ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS TO THE QUADRATIC NONLINEAR SCHRÖDINGER

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ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS TO THE QUADRATIC NONLINEAR SCHRÖDINGER

EQUATIONS IN TWO-DIMENSIONAL SPACES

NAKAO HAYASHI and PAVEL I. NAUMKIN Received 15 February 2001 and in revised form 20 May 2001

We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrödinger equation in two-dimensional spacesi∂tu+(1/2)∆u=ᏺ(u),(t, x)∈R×R²; u(0, x)=ϕ(x),x∈R², whereᏺ(u)=₂

j,k=1(λjk(∂x_ju)(∂x_ku)+µjk(∂x_ju)(∂x_ku)), where λjk, µjk∈C. We prove that if the initial dataϕsatisfy some analyticity and smallness conditions in a suitable norm, then the solution of the above Cauchy problem has the asymptotic representation in the neighborhood of the scattering states.

2000 Mathematics Subject Classification: 35Q35.

1. Introduction. We consider the large time asymptotic behavior of small analytic solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in two-dimensional spaces

i∂tu+1

2∆u=ᏺ(u), (t, x)∈R×R², u(0, x)=ϕ(x), x∈R²,

(1.1)

with quadratic nonlinearity

ᏺ(u)= 2 j,k=1

λjk

∂x_ju

∂x_ku +µjk

∂x_ju

∂x_ku

, (1.2)

whereλjk, µjk∈C. In [8], we proved the global in time existence of small analytic solutions to the Cauchy problem (1.1) and showed that the usual scattering states exist. In [3], a global existence theorem of small solutions to (1.1) withλjk=0 was shown in the usual weighted Sobolev space by using the method of normal forms by Shatah [12]. In the present paper, we continue to study the asymptotic behavior in time of solutions to the Cauchy problem (1.1) and obtain the asymptotic expansion of solutions in the neighborhood of the scattering states.

We use the following classification of the scattering problem. If the usual scattering states exist inL²sense, then we call the scattering problem a super-critical problem.

If the usual scattering states do not exist and theL²norm of the nonlinearity decays likeCt^−δ, then we call the problem a critical one, whenδ=1 and a sub-critical one,

when 0< δ <1. The problem under consideration is classified as super-critical since the usual scattering states were shown, in [8], to exist inL². In [10], the asymptotic expansion was obtained in the neighborhood of scattering states for small solutions to the nonlinear nonlocal Schrödinger equations with nonlinearities of Hartree type

ᏺ^(u)=u(t, x)

dy|x−y|^−δu(t, y)² (1.3) in the super-critical case 1< δ < n. The critical caseδ=1 was treated in [13], where the asymptotic expansion of small solutions in the neighborhood of the modified scatter- ing states was obtained. In the case of critical power nonlinearityᏺ(u)= |u|²uin one- dimensional spaces, the asymptotic expansion of solutions was constructed in [11]. In [5,6], the sub-critical scattering problem in one-dimensional spaces was studied for the nonlinear Schrödinger equation with power nonlinearityᏺ(u)=t^1−δ|u|²uand Hartree type nonlinearity (1.3) with 0< δ <1. Roughly speaking, they used the asymp- totic expansion in the neighborhood of the final states to the transformed equations for the new dependent variable

w=Ᏺᐁ(−t)u(t)exp

i t

1t^−δᏲᐁ(−t)u(t)²dt

(1.4)

(in the case of the power type nonlinearity).

Thus the asymptotic expansions of solutions to the nonlinear Schrödinger equa- tions were studied extensively in the case of the nonlinear terms without derivatives of unknown function and satisfying the gauge condition (i.e., having the self-conjugate propertyᏺ(u)=e^−iθᏺ(e^iθu)for anyθ∈R). The present paper is concerned with the derivative nonlinear Schrödinger equations which do not satisfy the gauge condition.

The presence of derivatives in the nonlinear term implies the so-called derivative loss and the absence of the gauge condition makes it difficult to estimate the norm involv- ing the operator᏶=x+it∇, which plays a crucial role in the large time asymptotic behavior of solutions to the nonlinear Schrödinger equations. To overcome these ob- stacles, we use the analytic function spacesA^m,pb defined in (1.9) and the operators ᏼ=x·∇+2t∂tandᏽ=x·∇+it∆.

To state our result precisely, we now give notation and function spaces. We de- note ∂x_j =∂/∂x_j and ∂^α= ∂^αx₁¹∂x^α₂², where α∈ (N∪ {0})². We define the following differential operators ᏼ =x· ∇ +2t∂t, ᏽ =x· ∇ +it∆, ᏶=x+it∇and the vec- torΩ=(Ω^(j,k))(j,k=1,2), where the operatorsΩ^(j,k)=xj∂k−xk∂j act as the angular derivatives. These operators help us to obtain the time decay properties of the linear Schrödinger evolution group

ᐁ^(t)φ= 1 2π it

e(i/2t)(x−y)²φ(y)dy=Ᏺ^{−1e−(it/2)ξ2}Ᏺ^{φ, (1.5)}

whereᏲ^φ≡φ(ξ)ˆ =(1/2π ) e^−i(x·ξ)φ(x)dx denotes the Fourier transform of the function φ(x), and Ᏺ⁻¹ is the inverse Fourier transformation defined by Ᏺ⁻¹φ≡ φ(x)ˇ =(1/2π ) e^i(x·ξ)φ(ξ)dξ. Note that the free Schrödinger evolution groupᐁ(t)

also can be represented asᐁ(t)=ᏹ(t)Ᏸ(t)Ᏺᏹ(t), whereᏹ(t)=exp(ix²/2t), the di- lation operator is(Ᏸ(t)φ)(x)=(i/t)φ(x/t), then the inverse free Schrödinger evolu- tion group is written asᐁ⁽−t)= −ᏹ⁽−t)iᏲ⁻¹Ᏸ^(1/t)ᏹ⁽−t), whereᏰ^−1(t)= −iᏰ^(1/t) is the inverse dilation operator. We define the extended vectorsΓ =(ᏼ,Ω,∇),Γ= (ᏼ+2,Ω,∇), andΘ=(ᏽ,Ω,∇). We have the following relations:

ᏽ=ᏼ−2itᏸ=᏶·∇ =ᐁ^(t)xᐁ⁽−t)·∇ =itᏹ^(t)∇ᏹ^(t)·∇, (1.6)

whereᏹ(t)=e^ix2/2t,ᏸ=i∂t+(1/2)∆. The commutation relations [ᏽ^,∇]=[ᏼ^,∇]= −∇, [ᏽ^,᏶^]=[ᏼ^,᏶^]=᏶^, [ᏼ,ᏽ]=[Ω,ᏼ]=[Ω,ᏽ]=0,

∂k,᏶l

=δ^(k)_l , Ω^(j,k)x , ∂l

=δ^(k)_l ∂j−δ^(j)_l ∂k,

(1.7)

whereδ^(k)_j =1 ifj=kandδ^(k)_j =0 ifj≠kare used freely in the paper. We denote the usual Lebesgue space byL^p(R²)with the normφp=(_R2|φ(x)|^pdx)^1/pif 1≤p <∞ andφ∞=ess sup{|φ(x)|;x∈R²}ifp= ∞. For simplicity we write· = ·2. The weighted Sobolev space is defined by

H^m,kp

R²

= φ∈L^p

R²

:x ^ki∇ ^mφ_p<∞

, (1.8)

where m, k ∈ R⁺, 1≤p ≤ ∞, x =√

1+x². We write for simplicity H^m,k(R²)= H^m,k2 (R²)and the normφm,k= φm,k,2. Now we define the analytic function space

A^mb =

φ∈L² R²

; φA^m_b =

|β|≤m

α

b^|α|

α! Γ^α+βφ<∞

, (1.9)

where the vectorΓ=Γ(t)=(ᏼ,Ω,∇),b=b(t)=b0+(a−b0)(log(e+t))^−γ, 0< b0<

a <1,γ >0 is sufficiently small. Similarly, we write

A^mb =

φ∈L² R²

; φA^m_b =

|β|≤m

α

b^|α|

α! Γ^α+βφ<∞

. (1.10)

Here the summation is over all admissible multi-indicesα. We often use the summa- tions convention if it does not cause confusion. By[s]we denote the largest integer less than or equal tos. LetC(I;B)be the space of continuous functions from a time intervalIto a Banach spaceB. We denote different positive constants by the same letterC. We introduce the following functional spaces

Xb= u∈C

R;L² R²

; uXb<∞ , Yb=

u∈C R;L²

R²

; sup

t>0

u(t)

Y_b<∞

, (1.11)

where

uXb=sup

t>0

u(t)

A³_b+sup

t>0

t^−1−η

|γ|≤1

᏶^γu(t)

A²_b

+

|γ|=1

_∞

0

Θ^γu

A³_bbdt+

|γ|=1,|σ|≤1

_∞

1

Θ^γ᏶^σu_A³

b

bdt t^1+η +sup

t>0

t^1−2η

α

b^|α|

α! ∂tᏲᐁ(−t)Γ^α∇u(t)_∞ +

|δ|≤3

_∞

1

α

b^|α|

α! ∂tᏲᐁ(−t)Γ^α+δu(t)t^2η−1/2dt, u(t)

Y_b=u(t)

A²_b+

|β|+|γ|≤1

t^{−|β|−|γ|−η}᏶^γΘ^βu(t)

A¹_b

+t^1−η

|γ|+|δ|≤1

α

b^|α|

α! ∂tᏲᐁ⁽−t)Γ^α+γΘ^δu(t),

(1.12)

whereη >0 is sufficiently small. We define the constants{bn}such that

0< bn< bn−1<···< b1< b0< a <1. (1.13) Let u0(t)=ᐁ(t)u⁺ with some final state u⁺ ∈L² and un(t), n= 1,2, . . . , be the solution to the final problem for the linear Schrödinger equations

ᏸ^un=

n−1

m=0

ᏺ^un−1−m, um

, (1.14)

such that lim_t→∞un(t)=0 inL², whereᏸ=i∂t+(1/2)∆and ᏺ(φ, ψ)=

2 j,k=1

λjk

∂x_jφ

∂x_kψ +µjk

∂x_jφ¯

∂x_kψ¯

. (1.15)

From [8] we see that if the initial dataϕ∈A³aare such thatxjϕ∈A²aforj=1,2 and the normϕ_A³_a+ x1ϕ_A²_a+ x2ϕ_A²_a =εis sufficiently small, then the final state u⁺∈A²a₁, whereb0< a1< a, henceu0∈Yb₀ andu−u0Yb0≤Cε²t^−w for allt≥1, wherew∈(0,1/2).

Now we state the main result in this paper.

Theorem1.1. We assume that the initial dataϕ∈A³a are such thatxjϕ∈A²afor j=1,2and the normϕ_A³_a+x1ϕ_A²_a+x2ϕ_A²_a=εis sufficiently small. Then there exists a unique global solutionu(t, x)∈A³b(t)of the Cauchy problem (1.1). Moreover, the estimates

un(t)

Y_bn≤Cnεⁿ⁺¹t^−nw, n=0,1,2, . . . (1.16) and the asymptotics

u(t)−

n−1 m=0

um(t)

Y_bn

≤Cnεⁿ⁺¹t^−nw, n=1,2, . . . (1.17)

are valid for allt≥1, wherew∈(0,1/2)and

Cn=C(n+1)²ⁿ



ⁿ

j=0

log bj

b_j+1 −1



2n

, (1.18)

whereCis a positive constant independent ofnandbj.

We assume inTheorem 1.1that 0< a <1. This ensures that the function spaceA³a

for the initial data is not empty, as in [1,2], we can see that our result is valid for the initial functionφ, which has analytic continuationΦto the domain

=

z∈C²; zj=xj+iyj, xj∈R,−C1−xjtanϑ < yj< C1+xjtanϑ, j=1,2 , (1.19) such that

Φ(z)²dx dy <∞, (1.20) whereϑ∈(0, π /2), sinϑ=C2, andC1, C2∈(a,1). For example, we can take 1/(1+x⁴), e^−x2 as the initial data for the Cauchy problem (1.1).

Denoteu⁺₀(t, ξ)=u⁺(ξ)and

u⁺n(t, ξ)= −1 4

n−1 m=0

u⁺_n−1−m

t,ξ 2

u⁺m

t,ξ

2 ²

j,k=1

λjkξjξk

t

∞e^itξ2/4dτ iτ

−1 4

n−1

m=0

u⁺_n−1−m

t,−ξ 2

u⁺m

t,−ξ

2 2

j,k=1

µjkξjξk

t

∞e^3itξ2/4dτ iτ.

(1.21)

Corollary1.2. Let the conditions ofTheorem 1.1be fulfilled. Then the following asymptotics inL²sense

Ᏺᐁ(−t)u(t)=u⁺(ξ)+

n−1 j=1

u⁺_j(t, ξ)+O t^−nw

(1.22)

are valid for large timet≥1, wheren=1,2, . . . .

For the convenience of the reader we now give the outline of the proof ofTheorem 1.1. As in [8] we apply the operatorᏲᐁ⁽−t)to (1.1) to get

i∂tᏲᐁ(−t)u=I(t, ξ)+R(t, ξ), (1.23) where

I(t, ξ)= 1 it

2 j,k=1

λjkᏰ2E²

Ᏺᐁ(−t)∂x_ju

Ᏺᐁ(−t)∂x_ku

+µjkᏰ−2E⁶

Ᏺᐁ⁽−t)∂x_ju

Ᏺᐁ⁽−t)∂x_ku ,

(1.24)

E =e^itξ2/2, andR is a remainder term since in [8] we proved the estimate R ≤ Ct^−1−w

|α|≤1Θ^αu²,Θ=(Q,Ω,∇), 0< w <1/2. Then we show that the first term of the integral I(t, ξ)dtis also convergent in view of the oscillating factorE. Roughly speaking, in [8] the following estimate was shown:

u(t)≤u(1)+C t

1τ^−1−w

|α|≤1

Θ^αu²dτ. (1.25)

Similarly, we have

|β|≤1

Γ^βu(t)≤

|β|≤1

Γ^βu (1)+C

t

1τ^−1−w

|α|≤1,|β|≤1

Θ^αΓ^βu²dτ. (1.26) However, the right-hand sides of (1.25) and (1.26) contain an additional operatorΘ(de- rivative loss with respect to derivativeΘ). This is the reason why we used the analytic function spaces involving generalized derivativeΓ, enabling us to get an additional regularity with respect to operatorΓ, hence we obtain the estimate

|β|≤1

b(t)^|β|

β! Γ^βu(t)< Cε. (1.27)

Similarly, we have

|β|

≤1b₁^|β|

β! Γ^βu(t)−Γ^βu(s)< Cε²|t|^−w (1.28) for all t > s >0. The last estimate implies existence of the usual scattering states u⁺. Method of analytic function spaces involving usual derivatives was used by many authors (e.g., see [4,9]) and analytic function spaces involving the generalized deriva- tives was used in [7]. By the definition ofun(t)we have withᏸ=i∂t+(1/2)∆

ᏸ



u−

n−1 m=0

um



=ᏺ



u−

n−1 m=0

um,

n−1 m=0

um



+ᏺ



ⁿ⁻¹

m=0

um, u−

n−1 m=0

um





+ᏺ



u−

n−1

m=0

um, u−

n−1

m=0

um



+R,

(1.29)

whereRis the remainder term sinceRY_bn+1≤Cεⁿ⁺²|t|^{−1−(n+1)w}. InSection 3, we will prove that the other three terms are estimated by Cεⁿ⁺¹|t|^−1−nw in the norm Yb_n+1. Then via the inequality

Γ^αu_Y

bn+1≤C



ⁿ

j=1

log bj

bj+1

₋1



|α|

uY_bn, (1.30)

for any|α|we obtain the desired result.

The rest of the paper is organized as follows. InSection 2, we state some preliminary estimates concerning the analytic functional spacesA^mb.Section 3is devoted to the proof ofTheorem 1.1andCorollary 1.2.

2. Preliminary estimates. We summarize some lemmas proved in [7,8], which are necessary to prove the theorem.

Lemma2.1. Letφ∈A^m,pb , then

φA^m,p_b ≤e^2bφ_A^m,p

b , (2.1)

where

A^m,pb =

φ∈L^p R²

; φ_A^m,p

b =

|β|≤m

α

b^|α|

α! Γ^α+βφ_p<∞

,

A^m,pb =

φ∈L^p R²

;φA^m,p_b =

|β|≤m

α

b^|α|

α! Γ^α+βφ_p<∞

,

(2.2)

and2≤p≤ ∞.

Lemma2.2. The following commutation relations are valid:

∂_x^l_j,᏶x_j

=l∂^l_x⁻_j¹,

᏶^l_j, ∂x_j

= −l᏶^lx⁻_j¹,

ᏼ^l᏶x_j=

0≤m≤l

C_l^m᏶x_jᏼ^l−m, ᏼ^l∂x_j=

0≤m≤l

C_l^m(−1)^m∂x_jᏼ^l−m

᏶x_jᏼ^l=

0≤m≤l

C_l^m(−1)^mᏼ^l−m᏶x_j, ∂x_jᏼ^l=

0≤m≤l

C_l^mᏼ^l−m∂x_j

Ω^lx_jx_k∂x_j=

0≤2m≤l

(−1)^mC_l^2m∂x_jΩ^l−2mx_jx_k +

0≤2m+1≤l

(−1)^m+1C_l^2m+1∂x_kΩ^l−2m−1x_jx_k ,

∂x_jΩ^lx_jx_k=

0≤2m≤l

(−1)^m+1C_l^2mΩ^lx⁻_j^2mx_k∂x_j+

0≤2m+1≤l

(−1)^mC_l^2m+1Ωx^l−_j^2mx_k⁻¹∂x_k, (2.3)

whereC_l^m=l!/(l−m)!m!is the binomial coefficient.

Lemma2.3. The estimate

᏶∇φ_A^m,p

b ≤C

|α|=1

Θ^αφ

A^m,p_b (2.4)

is true.

Lemma2.4. The inequalities C1∂x_jφ

A^m,p_b ≤

|β|≤m

α

b^|α|

α!

∂x_jΓ^α+βφ_p≤C2∂x_jφ

A^m,p_b , C1᏶x_jφ

A^m,p_b ≤

|β|≤m

b^|α| α!

᏶x_jΓ^α+βφ_p+φ_A^m,p

b

≤C2

᏶x_jφ

A^m,p_b +φ_A^m,p

b

(2.5)

are true for allt >0, whereC1, C2>0.

We define the evolution operator

ᐂ^(t)φ=Ᏺ^−1eiξ2/2tᏲ^φ= t 2iπ

e(−it/2)(ξ−y)²φ(y)dy (2.6)

and᏷=Ᏺᏹᐁ(−t). By a direct calculation we see that ᐂ(−t)

E^ν−1φ

=ᏰνE^ν(ν−1)ᐂ(−νt)φ, (2.7) withᏰνφ=(1/ν)φ(ξ/ν)andE=e^itξ2/2, whereν≠0. We need the following lemma to get the decay estimates of the solution for large time.

Lemma2.5. The estimate ᏰνE^ν(ν−1)

ᐂ⁽−νt)−1

(᏷^φ)(᏷^ψ) +ᏰνE^ν(ν−1)(᏷ψ)

ᐂ(−νt)−1 (᏷φ)

≤Ct^η−1/2

|α|≤1,|β|≤1

᏶^αφ᏶^βψ (2.8)

is valid for allt >0, whereν≠0,η >0is sufficiently small.

3. Proof of Theorem1.1. We consider the linear Schrödinger equation

ᏸun=

n−1

m=0

ᏺun−1−m, um

. (3.1)

Sinceu0(t)is a solution of linear Schrödinger equation it is easy to see thatu0(t)Y_b

≤C0ε. Then by induction we assume that 0

uj(t)_Y

bj ≤Cjε^j+1|t|^−jw, 0≤j≤n−1. (3.2) Multiplying both sides of (3.1) by᏷Γ^α+δ, where᏷=Ᏺᏹᐁ(−t), we get

ᏸξ᏷Γ^α+δun

= 1 it

n−1 m=0

β≤α

γ≤δ

Cα^βC_δ^γ 2 j,k=1

λjkE

᏷Γ^δ−γf

᏷Γ^γg+µjkE¯³

᏷Γ^δ−γf

᏷Γ^γg ,

(3.3)

whereᏸξ =i∂t+(1/2t²)∆ξ, f =Γ^α−β∂x_ju_n−1−m, g=Γ^β∂x_kum, Cα^β =α!/(α−β)!β!,

|δ| ≤2. Applying the operatorᐂ(−t)=Ᏺ⁻¹ᏹ(t)Ᏺto both sides of (3.3) we obtain by virtue of identity (2.7)

i∂tᐂ(−t)᏷Γ^α+δun(t)

= 1 it

n−1

m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

ᐂ(−t) λjkE

᏷Γ^δ−γf

᏷Γ^γg+µjkE¯³

᏷Γ^δ−γf

᏷Γ^γg

= 1 it

n−1 m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

λjkᏰ2E²ᐂ(−2t)

᏷Γ^δ−γf

᏷Γ^γg +µjkᏰ−2E⁶ᐂ(2t)

᏷Γ^δ−γf

᏷Γ^γg .

(3.4) Then we write the identity

ᐂ⁽−2t)

᏷Γ^δ−γf

᏷Γ^γg

=

ᐂ⁽−2t)᏷Γ^δ−γf

ᐂ⁽−2t)᏷Γ^γg−

᏷Γ^γg

ᐂ⁽−2t)−1

᏷Γ^δ−γf

−

ᐂ⁽−2t)᏷Γ^δ−γf

ᐂ⁽−2t)−1

᏷Γ^γg+

ᐂ⁽−2t)−1

᏷Γ^δ−γf

᏷Γ^γg.

(3.5)

ByLemma 2.5, we have the estimate Ᏸ2E²

ᐂ⁽−2t)−1

᏷Γ^δ−γf

᏷Γ^γg +Ᏸ2E²

᏷Γ^γg

ᐂ(−2t)−1

᏷Γ^δ−γf

≤C|t|^η−1/2





|σ|≤1

᏶^σΓ^σ−γf









|σ|≤1

᏶^σΓ^γg



.

(3.6)

Thus we can rewrite (3.4) in the form i∂tᐂ(−t)᏷Γ^α+δun(t)

= 1 it

n−1 m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

λjkᏰ2E²

ᐂ(−t)᏷Γ^δ−γf

ᐂ(−t)᏷Γ^γg

+µjkᏰ−2E⁶

ᐂ(−t)᏷Γ^δ−γf

ᐂ(−t)᏷Γ^γg

+R1(t), (3.7) where the remainder termR1(t)can be estimated by virtue of (3.6) and Lemmas2.1, 2.2,2.3, and2.4as follows:

α

bn

_|α|

α! R1(t)

≤C|t|^η−3/2

n−1

m=0

γ≤δ

α

β≤α

C_δ^γCα^β

bn

_|α|

α!





|σ|≤1

᏶^σΓ^δ−γf



|σ|≤1

᏶^σΓ^γg

≤C|t|^η−3/2

n−1 m=0

|σ|≤1

Θ^σun−1−m(t)

A²_bn

|σ|≤1

Θ^σum(t)

A²_bn

≤C|t|^η−3/2



ⁿ

j=0

log bj

b_j+1 −1



2n−1

m=0

un−1−m(t)

A²_bn

−1−m

um(t)

A²_bm

≤C



ⁿ

j=0

log bj

bj+1

₋1



2

ⁿ⁻¹

m=0

C_n−1−mCm



εⁿ⁺¹|t|η−3/2−(n−1)w.

(3.8)

Since

n−1

m=0

Cn−1−mCm≤

n−1

m=0

(n−m)^2(n−1−m)



^n−1−m

j=0

log bj

b_j+1 −1



2(n−1−m)

×(m+1)^2m



^m

j=0

log bj

b_j+1 −1



2m

≤



ⁿ⁻¹

j=0

log bj

bj+1

₋1



2(n−1)

(n+1)²ⁿ,

(3.9)

we have

α

bn|α|

α! R1(t)≤C



ⁿ

j=0

log bj

b_j+1 −1



2n

(n+1)²ⁿεⁿ⁺¹|t|^{η−3/2−(n−1)w}

≤Cnεⁿ⁺¹|t|^{η−3/2−(n−1)w}.

(3.10)

By virtue of the identityᐂ(−t)᏷=Ᏺ⁻¹ᏹᏲᏲᏹᐁ(−t)=Ᏺᐁ(−t), we haveiξjᐂ(−t)᏷= ᐂ(−t)᏷∂x_j. Hence by (3.7) we get

i∂tᏲᐁ(−t)Γ^α+δun(t)

=1 t

n−1

m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

λjkᏰ2E²ξj

Ᏺᐁ⁽−t)∂⁻¹_x

jΓ^δ−γf

Ᏺᐁ⁽−t)Γ^γg

−µjkD₋₂E⁶ξj

Ᏺᐁ(−t)∂⁻x_j¹Γ^δ−γf

Ᏺᐁ(−t)Γ^γg +R1(t).

(3.11) If|δ−γ|<|γ|we exchangef andgin the right-hand side of (3.11). By virtue of the equalityE^ν=(1+(it/2)νξ²)⁻¹∂t(tE^ν)we obtain the identity

φ tE^ν=∂t

φE^ν 1+(it/2)νξ²

− E^ν∂tφ

1+(it/2)νξ²+ 1+itνξ² t

1+(it/2)νξ²2φE^ν. (3.12) Therefore, we get from (3.11)

i∂tΨ=R2, (3.13)

where

Ψ=Ᏺᐁ(−t)Γ^α+δun(t) +

n−1

m=0

γ≤δ

β≤α

C_δ^γCα^β

× 2 j,k=1

λjkᏰ2

ξjE² 1+itξ²

Ᏺᐁ(−t)∂_x⁻_j¹Γ^δ−γf

Ᏺᐁ(−t)Γ^γg

+µjkᏰ−2 ξjE⁶ 1+3itξ²

Ᏺᐁ(−t)∂x⁻_j¹Γ^δ−γf

Ᏺᐁ(−t)Γ^γg

,

(3.14)

R2=R1+3

j=1Ij, and

I1= 1 it

n−1

m=0

γ≤δ

β≤α

C_δ^γCα^β

× 2 j,k=1

λjkᏰ2

1+2itξ² E² 1+itξ²2 ξj

Ᏺᐁ⁽−t)∂_x⁻¹

jΓ^δ−γf

Ᏺᐁ⁽−t)Γ^γg

−µjkᏰ−2

1+6itξ² E⁶ 1+3itξ²2 ξj

Ᏺᐁ(−t)∂⁻x_j¹Γ^δ−γf

Ᏺᐁ(−t)Γ^γg

,

I2= −1 it

n−1 m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

λjkᏰ2

ξjE² 1+itξ²

×

∂tᏲᐁ⁽−t)∂_x⁻¹

jΓ^δ−γf

Ᏺᐁ⁽−t)Γ^γg +

Ᏺᐁ(−t)∂_x⁻_j¹Γ^δ−γf

∂tᏲᐁ(−t)Γ^γg ,

I3= −1 it

n−1 m=0

γ≤δ

β≤α

C_δ^γCα^β

2 j,k=1

µjkᏰ−2 ξjE⁶ 1+3itξ²

×

∂tᏲᐁ⁽−t)∂x⁻¹_jΓ^δ−γf

Ᏺᐁ⁽−t)Γ^γg

+

Ᏺᐁ(−t)∂x⁻_j¹Γ^δ−γf

∂tᏲᐁ(−t)Γ^γg .

(3.15)

By Hölder’s inequality, the identity᏶j=ᐁ(t)xjᐁ(−t), and Lemmas2.1,2.2,2.3, and 2.4we get the estimates

α

bn

_|α|

α! I1(t)

≤C|t|^−3/2

n−1

m=0

α

β≤α

Cα^β

bn

_|α|

α!

2 j,k=1





|δ|≤2

∂_x⁻¹

jΓ^δf





×

|γ|≤1

ᐂ⁽−t)᏷Γ^γg_∞

≤C|t|^−3/2

n−1 m=0

α

β≤α

Cα^β

bn

|α|

α!





|δ|≤2

∂x⁻_j¹Γ^δf





×

|γ|≤1





|σ|≤1

᏶^σΓ^γg^1−η



|σ|≤2

᏶^σΓ^γg^η

≤Cnεⁿ⁺¹|t|−3/2−(n−1−m)w−mw+(1−mw)η

≤Cnεⁿ⁺¹|t|η−3/2−(n−1−mη)w

(3.16)