Cubic nonlinear Dirac equation in a quarter plane

J. Math. Anal. Appl. 434 (2016) 1633–1664

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Cubic nonlinear Dirac equation in a quarter plane I.P. Naumkin Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-726, México DF 01000, Mexico

a r t i c l e

i n f o

Article history: Received 16 April 2015 Available online 13 October 2015 Submitted by P. Exner Keywords: Nonlinear Dirac equation Initial–boundary value problem Thirring model Critical nonlinearity

a b s t r a c t We study the initial–boundary value problem (IBV) for the cubic nonlinear Dirac equation in one space dimension 

i (∂t + α∂x ) ψ + βψ = βψ, ψ βψ, x > 0, t > 0, ψ (x, 0) = ψ0 (x) , ψ (0, t) = h0 (t) ,

where ψ = ψ (t, x) ∈ C2 is a two-spinor field, α, β are hermitian (2 × 2)-matrices satisfying β 2 = α2 = I, αβ + βα = 0, ·, · denotes the C2 -scalar product. We prove the global in time existence of solutions of IBV problem for cubic Dirac equations with inhomogeneous Dirichlet boundary conditions. We obtain the sharp time decay of solutions in the uniform norm. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Consider the initial–boundary value problem for the cubic nonlinear Dirac equation with inhomogeneous Dirichlet boundary conditions ⎧ ⎪ ⎨ i (∂t + α∂x ) ψ + βψ = βψ, ψ βψ, x > 0, t > 0, ψ (x, 0) = ψ0 (x) , x > 0, ⎪ ⎩ ψ (0, t) = h (t) , t > 0,

(1.1)

 ψ1 (x, t) where ψ = ψ (x, t) = ∈ C2 is a two-spinor field, α, β are hermitian (2 × 2)-matrices satisfying ψ2 (x, t) β 2 = α2 = I, and the anticommutation relation 

αβ + βα = 0, E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.09.049 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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and ·, · denotes the C2 -scalar product. A particular representation of the matrices α and β is given by  α=

 0 1 1 0

 , β=

 1 0

0 −1

.

The nonlinear Dirac equation (1.1) is known as the Thirring model [29] (see [26] for three space dimensions), which is modeling selfinteracting Dirac fermions in the quantum field theory (see also papers [8,7,11]). Our purpose is to prove the global in time existence of small solutions to the initial–boundary value problem (1.1) and to find a sharp time decay of the solutions in the L∞ norm. There are several papers that consider the Cauchy problem for the nonlinear Dirac equation in different space dimensions and various types of nonlinearities ([3–5,30,18,15,17,16,19,25,2,21,1], and the references cited therein). To our knowledge there are no results on the well posedness of the initial–boundary value problem (1.1). We fill this gap in the present work. The nonlinear scattering problem consists in comparing the large time behavior of the solutions to nonlinear equations with the asymptotic behavior of the solutions to the corresponding linear equations. Then, the sharp time decay of the solutions of nonlinear equations is an important problem in the study of the scattering properties of these equations. For a general approach to the scattering problem for nonlinear equations we refer to [22–24,27,28]. In the nonlinear scattering problem one can consider the nonlinearity as a perturbation of the linear equation and establish an analogy of the nonlinear scattering theory with the linear potential scattering theory (see, e.g., [20]). Then, there appears the notion of scattering-critical nonlinearity or long range scattering: a nonlinearity that has the same large time behavior as the linear part. For nonlinear equations with a scattering-critical nonlinearity the effect of the nonlinearity is no longer negligible and the solutions to such equations are affected by the nonlinearity. As a consequence, there are no asymptotically free solutions (that have the same asymptotics for large time as the solutions to the corresponding linear equation) and there appear some corrections in the large time behavior of the solutions to these equations. For example, cubic nonlinearities in one space dimension often are scattering-critical. That is the case of the Schrödinger equation in one space dimension with gauge invariant nonlinearity of power 3: a logarithmic correction appears in the asymptotic expansion of the solution [20]. We note that there are cases when it is important to know this correction even if one only considers the global well posedness problem [14]. In the case of the Cauchy problem for the Thirring model there are some results concerning the large time behavior of the solutions. For example, for the one dimensional Thirring model, an a priori bound on the uniform norm of the solution was obtained in [3]. On the other hand, in the case of the three dimensional Thirring model, the nonlinearity is not scattering-critical, all the solutions scatters to the free solutions [1]. As far as we know, there are no results on the sharp time decay of solutions to the one dimensional Thirring model in the uniform norm. In the present paper we study the sharp time decay of the solutions to (1.1). From the point of view of applications, the boundary value problems are more natural than the Cauchy problems. However, in the case of the initial–boundary value problems there appear some difficulties compared with the Cauchy problems due to the boundary. For example, in the case of the initial–boundary value problems it is not clear the quantity of the boundary conditions that are required for the well posedness of the boundary problem. Following the ideas of [12] we answer this question by constructing the Green function for the linear Dirac equation. This construction is interesting on its own because the Green function for the inhomogeneous Dirichlet boundary problem for Dirac equation was not constructed previously. Also, in the case of the initial–boundary value problems, it is necessary to take into an account the boundary effects which have influence on the behavior of the solutions: in some Dirichlet problems for dissipative equations, such as Korteweg–de Vries (KdV) and intermediate long-wave equations, posed on the positive half-line, the solutions have some extra time decay compared with the solutions of these equations in the case of the Cauchy problem (see [12]). However, as we will see, this is not the case for the initial–boundary

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value problem for the Thirring model. The cubic nonlinearity in this case is scattering-critical. Besides, in the study of the initial–boundary value problems there appears a question of quantity of compatibility conditions on the initial and boundary data needed for the well-posedness of the problem. These compatibility conditions are related to the regularity of the solution. Therefore, in order to avoid many compatibility conditions, one needs to ask for as less regularity on the initial and boundary data as possible. We observe that in the case of the Cauchy problem one disposes of more regularity on the initial data in the study of the time behavior of the solutions. Our method is based on a new decomposition of the nonlinearity, which together with some estimates of the corresponding integral equation in the Sobolev spaces allows us to prove the global in time existence and to obtain a sharp time decay of the solutions in the uniform norm. We observe that our method can be used to treat general non-integrable equations with arbitrary nonlinearizable boundary conditions. It is worth noting that there is a method based in the so-called Inverse Scattering Transform that can be used to establish existence and asymptotics of solutions to the initial–boundary value problems for integrable nonlinear evolution PDEs (such as the KdV and Schrödinger equations) with a particular class of boundary conditions, called linearizable [9]. Recently, this method was used to study the nonlinear Schrödinger equation on the half-line [10]. As far as we know, this method was not applied to the initial–boundary value problem (1.1). In order to state our results precisely, we introduce some notation. The Fourier transform F and the inverse Fourier transform F −1 are defined as 1 1 −ixξ −1 Fφ(ξ) = √ e φ (x) dx, and F φ(x) = √ eixξ φ (ξ) dξ. 2π 2π R

R

We also introduce the Fourier sine transform Fs and the Fourier cosine transform Fc as follows

∞

∞ 2 2 Fs φ(p) = φ(x) sin pxdx, Fc φ(p) = φ(x) cos pxdx. π π 0

0

We denote by Lp = Lp (R+ ), 1 ≤ p ≤ ∞, the Lebesgue space on R+ := (0, ∞) and the weighted Lebesgue space is given by ⎧ ⎫ k ⎨ ⎬  j Lp,k := φ : φLp,k := (·) φ (·) p < ∞ . ⎩ ⎭ L j=0

Moreover, for any positive integer m and k ∈ R we define the weighted Sobolev spaces by ⎧ ⎫ m ⎨ ⎬  ∂ j φ 2,k < ∞ . Hm,k = φ : φHm,k := L ⎩ ⎭ j=0

Note that for k = 0 we obtain the usual Sobolev spaces Hm,0 . We omit the index 0 and write Hm instead of 2  2   2  Hm,0 where there is no place to confusion. We denote by (Lp ) , Lp,k and Hm,k , the spaces Lp R+ ; C2 ,     Lp,k R+ ; C2 and Hm,k R+ ; C2 , respectively. For simplicity we often will omit these higher indexes. Also we write ϕ∞ and ϕ in place of ϕL∞ and ϕ L2 , respectively. Finally, we denote by C different positive constants. Our main result is the following  2  2 Theorem 1.1. Let ψ0 ∈ H2,1 , h(t) ∈ H1,2 and the norm ψ0 (H2,1 )2 + h(H1,2 )2 = ε. Also suppose that the compatibility condition ψ0 (0) = h(0) is fulfilled. Then, there exists ε0 > 0 such that for all 0 < ε < ε0 the initial–boundary value problem (1.1) has a unique global solution

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  2  ψ (t) ∈ C [0, ∞) ; H2,1 with the following time decay estimate − 12

ψ (t)(L∞ )2 ≤ C (1 + t)

.

 2 Remark 1.2. In Theorem 1.1 we ask that the initial data belongs to H2,1 . Also a smallness condition is imposed. We believe that, as in the case of the Cauchy problem for the one dimensional Thirring model (see [2]), the assumption about the decay of the initial data can be relaxed (but anyway the smallness condition may be required). However, it seems that in order to obtain a time decay estimate one needs to impose some decay and regularity conditions on the initial data. On the other hand, we can deduce from Proposition 2.1 and in particular from equation (2.5) that one cannot obtain a global well posedness result for the problem (1.1) with more general initial data at the expense of imposing stronger assumptions on the boundary data. We now present a sketch of our proof. We define   ∂x  φ := Fs p Fs φ − e−x φ(0) , where x :=

√

(1.2)

1 + x2 , and −1

∂x 

:= Fs

1 Fs . p

(1.3)

Here we note that as φ (x) − e−x φ(0) = 0, at x = 0, for any φ ∈ H1 , the operator ∂x  : H1 → L2 is well −1 defined. The operator ∂x  : L2 → H1 is the right inverse of ∂x , since    1 1 −x Fs φ − e Fs Fs φ (0) = ∂x  ∂x  φ = Fs p Fs Fs p p   1 1 = Fs p Fs Fs Fs φ = Fs p Fs φ = φ, p p 

−1





  1 1 for all φ ∈ L2 . (Here we used that p Fs φ ∈ L1 , if φ ∈ L2 , and then Fs p Fs φ (0) = 0.) We will prove that the initial–boundary value problem for nonlinear Dirac equation (1.1) can be rewritten as the following first order in time 4 × 4 system ⎧ −1 ⎪ ⎨ Lu = ∂x  N (u) , t > 0, x > 0, u(x, 0) = u0 , x > 0, ⎪ ⎩ u(0, t) = 12 Bh(t), t > 0, where L := I4 ∂t + iγ ∂x  , I4 := I4×4 is the 4 × 4 identity matrix, ⎛

1 ⎜0 γ := ⎝ 0 0

0 1 0 0

⎞ 0 0 0 0 ⎟ , −1 0 ⎠ 0 −1

(1.4)

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the C4 -vector valued functions u and u0 are given by   1 −1 Bψ + iC ∂x  ∂t ψ , 2   1 −1 u0 := Bψ0 + iC ∂x  ψ1 , 2

u :=

and N (u) :=

i C (D− (βAu, Au βAu)) , 2

with  A :=

1 0 1 0 0 1 0 1



⎛

1 ⎜0 , B:=⎝ 1 0

⎞ ⎛ 1 0 1⎟ ⎜ 0 , C:=⎝ −1 0⎠ 0 1

⎞ 0 1 ⎟ . 0 ⎠ −1

Note that u and ψ are related by the equation ψ = Au. We move equation (1.4) into another one by making the change u (x, t) = Fs e−iγpt Fs φ(p, t)

(1.5)

and use the large time asymptotics for the evolution group G (t) (see (2.1)). Next we consider two regions p ≤ t2σ and p ≥ t2σ , with σ > 0 small enough. In the case p ≤ t2σ , with σ > 0 small enough, we obtain from (1.4) −1

−1

∂t (χ(p, t)) = 2ti p  E (p; χ(p, t)) χ(p, t) + eiγpt p Λ (t, p; χ(p, t)) 1+δ  3 3 −1−δ +O t J uH2 + uH2 + t h ∞ , δ > 0, 3/2

where χ(p, t) := p

(1.6)

Fs φ(p, t) and J := x + itγ∂x

1 . ∂x 

(1.7)

Thus, as we shall see, the nonlinearity in the R.H.S. of (1.4) is decomposed into the resonant term −1 −1 i iγpt χ(p, t)) χ(p, t), nonresonant cubic p Λ (t, p; χ(p, t)) and the remain2t p  E (p;   nonlinearities e 3 3 . This fact is exploited in Lemma 2.5 and Lemma 2.6 below. der O t−1−δ J u 2 + u 2 + t1+δ h H

H

∞

Then, in Lemma 2.8, we will show that the nonresonant terms have a better time decay property through the integration by parts. Furthermore, the resonant term of the R.H.S. of (1.6) can be removed by a change of dependent variables. Hence, the L∞ -estimate of Fs φ(p, t) follows for p ≤ t2σ . In the case of p ≥ t2σ we have  3    sup p 2 Fs φ(p, t) ≤ Ct−σ (J uH2 + uH2 ) . |p|>t2σ

The estimates of the integral equation in the Sobolev spaces yield a priori estimate of J uH2 and uH2 , which again implies the L∞ -estimate of Fs φ(p, t). Therefore, using the large time asymptotics for the evolution group G (t) in (1.5) we attain the result of Theorem 1.1.

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We note that a similar operator J was also used in [13]. In our case the definition of J is not so natural. The difficulty consists in the definition of the pseudodifferential operators for the boundary problems which is not as intuitive as in the case of the whole line. Also, we need more delicate estimates for the operator J . This is because, as we mentioned above, the regularity of the solution, in the case of the boundary problems, is related to the compatibility conditions on the initial and boundary data. For the Cauchy problem there is no such complication and usually one disposes of more regularity on the initial data. The paper is organized as follows. In Section 2, we prove some preliminary results involved in the proof of Theorem 1.1. In the first subsection we construct the Green function for the initial–boundary value problem (1.4) and we prove that the equation (1.1) can be rewritten as (1.4). In the second subsection we obtain several lemmas that we use in the proof of the main result. Finally, Section 3 is devoted to the proof of Theorem 1.1. 2. Preliminaries 2.1. Construction of the free evolution group Denote by G(t)φ := Fs e−iγpt Fs φ = Fs (cos p t − iγ sin p t) Fs φ,

(2.1)

and D− := −i∂t + iα∂x + β. We consider the linear initial–boundary value problem on half-line ⎧ ⎪ ⎨ i (∂t + α∂x ) ψ + βψ = g(x, t), x > 0, t > 0, ψ (x, 0) = ψ0 (x) , x > 0, ⎪ ⎩ ψ (0, t) = h (t) , t > 0.

(2.2)

We prove the following result  2 Proposition 2.1. Let the initial data and boundary data ψ0 , ψ1 , h ∈ L1 (R+ ) and the force g ∈ 2  1 + H (R × R+ ) . Then the solution ψ(x, t) of the initial–boundary value problem (2.2) is given by ψ(x, t) = Au,

(2.3)

where the C4 -vector valued function u is the solution of the following initial–boundary value problem ⎧ −1 i ⎪ ⎨ Lu = 2 ∂x  Cf (x, t), t > 0, x > 0, (2.4) u (x, 0) = u0 (x) , x > 0, ⎪ ⎩ u(0, t) = 12 Bh(t), t > 0,    −1 with f (x, t) := D− g(x, t) and initial data u0 (x) := 12 Bψ0 (x) + iC ∂x  ψ1 (x) . Moreover, the function u has the following integral representation ⎫ ⎧

t ⎬ t ⎨ 2 p i −1 i iγpτ Fs Ch(τ )dτ + G(t − τ ) ∂x  Cf (x, τ )dτ, (2.5) u(x, t) = G(t) u0 + e ⎭ ⎩ π p 2 2 0

where the operator G(t) is given by (2.1).

0

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Proof. We multiply from the left side the both sides of the linear Dirac equation (2.2) i (∂t + α∂x ) ψ + βψ = g(x, t) by the operator D− . Then, we get 

 ∂t2 − ∂x2 + I ψ = f (x, t)

(I := I2×2 is the 2 × 2 unit matrix). Hence we arrive to the following initial–boundary value problem for a linear system of Klein–Gordon type equations posed on a half-line ⎧ ⎪ ⎨

ψtt − ψxx + ψ = f (x, t), t > 0, x > 0, ψ (x, 0) = ψ0 (x), ψt (x, 0) = ψ1 (x), x > 0, ⎪ ⎩ ψ(0, t) = h(t), t > 0,

(2.6)

where ψ1 (x) := −αψ0 (x) + iβψ0 (x) − ig (x, 0). Applying the Laplace transforms with respect to space and time variables to problem (2.6) we find for Re p > 0, Re ξ > 0,   ξ) = ψ(p,

1 2 ξ + 1 − p2



       −pψ(0, ξ) − ψx (0, ξ) + f (p, ξ) + ξ ψ0 (p) + ψ1 (p) .

(2.7)

  ξ), ψx (0, ξ), f  ξ), ψ(0, (p, ξ), ψ0 (p) and ψ1 (p) are the Laplace transforms of ψ(x, t), Here the functions ψ(p, ψ(0, t), ψx (0, t), f (x, t), ψ0 (x) and ψ1 (x), respectively. There exists only one root p = k(ξ) = ξ of equation   ξ), the ξ 2 +1 −p2 = 0 such that Re k(ξ) > 0 for all Re ξ > 0. Therefore, in the expression for the function ψ(p, 1 factor ξ2 +1−p 2 has a pole in the point p = k(ξ). This implies that for the solubility of the non-homogeneous problem (2.6) it is necessary and sufficient that the following condition is satisfied

 ξ) − ψx (0, ξ) + f (k(ξ), ξ) + ξ ψ0 (k(ξ)) + ψ1 (k(ξ)) = 0. −k(ξ)ψ(0,

(2.8)

Therefore, we need to fix in the problem (2.6) one boundary data and the rest of boundary data can be found from equation (2.8). Thus, for example, if we put the condition ψ(0, t) = h(t),  h(ξ). ψx (0, ξ) = f(k(ξ), ξ) + ξ ψ0 (k(ξ)) + ψ1 (k(ξ)) − k(ξ)

(2.9)

Then, substituting (2.9) in (2.7) and taking inverse Laplace transforms with respect to time and to space variables we obtain ∞ ψ(x, t) = t +

(Gt (x, y, t)ψ0 (y) + G(x, y, t)ψ1 (y)) dy 0

⎞ ∞ ⎝Gy (x, 0, t − τ )h(τ ) + dyf (y, τ )G(x, y, t − τ )⎠ dτ, ⎛

0

0

where the function G(x, y, t) is defined by formula 1 G(x, y, t) = 4π 2

i∞+ε

i∞ dξe

−i∞+ε

ξt −i∞

epx

  1 −k(ξ)y −py − e e dp. ξ 2 + 1 − p2

(2.10)

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Now we simplify the representation of G(x, y, t). Taking residues in the poles ξ = ±i ip by the Cauchy Theorem we get

1 − 2 4π

i∞+ε

i∞ dξe

ξt

−i∞+ε

e

p(x−y)

−i∞

1 1 dp = ξ 2 + 1 − p2 2πi

i∞

1 (eiipt − e−iipt )ep(x−y) dp. 2i ip

−i∞

Taking the residue in the point p = −k(ξ) we find i∞+ε

− 4π1 2

ξt −k(ξ)y

dξe e

−i∞+ε i∞+ε

=

i∞

i∞

1 dξeξt 2πi

1 2πi −i∞+ε

−i∞

=

i+ε e

px

1 ξ 2 +1−p2 dp

−i∞ e−p(x+y) ξ 2 +1−p2 dp

1 2πi

=

1 dξeξt e−k(ξ)(x+y) 2k(ξ)

−i+ε

i∞

1 iipt 2iip (e

− e−iipt )e−p(x+y) dp

−i∞

i∞

1 2πi

=

1 2πi

1 iipt 2iip (e

− e−iipt )ep(x+y) dp.

−i∞

Therefore the representation of G(x, y, t) takes the form G(x, y, t) i∞ =

1 2πi

1 iipt 2iip (e

−i∞ ∞

=

1 2πi

e

ipx

−e

(−2i) ipt (e 2p

−iipt

−e

∞ p(x−y)

)(e

−ipt

−e

p(x+y)

)dp =

) sin pydp =

−∞

∞ =

e

ipx

(−2i)2i 2p

∞

sin p t sin pydp =

−∞

sin px p

2 π

− e−ipt )(eip(x−y) − eip(x+y) )dp

−∞

∞ 1 2πi

1 ipt 2p (e

1 2πi

1 πi

eipx p

sin p t sin pydp

−∞

sin p t sin pydp,

0

and thus,

G(x, y, t) =

2 sin p t Fs sin py. π p

Substituting this relation into (2.10) we get sin p t Fs ψ1 + ψ(x, t) = Fs cos p tFs ψ0 + Fs p

t Fs 0

sin p (t − τ ) Fs f (p, τ )dτ p

t 2 p sin p (t − τ ) h(τ )dτ. Fs + π p 0

Observe that ψ(x, t) = Au(x, t), where u is represented by (2.5). By a direct calculation we find that u(x, t) is solution of the initial boundary value problem (2.4). 2

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2.2. Large time asymptotics for the evolution group G (t) and the nonlinearity Lemma 2.2. For φ ∈ L1 ∩ L2 , the following relation J G (t − τ ) φ = Fc e−iγp(t−τ ) (iτ γ

p + ∂p )Fs φ, p

(2.11)

holds. Moreover, if φ is such that Fs φH1,2 < ∞, the identities ∂p Fs φL2,2 = J G (t) φH2

(2.12)

Fs φH1,2 = J G (t) φH2 + G (t) φH2

(2.13)

and

are valid. Finally, if φ ∈ H2 and φ(0) = 0 G (t) φH2 ≤ C φH2 .

(2.14)

Proof. We have  ∞

xG (t − τ ) φ = x  =

2 π

−iγpt iγpτ

sin pxe

e

 ∞ Fs φdp = − π2 e−iγpt eiγpτ Fs φd cos px

0 2 π

0

p limp→0 e−iγpt eiγpτ cos pxFs φ + Fc e−iγp(t−τ ) (−i (t − τ ) γ p + ∂p )Fs φ.

Since φ ∈ L1 , limp→0 Fs φ = 0, and therefore xG (t − τ ) φ + itγFc p

1 p Fs G (t − τ ) φ = Fc e−iγp(t−τ ) (iτ γ + ∂p )Fs φ. p p

(2.15)

Hence, as Fc p

1 1 Fs G (t − τ ) φ = ∂x Fs Fs G (t − τ ) φ, p p

for φ ∈ L2 , using (1.3), we obtain (2.11). Using the identities ∂x J G (t) = −Fs e−iγpt p∂p Fs , ∂x2 J G (t) = −Fc e−iγpt p2 ∂p Fs and applying Plancherel Theorem we get relation (2.12). Moreover, as Fs φL2,2 < ∞, G (t) φH2 = Fs φL2,2 , and thus, equality (2.13) follows from the last relation and (2.12). Since for p = 0 Fs φ =

1 φ(0) 1 − 2 lim (φ (x) sin px) − 2 Fs ∂x2 φ, p p x→0 p

then, if φ(0) = 0, it follows that G (t) φH2 ≤ C φH2 .

2

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Lemma 2.3. Estimate t iγpτ Fs p e g(τ )dτ p2

≤ C gL1,δ ,

(2.16)

H2

0

for δ > 0, is valid, provided the right hand side is finite. Proof. As eiγx = Aeix + Be−ix ,

(2.17)

with ⎛

1 ⎜0 A := ⎝ 0 0

0 1 0 0

⎞ ⎛ 0 0 0⎟ ⎜0 , B := ⎝ 0 0⎠ 0 0

0 0 0 0

0 0 0 0

⎞ 0 0⎟ , 0⎠ 1

0 0 1 0

(2.18)

then t

t e

iγpτ

t

g(τ )dτ = A e

0

ipτ

g(τ )dτ + B

0

e−ipτ g(τ )dτ,

0

and thus, t iγpτ Fs p e g(τ )dτ p2

H2

0

t p ipτ F ≤ e g(τ )dτ s p2

H2

0

t p −ipτ F + e g(τ )dτ s p2

(2.19)

H2

0

Let us prove that t p iγpτ Fs g(τ )dτ p2 e

≤ C g

1

L2, 2 +δ

.

H2

0

The other term in the R.H.S. of (2.19) is estimated similarly. Using that p − p = p τ + px = p(τ + x) + τ

1 p+p

we have

1 . p + p

Therefore, since x > 0, τ > 0, using the analytic properties of the integrand function, by Cauchy Theorem we have ∞ p ipτ Fs p 2e

=

i(1+ε)

e −∞

ipx+ipτ

 ipτ  p eipx 1+p e − e−ipτ dp 2

= limε→0 0

1 p iipτ − e−iipτ )dp. = i e−px ip 2 (e 0

As

p dp p2

(2.20)

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   iipτ  δ − e−iipτ  ≤ C ip τ δ , 0 ≤ δ ≤ 1, e it follows that p ipτ Fs 2 e p

1 ≤ Cτ

p3

δ

H2

2−δ

0

ip

dp ≤ Cτ δ ,

and therefore, t p ipτ Fs g(τ )dτ p2 e

t ≤C

H2

0

τ δ |g(τ )| dτ ≤ C gL1,δ .

2

0

Lemma 2.4. We have t J G(t − τ ) ∂x −1 N (u)dτ

H2

0

t 2

≤C

u∞ (N (u)L2,1 + ∂τ u + J uH2 + uH1 + th∞ + hH1 ) dτ 0

3

3

2

2

+ hH1,1 + u0 H1,1 + u∞ uH1,1 + t u∞ uH1 .

(2.21)

Proof. We write N (u) = C ((−i∂t + iα∂x + β) n (u)) with n (u) :=

i 2

t

(βAu, Au βAu). Let us consider the part (−i∂t ) Cn (u). Using (2.11) we get −1

J G(t − τ ) ∂x 

(−i∂τ ) Cn (u) dτ =

1 Fc e−iγpt p 2

0 p − Fc e−iγpt p 3

t

t eiγpτ (iτ γp + p ∂p )Fs ((−i∂τ ) Cn (u)) dτ 0

eiγpτ Fs (−i∂τ ) Cn (u) dτ.

(2.22)

0

Observe that t iγpτ Fc e−iγpt p e F (−i∂ ) Cn (u) dτ s τ 3 p 0

t 2

≤C H2

u∞ ∂τ u dτ.

(2.23)

0

Integrating by parts we have t eiγpτ (iτ γp + p ∂p )Fs ((−i∂τ ) Cn (u)) dτ = −ieiγpt (itγp + p ∂p )Fs Cn (u) 0

t

+ i p ∂p Fs Cn (u0 ) − γ p

e 0

It follows from the relations

t iγpτ

(iτ γp + p ∂p )Fs Cn (u) dτ −

eiγpτ γpFs Cn (u) dτ. 0

(2.24)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1644

t −γ p



t e

iγpτ

(iτ γp) Fs Cn (u) dτ = −i p

0

e

iγpτ

Cτ n

 t 1 Bh(τ ) dτ − i p eiγpτ Fc Cτ ∂x n (u) dτ 2

0

0

and t −γ p

eiγpτ p ∂p Fs Cn (u) dτ = iγ p eiγpt Fc xCn (u) − iγ p Fc xCn (u0 ) 0

t eiγpτ Fc Cx∂τ n (u) dτ

− i p 0

that t −γ p

t e

iγpτ

(iτ γp + p ∂p )Fs Cn (u) dτ −

0

t

0

t

eiγpτ Fc CPn (u) dτ − i p

= −i p 0

eiγpτ Cτ n

1

2 Bh(τ )

0

t

− iγ p Fc xCn (u0 ) − iγ

eiγpτ γpFs Cn (u) dτ

e

iγpτ

Cn

1

2 Bh(τ )





dτ + iγ p eiγpt Fc xCn (u)

t dτ − iγ

0

eiγpτ Fc C∂ x n (u) dτ 0

where P := (x∂t + t∂x ) I, and thus, by (2.11), ⎞ ⎛ t t Fc e−iγpt 1 2 ⎝−γ p eiγpτ (iτ γp + p ∂p )Fs Cn (u) dτ − eiγpτ iγpFs Cn (u) dτ ⎠ p 2 0 0 H t t   2 p iγpτ ≤ C u∞ (PuH1 + uH1 ) dτ + Cτ n 12 Bh(τ ) dτ Fs p2 e 2 0

3

2

0

2

+ hH1,1 + u0 ∞ u0 H1,1 + u∞ uH1,1 .

H

Moreover, using (2.16) we have ⎞ ⎛ t t Fc e−iγpt 1 2 ⎝−γ p eiγpτ (iτ γp + p ∂p )Fs Cn (u) dτ − eiγpτ iγpFs Cn (u) dτ ⎠ p 2 0 0 H t 2 3 2 2 ≤ C u∞ (PuH1 + uH1 ) dτ + hH1,1 + u0 ∞ u0 H1,1 + u∞ uH1,1 . (2.25) 0 −1

Recall that Lu = I∂t u + iγ ∂x  u = ∂x 

N (u) (see (1.4)). Then, we use the relations

1 Pu = xLu − iγJ ∂x  u − ite−x Bh(t), 2 and

  −x 1 J G(t)G(−t)F u − e Bh(t) J ∂x  uH1 = p F s s 1 2 H    1 −iγpt iγpt −x Bh(t) = Fc e ∂p e p Fs u − e 1 2 H

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1645

    1 −x 1 −x ≤ C J u − e Bh(t) + u − e Bh(t) 1 2 2 H2 H ≤ C (J uH2 + uH1 + th∞ + hH1 ) , to get  −1 PuH1 ≤ C x ∂x  N (u)

H1

 + J uH2 + uH1 + th∞ + hH1 ,

and hence, ⎞ ⎛ t t Fc e−iγpt 1 2 ⎝−γ p eiγpτ (iτ γp + p ∂p )Fs Cn (u) dτ − eiγpτ iγpFs Cn (u) dτ ⎠ p 2 0 0 H t   2 −1 ≤ C u∞ x ∂x  N (u) + J uH2 + uH1 + th∞ + hH1 dτ H1

0

3

3

2

+ hH1,1 + u0 H1,1 + u∞ uH1,1 .

(2.26)

Finally, introducing (2.24) in (2.22) and using estimates (2.26), (2.23), 1 3 2 2 Fc 2 (itγp + p ∂p )Fs Cn (u) ≤ hH1,1 + t u∞ uH1 + u∞ uH1,1 , p 2 H 3 1 ∂p Fs Cn (u0 ) ≤ C u0 H1,1 Fc e−iγpt p H2

and −1 x ∂x  N (u)

H1

≤ N (u)L2,1 ,

we conclude that t J G(t − τ ) ∂x −1 (−i∂τ ) Cn (u) dτ 0

t

H2

2

≤C

u∞ (∂τ u + N (u)L2,1 + J uH2 + uH1 + th∞ + hH1 ) dτ 0

3

3

2

2

+ hH1,1 + u0 H1,1 + u∞ uH1,1 + t u∞ uH1 .

Proceeding similarly for the part in J arrive to (2.21). 2

t 0

G(t − τ ) ∂x 

−1

N (u)dτ corresponding to C ((iα∂x + β) n (u)) we

In the next lemma we obtain the large time asymptotics for the evolution group G (t). Lemma 2.5. Let θ(x) be the step function

θ(x) := Then, the following asymptotics

1, 0,

0 ≤ x < 1, x > 1.

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1646

 ! −1 " π  "− 3 i x ! xt−1 ) + R, (G (t) φ) (x) = − √ θ( ) ixt−1 2 γe−iγ t ixt + 4 Fs φ( ixt−1  t t

(2.27)

hold, where R satisfies the estimate R∞ ≤ Ct− 2 −ε Fs φH1,2 , ε > 0. 1

Proof. We decompose G (t) φ as G (t) φ = V (t) φ + W (t) φ,

(2.28)

where i x (V (t) φ) (x) = − √ θ( ) 2π t

∞

x

eit(−γp+p t ) Fs φ(p)dp

−∞

and i x (W (t) φ) (x) = − √ (1 − θ( )) t 2π

∞

x

eit(−γp+p t ) Fs φ(p)dp.

−∞

Taking x =

ξt ξ

in the relation for (V (t) φ) (x) and using (2.17) we have ξt (V (t) φ) ( ) = ξ



   ξt ξt (V− (t) φ) ( ) A + (V+ (t) φ) ( ) B, ξ ξ

(2.29)

where i ξt (V± (t) φ) ( ) := − √ ξ 2π

∞ eitf± (p,ξ) Fs φ(p)dp,

−∞

ξ with f± (p, ξ) := ± p + p ξ . Let us show that 

i ξt 3/2 ±i ) = − √ ξ e (V± (t) φ) ( ξ t

t π ξ + 4



Fs φ(∓ξ) + R± ,

(2.30)

where R± ∞ ≤ Ct− 2 −ε Fs φH1,2 , ε > 0. 1

Note that i ξt (V± (t) φ) ( ) = − √ ξ 2π

∞ eitf± (p∓ξ,ξ) Fs φ(p ∓ ξ)dp.

−∞

ξt Suppose that ξ ≤ 1. We decompose (V± (t) φ) ( ξ ) as

(V± (t) φ) (

ξt 1 2 3 ) = V± (t) + V± (t) + V± (t) , ξ

(2.31)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1647

with ⎞ ⎛ 3 i 1 V± (t) := − √ ⎝ eitf± (p∓ξ,ξ) dp⎠ Fs φ(∓ξ) 2π ⎛ i 2 V± (t) := − √ ⎝ 2π

−3

3

⎞

eitf± (p∓ξ,ξ) (Fs φ(p ∓ ξ) − Fs φ(∓ξ)) dp⎠

−3

and i (t) := − √ 2π

3 V±

∞ −3 i itf± (p∓ξ,ξ) e Fs φ(p ∓ ξ)dp − √ eitf± (p∓ξ,ξ) Fs φ(p ∓ ξ)dp. 2π −∞

3

3 3 We consider first V± (t). Integrating by parts in the first term in the R.H.S. in the definition of V± (t) we get

− √i2π

∞ ∞  ξp∓ξ eitf± (p∓ξ,ξ) Fs φ(p ∓ ξ)dp = − t√12π eitf± (p∓ξ,ξ) ±ξ(p∓ξ)+ξp∓ξ Fs φ(p ∓ ξ) 3

3

∞ +

√1 t 2π

eitf± (p∓ξ,ξ) ∂p



ξp∓ξ ±ξ(p∓ξ)+ξp∓ξ Fs φ(p

 ∓ ξ) dp.

3

Since the functions

ξp∓ξ ±ξ(p∓ξ)+ξp∓ξ

ξp∓ξ and ∂p ±ξ(p∓ξ)+ξp∓ξ are bounded for |p| ≥ 5/2 and ξ ≤ 1, we have

∞     itf (p∓ξ,ξ)  C  e ± Fs φ(p ∓ ξ)dp ≤ Fs φH1 .  t   3

Similarly we get  −3      C itf± (p∓ξ,ξ)   ≤ Fs φ 1 , e F φ(p ∓ ξ)dp s H   t   −∞

and thus,   3 V± (t) ≤ C Fs φ 1 , H t

(2.32)

uniformly on ξ ≤ 1. 1 Next we consider V± (t). Let θ1 (p) ∈ C0∞ ([−3, 3]) be such that θ1 (p) = 1 for |p| ≤ 5/2. Then, 1 V±

i (t) = − √ 2π

3

−3

e

itf± (p∓ξ,ξ)

i θ1 (p) Fs φ(∓ξ)dp − √ 2π

3 eitf± (p∓ξ,ξ) (1 − θ1 (p)) Fs φ(∓ξ)dp.

(2.33)

−3

It follows from Theorem 1.6, p. 167, of [6] that i −√ 2π

3

−3



i 3/2 ±i eitf± (p∓ξ,ξ) θ1 (p) Fs φ(∓ξ)dp = − √ ξ e t

t π ξ + 4



  Fs φ(∓ξ) + O t−1 Fs φ(∓ξ),

(2.34)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1648

  3 where O t−1 is uniform on ξ ≤ 1. Similarly to the case of V± (t) we obtain for the second term in the R.H.S. of (2.33)  3       eitf± (p∓ξ,ξ) (1 − θ1 (p)) Fs φ(∓ξ)dp ≤ C Fs φ 1 H   t  

(2.35)

−3

uniformly on ξ ≤ 1. Observe now that ⎞ ⎛ 3  itf (p∓ξ,ξ)  ± ∂ pe i p 2 (Fs φ(p ∓ ξ) − Fs φ(∓ξ)) dp⎠ . V± (t) = − √ ⎝ 1 + itp∂p f± (p ∓ ξ, ξ) 2π −3

Integrating by parts we get peitf± (p∓ξ,ξ) 1+itp∂p f± (p∓ξ,ξ) 3

2 V± (t) = − √i2π

+

√i 2π −3

3  (Fs φ(p ∓ ξ) − Fs φ(∓ξ))

−3

s φ(p∓ξ)−Fs φ(∓ξ)) peitf± (p∓ξ,ξ) ∂p (F1+itp∂ dp. p f± (p∓ξ,ξ)

(2.36)

Note that ⎛ ⎝

3

⎞1/2 p2 dp⎠ 1+t2 |p∂p f± (p∓ξ,ξ)|2

−3

≤C



ξ3/2 √ t

2ε

⎛ ⎝

3

⎛ ≤C⎝

p2−4ε dp⎠



−3

⎞1/2 ≤C

⎞1/2

3



1+

ξ3/2 √ t

p2 √ p t ξ3/2

4 dp⎠

2ε , ε < 3/4,

−3

and 3

3/2

t |p|

  ∂p f± (p ∓ ξ, ξ) + p∂p2 f± (p ∓ ξ, ξ) 2

−3

1 + t2 |p∂p f± (p ∓ ξ, ξ)|

 dp ≤ C

3/2

ξ √

t

4ε−2 3

dp 4ε−5/2

−3

|p|

 ≤C

3/2

ξ √

4ε−2

t

, ε < 7/8.

Thus, we obtain  3    p∂p (Fs φ(p ∓ ξ) − Fs φ(∓ξ))  dp ≤ Ct−1/2−ε Fs φ 1,2 , ε > 0, H  1 + itp∂p f± (p ∓ ξ, ξ) 

−3

and then it follows from (2.36) that  2  V± (t) ≤



3/2

ξ √

t

1+ε Fs φH1,2 , ε > 0.

Therefore, using (2.33), (2.34), (2.35), (2.37) and (2.32) in (2.31) we obtain (2.30) for ξ ≤ 1.

(2.37)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1649

Suppose now that ξ ≥ 1. Making the change p = yξ and using that ⎞ ⎛ # 4 2 ξ2 ⎝ ξ ξ ξ = ± yξ ∓ ξ + (yξ ∓ ξ) + 2 (y 2 ∓ 2y) + (y ∓ 1)⎠ ± ξ ξ ξ4 ξ we get i ξt ) = −√ ξ (V± (t) φ) ( ξ 2π

∞ e

 $  iλ ± η 2 +η(y 2 ∓2y)+(y∓1)

Fs φ(ξ (y ∓ 1))dy,

−∞

ξ where λ := t ξ and η := ξ ξ 2 . Note that η is bounded for ξ ≥ 1. Then, similarly to the case ξ ≤ 1, we obtain (2.30) for ξ ≥ 1. Thus, we proved (2.30). Observe that introducing (2.30) in (2.29) and using that      2

2

Fs φ(∓ξ) = ∓Fs φ(ξ) and Ae

−i

t π ξ + 4

i

− Be

−iγ

t π ξ + 4

t π ξ + 4

= γe

3 i ξt ξ −iγ (V± (t) φ) ( ) = √ θ( ) ξ 2 γe ξ ξ t



in the resulting expression we get

t π ξ + 4



Fs φ(ξ) + R1 ,

(2.38)

where R1 ∞ ≤ Ct− 2 −ε Fs φH1,2 , ε > 0. 1

Similarly to (2.29) we obtain the following decomposition for (W (t) φ) (x), (W (t) φ) (x) = ((W− (t) φ) (x)) A + ((W+ (t) φ) (x)) B, where i x (W± (t) φ) (x) = − √ (1 − θ( )) t 2π

∞ eit



±p+p xt



Fs φ(p)dp.

−∞

Integrating by parts in the R.H.S. of the last equation we get ∞ (W± (t) φ) (x) =

± √12π (1

−

θ( xt )) 1t

√1 (1 2π

− θ( xt )) 1t



±p+p xt

−∞

∞ +

eit

eit



±p+p xt





 1 2 Fs φ(p)dp p p3 ± p + xt

1 p ± p + xt

∂p Fs φ(p)dp.

(2.39)

−∞

Noting that the terms in the R.H.S. of (2.39) are equal to 0 for xt < 1 (due to the function 1 − θ( xt )) we have    p |p| x  1 ±  p + t  ≥ 1 − p = p (p + |p|) . Thus, we conclude that (W± (t) φ) (x)∞ ≤

C Fs φH1,2 . t

Finally, using (2.38) and (2.40) in (2.28) we obtain (2.27). 2

(2.40)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1650

We define %  1 N1 (f ) := β 0 %  1 N2 (f ) := β 0

0 1

0 0

0 0

0 0 0 1 0 0



 f,



 f,

1 0

0 1

0 0

0 0

0 0 1 0 0 0 0 1

 & f ,  & f ,

and N3 (f ) :=

%     & 0 0 1 0 0 0 1 0 β f, f . 0 0 0 1 0 0 0 1

We are now in position to prove the asymptotic representation for the nonlinearity. We have Lemma 2.6. Suppose that φ ∈ H2,1 . Then the following asymptotic formula for large time t holds Fs N (Gφ) =

i −iγtp e p E (p; p Fs φ(p)) Fs φ(p) + Λ (t, p; p Fs φ(p)) + R, 2t

(2.41)

where 

   Ω 0 0 + N2 (f ) , 0 Ω 0 !p"  !p"  i i p  p  i Λ (t, p; χ(p)) := − eiγtp F (χ(p)) − e−3iγt 3 I χ( ) + e3iγt 3 J χ( ) , 2t 2t 3 2t 3   3 R = O t−1−δ Fs φH1,2 in L∞ , for δ > 0, and E (p; f ) := (N1 (f ) + N3 (f ))

Ω 0

0 Ω





+ N2 (f )

0 0



    0 Ω 0 0 Ω F := (N1 (p Fs φ(p)) + N3 (p Fs φ(p))) + N2 (p Fs φ(p)) + N2 (p Fs φ(p)) Ω 0 0 Ω 0     ' p ( ' p ( p  Ω 0 p  0 0 + N2 , I :=N2 Fs φ( ) Fs φ( ) 0 0 0 Ω 3 3 3 3     ' p ( ' p ( p  0 0 p  0 Ω J :=N2 + N2 , Fs φ( ) Fs φ( ) Ω 0 0 0 3 3 3 3 $  p2 + 1 − 1 − |p| $ Ω := . |p| − p2 + 1 − 1

0 0

 ,

Proof. Let us define   √ ξt iγ t + π −3 W (ξ) := −i t ξ 2 γe ξ 4 (G (t) φ) ( ), ξ

(2.42)

so that (G (t) φ) (

3 i ξt −iγ ) = √ ξ 2 γe ξ t



t π ξ + 4



W (ξ).

(2.43)

Note that Fs N (Gφ) =

i 1 C ((− p + β) βFs (βAGφ, AGφ AGφ)) + C (αβpFc (βAu, Au Au)) . 2 2

We consider the expression 2i Fs (βAGφ, AGφ βAGφ). Using (2.43) we have

(2.44)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664



3

i 2 Fs

%

× ξ 2

(βAGφ, AGφ AGφ)

βAγe

−iγ



t π ξ + 4



∞  ip ξt = √i t √12π e ξ 0 &   t −iγ ξ +π 4

W (ξ), Aγe

ξt

1651

− e−ip ξ −iγ



W (ξ) Aγe



t π ξ + 4





W (ξ)

.

Noting that Aγe

−iγ



t π ξ + 4





=

1 0

0 1

0 0

0 0





e

−i

t π ξ + 4





−

0 0 1 0 0 0 0 1





e

i

t π ξ + 4



we get  i Fs (βAGφ, AGφ AGφ) = Mj (p) , 2 j=1 8

(2.45)

where ⎛  ∞   ξ    ξ 1 1 3 i −i π ⎝ 1 1 it p ξ − ξ −it p ξ + ξ √ e ξ 2 N1 (W (ξ)) M1 (p) := √ e 4 −e 0 t 2 2π 0 ⎛  ∞   ξ    ξ 1 1 3 i iπ ⎝ 1 0 it p ξ + ξ −it p ξ − ξ 2 4 √ M2 (p) := − √ e e ξ N1 (W (ξ)) −e 0 t 2 2π 0 ⎛  ∞   ξ    ξ 3 3 3 i −i π ⎝ 1 1 it p ξ − ξ −it p ξ + ξ √ e ξ 2 N2 (W (ξ)) M3 (p) := − √ e 4 −e 0 t 2 2π 0 ⎛  ∞   ξ    ξ 1 1 3 i −i π ⎝ 1 0 it p ξ − ξ −it p ξ + ξ 2 4 √ e ξ N2 (W (ξ)) M4 (p) := √ e −e 0 t 2 2π 0 ⎛  ∞   ξ    ξ 1 1 3 i π 1 1 it p ξ + ξ −it p ξ − ξ M5 (p) := − √ ei 4 ⎝ √ e ξ 2 N2 (W (ξ)) −e 0 t 2 2π ⎛

0 0 0 1 0 0 0 1 0 0 0 1





0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0

W (ξ)dξ ⎠ , ⎞ W (ξ)dξ ⎠ ,







0

 ∞   ξ    ξ 3 3 3 i −i π ⎝ 1 0 it p ξ + ξ −it p ξ − ξ 2 √ e ξ N2 (W (ξ)) −e M6 (p) := √ e 4 0 t 2 2π 0 ⎛  ∞   ξ    ξ 1 1 3 i −i π ⎝ 1 1 it p ξ − ξ −it p ξ + ξ 2 4 √ √ e ξ N3 (W (ξ)) M7 (p) := −e e 0 t 2 2π

⎞





⎞ W (ξ)dξ ⎠ , ⎞

W (ξ)dξ ⎠ , ⎞ W (ξ)dξ ⎠ , ⎞ W (ξ)dξ ⎠ , ⎞ W (ξ)dξ ⎠ ,

0

and ⎞ ⎛   ∞   ξ    ξ 1 1 3 i iπ ⎝ 1 0 0 1 0 it p ξ + ξ −it p ξ − ξ √ e ξ 2 N3 (W (ξ)) M8 (p) := − √ e 4 −e W (ξ)dξ ⎠ . 0 0 0 1 t 2 2π 0

We consider M1 (p). Similarly to (2.27) we show that M1 (p) = ∓ where

i −itp 3 e p (CN1 (W (±p))) 2t



1 0 0 0 0 1 0 0

 W (±p) + R3 ,

±p≥0

(2.46)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1652

R3 ∞ ≤

t

C 32 N (W (ξ)) W (ξ) ξ 1 1+δ

H1,2

≤

t

2 C 32 −η W (ξ) ξ W H1, 12 +2η , δ > 0, 1+δ ∞

(2.47)

and η > 0 small enough. It follows from (2.27) that W (ξ) = Fs φ(ξ) +

√

t ξ

− 32

 iγ

γe

t π ξ + 4



R,

(2.48)

3 −η ≤ C ξ 2 Fs φ(ξ)

+ Ct−δ Fs φH1,2 ≤ C Fs φH1,2 ,

(2.49)

1 +2η ≤ C ξ 2 Fs φ

+ Ct−δ Fs φH1,2 ≤ C Fs φH1,2 .

(2.50)

and then, 3 2 −η W ξ

∞

∞

and W (ξ)

1 L2, 2 +2η

L2

Now we show that ∂ξ W 

1

L2, 2 +2η

≤ C Fs φH1,2 .

(2.51)

From (2.42) and (2.17) we have W (ξ) = W11 (ξ) + W12 (ξ), where π 1 √ −3 t ξ 2 ei 4 A W11 (ξ) := − √ 2π

∞

e−itg(z,ξ) Fs φ(z)dz

−∞

∞

π 1 √ −3 t ξ 2 e−i 4 B W12 (ξ) := √ 2π

eith(z,ξ) Fs φ(z)dz,

−∞

ξ 1 and g (z, ξ) := z − ξ − z ξ , h (z, ξ) := z − is considered similarly. Note that

1 ξ

ξ + z ξ . We consider the term ∂ξ W11 (ξ). The other term

1 2 ∂ξ W11 (ξ) = W11 (ξ) + W11 (ξ)

with 1 W11

π 3 1 √ −7 (ξ) := √ tξ ξ 2 ei 4 A 2 2π

∞

e−itg(z,ξ) Fs φ(z)dz

−∞

and 2 W11

π i −3 (ξ) := √ t3/2 ξ 2 ei 4 A 2π

∞

e−itg(z,ξ) gξ (z, ξ) Fs φ(z)dz,

−∞

−3

where gξ (z, ξ) = ξ

1 (ξ − z). For W11 (ξ), similarly to (2.27), we get 1 W11 (ξ) =

3 −2 −2 ξ ξ AFs φ(ξ) + ξ R4 , 2

(2.52)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1653

where R4 ∞ ≤ Ct−δ Fs φH1,2 , δ > 0. Hence, 1 W11

1

L2, 2 +2η

≤ C Fs φH1,2 .

(2.53)

2 In order to treat W11 (ξ) we consider two regions, ξ ≤ 1/2 and ξ ≥ 1/2. We estimate first the norm 2 W11 (ξ) 2, 1 +2η . Integrating by parts we have 2 L

(ξ≤1/2)

2 21 22 W11 (ξ) = W11 (ξ) + W11 (ξ) ,

(2.54)

where 21 W11

(ξ) =

− √12π

√

te

iπ 4

− 32

A ξ

e

1 it ξ

∞

ξ

eitz ξ e−itz Θ(0, z)dz

−∞

and 22 W11

1 √ iπ −3 (ξ) = − √ te 4 A ξ 2 2π

∞

e−itg(z,ξ) Θ1 (ξ, z)dz,

−∞

with gξ Θ(ξ, z) := gz gz :=



 gzz gξz ( − )Fs φ(z) + ∂z Fs φ(z) , gξ gz

z ξ −3 −3 −3 − , gξ := ξ (ξ − z) , gzz := z and gξz := − ξ , z ξ

and Θ1 (ξ, z) := Θ(ξ, z) − Θ(0, z). Note that ⎛ ∞ ⎞1/2 21 2  1/√2 W11 (ξ) 2, 1 +2η ⎝ eizξ1 e−itz Θ(0, z)dz ⎠ dξ1 ≤C 0 (ξ≤1/2) L 2 −∞   2 2 2 ≤ C F e−itz Θ(0, z) L2 = C Θ(0, z)L2 ≤ C Fs φH1,2 .

(2.55)

Since      gξ  z  (z ξ + ξ z)  z  =    gz  ξ  ξ (z + ξ)  ≤ C ξ

(2.56)

and       gξz   (z − ξ)2 g ξ ξ (ξ − z) ξ  zz    gξ − gz  =  z2 (z + ξ) + z2 (z 2 − ξ 2 )  ≤ C z

(2.57)

we get |Θ1 (ξ, z)| ≤ Cξ (|Fs φ(z)| + |∂z Fs φ(z)|) .

(2.58)

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Then, we have 2 ∞ −2+4η 22 2 −itg(z,ξ) ξ W11 2, 1 +2η ≤ Ct e Θ (ξ, z)dz 1 (ξ≤1/2) L 2 −∞ L2 (ξ≤1/2) ⎛ ⎞ ∞ ∞  1/2 −2+4η = Ct 0 dξ ⎝ξ e−itg(z1 ,ξ) Θ1 (ξ, z1 )dz1 × eitg(z2 ,ξ)) Θ∗1 (ξ, z2 )dz2 ⎠ ∞ = Ct

e−iz1 t dz1

−∞

∞

−∞

eitz2  dz2

−∞



1/2 0

−2+4η

ξt

Θ1 (ξ, z1 )Θ∗1 (ξ, z2 )ei(z1 −z2 ) ξ ξ

 dξ .

−∞ −2+4η

Since Θ1 (ξ, z1 )Θ∗1 (ξ, z2 ) ξ is analytic for all  ξ ∈ C  {ξ | Re ξ = 0}, we can move the contour of ξt integration in the ξ variable in order to make Re i(z1 − z2 ) ξ > 0, and hence, we get the estimate   ξt   i(z1 −z2 ) ξ e  ≤ C

1 ξt 1 + |z1 − z2 | ξ

1+δ , δ > 0.

Thus, via estimate (2.58), using the Young’s inequality, we obtain 22 2 W11 2, 1 +2η (ξ≤1/2) L 2 ⎛ ⎞  ∞ ∞   1/2 ξ−2+4η tξdξ   ≤C (|∂z Fs φ(z2 )| + |Fs φ(z2 )|) ⎝ (|∂z Fs φ(z1 )| + |Fs φ(z1 )|) dz1 ⎠ dz2 0 ξt 1+δ −∞

⎛

∞

≤C⎝

1+|z1 −z2 | ξ

⎞

−∞

 1/2 0

−∞

2 ξ−2+4η tξdξ   dz ⎠ Fs φH1,2 (ξ≤1/2) ξt 1+δ 1+|z| ξ

2

≤ C Fs φH1,2 .

(2.59)

Using (2.55) and (2.59) in (2.54) we see that 2 W11 (ξ) 2 Let us now estimate W11 (ξ)

1

1

L2, 2 +2η (ξ≤1/2)

L2, 2 +2η (ξ≥1/2)

≤ C Fs φH1,2 .

. It follows from (2.56) and (2.57) we get

|Θ(ξ, z)| ≤ C (|Fs φ(z)| + z ∂z (Fs φ(z))) . Thus, similarly to (2.59), we have 2 2 W11 2, 1 +2η (ξ≥1/2) L 2 ⎛ ∞ ⎞ ∞ t ≤C (|Fs φ(z2 )| + |z2  ∂z Fs φ(z2 )|) ⎝ (|Fs φ(z1 )| + |z1  ∂z Fs φ(z1 )|) dz1 ⎠ dz2 (1+t(z −z ))1+δ 1

−∞

⎛ ≤C⎝

∞

2

⎞−∞ 2 t dz ⎠ Fs φH1,2 (ξ≥1/2) (1+t|z|)1+δ

2

≤ C Fs φH1,2 (ξ≥1/2) ,

−∞

and hence, 2 W11 (ξ)

1

L2, 2 +2η

≤ C Fs φH1,2 .

(2.60)

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1

Taking the L2, 2 +2η norm in (2.52) and using (2.53) and (2.60) we obtain ∂ξ W11 

1

L2, 2

≤ C Fs φH1,2 .

Thus, we prove estimate (2.51). Using (2.49), (2.50) and (2.51) in (2.47) we get R3 ∞ ≤

C Fs φH1,2 , δ > 0. t1+δ

Furthermore, making use of (2.48) in (2.46) we arrive to i M1 (p) = − e−itp N1 (p Fs φ(p)) 2t



1 0

0 1

0 0

0 0

 ˜1 (p Fs φ(p)) + R

with ˜1 R

∞

≤

C Fs φH1,2 , δ > 0. t1+δ

Similarly for Mj (ξ), j = 2, 3, 4, 5, 6, 7, 8, we obtain   i itp 0 0 1 0 ˜2, e N1 (p Fs φ(p)) (p Fs φ(p)) + R 0 0 0 1 2t   i −3it! p " ' p ( p  1 0 0 0 'p( p ˜3, 3 M3 (p) = e N2 Fs φ( ) Fs φ( ) + R 0 1 0 0 2t 3 3 3 3   i 0 0 1 0 ˜4, M4 (p) = − e−itp N2 (p Fs φ(p)) (p Fs φ(p)) + R 0 0 0 1 2t   i 1 0 0 0 ˜5, M5 (p) = eitp N2 (p Fs φ(p)) (p Fs φ(p)) + R 0 1 0 0 2t   i 3it! p " ' p ( p  0 0 1 0 'p( p ˜6, 3 M6 (p) = − e N2 Fs φ( ) Fs φ( ) + R 0 0 0 1 2t 3 3 3 3   i 1 0 0 0 ˜7 M7 (p) = − e−itp N3 (p Fs φ(p)) (p Fs φ(p)) + R 0 1 0 0 2t M2 (p) =

and M8 (p) =

i itp e N3 (p Fs φ(p)) 2t



0 0

0 0

1 0

0 1

 ˜8, (p Fs φ(p)) + R

where ˜j R

∞

≤

C Fs φH1,2 , δ > 0, j = 2, 3, 4, 5, 6, 7, 8. t1+δ

Note that, similarly to (2.45), for 12 Fc (βAu, Au βAu) we get  1 Fc (βAu, Au βAu) = M1j (p) , 2 j=1 8

with

(2.61)

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  i −itp 1 0 0 0 ˜1, (p) = ± e N1 (p Fs φ(p)) (p Fs φ(p)) + R 1 0 1 0 0 2t   i 0 0 1 0 ˜ 21 , (p Fs φ(p)) + R M12 (p) = ∓ eitp N1 (p Fs φ(p)) 0 0 0 1 2t   i −3it! p " ' p ( p  1 0 0 0 'p( p 1 ˜1, 3 Fs φ( ) Fs φ( ) + R M3 (p) = ∓ e N2 3 0 1 0 0 2t 3 3 3 3   i 0 0 1 0 ˜1, M14 (p) = ± e−itp N2 (p Fs φ(p)) (p Fs φ(p)) + R 4 0 0 0 1 2t   i 1 0 0 0 ˜1, M15 (p) = ∓ eitp N2 (p Fs φ(p)) (p Fs φ(p)) + R 5 0 1 0 0 2t   ' p ( !p" i p  0 0 1 0 'p( p ˜1, M16 (p) = ± e3it 3 N2 Fs φ( ) Fs φ( ) + R 6 0 0 0 1 2t 3 3 3 3   i 1 0 0 0 ˜ 71 (p Fs φ(p)) + R M17 (p) = ± e−itp N3 (p Fs φ(p)) 0 1 0 0 2t M11

and M18 (p) = ∓

i itp e N3 (p Fs φ(p)) 2t



0 0

0 0

1 0

0 1

 ˜1, (p Fs φ(p)) + R 8

for ±p ≥ 0, where 1 ˜ R j

∞

≤

C t1+δ

Fs φH1,2 , δ > 0, j = 1, 2, 3, 4, 5, 6, 7, 8.

Therefore, using (2.45) and (2.61) in (2.44) and ⎛

1 ix iγx ⎜ 0 e C=e ⎝ 0 0

⎞ ⎛ 0 0 1⎟ −iγx ⎜ 0 +e ⎝ −1 0⎠ 0 0

⎞ 0 0 ⎟ 0 ⎠ −1

we arrive to i  2 M (p) , 2t j=1 j 6

Fs N (Gφ) =

(2.62)

where M21

−iγtp

(p) = e



 (N1 (p Fs φ(p)) + N3 (p Fs φ(p)))

−iγtp

0 Ω 0 0





0 Ω

 ˜ 12 , (p Fs φ(p)) + R

 0 0 ˜2, N2 (p Fs φ(p)) (p) = e + N2 (p Fs φ(p)) (p Fs φ(p)) + R 2 Ω 0   0 Ω ˜ 32 , M23 (p) = −eiγtp (N1 (p Fs φ(p)) + N3 (p Fs φ(p))) (p Fs φ(p)) + R Ω 0      0 0 Ω 0 2 iγtp ˜2, N2 (p Fs φ(p)) M4 (p) = −e + N2 (p Fs φ(p)) (p Fs φ(p)) + R 4 0 Ω 0 0      ' p ( ' p ( ' p ( !p" p  Ω 0 p  0 0 p  ˜2 + N2 M25 (p) = −e−3iγt 3 N2 Fs φ( ) Fs φ( ) Fs φ( ) + R 5, 0 0 0 Ω 3 3 3 3 3 3     ' (  ' ( ' p ( ! " p p  0 0 p  0 Ω p  ˜2 p 2 3iγt p 3 + N2 N2 Fs φ( ) Fs φ( ) Fs φ( ) + R M6 (p) = e 6, Ω 0 0 0 3 3 3 3 3 3 M22



Ω 0

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

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with ˜j R

∞

≤

C Fs φH1,2 , δ > 0, j = 1, 2, 3, 4, 5, 6. t1+δ

Hence, expansion (2.41) follows from (2.62). 2 Remark 2.7. The important property of the matrix E (p; p Fs φ(p)) in the asymptotic formula (2.41) is that E has only real eigenvalues and 4 linearly independent eigenvectors for any p. We now consider the following ordinary differential equation for t ≥ 1 depending on a parameter p ∈ R dχ dt

=

i 2t

−1

p

−1

1/2

E (p; χ(p)) (χ(p)) + p eiγtp Λ (t, p; χ(p)) + p χ (1, p) = χ0 ,

R,

(2.63)

where RL1,δ2 ≤ ε, for some δ2 > 0. We have the following result Lemma 2.8. Let the initial data χ0 ∈ L∞ with a norm χ0 ∞ = ε. Then, there exists ε0 > 0 such that    for all 0 < ε < ε0 the initial value problem (2.63) has a unique solution χ ∈ C [1, ∞) ; L∞ p ≤ tδ1 , δ1 < 23 δ2 , satisfying the estimate sup |χ(p)| ≤ Cε

p≤tδ1

uniformly on t. Proof. Let us define the set  )    ∞ δ1     Xε := g ∈ C [1, ∞) ; L p ≤ t : gXε = sup g (t)L∞ p≤tδ1 + sup t ∂t g (t)L∞ p≤tδ1 ≤ 6ε , t≥1

t≥1

where ε = χ0 ∞ . We assume that χn−1 ∈ Xε and consider the linearized version of (2.63) dχn dt

=

i 2t

−1

p

−1

1/2

E (p; χn−1 (p)) χn (p) + p eiγtp Λ (t, p; p χn−1 (p)) + p χn (1, p) = χ0 (p) ,

R,

(2.64)

where n ∈ {0} ∪ N and χ0 (p, t) = χ0 (p) . Multiplying both sides of (2.64) by ⎛ 1 −1 Dn−1 (p, t) := exp i ⎝ p 2

t

⎞ 1 E (p; χn−1 (p)) dt⎠ t

1

we obtain d −1 1/2 (Dn−1 (p, t)χn ) = Dn−1 (p, t) p eiγtp Λ (t, p; χn−1 (p)) + p Dn−1 (p, t)R. dt

(2.65)

Since χn−1 ∈ Xε , from the fact that E has only real eigenvalues and 4 linearly independent eigenvectors for any p we have

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1658

|Dn−1 (p, t)| ≤ C p ,

(2.66)

uniformly on t. Observe that  t      −1  Dn−1 (p, t) p eiγtp Λ (t, p; χn−1 (p)) dt     1  t  t        ! p "  1      −1 −1 p 1 2iγtp iγt p−3 3    ≤ C  t Dn−1 (p, t)e p F (χn−1 ) (p) dt + C  t Dn−1 (p, t)e p I χn−1 ( 3 ) dt     1 1  t     ! p "    −1  + C  1t Dn−1 (p, t)eiγt p+3 3 p J χn−1 ( p3 ) dt . (2.67)   1

3

Let us prove that each term in the R.H.S. of the last inequality is bounded by C χn−1 Xε . We consider only the second term. The other terms are treated similarly. Using that eiγt



! " p−3 p 3

= Aeit



! " p−3 p 3

+ Be−it



! " p−3 p 3

,

we obtain t

 ! "   1 iγt p−3 p 3 I χn−1 ( p3 ) dt t Dn−1 (p, t)e

1

t +

t eit

=

! "   p 1 p−3 p 3 t Dn−1 (p, t)AI χn−1 ( 3 ) dt



1 ! p "   e−it p−3 3 1t Dn−1 (p, t)BI χn−1 ( p3 ) dt. 

(2.68)

1

Noting that e

 ! " it p−3 p 3



p + 3 =i 8

! p " 3

∂t eit



! " p−3 p 3

,

integrating by parts in the first term of the R.H.S. of (2.68) and using (2.66) we have  t     t    it p−3! p " 1  p p    p 3 1  −1  e  3 Dn−1 (p, t) p AI χn−1 ( ) dt ≤ C t2−δ1 χn−1 ( ) + ∂t χn−1 ( )  dt  t 3 3 3   1

1

≤C

3 χn−1 Xε

,

and similarly  t       −itp−3! p " 1  p −1 3  e 3 Dn−1 (p, t) p BI χn−1 ( ) dt ≤ C χn−1 Xε .  t 3   1

Thus, it follows from (2.67) that  t       Dn−1 (p, t) p−1 eiγtp Λ (t, p; χn−1 (p)) dt ≤ C χn−1 3 . Xε     1

(2.69)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1659

Integrating equation (2.65) we get

χn =

−1 Dn−1 (p, t)χ0

t

(p) +

−1 Dn−1 (p, t)

Dn−1 (p, t) p

−1 iγtp

e

t 1/2

p

Λ (t, p; χn−1 (p)) dt +

1

Dn−1 (p, t)Rdt.

1

(2.70) Hence, using (2.69), we obtain for ε sufficiently small

sup χn L∞ p≤tδ1  t≥1

3 ≤ χ0 ∞ + C χn−1 Xε +

t

3

t 2 δ1 |R| dt ≤ 3ε. 1

In the same way, we have sup t ∂t χn L∞ p≤tδ1  ≤ 3ε. t≥1

Therefore, χn ∈ Xε for all n. Moreover, similarly, it follows from (2.70) that −1 supt≥1 χn+1 − χn L∞ p≤tδ1  ≤ χ0 ∞ supt≥1 Dn−1 (p, t) − Dn−1 (p, t) L∞ p≤tδ1  t t −1 −1 iγtp −1 iγtp −1 D + supt≥1 D p e Λ (t, p; χ (p)) dt −D D p e Λ (t, p; χ (p)) dt n n n−1 n−1 n−1 n ∞  1 1 L p≤tδ1 t t + supt≥1 D (p, t)Rdt − Dn−1 (p, t)Rdt ≤ Cε2 χn − χn−1 Xε n ∞  δ 1

1

L

p≤t

1

(Dn := Dn (p, t)) and sup t ∂t χn+1 − ∂t χn L∞ p≤tδ1  ≤ Cε2 χn − χn−1 Xε . t≥1

Thus, χn+1 − χn Xε ≤ which means that {φn } is a Cauchy sequence in Xε .

1 χn − χn−1 Xε , 2 2

3. Proof of Theorem 1.1 Let us consider the following initial–boundary value problem ⎧ −1 ⎪ ⎨ Lu = ∂x  N (u) , t > 0, x > 0, u (x, 0) = u0 (x), x > 0, ⎪ ⎩ u(0, t) = 12 Bh(t), t > 0, where N (u) :=

i C (D− (βAu, Au βAu)) . 2

(3.1)

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1660

It follows from Proposition 2.1 that the solution of (3.1) is given by t u = G(t)φ0 + Fs

eiγp(τ −t)

0

1 Fs N (u)dτ, p

(3.2)

where

φ0 (x, t) = u0 +

2 p Fs π p

t

i eiγpτ Ch(τ )dτ. 2

0

We introduce the function space  *  + XT := φ ∈ C [0, T ] ; L2 ; φXT < ∞ , where −σ

φXT = sup t t∈[0,T ]

  1 −1 +σ J φ (t)H2 + φ (t)H2 + t φ (t)H2,1 + t 2 φ (t)∞ .

Here 0 < σ < min{δ/7, 1/7}, with δ defined in Lemma 2.6, and J is given by (1.7). The local existence in the function space XT can be proved by a standard contraction mapping principle. We state it without a proof. Theorem 3.1. Let u0 ∈ H2,1 , h(t) ∈ H1,2 and the norm u0 H2,1 + hH1,2 = ε. Also suppose that u0 (0) = h(0). Then, there exist ε0 > 0 and T > 1 such that for all 0 < ε < ε0 the initial value problem (3.1) has a   √ unique local solution u ∈ C [0, T ] ; H2,1 with the estimate uXT < ε. Remark 3.2. For our convenience, in Theorem 3.1, we suppose that the initial and boundary data are small. √ We can relax this assumption if we do not need that the time T > 1 and uXT < ε. Let us prove that the existence time T can be extended to infinity which then yields the result of Theorem 1.1. Suppose that this is false. That is, assume that there exists a minimal time T > 0 such that √ √ the a priori estimate uXT < ε does not hold, namely, we have uXT = ε. Let us first prove that σ

uH2 ≤ Cε t . It follows from (3.2) and uXT =

√

(3.3)

ε that in order to get (3.3), it is enough to show that

G(t)φ0 H2 ≤ C (hH1,1 + u0 H2 )

(3.4)

and t Fs eiγp(τ −t) 1 Fs N (u)dτ p

t ≤C

3 hH1,1

H2

0

p 2 π Fs p2

= e−x we have

⎛ G(t)φ0 =

(3.5)

0

 Integrating by parts in the expression for φ0 and using that

2

u∞ uH1 dτ.

+C

1 1 −x 1 p e Bh(t) + G(t) ⎝u0 − e−x Bh(0) − √ Fs 2 2 2 2π p

t 0

⎞ eiγpτ Bh (τ )dτ ⎠ .

(3.6)

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1661

Therefore, since u0 (0) = 12 Bh(0), applying estimate (2.14) of Lemma 2.2 and estimate (2.16) of Lemma 2.3 we get (3.4). Since t t 1 2 iγp(τ −t) Fs e Fs N (u)dτ ≤ C u∞ u dτ, p 0

0

to prove (3.5) we only need to show that t t 2 1 3 2 ∂x Fs eiγp(τ −t) Fs N (u)dτ ≤ C hH1,1 + C u∞ uH1 dτ. p 0

Using that t ∂x2 Fs

p2 p

 = 1−

(3.7)

0

1 p(p+p)



p and pFs N (u) = N ( 12 Bh(t)) + Fc ∂x N (u) we get

1 eiγp(τ −t) p Fs N (u)dτ

0

t t −iγp(t−τ ) = − Fs e pFs N (u)dτ + Fs e−iγp(t−τ ) O(1)Fs N (u)dτ 0

0

t t t 1 −iγp(t−τ ) −iγp(t−τ ) = − Fs e Fs N ( 2 Bh(t))dτ − Fs e Fc ∂x N (u)dτ + Fs e−iγp(t−τ ) O(1)Fs N (u)dτ. 0

0

0

(3.8) Observe that t t t Fs e−iγp(t−τ ) Fc ∂x N (u)dτ + Fe−iγp(t−τ ) O(1)Fs N (u)dτ ≤ C u2 u 1 dτ. H ∞ 0

0

(3.9)

0

t eiγpτ N ( 12 Bh(τ ))dτ we get

Integrating by parts 0

t e

iγpτ

0

1 iγpt 1 1 1 1 1 e N ( Bh(0)) + iγ N ( Bh(τ ))dτ = −iγ N ( Bh(t)) + iγ 2 p 2 p 2 p

t

1 eiγpτ ∂τ N ( Bh(τ ))dτ, 2

0

and hence, t 1 Fs e−iγp(t−τ ) Fs N ( Bh(t))dτ ≤ C h3 1,1 H 2

(3.10)

0

Therefore, we see that estimate (3.7) follows from (3.8), (3.9) and (3.10). Next we prove that σ

J uH2 ≤ Cε t

(3.11)

and σ+1

uH2,1 ≤ Cε t

.

(3.12)

I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

1662

Using (2.11) of Lemma 2.2 in (3.6) we have ⎞ ⎛ )  t 1 1 1 p J G(t)φ0 = Bh(t)J e−x + J G(t) u0 − Bh(0)e−x − √ Fc e−iγpt ∂p ⎝ 2 eiγpτ Bh (τ )dτ ⎠ . 2 2 2π p 0

Note that, by (2.16) of Lemma 2.3, ⎞ ⎛ t Fc e−iγpt ∂p ⎝ p 2 eiγpτ Bh (τ )dτ ⎠ p 2 0 H t t p p iγpτ  iγpτ  ≤ Fs p2 e Bh (τ )dτ + Fs p2 e τ Bh (τ )dτ 2 H

0

≤ C hH1,2 ,

H2

0

and moreover, (2.13) and u0 (0) = 12 Bh(0) imply   )  u0 − 1 Bh(0)e−x J G(t) u0 − 1 Bh(0)e−x ≤ Fs u0 − 1 Bh(0)e−x ≤ 2,1 . 2 2 2 H2 H1,2 H Then, as −x J e

H2

≤ C t ,

we obtain J Gφ0 H2 ≤ C u0 H2,1 + C hH1,2 . Thus, using (3.2), Lemma 2.4 and uXT = J uH2

√

t −1 ≤ J G(t − τ ) ∂x  N (u)(τ )dτ

(3.13)

ε we see that t σ

H2

0

2

+ J G (t) φ0 H2 ≤ Cε t + C

u∞ N (u)L2,1 dt. 0

(3.14) Note that 2

N (u)L2,1 ≤ C u∞ (∂t uL2,1 + uH1,1 ) . From the equation (3.1) we get   −1 ∂t uL2,1 ≤ C ∂x  uL2,1 + ∂x  N (u) L2,1   2 2 ≤ C uH1,1 + h∞ + u∞ uH1,1 + u∞ ∂t uL2,1 , and then,     2 2 ∂t uL2,1 1 − C u∞ ≤ C uL2,1 + h∞ + u∞ uH1,1 , what implies

(3.15)

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1663

 C  2 uL2,1 + h∞ + u∞ uH1,1 . 1 − Cε √ Hence, it follows from (3.15) and uXT = ε that ∂t uL2,1 ≤

t 2

σ

u∞ N (u)L2,1 dt ≤ Cε5/2 t , 0

for ε small enough. Therefore, using the last inequality in (3.14) we attain (3.11). Relation (3.12) follows √ from the identity xu = J u − itγ∂x ∂1x  u, (3.11), (3.3) and uXT = ε. We now estimate the solution u in L∞ space. We write the solution u in the following form u (x, t) = Gφ(p, t).

(3.16)

Substituting (3.16) in (3.1) we obtain for the new function φ(p, t) the equation ∂t (p Fs φ(p, t)) = eiγpt Fs N (Gφ) .

(3.17)

Note that Lemma 2.6 implies i E (p; p Fs φ(p)) (p Fs φ(p)) + eiγpt Λ (t, p; p Fs φ(p)) + eiγpt R, 2t   3 where R = O t−1−δ Fs φH1,2 in L∞ , δ > 0. Using (2.12) we get eiγpt Fs N (Gφ) =

t−σ Fs φH1,2 ≤ Ct−σ (J uH2 + uH2 ) ≤ Cε.

(3.18)

(3.19) 1

Then, R∞ ≤ εt−1−(δ−3σ) . Suppose that p ≤ t2σ . Multiplying (3.17) by p 2 and using (3.18) we get ∂t (χ(p, t) = 3/2

with χ(p, t) := p

i −1 −1 1/2 p E (p; χ(p, t)) χ(p, t) + eiγpt p Λ (t, p; χ(p, t)) + p eiγpt R, 2t Fs φ(p, t). Thus, from Lemma 2.8 (with δ2 < δ − 4σ) we get    3/2  sup p Fs φ(p, t) ≤ Cε.

(3.20)

p≤t2σ

In the case of p ≥ t2σ , using (3.19), we have  3      2   sup p 2 Fs φ = t−σ sup p Fs φ ≤ Ct−σ Fs φH1,2 ≤ Cε.

p≥t2σ

(3.21)

p≥t2σ

Therefore, from (3.16), (3.20), (3.21), Lemma 2.5 and Theorem 3.1 it follows that  3 1 sup t 2 u (t)∞ ≤ C sup p 2 Fs φ t≥1

t≥1

−δ

∞

+ t

Also, using that u∞ ≤ C uH1 and estimate (3.3) we obtain 1

sup t 2 u (t)∞ ≤ Cε. t≤1

 Fs φH1,2 ≤ Cε.

(3.22)

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I.P. Naumkin / J. Math. Anal. Appl. 434 (2016) 1633–1664

Hence, relations (3.3), (3.11), (3.12) and (3.22) imply uXT ≤ Cε <

√

ε,

for ε > 0, small enough, which leads to the desired contradiction. Thus, there exists a unique global solution   1 u ∈ C [0, ∞) ; H2,1 of (1.4) with the time decay estimate supt≥1 t 2 u (t)∞ ≤ C. Theorem 1.1 is now proved. Acknowledgments We would like to thank the referees for useful comments. References [1] I. Bejenaru, S. Herr, The cubic Dirac equation: small initial data in H1 (R3 ), Comm. Math. Phys. 335 (1) (2015) 43–82. [2] T. Candy, Global existence for an L2 critical nonlinear Dirac equation in one dimension, Adv. Differential Equations 16 (2011) 643–666. [3] V. Delgado, Global solutions of the Cauchy problem for the (classical) coupled Maxwell–Dirac and other nonlinear Dirac equations in one space dimension, Proc. Amer. Math. Soc. 69 (1978) 289–296. [4] J. Dias, M. Figueira, Time decay for the solutions of a nonlinear Dirac equation in one space dimension, Ric. Mat. 35 (2) (1986) 309–316. [5] M. Escobedo, L. Vega, A semilinear Dirac equation in Hs (R3 ) for s > 1, SIAM J. Math. Anal. 28 (1997) 338–362. [6] M.V. Fedoryuk, Asymptotics: Integrals and Series, Nauka, Moscow, 1987, 544 pp. (in Russian). [7] R. Finkelstein, C. Fronsdal, P. Kaus, Nonlinear spinor field, Phys. Rev. 103 (1956) 1571–1579. [8] R. Finkelstein, R. LeLevier, M. Ruderman, Nonlinear spinor fields, Phys. Rev. 83 (1951) 326–333. [9] A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002) 1–39. [10] A.S. Fokas, A.A. Himonas, D. Mantzavinos, The nonlinear Schrödinger equation on the half-line, Trans. Amer. Math. Soc. (2015), http://dx.doi.org/10.1090/tran/6734, in press. [11] D. Gross, A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235–3253. [12] N. Hayashi, E. Kaikina, Nonlinear Theory of Pseudodifferential Equations on a Half-Line, North-Holland Mathematics Studies, vol. 194, Elsevier Science B.V., Amsterdam, 2004, 319 pp. [13] N. Hayashi, P.I. Naumkin, The initial value problem for the cubic nonlinear Klein–Gordon equation, Z. Angew. Math. Phys. 59 (6) (2008) 1002–1028. [14] A.D. Ionescu, F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal. 266 (1) (2014) 139–176. [15] S. Machihara, One dimensional Dirac equation with quadratic nonlinearities, Discrete Contin. Dyn. Syst. 13 (2) (2005) 277–290. [16] S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension, Commun. Contemp. Math. 9 (3) (2007) 421–435. [17] S. Machihara, M. Nakamura, K. Nakanishi, T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal. 219 (2005) 1–20. [18] S. Machihara, K. Nakanishi, T. Ozawa, Small global solutions and the relativistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoam. 19 (2003) 179–194. [19] S. Machihara, K. Nakanishi, K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math. 50 (2) (2010) 403–451. [20] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (3) (1991) 479–493. [21] H. Pecher, Local well-posedness for the nonlinear Dirac equation in two space dimensions, Commun. Pure Appl. Anal. 13 (2) (2014) 673–685. [22] I.E. Segal, Non-linear semi-groups, Ann. of Math. 78 (1963) 339–364. [23] I.E. Segal, Quantization and dispersion for non-linear relativistic equations, in: Proceeding Conf. Math. Theory Elem. Part, MIT Press, Cambridge, Mass, 1966, pp. 79–108. [24] I.E. Segal, Dispersion for non-linear relativistic equations, II, Ann. Sci. Éc. Norm. Supér. 1 (4) (1968) 459–497. [25] S. Selberg, A. Tesfahun, Low regularity well-posedness for some nonlinear Dirac equations in one space dimension, Differential Integral Equations 23 (2010) 265–278. [26] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy, Phys. Rev. D 1 (1970) 2766–2769. [27] W. Strauss, Nonlinear scattering theory, in: J.A. La Vita, J.P. Marchards (Eds.), Scattering Theory in Mathematical Physics, Reidel, Dordrecht, 1974, pp. 53–78. [28] W.A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1) (1981) 110–133. [29] W.E. Thirring, A soluble relativistic field theory, Ann. Physics 3 (1958) 91–112. [30] N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math. 22 (1) (1998) 193–211.