IJMMS 29:9 (2002) 501–516
PII. S0161171202007652
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
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 spaces i∂t u + (1/2)∆u = ᏺ(u), (t, x) ∈ R × R2 ;
u(0, x) = ϕ(x), x ∈ R2 , where ᏺ(u) = 2j,k=1 (λjk (∂xj u)(∂xk u)+µjk (∂xj u)(∂xk u)), where
λjk , µjk ∈ C. We prove that if the initial data ϕ satisfy 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
1
i∂t u + ∆u = ᏺ(u),
2
u(0, x) = ϕ(x),
(t, x) ∈ R × R2 ,
(1.1)
x ∈ R2 ,
with quadratic nonlinearity
ᏺ(u) =
2 λjk ∂xj u ∂xk u + µjk ∂xj u ∂xk u ,
(1.2)
j,k=1
where λjk , 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) with λjk = 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 in L2 sense, then we call the scattering problem a super-critical problem.
If the usual scattering states do not exist and the L2 norm of the nonlinearity decays
like Ct −δ , then we call the problem a critical one, when δ = 1 and a sub-critical one,
502
N. HAYASHI AND P. I. NAUMKIN
when 0 < δ < 1. The problem under consideration is classified as super-critical since
the usual scattering states were shown, in [8], to exist in L2 . 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
2
ᏺ(u) = u(t, x) dy|x − y|−δ u(t, y)
(1.3)
in the super-critical case 1 < δ < n. The critical case δ = 1 was treated in [13], where the
asymptotic expansion of small solutions in the neighborhood of the modified scattering states was obtained. In the case of critical power nonlinearity ᏺ(u) = |u|2 u in onedimensional 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 nonlinearity ᏺ(u) = t 1−δ |u|2 u and
Hartree type nonlinearity (1.3) with 0 < δ < 1. Roughly speaking, they used the asymptotic expansion in the neighborhood of the final states to the transformed equations
for the new dependent variable
t
2
w = Ᏺᐁ(−t)u(t) exp i t −δ Ᏺᐁ(−t)u(t) dt
(1.4)
1
(in the case of the power type nonlinearity).
Thus the asymptotic expansions of solutions to the nonlinear Schrödinger equations 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
property ᏺ(u) = e−iθ ᏺ(eiθ u) for any θ ∈ 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 involving 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 obm,p
stacles, we use the analytic function spaces Ab defined in (1.9) and the operators
ᏼ = x · ∇ + 2t∂t and ᏽ = x · ∇ + it∆.
To state our result precisely, we now give notation and function spaces. We deα
α
note ∂xj = ∂/∂xj and ∂ α = ∂x11 ∂x22 , where α ∈ (N ∪ {0})2 . We define the following
differential operators ᏼ = x · ∇ + 2t∂t , ᏽ = x · ∇ + it∆, ᏶ = x + it∇ and the vector Ω = (Ω(j,k) )(j,k=1,2) , where the operators Ω(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
2
2
e(i/2t)(x−y) φ(y)dy = Ᏺ−1 e−(it/2)ξ Ᏺφ,
(1.5)
where Ᏺφ ≡ φ̂(ξ) = (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−1 φ ≡
φ̌(x) = (1/2π ) ei(x·ξ) φ(ξ)dξ. Note that the free Schrödinger evolution group ᐁ(t)
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
503
also can be represented as ᐁ(t) = ᏹ(t)Ᏸ(t)Ᏺᏹ(t), where ᏹ(t) = exp(ix 2 /2t), the dilation operator is (Ᏸ(t)φ)(x) = (i/t)φ(x/t), then the inverse free Schrödinger evolution group is written as ᐁ(−t) = −ᏹ(−t)iᏲ−1 Ᏸ(1/t)ᏹ(−t), where Ᏸ−1 (t) = −iᏰ(1/t)
is the inverse dilation operator. We define the extended vectors Γ = (ᏼ, Ω, ∇), Γ
=
(ᏼ + 2, Ω, ∇), and Θ = (ᏽ, Ω, ∇). We have the following relations:
ᏽ = ᏼ − 2it ᏸ = ᏶ · ∇ = ᐁ(t)x ᐁ(−t) · ∇ = it ᏹ(t)∇ᏹ(t) · ∇,
where ᏹ(t) = eix
2 /2t
(1.6)
, ᏸ = i∂t + (1/2)∆. The commutation relations
[ᏽ, ∇] = [ᏼ, ∇] = −∇,
[ᏽ, ᏶] = [ᏼ, ᏶] = ᏶,
(k)
∂k , ᏶l = δl ,
[ᏼ, ᏽ] = [Ω, ᏼ] = [Ω, ᏽ] = 0,
(j,k)
(j)
(k)
Ωx , ∂l = δl ∂j − δl ∂k ,
(k)
(1.7)
(k)
where δj = 1 if j = k and δj = 0 if j ≠ k are used freely in the paper. We denote the
usual Lebesgue space by Lp (R2 ) with the norm φp = ( R2 |φ(x)|p dx)1/p if 1 ≤ p < ∞
and φ∞ = ess sup{|φ(x)|; x ∈ R2 } if p = ∞. For simplicity we write · = ·2 . The
weighted Sobolev space is defined by
Hpm,k R2 = φ ∈ Lp R2 : xk i∇m φp < ∞ ,
(1.8)
√
where m, k ∈ R+ , 1 ≤ p ≤ ∞, x = 1 + x 2 . We write for simplicity Hm,k (R2 ) =
m,k
H2 (R2 ) and the norm φm,k = φm,k,2 . Now we define the analytic function space
b|α| 2
2
Γ α+β φ < ∞ ,
m =
Am
=
φ
∈
L
R
;
φ
Ab
b
α!
|β|≤m α
(1.9)
where the vector Γ = 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
b|α| 2
2
α+β m
Γ
=
φ
∈
L
=
φ
A
R
;
φ
<
∞
.
m
b
A
b
α!
|β|≤m α
(1.10)
Here the summation is over all admissible multi-indices α. We often use the summations convention if it does not cause confusion. By [s] we denote the largest integer
less than or equal to s. Let C(I; B) be the space of continuous functions from a time
interval I to a Banach space B. We denote different positive constants by the same
letter C. We introduce the following functional spaces
Xb = u ∈ C R; L2 R2 ; uXb < ∞ ,
Yb = u ∈ C R; L2 R2 ; sup u(t)Yb < ∞ ,
t>0
(1.11)
504
N. HAYASHI AND P. I. NAUMKIN
where
᏶γ u(t) 2
uXb = sup u(t)A3 + sup t −1−η
A
b
t>0
t>0
b
|γ|=1 0
+ sup t 1−2η
t>0
b
|γ|≤1
∞
Θγ u 3 b dt +
+
A
∞
|γ|=1, |σ |≤1 1
γ σ Θ ᏶ u
b dt
A3
b
t 1+η
b|α| ∂t Ᏺᐁ(−t)Γ α ∇u(t)
∞
α!
α
∞ b|α| ∂t Ᏺᐁ(−t)Γ α+δ u(t)t 2η−1/2 dt,
+
1 α α!
|δ|≤3
u(t)
Yb
= u(t)A2 +
b
+ t 1−η
|β|+|γ|≤1
(1.12)
t −|β|−|γ|−η ᏶γ Θβ u(t)A1
b
b|α| ∂t Ᏺᐁ(−t)Γ α+γ Θδ u(t),
α!
|γ|+|δ|≤1 α
where η > 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
ᏺ un−1−m , um ,
(1.14)
m=0
such that limt→∞ un (t) = 0 in L2 , where ᏸ = i∂t + (1/2)∆ and
ᏺ(φ, ψ) =
2
λjk ∂xj φ ∂xk ψ + µjk ∂xj φ̄ ∂xk ψ̄ .
(1.15)
j,k=1
From [8] we see that if the initial data ϕ ∈ A3a are such that xj ϕ ∈ A2a for j = 1, 2 and
the norm ϕA3a + x1 ϕA2a + x2 ϕA2a = ε is sufficiently small, then the final state
u+ ∈ A2a1 , where b0 < a1 < a, hence u0 ∈ Yb0 and u − u0 Yb ≤ Cε2 t −w for all t ≥ 1,
0
where w ∈ (0, 1/2).
Now we state the main result in this paper.
Theorem 1.1. We assume that the initial data ϕ ∈ A3a are such that xj ϕ ∈ A2a for
j = 1, 2 and the norm ϕA3a +x1 ϕA2a +x2 ϕA2a = ε is sufficiently small. Then there
exists a unique global solution u(t, x) ∈ A3b(t) of the Cauchy problem (1.1). Moreover,
the estimates
un (t)
Ybn
≤ Cn εn+1 t −nw ,
n = 0, 1, 2, . . .
(1.16)
and the asymptotics
n−1
u(t) −
um (t)
m=0
Ybn
≤ Cn εn+1 t −nw ,
n = 1, 2, . . .
(1.17)
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
505
are valid for all t ≥ 1, where w ∈ (0, 1/2) and

2n 
Cn = C(n + 1)
n
j=0
bj
log
bj+1
−1 2n
 ,
(1.18)
where C is a positive constant independent of n and bj .
We assume in Theorem 1.1 that 0 < a < 1. This ensures that the function space A3a
for the initial data is not empty, as in [1, 2], we can see that our result is valid for the
initial function φ, which has analytic continuation Φ to the domain
= z ∈ C2 ; zj = xj + iyj , xj ∈ R, −C1 − xj tan ϑ < yj < C1 + xj tan ϑ, j = 1, 2 ,
(1.19)
such that
Φ(z)2 dx dy < ∞,
(1.20)
where ϑ ∈ (0, π /2), sin ϑ = C2 , and C1 , C2 ∈ (a, 1). For example, we can take 1/(1+x 4 ),
2
e−x as the initial data for the Cauchy problem (1.1).
+
Denote u+
0 (t, ξ) = u (ξ) and
u+
n (t, ξ) = −
t
2
n−1
dτ
ξ
ξ 1 2
+
t,
t,
u+
λ
ξ
ξ
eitξ /4
u
m
jk j k
4 m=0 n−1−m
2
2 j,k=1
iτ
∞
t
2
n−1
dτ
ξ
ξ 1 2
+
t,
−
t,
−
−
u+
µ
ξ
ξ
e3itξ /4
u
.
m
jk
j
k
4 m=0 n−1−m
2
2 j,k=1
iτ
∞
(1.21)
Corollary 1.2. Let the conditions of Theorem 1.1 be fulfilled. Then the following
asymptotics in L2 sense
Ᏺᐁ(−t)u(t) = u+ (ξ) +
n−1
−nw u+
j (t, ξ) + O t
(1.22)
j=1
are valid for large time t ≥ 1, where n = 1, 2, . . . .
For the convenience of the reader we now give the outline of the proof of Theorem
1.1. As in [8] we apply the operator Ᏺᐁ(−t) to (1.1) to get
i∂t Ᏺᐁ(−t)u = I(t, ξ) + R(t, ξ),
(1.23)
where
I(t, ξ) =
2
1 λjk Ᏸ2 E 2 Ᏺᐁ(−t)∂xj u Ᏺᐁ(−t)∂xk u
it j,k=1
+ µjk Ᏸ−2 E Ᏺᐁ(−t)∂xj u Ᏺᐁ(−t)∂xk u ,
6
(1.24)
506
N. HAYASHI AND P. I. NAUMKIN
2
E = eitξ /2 , and R is a remainder term since in [8] we proved the estimate R ≤
Ct −1−w |α|≤1 Θα u2 , Θ = (Q, Ω, ∇), 0 < w < 1/2. Then we show that the first term
of the integral I(t, ξ)dt is also convergent in view of the oscillating factor E. Roughly
speaking, in [8] the following estimate was shown:
u(t) ≤ u(1) + C
t
Θα u2 dτ.
τ −1−w
1
(1.25)
|α|≤1
Similarly, we have
t
Γ β u(t) ≤
Γ β u (1) + C τ −1−w
|β|≤1
1
|β|≤1
α β 2
Θ Γ u dτ.
(1.26)
|α|≤1, |β|≤1
However, the right-hand sides of (1.25) and (1.26) contain an additional operator Θ (derivative loss with respect to derivative Θ). This is the reason why we used the analytic
function spaces involving generalized derivative Γ , enabling us to get an additional
regularity with respect to operator Γ , hence we obtain the estimate
b(t)|β| Γ β u(t) < Cε.
β!
|β|≤1
(1.27)
|β|
b1 Γ β u(t) − Γ β u(s) < Cε2 |t|−w
β!
(1.28)
Similarly, we have
≤1
|β|
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 derivatives was used in [7]. By the definition of un (t) we have with ᏸ = i∂t + (1/2)∆






n−1
n−1
n−1
n−1
n−1





ᏸ u−
um = ᏺ u −
um ,
um + ᏺ
um , u −
um 
m=0
m=0

+ ᏺ u −
m=0
n−1
m=0
um , u −
m=0
n−1
m=0
(1.29)

um  + R,
m=0
where R is the remainder term since RYb
n+1
≤ Cεn+2 |t|−1−(n+1)w . In Section 3, we
will prove that the other three terms are estimated by Cεn+1 |t|−1−nw in the norm
Ybn+1 . Then via the inequality
α Γ u

Yb
n+1
≤ C
n j=1
bj
log
bj+1
−1
|α|

uYbn ,
(1.30)
for any |α| we obtain the desired result.
The rest of the paper is organized as follows. In Section 2, we state some preliminary
estimates concerning the analytic functional spaces Am
b . Section 3 is devoted to the
proof of Theorem 1.1 and Corollary 1.2.
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
507
2. Preliminary estimates. We summarize some lemmas proved in [7, 8], which are
necessary to prove the theorem.
m,p
Lemma 2.1. Let φ ∈ Ab
, then
φA
m,p ≤ e2b φAm,p ,
b
(2.1)
b
where
m,p
Ab
m,p
A
b
b|α| Γ α+β φ < ∞ ,
= φ ∈ Lp R2 ; φAm,p =
p
b
α!
|β|≤m α
b|α| Γ
α+β φ < ∞ ,
= φ ∈ Lp R2 ; φA
m,p =
p
b
α!
|β|≤m α
(2.2)
and 2 ≤ p ≤ ∞.
Lemma 2.2. The following commutation relations are valid:
l
,
᏶
,
∂
= −l᏶l−1
∂xl j , ᏶xj = l∂xl−1
x
xj ,
j
j
j
ᏼl ᏶xj =
Clm ᏶xj ᏼl−m ,
ᏼl ∂xj =
Clm (−1)m ∂xj ᏼl−m
0≤m≤l
᏶ xj ᏼ =
l
0≤m≤l
Clm (−1)m ᏼl−m ᏶xj ,
0≤m≤l
Ωxl j xk ∂xj =
(−1)m Cl2m ∂xj Ωxl−2m
+
j xk
0≤2m≤l
∂xj Ωxl j xk =
∂xj ᏼl =
Clm ᏼl−m ∂xj
0≤m≤l
(2.3)
(−1)m+1 Cl2m+1 ∂xk Ωxl−2m−1
,
j xk
0≤2m+1≤l
(−1)m+1 Cl2m Ωxl−2m
∂ +
j xk xj
0≤2m≤l
(−1)m Cl2m+1 Ωxl−2m−1
∂xk ,
j xk
0≤2m+1≤l
where Clm = l!/(l − m)!m! is the binomial coefficient.
Lemma 2.3. The estimate
᏶∇φAm,p ≤ C
b
Θ α φ
|α|=1
m,p
Ab
(2.4)
is true.
Lemma 2.4. The inequalities
b|α| ∂xj Γ α+β φ ≤ C2 ∂xj φAm,p ,
C1 ∂xj φAm,p ≤
p
b
b
α!
|β|≤m α
b|α| ᏶xj Γ α+β φ + φAm,p
C1 ᏶xj φAm,p ≤
p
b
b
α!
|β|≤m
≤ C2 ᏶xj φAm,p + φAm,p
b
are true for all t > 0, where C1 , C2 > 0.
b
(2.5)
508
N. HAYASHI AND P. I. NAUMKIN
We define the evolution operator
ᐂ(t)φ = Ᏺ−1 eiξ
2 /2t
Ᏺφ =
t
2iπ
2
e(−it/2)(ξ−y) φ(y)dy
(2.6)
and ᏷ = Ᏺᏹᐁ(−t). By a direct calculation we see that
ᐂ(−t) E ν−1 φ = Ᏸν E ν(ν−1) ᐂ(−νt)φ,
(2.7)
2
with Ᏸν φ = (1/ν)φ(ξ/ν) and E = eitξ /2 , where ν ≠ 0. We need the following lemma
to get the decay estimates of the solution for large time.
Lemma 2.5. The estimate
Ᏸν E ν(ν−1) ᐂ(−νt) − 1 (᏷φ)(᏷ψ)
+ Ᏸν E ν(ν−1) (᏷ψ) ᐂ(−νt) − 1 (᏷φ)
α β ᏶ φ ᏶ ψ ≤ Ct η−1/2
(2.8)
|α|≤1,|β|≤1
is valid for all t > 0, where ν ≠ 0, η > 0 is sufficiently small.
3. Proof of Theorem 1.1. We consider the linear Schrödinger equation
ᏸ un =
n−1
ᏺ un−1−m , um .
(3.1)
m=0
Since u0 (t) is a solution of linear Schrödinger equation it is easy to see that u0 (t)Yb
0
≤ C0 ε. Then by induction we assume that
uj (t)
Yb
j
≤ Cj εj+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+δ un
=
2
n−1
(3.3)
1 β γ λjk E ᏷Γ
δ−γ f ᏷Γ γ g + µjk Ē 3 ᏷Γ
δ−γ f ᏷Γ γ g ,
Cα Cδ
it m=0 β≤α γ≤δ
j,k=1
β
where ᏸξ = i∂t + (1/2t 2 )∆ξ , f = Γ
α−β ∂xj un−1−m , g = Γ β ∂xk um , Cα = α!/(α − β)!β!,
|δ| ≤ 2. Applying the operator ᐂ(−t) = Ᏺ−1 ᏹ(t)Ᏺ to both sides of (3.3) we obtain by
virtue of identity (2.7)
i∂t ᐂ(−t)᏷Γ α+δ un (t)
=
2
n−1
1 γ β Cδ Cα
ᐂ(−t) λjk E ᏷Γ
δ−γ f ᏷Γ γ g + µjk Ē 3 ᏷Γ
δ−γ f ᏷Γ γ g
it m=0 γ≤δ β≤α
j,k=1
509
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
=
2
n−1
1 γ β Cδ Cα
λjk Ᏸ2 E 2 ᐂ(−2t) ᏷Γ
δ−γ f ᏷Γ γ g
it m=0 γ≤δ β≤α
j,k=1
+ µjk Ᏸ−2 E 6 ᐂ(2t) ᏷Γ
δ−γ f ᏷Γ γ g .
(3.4)
Then we write the identity
ᐂ(−2t) ᏷Γ
δ−γ f ᏷Γ γ g
= ᐂ(−2t)᏷Γ
δ−γ f ᐂ(−2t)᏷Γ γ g − ᏷Γ γ g ᐂ(−2t) − 1 ᏷Γ
δ−γ f
− ᐂ(−2t)᏷Γ
δ−γ f ᐂ(−2t) − 1 ᏷Γ γ g + ᐂ(−2t) − 1 ᏷Γ
δ−γ f ᏷Γ γ g.
(3.5)
By Lemma 2.5, we have the estimate
Ᏸ2 E 2 ᐂ(−2t) − 1 ᏷Γ
δ−γ f ᏷Γ γ g + Ᏸ2 E 2 ᏷Γ γ g ᐂ(−2t) − 1 ᏷Γ
δ−γ f 


σ σ −γ  σ γ 
η−1/2 
᏶ Γ
᏶ Γ g .
f
≤ C|t|
|σ |≤1
(3.6)
|σ |≤1
Thus we can rewrite (3.4) in the form
i∂t ᐂ(−t)᏷Γ α+δ un (t)
=
2
n−1
1 γ β Cδ Cα
λjk Ᏸ2 E 2 ᐂ(−t)᏷Γ
δ−γ f ᐂ(−t)᏷Γ γ g
it m=0 γ≤δ β≤α
j,k=1
+ µjk Ᏸ−2 E 6 ᐂ(−t)᏷Γ
δ−γ f ᐂ(−t)᏷Γ γ g + R1 (t),
(3.7)
where the remainder term R1 (t) can be estimated by virtue of (3.6) and Lemmas 2.1,
2.2, 2.3, and 2.4 as follows:
bn |α| R1 (t)
α!
α
≤ C|t|
η−3/2
n−1
γ β
Cδ Cα
m=0 γ≤δ α β≤α
≤ C|t|η−3/2
n−1

|α| 
σ γ bn
᏶σ Γ
δ−γ f 
᏶ Γ g 
α!
|σ |≤1
|σ |≤1
Θσ un−1−m (t) 2
A
bn
m=0 |σ |≤1

≤ C|t|
η−3/2 
n j=0

≤ C
n j=0
bj
log
bj+1
bj
log
bj+1
−1
−1
2 
 
2

n−1
Θσ um (t)
un−1−m (t) 2
A
bn−1−m
m=0
n−1
m=0
A2
b
|σ |≤1
(3.8)
n
um (t) 2
A

Cn−1−m Cm εn+1 |t|η−3/2−(n−1)w .
bm
510
N. HAYASHI AND P. I. NAUMKIN
Since
n−1
Cn−1−m Cm ≤
m=0
n−1

2(n−1−m) 
(n − m)
n−1−m
m=0
j=0

× (m + 1)2m 
m j=0

≤
n−1
j=0
bj
log
bj+1
−1
bj
log
bj+1
bj
log
bj+1
−1
−1
2(n−1−m)

2m

(3.9)
2(n−1)

(n + 1)2n ,
we have


−1 2n
n bn |α| b
j
R1 (t) ≤ C 
 (n + 1)2n εn+1 |t|η−3/2−(n−1)w
log
α!
bj+1
α
j=0
(3.10)
≤ Cn εn+1 |t|η−3/2−(n−1)w .
By virtue of the identity ᐂ(−t)᏷ = Ᏺ−1 ᏹᏲᏲᏹᐁ(−t) = Ᏺᐁ(−t), we have iξj ᐂ(−t)᏷ =
ᐂ(−t)᏷∂xj . Hence by (3.7) we get
i∂t Ᏺᐁ(−t)Γ α+δ un (t)
=
2
n−1
1 γ β Γ
δ−γ f Ᏺᐁ(−t)Γ γ g
Cδ Cα
λjk Ᏸ2 E 2 ξj Ᏺᐁ(−t)∂x−1
j
t m=0 γ≤δ β≤α
j,k=1
δ−γ f Ᏺᐁ(−t)Γ γ g +R1 (t).
−µjk D−2 E 6 ξj Ᏺᐁ(−t)∂x−1
j Γ
(3.11)
If |δ − γ| < |γ| we exchange f and g in the right-hand side of (3.11). By virtue of the
equality E ν = (1 + (it/2)νξ 2 )−1 ∂t (tE ν ) we obtain the identity
1 + itνξ 2
φE ν
E ν ∂t φ
φ ν
ν
E = ∂t
+ −
2 φE .
2
2
t
1 + (it/2)νξ
1 + (it/2)νξ
t 1 + (it/2)νξ 2
(3.12)
Therefore, we get from (3.11)
i∂t Ψ = R2 ,
(3.13)
where
Ψ = Ᏺᐁ(−t)Γ α+δ un (t)
+
n−1
γ
β
Cδ Cα
m=0 γ≤δ β≤α
×
2
j,k=1
λjk Ᏸ2
ξj E 2 Γ
δ−γ f Ᏺᐁ(−t)Γ γ g
Ᏺᐁ(−t)∂x−1
j
2
1 + itξ
ξj E 6 −1 δ−γ
γ
Ᏺᐁ(−t)∂xj Γ
f Ᏺᐁ(−t)Γ g ,
+ µjk Ᏸ−2
1 + 3itξ 2
(3.14)
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
R2 = R1 +
I1 =
3
j=1 Ij ,
511
and
n−1
1 γ β
C Cα
it m=0 γ≤δ β≤α δ
2
×
j,k=1
1 + 2itξ 2 E 2 −1 δ−γ
f Ᏺᐁ(−t)Γ γ g
λjk Ᏸ2 2 ξj Ᏺᐁ(−t)∂xj Γ
1 + itξ 2
1 + 6itξ 2 E 6 −1
− µjk Ᏸ−2 2 ξj Ᏺᐁ(−t)∂xj Γ
δ−γ f Ᏺᐁ(−t)Γ γ g ,
1 + 3itξ 2
I2 = −
2
n−1
ξj E 2
1 γ β Cδ Cα
λjk Ᏸ2
it m=0 γ≤δ β≤α
1 + itξ 2
j,k=1
Γ
δ−γ f Ᏺᐁ(−t)Γ γ g
× ∂t Ᏺᐁ(−t)∂x−1
j
(3.15)
Γ
δ−γ f ∂t Ᏺᐁ(−t)Γ γ g ,
+ Ᏺᐁ(−t)∂x−1
j
I3 = −
2
n−1
ξj E 6
1 γ β Cδ Cα
µjk Ᏸ−2
it m=0 γ≤δ β≤α
1 + 3itξ 2
j,k=1
×
δ−γ f Ᏺᐁ(−t)Γ γ g
∂t Ᏺᐁ(−t)∂x−1
j Γ
δ−γ f ∂t Ᏺᐁ(−t)Γ γ g .
+ Ᏺᐁ(−t)∂x−1
j Γ
By Hölder’s inequality, the identity ᏶j = ᐁ(t)xj ᐁ(−t), and Lemmas 2.1, 2.2, 2.3, and
2.4 we get the estimates
bn |α| I1 (t)
α!
α
≤ C|t|
−3/2
n−1
β
Cα
m=0 α β≤α

|α| 2 
bn
−1
δ

∂ Γ
f 
α! j,k=1 |δ|≤2 xj
ᐂ(−t)᏷Γ γ g ×
∞
|γ|≤1
≤ C|t|
−3/2
n−1
m=0 α β≤α
×
|γ|≤1
β
Cα

|α| 
bn
−1 δ 

∂xj Γ f α!
|δ|≤2


1−η σ γ η
σ
γ
᏶ Γ g ᏶ Γ g 

|σ |≤1
|σ |≤2
≤ Cn εn+1 |t|−3/2−(n−1−m)w−mw+(1−mw)η
≤ Cn εn+1 |t|η−3/2−(n−1−mη)w
(3.16)
512
N. HAYASHI AND P. I. NAUMKIN
for |δ| ≤ 3. In the same way we obtain
bn |α| I2 (t) + I3 (t)
α!
α
≤ C|t|
−1/2
n−1
γ β
Cδ Cα
m=0 α β≤α |γ|≤1
×
2
|α|
bn
α!
∂t Ᏺᐁ(−t)∂ −1 Γ
δ−γ f Ᏺᐁ(−t)Γ γ g xj
j,k=1
∞
Γ
δ−γ f + ∂t Ᏺᐁ(−t)Γ γ g ∞ Ᏺᐁ(−t)∂x−1
j
(3.17)
bn |α|
≤ C|t|−1/2
α!
m=0 α
n−1
∂t Ᏺᐁ(−t)Γ
α+δ un−1−m (t)
× εm+1 |t|−mw
|δ|≤3
+ εn−m |t|−(n−m−1)w
∂t Ᏺᐁ(−t)Γ α+γ ∇um (t)
γ≤1
≤ Cn ε
n+1
|t|
η−3/2−(n−1)w
.
In view of (3.10), (3.16), and (3.17) we have the estimate
bn |α| R2 (t) ≤ Cn εn+1 |t|−1−nw .
α!
α
(3.18)
Multiplying both sides of (3.13) by Ψ (t), integrating with respect to the space variables,
and taking the imaginary part of the result, we obtain the inequality (d/dt)Ψ (t) ≤
R2 (t), hence
|α|
bn |α| d bn
Ψ (t) ≤
R2 (t) ≤ Cn εn+1 |t|−1−nw .
dt α
α!
α!
α
(3.19)
Then integration with respect to t in view of (3.18) yields
un (t) 2 ≤ Cn εn+1 |t|−nw .
A
bn
(3.20)
Applying the operator ᏶xl to both sides of (3.1), we get
ᏸ᏶xl Γ α+δ un =
γ≤δ β≤α
γ
β
Cδ Cα
2 λjk Γ
δ−γ f ᏶xl Γ γ g + itλjk ∂xl Γ
δ−γ f Γ γ g
j,k=1
+ µjk Γ
δ−γ f ᏶xl Γ γ g − itµjk ∂xl Γ
δ−γ f Γ γ g ,
(3.21)
hence by the classical energy method, via the inequality
∂x φ ∂x ψ ≤ C ΘφΘψ,
j
k
t
(3.22)
513
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
and Lemmas 2.1, 2.2, 2.3, and 2.4 we obtain
bn |α| d ᏶x Γ α+δ un (t) ≤ Cn εn+1 |t|−1/2−(n−1)w+η
l
α!
dt
α
(3.23)
for |δ| ≤ 1. Multiplying both sides of the last inequality by t −1−η and integrating with
respect to t, we have
᏶γ un (t)
t −1−η
|γ|=1
≤ Cn εn+1 |t|−nw .
A2
b
n
(3.24)
By the identity ᏼu = ᏽu + 2it ᏺ(ν), and Lemmas 2.1, 2.2, 2.3, and 2.4 we see that
ᏽ un n−1
ᏽun−1−m 1 ᏽum 1
≤ ᏼun A1 + C
A
A
A1
bn
bn
≤ ᏼun A1
b
n
bn
m=0
+ Cn ε
n+1
|t|
−nw
bn
(3.25)
.
Therefore, by virtue of (3.24) and (3.25) we get
t −1−η
γ δ
᏶ Θ un (t)
|γ|+|δ|=1
A2
b
n
≤ Cn εn+1 |t|−nw .
(3.26)
In the same way as above by virtue of (3.11) and (3.26) we obtain
t 1−η
bn |α| ∂t Ᏺᐁ(−t)Γ α+γ Θδ un (t) ≤ Cn εn+1 |t|−nw .
α!
|γ|+|δ|≤1 α
(3.27)
By (3.20), (3.26), and (3.27) we have the first part of the theorem
un (t)
Ybn
≤ Cn εn+1 |t|−nw
(3.28)
for any n ∈ N. We next prove the last part of the theorem by induction. By the definition of un (t) we have with ᏸ = i∂t + (1/2)∆
ᏸ u−
n
um
= ᏺ(u, u) −
m=0
n k−1
ᏺ uk−1−m , um = I + R,
(3.29)
k=1 m=0
where

I = ᏺu −
n−1
um ,
m=0

+ ᏺ u −
n−1
m=0
n−1


um  + ᏺ 
m=0
um , u −
n−1
m=0
n−1

um , u −
n−1

um 
m=0
(3.30)
um 
m=0
and R consists of quadratic nonlinearities involving uk ul and ūk ūl with k+l ≥ n. We
have
n
α+δ
u−
um = Ᏺᐁ(−t)Γ α+δ (I + R).
(3.31)
i∂t Ᏺᐁ(−t)Γ
m=0
514
N. HAYASHI AND P. I. NAUMKIN
In the same way as in the proof of (3.7) we estimate
Ᏺᐁ(−t)Γ α+δ ᏺ(φ, ψ)
=
2
n−1
1 γ β Cδ Cα
λjk Ᏸ2 E 2 Ᏺᐁ(−t)Γ
δ−γ f Ᏺᐁ(−t)Γ γ g
it m=0 γ≤δ β≤α
j,k=1
(3.32)
+ µjk Ᏸ−2 E 6 Ᏺᐁ(−t)Γ
δ−γ f Ᏺᐁ(−t)Γ γ g + R3 (t),
f = Γ
α−β ∂xj φ, g = Γ β ∂xk ψ. We have by (3.10) and the fact that bn < bn−1
bn |α| R3 (t) ≤ C|t|η−3/2
Θσ φ(t) 3
Θσ ψ(t) 3 .
Ab
Ab
n
n
α!
α
|σ |≤1
|σ |≤1
(3.33)
Hence by the assumption and (3.28)
bn+1 |α| Γ α+δ I(t) ≤ Cn εn+2 |t|−1−(n+1)w .
α!
α
(3.34)
We also have by (3.28)
bn+1 |α| Γ α+δ R(t) ≤ C uk (t)ul (t)
≤ Cn εn+2 |t|−1−(n+1)w .
Yb
n+1
α!
α
Thus in view of (3.28), (3.31), (3.32), (3.34), and (3.35) we obtain
n
u −
um ≤ Cn εn+2 |t|−1−(n+1)w
m=0
(3.35)
(3.36)
Yb
n+1
which yields the second part of the theorem. Theorem 1.1 is proved.
Proof of Corollary 1.2. By (1.1) we have
i∂t Ᏺᐁ(−t)u(t) =
2
1 λjk Ᏸ2 E 2 ξj Ᏺᐁ(−t)u Ᏺᐁ(−t)∂xk u
t j,k=1
− µjk Ᏸ−2 E 6 ξj Ᏺᐁ(−t)u Ᏺᐁ(−t)∂xk u + R(t)
(3.37)
and the property of the solution of (1.1) we see that R ≤ Cε2 |t|−1−wt . In the same
way as in the proof of Theorem 1.1 we have
Ᏺᐁ(−t)u(t) − u+ (ξ) ≤ C1 ε2 |t|−wt .
(3.38)
By the definition of u we see that
Ᏺᐁ(−t)u(t) − u+ (ξ)
2 ξ
ξ
1 t λjk 1/2
+
=−
E dτξj ξk u
u+
4 j,k=1 ∞ iτ
2
2
2
1 t µjk 3/2
ξ
ξ
E dτξj ξk u+ −
+ I,
−
u+ −
4 j,k=1 ∞ iτ
2
2
(3.39)
ASYMPTOTIC EXPANSION OF SMALL ANALYTIC SOLUTIONS
515
where
2 2
ξ
1 t λjk 1/2 t,
E
dτξj ξk
Ᏺᐁ(−t)u
4 j,k=1 ∞ iτ
2
2 2
1 t µjk 3/2 ξ
E
dτξj ξk
−
Ᏺᐁ(−t)u
t, −
4 j,k=1 ∞ iτ
2
I=−
2 ξ
ξ
1 t λjk 1/2
E dτξj ξk u+
u+
+
4 j,k=1 ∞ iτ
2
2
+
(3.40)
2 1 t µjk 3/2
ξ
ξ
E dτξj ξk u+ −
.
u+ −
4 j,k=1 ∞ iτ
2
2
From (3.38) the L2 norm of I is estimated as I(t) ≤ Cε3 |t|−2w . Therefore we get
2 1 t λjk 1/2
ξ
ξ
Ᏺᐁ(−t)u(t) − u+ (ξ) +
E dτξj ξk u+
u+
4 j,k=1 ∞ iτ
2
2
(3.41)
2 1 t µjk 3/2
ξ
ξ 3
−2w
+
+
≤ C3 ε |t|
E dτξj ξk u −
.
+
u −
4 j,k=1 ∞ iτ
2
2 We iterate this procedure to get
Ᏺᐁ(−t)u(t) − u+ (ξ)
n l−1
2 1 t λjk 1/2
ξ
ξ
+
+
E dτξj ξk ul−1−m t,
+
um t,
4 l=1 m=0 j,k=1 ∞ iτ
2
2
n l−1
2 1 t µjk 3/2
ξ
ξ +
+
u
+
t,
−
t,
−
E dτξj ξk u
m
l−1−m
4 l=1 m=0 j,k=1 ∞ iτ
2
2 (3.42)
≤ Cn+1 εn+2 |t|−(n+1)w
+
with u+
0 (t, ξ) = u (ξ). This completes the proof of Corollary 1.2.
Acknowledgments. The authors would like to thank the referees for their useful
comments. The works of Pavel I. Naumkin are partially supported by Consejo Nacional
de Ciencia y Tecnologia (CONACYT).
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Nakao Hayashi: Department of Mathematics, Graduate School of Science, Osaka
University, Osaka 560-0043, Japan
E-mail address: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin: Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari),
Morelia, CP 58089, Michoacán, Mexico
E-mail address: pavelni@matmor.unam.mx