(𝑡)_𝐻1/2=0for𝑙≥0. (A.8) Note that the identity 𝒜^𝐷_±−ℬ^𝐷_±𝑖=±2⟨∇⟩_𝑀⁻¹𝒟₊ (A.9) holds. From the Dirac part of (55), we have ℬ^𝐷_±Ψ^𝑙+1=𝒜^𝐷_±Ψ^𝑙+1−⟨∇⟩_𝑀⁻¹𝐹^𝑙for𝑙≥0. (A.10) Thus, it is sufficient to show that lim_𝑡→∞𝐹^𝑙_𝐻−1/2=0for𝑙≥0. (A.11) By the Sobolev inequality and the Hölder inequality, we have, for 𝑙≥1, 𝐹^𝑙_𝐻−1/2≲𝜙^𝑙_𝐻1/2𝜓^𝑙_𝐻1/2+𝜙^𝑙_𝐻1/2𝜓⁺_𝐻1/2+𝜙⁺_1𝐻1/2+𝜙⁺_2𝐻−1/2𝜓^𝑙_𝐻1/2+𝜙_0𝐻28𝜓_0𝐻28/3. (A.12) By Remark 5 with 𝑝=8,8/3, we get 𝜙_0𝐻28≲𝑡^−3/4𝐵,𝜓_0𝐻28/3≲𝑡^−1/2𝐵. (A.13) Thus, by assumptions and estimates (A.12)-(A.13), we obtain (A.11) for 𝑙≥1. In the case of 𝑙=0, it is easy to see (69). We omit the details. Conversely, assume (69) and will prove (52). By the decomposition 𝐼=∑_±ℬ^𝐷_±, we have only to show that lim_𝑡→∞±ℬ^𝐷_±𝑓_𝐷𝐻1/2=0. (A.14) We have ℬ𝑓_𝐷𝐻1/2≲ℬ𝒬^𝐷₀𝒟𝜙₀,𝜓_0𝐻1/2+remainder,ℬ𝒬^𝐷₀𝒟𝜙₀,𝜓_0𝐻1/2≲𝒬^𝐷_0𝐻1/2+𝜕_𝑡𝒬^𝐷_0𝐻1/2. (A.15)(A.16) By the Hölder inequality and Remark 5 with 𝑝=8,8/3, we obtain 𝜕_𝑡𝒬₀𝒟𝜙₀,𝜓_{0,𝑗𝐻−1/2}≲𝜙_0𝐻38/3𝜓_0𝐻18+𝜙_0𝐻28𝜓_0𝐻28/3≲𝑡⁻¹𝐵². (A.17) Since the remainder terms in (A.15) can be estimated in the same manner as the proof of (A.17), we obtain ℬ𝑓_𝐷𝐻1/2≲𝑡⁻¹𝐵², (A.18)
from which (A.14) follows.
Next, we consider the KG part. By the identity 𝑓+𝑔²_𝐻𝜅+𝑓−𝑔²_𝐻𝜅=2𝑓²_𝐻𝜅+𝑔²_𝐻𝜅, (A.19) we can see that (53) is equivalent to _±ℬ^𝐾_±𝜙^𝑙(𝑡)−𝜙₀(𝑡)_𝐻1. (A.20) In the same manner as the proof of estimate (A.18), we can obtain ℬ𝑓_𝐾𝐻1≲𝑡⁻¹𝐵², (A.21) which completes the proof of the lemma.

Acknowledgment

The author would like to express deep gratitude to an anonymous referee for the useful suggestions and comments.