Electronic Journal of Differential Equations, Vol. 2008(2008), No. 15, pp. 1–38.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ASYMPTOTIC BEHAVIOR FOR A QUADRATIC NONLINEAR
SCHRÖDINGER EQUATION
NAKAO HAYASHI, PAVEL I. NAUMKIN
Abstract. We study the initial-value problem for the quadratic nonlinear
Schrödinger equation
1
iut + uxx = ∂x u2 , x ∈ R, t > 1,
2
u(1, x) = u1 (x), x ∈ R.
For small initial data u1 ∈ H2,2 we prove that there exists a unique global
solution u ∈ C([1, ∞); H2,2 ) of this Cauchy problem. Moreover we show that
the large
√ time asymptotic behavior of the solution is defined in the region
|x| ≤ C t by the self-similar solution √1 M S( √x ) such that the total mass
t
t
Z
Z
1
x
√
M S( √ )dx =
u1 (x)dx,
t R
t
R
√
and in the far region |x| > t the asymptotic behavior of solutions has rapidly
oscillating structure similar to that of the cubic nonlinear Schrödinger equations.
1. Introduction
We consider the quadratic nonlinear Schrödinger equation
1
iut + uxx = ∂x u2 , x ∈ R, t > 1,
2
u(1, x) = u1 (x), x ∈ R.
(1.1)
In general, the quadratic type nonlinearities in the one dimensional case are considered to be subcritical with respect to the large time asymptotic behavior of solutions. Different types of the quadratic nonlinearities, including derivatives of the
unknown function were considered previously (see [6, 7, 12, 15, 20] and references
cited therein). We choose the initial time value t = 1 for the convenience of the
forthcoming calculations (note that by the change t0 = t − 1 it can be transformed
to the usual case of the initial time value t0 = 0).
2000 Mathematics Subject Classification. 35B40, 35Q55.
Key words and phrases. Nonlinear Schrodinger equation; large time asymptotic;
self-similar solutions.
c 2008 Texas State University - San Marcos.
Submitted March 19, 2007. Published February 1, 2008.
P. I. Naumkin is partially supported by CONACYT.
1
2
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
If we replace the nonlinear term of (1.1) by ∂x u2 , then we can represent the
solution u by the formula (see [17])
Rx
1 (2s(t, x)ψ(t, x) + ∂x ψ(t, x)) exp 2 0 s(t, y)dy
Rx
u(t, x) = s(t, x) −
2
1 + ψ(t, x) exp 2 0 s(t, y)dy
through the Hope-Cole transformation, where
1 ix2
s(t, x) = √ e 2t
t
1
√
,
√ R √xt − 2i
C − 2 2i 0
e−y2 dy
C is a constant determined by
Z
Z
Z
iξ2
u1 (x)dx = s(t, x)dx = e 2
1
dξ
√ R ξ √− i
2
C − 2 2i 0
e−y2 dy
and ψ is the solution of the linear Schrödinger equation iψt + 21 ψxx = 0 with the
initial data
Z x
Z x
ψ(1, x) = exp − 2
(u1 (y) − s(1, y))dy − 1 exp − 2
s(1, y)dy .
0
0
See Appendix 8 for details. However the Hope-Cole transformation can not be
applied to our problem. Recently in [6] we considered the nonlinear Schrödinger
equation
1
iut + uxx = λ(ux )2 + µu2x ,
2
with λ,µ ∈ C. We applied a method similar to the normal forms of Shatah [18],
making a transformation of the original equation with quadratic nonlinearity to a
nonlinear Schrödinger equation with critical cubic nonlinearity
µ
L(u − u2 − λG(u, u)) = −µu(Lu) + 2λG(Lu, u)
2
= −λµu(ux )2 − λµuu2x + 2|λ|2 G(u, u2x ) + 2λµG(u, u2x ),
where L = i∂t + 21 ∂x2 , G is a symmetric bilinear operator. Then for small initial
data u0 ∈ H3,1 we obtained the large time asymptotic behavior Rof small solutions
which has an additional logarithmic oscillation.
R xWe note that if u(t, x)dx = 0 in
(1.1), then by introducing a new variable v = −∞ udx, we have for v,
1
ivt + vxx = (v x )2 .
2
R
Therefore in the case of u1 (x)dx = 0, asymptotic behavior of solutions has been
shown in [6].
In [7] we studied the one dimensional quadratic nonlinear Schr ödinger equation
iut + 21 uxx = t−α |ux |2 with α ∈ (0, 1). Heuristically the solution should have
a quasilinear character if α ∈ ( 12 , 1). However we showed that the asymptotic
behavior of solutions does not have a quasilinear character for all range α ∈ (0, 1)
due to the special structure of the nonlinear term. For the case α ∈ [ 21 , 1) we
proved that if the initial data u0 ∈ H3,0 ∩ H2,2 are small then the solution has a
α
slow time-decay as t− 2 . And the derivative ux of the solution has a quasilinear
behavior t−1/2 as t → ∞. When α ∈ (0, 21 ), if we assume that the initial data u0
are analytic and small, then the same result as for the case α ∈ [ 21 , 1) holds.
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
3
The aim of the present paper is to prove existence of global solutions and large
time behavior of solutions to the Cauchy problem (1.1). Here we use the method
similar to the normal forms of Shatah and the transformation in [11]. Also we use
the following factorization formulas for the free Schrödinger evolution group
U(t)F −1 = M (t)Dt V(t),
FU(−t) = iV(−t)E(t)D1/t .
Here we denote
i
2
it
2
M (t) = e 2t x , E(t) = e 2 ξ ,
the dilation operator (Da φ)(x) = √1ia φ( xa ) and V(t) = FM (t)F −1 . Note that
D1/t M (t) = E(t)D1/t . So we represent the solution
u(t) = U(t)F −1 w(t) = Dt E(t)v(t),
where w = V(−t)v(t). The direct Fourier transform φ̂(ξ) of the function φ(x) is
defined by
Z
1
e−ixξ φ(x)dx,
Fφ = φ̂ = √
2π R
then the inverse Fourier transformation is given by
Z
1
eixξ φ(ξ)dξ.
F −1 φ = √
2π R
Denote the
space Lp = {φ ∈ S0 ; kφkLp < ∞}, where the norm
R usual pLebesgue
1/p
p
kφkL = ( R |φ(x)| dx)
if 1 ≤ p < ∞ and kφkL∞ = ess. supx∈R |φ(x)| if p = ∞.
The weighted Lebesgue norm is kφkLp,a = kh·ia φkLp . Weighted Sobolev space is
Hm,a = {φ ∈ S0 : kφkHm,a ≡ khi∂im φkL2,a < ∞},
√
where m, a ∈ R, hxi = 1 + x2 . The usual Sobolev space is Hm = Hm,0 , so the
index 0 we usually omit if it does not cause a confusion. Different positive constants
we denote by the same letter C.
Denote Y = {φ ∈ L∞ , φ0 ∈ H1,1 }.
i
2
Theorem 1.1. Let the initial data M (1)u1 = e 2 x u1 ∈ Y with a norm ku1 kY ≤
ε, where ε > 0 is sufficiently small. Then there exists a unique solution u ∈
C([1, ∞); Y) of the Cauchy problem (1.1).
We denote by √1t M Ψ( √xt ) a self-similar solution of the quadratic nonlinear
Schrödinger equation ( 1.1) such that the total mass
Z
Z
1
x
√
M Ψ( √ )dx =
u1 (x)dx.
t R
t
R
The following theorem states that the large time asymptotic behavior
√ of solution
of (1.1) is defined
by
this
self-similar
solution
in
the
region
|x|
≤
C
t and in the
√
far region |x| > t it has rapidly oscillating structure similar to that of the cubic
Schrödinger equations.
i
2
Theorem 1.2. Let the initial data M (1)u1 = e 2 x u1 ∈ Y with a norm ku1 kY
≤ ε, where ε > 0 is sufficiently small. Then there exist unique functions Hj and
Bj ∈ L∞ (Bj are real-valued), j = 1, 2, such that the following asymptotic formula
is valid
1
x
u(t, x) = √ M (t)Ψ( √ )
t
t
4
N. HAYASHI, P. I. NAUMKIN
2
X
1+
1
x
x
+ √ M (t)
Hj ( ) exp iBj ( ) log
t
t
t
1+
j=1
EJDE-2008/15
|x|
√
t
|x|
t
1
+ O(t− 2 −κ )
for t → ∞ uniformly with respect to x ∈ R, where κ > 0.
We organize the rest of our paper as follows. In Sections 2–6 we prove some
preliminary estimates. Section 7 is devoted to the proof of Theorem 1.1. We prove
Theorem 1.2 in Section 8 . Section 10 is devoted to the proof of existence of the
self-similar solution.
2. Transformation of equation
We represent the solution u(t) = U(t)F −1 w(t), where the free Schrödinger evoit 2
lution group U(t) = F −1 e 2 ξ F. Applying the Fourier transformation to equation
it 2
(1.1), changing the dependent variable w(t, ξ) = e 2 ξ u
b(t, ξ), we get
Z
ξ
eitS w(t, η − ξ)w(t, −η)dη,
wt (t, ξ) = √
(2.1)
2π R
where S = 12 (ξ 2 + (ξ − η)2 + η 2 ). Using the identity
d 2 + itS
itS − 1 itS
eitS =
teitS +
e
2
dt (1 + itS)
(1 + itS)3
in view of the symmetry η ↔ ξ − η we rewrite equation (2.1) as
Z
∂t (w(t, ξ) − eitS w(t, η − ξ)w(t, −η)Adη)
R
Z
e
= t−1 eitS w(t, η − ξ)w(t, −η)Adη
R
r Z
2
+
eitQ w(t, η − ξ)w(t, η − ζ)w(t, ζ)Aηdηdζ,
π R2
(2.2)
where
1 2
(ξ + (ξ − η)2 − (η − ζ)2 − ζ 2 ),
2
ξt(2 + itS)
iξ
1
A= √
= − √ (1 + t∂t )( + S)−1 ,
2
it
2π(1 + itS)
2π
ξt(itS
−
1)
iξ
1
e= √
A
= − √ (2 + t∂t )t∂t ( + S)−1 .
3
it
2π(1 + itS)
2π
Q=
Now we return to the function u = U(t)F −1 w(t) to get, from (2.2),
1
e u) + Q(u, ∂x u2 ),
(i∂t + ∂x2 )(u − Q(u, u)) = t−1 Q(u,
2
where
Z
Q(u, u) = U(t)F −1 eitS w(t, η − ξ)w(t, −η)Adη
Z R
1
1
= − √ ∂x F −1 u
b(η − ξ)b
u(−η)(1 + t∂t )( + S)−1 dη,
it
2π
R
(2.3)
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
5
Z
e u) = U(t)F −1 eitS w(t, η − ξ)w(t, −η)Adη
e
Q(u,
R
Z
1
1
−1
√
=−
∂x F
u
b(η − ξ)b
u(−η)(3 + t∂t )∂t ( + S)−1 dη.
it
2π
R
In the next lemma we give an x-representation of the operator
Z
1
b − η)ψ(η)(
b
I(φ, ψ) = F −1 φ(ξ
+ S)−1 dη.
it
R
Denote the convolution with a kernel g
Z
(ψ ∗ φ)g ≡
g(t, y, z)ψ(x − y)φ(x − z)dydz.
R2
Lemma 2.1. The representation I(φ, ψ) = (ψ ∗ φ)g is true, where
r
r 4
2
K0
(y 2 − yz + z 2 )
g(t, y, z) ≡
3π
3it
and K0 is the Macdonalds function.
Proof. We substitute the Fourier transformation
Z
Z
1
b ξ + η) 1 + 3 ξ 2 + η 2 −1
b ξ − η) dξeixξ φ(
I(φ, ψ) = √
dη ψ(
2
2
it 4
2π R
R
Z
Z
Z
8
=
dη dye−iyη ψ(x − y) dzeizη φ(x − z)
3
3(2π) 2 R
R
R
Z
4
4i
y+z
ξ)(ξ 2 + η 2 − )−1 .
× dξ cos(
2
3
3t
R
By [2] we find
Z
∞
0
cos(zξ)
π −a|z|
dξ =
e
2
2
ξ +a
2a
for Re a > 0. Hence
Z
2
dydzψ(x − y)φ(x − z)
I(φ, ψ) = √
2
6π
Z ∞R
√
cos((z − y)η) − √1 |y+z| η2 − ti
q
×
e 3
dη.
0
η 2 − ti
We compute by [2] for Re a > 0, Re β > 0
√ 2 2
Z ∞
p
cos((z − y)η)e−β η +a dη
p
= K0 (a β 2 + (z − y)2 ).
2
2
η +a
0
Then taking a = √1it , β =
proof is complete.
√1 |y
3
(2.4)
+ z| we find the representation of the lemma. The
e in equation (2.3) as
By Lemma 2.1 we can rewrite the operators Q and Q
e
Q(φ, ψ) = (ψ ∗ φ)q and Q(φ, ψ) = (ψ ∗ φ)qe with
1
q = − √ (1 + t∂t )(∂y + ∂z )g(t, y, z),
2π
1
qe = − √ (2 + t∂t )t∂t (∂y + ∂z )g(t, y, z).
2π
6
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
it
2
In the same way as in paper [11] we change the variables u(t, x) = t−1/2 e 2 ξ v(t, ξ)
and ξ = xt in equation (2.3) to obtain
L(v − E(Ev ∗ Ev)h ) = P1 − P2 ,
(2.5)
where
1 2
∂ , P1 = t−1 E(Ev ∗ Ev)eh , P2 = t−1 E(Ev ∗ ∂ζ (E 2 v 2 ))h ,
2t2 ξ
and by changing x = ξt, y = ηt, z = ζt. Denote the operators
L = i∂t +
a+b
(E a φ ∗ E b ψ)h
Z
2
2
a
b
a+b
=E
h(t, η, ζ)e 2 it(ξ−η) φ(ξ − η)e 2 it(ξ−ζ) ψ(ξ − ζ)dηdζ,
Ga,b (φ, ψ) = E
R2
a+b
(E a φ ∗ E b ψ)eh
Gea,b (φ, ψ) = E
Z
2
2
a
b
a+b
e
h(t, η, ζ)e 2 it(ξ−η) φ(ξ − η)e 2 it(ξ−ζ) ψ(ξ − ζ)dηdζ,
=E
R2
a+b
(E a φ ∗ ∂ζ (E b ψ))h
Z
2
2
a
b
a+b
h(t, η, ζ)e 2 it(ξ−η) φ(ξ − η)∂ζ e 2 it(ξ−ζ) ψ(ξ − ζ)dηdζ
=E
Ha,b (φ, ψ) = E
R2
with
√
r 4t
t
h(t, η, ζ) = − √ (1 + t∂t )(∂η + ∂ζ )K0
(η 2 − ηζ + ζ 2 ) ,
3i
π 3
r
√
t
4t 2
e
(η − ηζ + ζ 2 ) .
h(t, η, ζ) = − √ (2 + t∂t )t∂t (∂η + ∂ζ )K0
3i
π 3
3. Preliminary estimates
Define the weighted Lebesgue norm kφkLp,a = kh·ia φkLp and define the linear
operator
Z
Kφ =
K(ξ, η)(φ(ξ − η) − φ(ξ))dη.
R
Lemma 3.1. Suppose that
K(ξ, η) = O(e−hηi hξηi−α )
for all ξ ∈ R, η ∈ R\{0}, where α > 1. Then the estimates are true
kKφkLp,θ ≤ Ck∂ξ φkL2,λ
if α >
3
2
+
1
p,
where θ =
3
2
+
1
p
+ λ, λ ≥ 0, p = 2, ∞, and
kKφkL2,1+λ ≤ CkφkL2,λ ,
if α > 1, where λ ≥ 0.
Proof. Since
Z
kφ(· − η) − φ(·)kLp,λ =
η
∂z φ(· − z)dz
0
1
Lp,λ
1
≤ C|η| 2 + p hηiλ k∂ξ φkL2,λ ,
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
7
for p = 2, ∞, we find
Z
1
1
e−hηi hηiλ hξηi−α |η| 2 + p dηkL∞,θ−λ ≤ Ck∂ξ φkL2,λ
kKφkLp,θ ≤ Ck∂ξ φkL2,λ k
R
if α >
3
2
+ p1 . We now prove the second estimate. For |ξ| < 1 we have
Z
1/2
|Kφ| ≤ CkφkL2
e−hηi dη
≤ CkφkL2 .
R
For |ξ| ≥ 1 we write
1+λ
||ξ|
Z
Kφ| ≤ C|ξ| |φ(ξ)||ξ| hξηi−α e−hηi dη
R
Z
hξηi−α hξ − ηiλ |φ(ξ − η)|dη
+ C|ξ|
λ
|η|>
|ξ|
2
Z
+ C|ξ|
|η|<
|ξ|
2
hξηi−α hξ − ηiλ |φ(ξ − η)|dη.
By the Cauchy-Schwarz inequality we find for the second summand
Z
Z
1/2
|ξ|
hξηi−α hξ − ηiλ |φ(ξ − η)|dη ≤ |ξ|kφkL2,λ
hξηi−2α dη
|η|>
|ξ|
2
|η|>
≤ C|ξ|
3
2 −2α
|ξ|
2
kφkL2,λ
x
|ξ| ,
and for the third summand we change η =
Z
|ξ|
hξηi−α hξ − ηiλ |φ(ξ − η)|dη
|η|<ξ|/2
Z
x λ
x
=C
hxi−α hξ −
i |φ(ξ −
)|dx.
|ξ|
|ξ|
2
|x|<ξ /2
Then taking the L2 - norm we have
3
kξKφkL2 (|ξ|≥1) ≤ C φkL2,λ + CkφkL2,λ k|ξ| 2 −2α L2
Z
x λ
x
+ Ck
hxi−α hξ −
i |φ(ξ −
)|dxkL2 (|ξ|≥1)
ξ2
|ξ|
|ξ|
|x|< 2
and
Z
k
2
|x|< ξ2
hxi−α hξ −
Z
Z
=
dξ
2
|x|< ξ2
|ξ|≥1
Z
×
Z
=
2
|y|< ξ2
hxi−α hξ −
hyi−α hξ −
−α
Z
dxhxi
R
x λ
x
i |φ(ξ −
)|dxk2L2 (|ξ|≥1)
|ξ|
|ξ|
R
x λ
x
i |φ(ξ −
)|dx
|ξ|
|ξ|
y λ
y
i |φ(ξ −
)|dy
|ξ|
|ξ|
Z
dyhyi−α
hξ −
|ξ|≥1,|ξ|≥2|x|,|ξ|≥2|y|
y λ
y
× hξ −
i |φ(ξ −
)|dξ
|ξ|
|ξ|
Z
2
≤ Ckφk2L2,λ
hxi−α dx ≤ Ckφk2L2,λ ,
R
x λ
x
i |φ(ξ −
)|
|ξ|
|ξ|
8
N. HAYASHI, P. I. NAUMKIN
x
|ξ|
since changing z = ξ −
in the domain |ξ| ≥ 1, |ξ| ≥ 2|x| we have
|
Therefore,
Z
hξ −
|ξ|≥1,|ξ|≥2|x|
EJDE-2008/15
dξ
1
≤ 2.
|=
dz
1 + ξx2
x 2λ 2
x
i φ (ξ −
)dξ ≤ 2
|ξ|
|ξ|
Z
hzi2λ φ2 (z)dz = Ckφk2L2,λ .
R
The proof is complete.
We now estimate the bilinear operator
Z
Aa,b (φ, ψ) =
e−iξ(aη+bζ) A(η, ζ)(φ(ξ − η) − φ(ξ))(ψ(ξ − ζ) − ψ(ξ))dηdζ.
R2
Lemma 3.2. Suppose that
|∂ηk ∂ζl A(η, ζ)| ≤ C(|η| + |ζ|)−s−k−l e−C|η|−C|ζ|
for all η, ζ ∈ R, k, l = 0, 1, where s ≥ 1. Then the estimates are valid
kAa,b (φ, ψ)kLp,θ ≤ Ck∂ξ φkL2,λ k∂ξ ψkL2,δ
if s < 3, a, b 6= 0, where θ = 2 + λ + δ − , p = 2, ∞, λ, δ ∈ R, and
kAa,b (φ, ψ)kL2,σ ≤ CkφkL2,λ k∂ξ ψkL2,δ
if s <
5
2,
b 6= 0, where σ = 1 + λ + δ − , λ, δ ∈ R, > 0 is small.
Proof. We integrate by parts with respect to η and ζ via identities e−iaξη =
P ∂η (ηe−iaξη ) and e−ibξζ = Q∂ζ (ζe−ibξζ ), where
P = (1 − iaξη)−1 , Q = (1 − ibξζ)−1 ,
to get
Z
Aa,b (φ, ψ) =
dηdζe−iξ(aη+bζ) ηζP QA∂η φ(ξ − η)∂ζ ψ(ξ − ζ)
R2
+ ζη∂η (P QA)(φ(ξ − η) − φ(ξ))∂ζ ψ(ξ − ζ)
+ ηζ∂ζ (P QA)(ψ(ξ − ζ) − ψ(ξ))∂η φ(ξ − η)
+ (φ(ξ − η) − φ(ξ))(ψ(ξ − ζ) − ψ(ξ))η∂η ζ∂ζ (P QA) .
Using the estimates
|P | ≤ Chξηi−1 , |Q| ≤ Chξζi−1 ,
1
|φ(ξ − η) − φ(ξ)| ≤ Chξi−λ |η| 2 hηi|λ| k∂ξ φkL2,λ ,
1
|ψ(ξ − ζ) − ψ(ξ)| ≤ Chξi−δ |ζ| 2 hζi|δ| k∂ξ ψkL2,δ
and by the condition of the lemma
|P QA| + |η∂η (P QA)| + |ζ∂ζ (P QA)| + |η∂η ζ∂ζ (P QA)|
≤ Chξζi−1 hξηi−1 (|η| + |ζ|)−s e−C|η|−C|ζ| ,
we obtain
|Aa,b (φ, ψ)|
Z
≤C
dηdζhξηi−1 hξζi−1 (|η| + |ζ|)−s e−C|η|−C|ζ|
R2
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
9
× |ηζ||∂η φ(ξ − η)||∂ζ ψ(ξ − ζ)| + hξi−λ k∂ξ φkL2,λ |η|1/2 |ζ||∂ζ ψ(ξ − ζ)|
+ hξi−δ k∂ξ ψkL2,δ |η||ζ|1/2 |∂η φ(ξ − η)| + hξi−λ−δ |ηζ|1/2 k∂ξ φkL2,λ k∂ξ ψkL2,δ .
Hence by the Cauchy-Schwarz inequality
|Aa,b (φ, ψ)| ≤ Chξi−λ−δ k∂ξ φkL2,λ k∂ξ ψkL2,δ
Z
1/2 Z
1/2
×
|η|2−s hξηi−2 e−C|η| dη
|ζ|2−s hξζi−2 e−C|ζ| dζ
R
Z
Z R
1/2
1−s
−1
−C|η|
2−s
dη
|ζ| hξζi−2 e−C|ζ| dζ
+ |η| 2 hξηi e
ZR
Z R
1−s
1−s
+ |η| 2 hξηi−1 e−C|η| dη |ζ| 2 hξζi−1 e−C|ζ| dζ
R
R
≤ Chξi−θ k∂ξ φkL2,λ k∂ξ ψkL2,δ
with θ = 2 + λ + δ − , s < 3, > 0. For the case p = 2 we use the inequality
kφ(· − η) − φ(·)kL2,λ ≤ C|η|hηi|λ| k∂ξ φkL2,λ
to get
kAa,b (φ, ψ)kL2,θ
Z
≤ sup
dηdζhξηi−1 hξζi−1 (|η| + |ζ|)−s e−C|η|−C|ζ|
ξ∈R
R2
× k∂ξ φkL2,λ hξiθ−λ |ηζ||∂ζ ψ(ξ − ζ)| + k∂ξ ψkL2,δ hξiθ−δ |ηζ||∂η φ(ξ − η)|
+ k∂ξ φkL2,λ k∂ξ ψkL2,δ hξiθ−λ−δ |η||ζ|1/2 .
Hence
kAa,b (φ, ψ)kL2,θ ≤ Ck∂ξ φkL2,λ k∂ξ ψkL2,δ suphξiθ−λ−δ
ξ∈R
×
Z
2s
|η|1− 3 hξηi−1 e−C|η| dη
ZR
ZR
1/2
−1 −C|ζ|
hξζi
e
dζ
Z
2s
|η|2− 3 hξηi−2 e−C|η| dη
1/2
R
1− 2s
3
|η|
+
2s
|ζ|2− 3 hξζi−2 e−C|ζ| dζ
R
1− 2s
3
|ζ|
+
Z
hξηi−1 e−C|η| dη
Z
1
s
|ζ| 2 − 3 hξζi−1 e−C|ζ| dζ
R
R
≤ Ck∂ξ φkL2,λ k∂ξ ψkL2,δ
with θ = 2 + λ + δ − , s < 3, > 0. To prove the second estimate of the lemma as
above we integrate by parts with respect to ζ
Z
Aa,b (φ, ψ) =
dηdζe−biξζ ζQA(φ(ξ − η) − φ(ξ))∂ζ ψ(ξ − ζ)
R2
+ (φ(ξ − η) − φ(ξ))(ψ(ξ − ζ) − ψ(ξ))ζ∂ζ (QA) .
Using the estimates
1
|ψ(ξ − ζ) − ψ(ξ)| ≤ Chξi−λ |ζ| 2 hζi|λ| k∂ξ ψkL2,λ
and
|ζ∂ζ (QA)| + |QA| ≤ Chξζi−1 (|η| + |ζ|)−s e−C|η|−C|ζ| ,
10
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
we obtain
Z
|Aa,b (φ, ψ)| ≤
dηdζ|(φ(ξ − η) − φ(ξ))|hξζi−1 (|η| + |ζ|)−s e−C|η|−C|ζ|
× |ζ||∂ζ ψ(ξ − ζ)| + k∂ξ ψkL2,δ hξi−δ |ζ|1/2 .
R2
Hence
σ−λ−δ
Z
kAa,b (φ, ψ)kL2,σ ≤ CkφkL2,λ k∂ξ ψkL2,δ suphξi
ξ∈R
×
Z
e−C|η| dη
R
|ζ|2 hξζi−2 (|η| + |ζ|)−2s e−C|ζ| dζ
1/2
R
Z
|ζ|1/2 hξζi−1 (|η| + |ζ|)−s e−C|ζ| dζ
+
R
≤ CkφkL2,λ k∂ξ ψkL2,δ
if σ = 1 + λ + δ − , s < 52 , > 0. The proof is complete.
We next estimate the operator
Z
A(φ, ψ) ≡
e−iξ(aη+bζ) A(η, ζ)φ(ξ − η)ψ(ξ − ζ)dηdζ,
R2
where a, b ∈ R\{0}. Denote
Z
e−iξ(aη+bζ) A(η, ζ)dηdζ.
Λ(ξ) =
R2
Lemma 3.3. Suppose that
a1 η + b1 ζ −hηi−hζi ≤ C(|η| + |ζ|)−k−l e−hηi−hζi
e
∂ηk ∂ζl A(η, ζ) − 2
η − ζη + ζ 2
for all η, ζ ∈ R, k, l = 0, 1, 2, 3, where a1 , b1 ∈ R. Then
kA(φ, ψ) − ΛφψkLp,θ
≤ CkφkLp,α k∂ξ ψkL2,δ + CkψkLp,β k∂ξ φkL2,λ + Ck∂ξ φkL2,λ k∂ξ ψkL2,δ ,
with
3 1
3 1
θ = min( + + α + δ, + + β + λ, 2 + λ + δ − ),
2 p
2 p
p = 2, ∞
and
kA(φ, ψ)kL2,σ ≤ CkφkL2,λ (kψkL∞,β + k∂ξ ψkL2,δ )
with σ = min(1 + λ + β, 1 + λ + δ − ), α, β, λ, δ ∈ R > 0 is small.
Proof. We represent
A(φ, ψ) = Λφψ + φK1 ψ + ψK2 φ + Aa,b (φ, ψ),
where the kernels of the operators K1 and K2 are
Z
Z
−iξ(aη+bζ)
K1 (ξ, ζ) =
e
A(η, ζ)dη, K2 (ξ, η) =
e−iξ(aη+bζ) A(η, ζ)dζ.
R
R
ζ
2
For |ξζ| < 1, ζ 6= 0 changing η − = z we get
Z
ζ
a1 η + b1 ζ
−hζi
K1 (ξ, ζ) = e
e−iaξη−hη− 2 i 2
dη
η
− ζη + ζ 2
R
(3.1)
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
11
Z
ζ
+ O e−hζi ((|η| + |ζ|)−1 |e−hηi − e−hη− 2 i | + e−hηi )dη
ZR
i
dz
= e− 2 aξζ−hζi e−hzi (C1 iz sin(aξz) + C2 ζ cos(aξz)) 2 3 2 + O(e−hζi )
z
+
R
4ζ
= O(e−hζi ).
For |ξζ| ≥ 1 integrating three times by parts we obtain
Z
−3
|K1 (ξ, ζ)| = |(aξ)
e−iaξη ∂η3 A(η, ζ)dη|
R
Z
dη
≤ C|ξ|−3 e−hζi
4
R (|η| + |ζ|)
≤ C|ξζ|−3 e−hζi .
Hence the estimates are true for all ξ ∈ R, ζ, η ∈ R\{0}
|K1 (ξ, ζ)| ≤ Ce−hζi hξζi−3 ,
|K2 (ξ, η)| ≤ Ce−hηi hξηi−3 .
Then by Lemma 3.1, we find
kψK2 φk
L
kφK1 ψk
p, 3 + 1 +β+λ
2
p
≤ CkψkLp,β k∂ξ φkL2,λ ,
p, 3 + 1 +α+δ
2
p
≤ CkφkLp,α k∂ξ ψkL2,δ .
L
Applying Lemma 3.2 with s = 1 we have
kAa,b (φ, ψ)kLp,2+λ+δ− ≤ Ck∂ξ φkL2,λ k∂ξ ψkL2,δ
with > 0, p = 2, ∞. Hence the first estimate of the lemma follows. To prove the
second estimate of the lemma we note that by (3.1),
Z
Λ(ξ) =
e−iξ(aη+bζ) A(η, ζ)dηdζ
2
ZR
Z
a1 η + b1 ζ
=
dηe−iaξη−hηi e−ibξζ 2
dζ + O(hξi−2 )
η − ηζ + ζ 2
R
R
Z
√
3
b
= π (C1 + C2 signη)e−i(a+ 2 )ξη− 2 |η||bξ|−hηi dη + O(hξi−2 )
R
= O(hξi−1 ).
Hence
kΛφψkL2,1+λ+β ≤ CkφkL2,λ kψkL∞,β .
By virtue of estimates in Lemma 3.1, we get
kφK1 ψk
3
L2, 2 +λ+δ
≤ CkφkL2,λ k∂ξ ψkL2,δ ,
kψK2 φkL2,1+λ+β ≤ CkφkL2,λ kψkL∞,β .
Then applying the second estimate of Lemma 3.2 with s = 1, we have
kAa,b (φ, ψ)kL2,1+λ+δ− ≤ CkφkL2,λ k∂ξ ψkL2,δ
with > 0. The proof is complete.
2
Next we consider the L -estimates of the operator
Z
2
2
i
i
H(φ, ψ) =
A(η, ζ)e 2 a(ξ−η) φ(ξ − η)∂ζ e 2 b(ξ−ζ) ψ(ξ − ζ) dη dζ.
R2
12
N. HAYASHI, P. I. NAUMKIN
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Lemma 3.4. Suppose that condition (3.1) is fulfilled. Then the estimates
kH(φ, ψ)kL2 ≤ CkφkL2 (kψkL∞ + k∂ξ ψkL2,λ + k∂ξ2 ψkL2 ),
kH(φ, ψ)kL2 ≤ CkψkL2 (kφkL∞ + k∂ξ φkL2 )
are true provided that the right-hand sides are finite, λ > 0.
Proof. To prove the first estimate of the lemma we write
2
i
H(φ, ψ) = e 2 (a+b)ξ (ξA1 (φ, ψ) + A2 (φ, ψ) + A3 (φ, ∂ξ ψ))
where the kernels are
i
A1 (η, ζ) = −ibe 2 (aη
2
+bζ 2 )
A(η, ζ),
ib i (aη2 +bζ 2 )
ζA(η, ζ),
e2
2
2
2
i
A3 (η, ζ) = −e 2 (aη +bζ ) A(η, ζ)
A2 (η, ζ) =
respectively. Then by the second estimate of Lemma 3.3 we obtain
kH(φ, ψ)kL2 ≤ kξA1 (φ, ψ)kL2 + kA2 (φ, ψ)kL2 + kA3 (φ, ∂ξ ψ)kL2
≤ CkφkL2 kψkL∞ + k∂ξ ψkL2,λ + k∂ξ ψkL∞ + k∂ξ2 ψkL2 .
Thus the first estimate of the lemma is true. To prove the second estimate we
2
i
e
denote e 2 bξ ψ(ξ) = ψ(ξ)
and represent
i
2
2
i
i
2
e 2 a(ξ−η) φ(ξ − η) = φ(ξ)e 2 a(ξ−η) + e 2 a(ξ−η) (φ(ξ − η) − φ(ξ)).
Then we integrate by parts with respect to ζ,
Z
2
i
e − ζ) − ψ(ξ))dηdζ
e
H(φ, ψ) = −
e 2 a(ξ−η) ∂ζ A(η, ζ)φ(ξ − η)(ψ(ξ
R2
2
i
e
= −φK1 ψe − e 2 aξ Aea,0 (φ, ψ),
where the kernels are respectively
Z
K1 (ξ, ζ) =
i
2
e 2 a(ξ−η) ∂ζ A(η, ζ)dη,
R
e ζ) = e 2i aη2 ∂ζ A(η, ζ).
A(η,
As above we have the estimate
|K1 (ξ, ζ)| ≤ Chξie−hζi hξζi−3 .
Thus by Lemma 3.1 we find
e L2 ≤ CkφkL∞ kψkL2 .
kφK1 ψk
Finally by the second estimate of Lemma 3.2 with s = 2, λ = 0 we estimate
2
i
e L2 ≤ Ck∂ξ φkL2 kψkL2
ke 2 aξ Aea,0 (φ, ψ)k
Thus the second estimate of the lemma is true. The lemma is proved.
As a consequence of Lemmas 3.3–3.4 we obtain the estimates of the operators
Z
2
2
a
b
a+b
Ga,b (φ, ψ) = E
h(t, η, ζ)e 2 it(ξ−η) φ(t, ξ − η)e 2 it(ξ−ζ) ψ(t, ξ − ζ)dηdζ
R2
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
13
and
Ha,b (φ, ψ) = t−1/2 E
a+b
Z
a
2
h(t, η, ζ)e 2 it(ξ−η) φ(t, ξ − η)
R2
× ∂ζ (e
2
b
2 it(ξ−ζ)
ψ(t, ξ − ζ))dηdζ.
√
Denote Bλ = hξ tiλ and
Z
Qa,b (t, ξ) =
e−itξ(aη+bζ) h(t, η, ζ)dηdζ.
R2
Lemma 3.5. Suppose that
√
√
√
t(a1 η + b1 ζ) −h√tηi−h√tζi
|∂ηk ∂ζl (h(t, η, ζ) − 2
e
)| ≤ C(|η| + |ζ|)−k−l e−Ch tηi−Ch tζi
2
η − ζη + ζ
(3.2)
for all t ≥ 1, η, ζ ∈ R, k, l = 0, 1, 2, 3. Then the following estimates are true for all
t ≥ 1:
kBθ (Ga,b (φ, ψ) − Qa,b φψ)kLp
1
1
≤ Ct− 4 kBα φkLp kBδ ∂ξ ψkL2 + Ct− 4 kBβ ψkLp kBλ ∂ξ φkL2
+ Ct−1/2 kBλ ∂ξ φkL2 kBδ ∂ξ ψkL2 ,
where
3 1
3 1
θ = min( + + α + δ, + + β + λ, 2 + λ + δ − ),
2 p
2 p
p = 2, ∞
and
1
kBσ Ga,b (φ, ψ)kL2 ≤ CkBλ φkL2 (kBβ ψkL∞ + t− 4 kBδ ∂ξ ψkL2 )
for σ = min(1 + λ + β, 1 + λ + δ − ), α, β, λ, δ ∈ R, > 0 is small. Also for all
t ≥ 1,
3
1
kHa,b (φ, ψ)kL2 ≤ CkφkL2 kψkL∞ + t− 4 kBλ ∂ξ ψkL2,λ + t− 4 k∂ξ2 ψkL2
and
1
kHa,b (φ, ψ)kL2 ≤ CkψkL2 kφkL∞ + t− 4 k∂ξ φkL2 ,
where λ > 0.
√
√
√
Proof. We make a change tξ = ξ 0 , tη = η 0 and tζ = ζ 0 ,
Z
2
2
a
b
a+b
Ga,b (φ, ψ) = E
h(t, η, ζ)e 2 it(ξ−η) φ(t, ξ − η)e 2 it(ξ−ζ) ψ(t, ξ − ζ)dηdζ
R2
Z
−iξ 0 (aη 0 +bζ 0 ) e 0 0 e
e ξ 0 − ζ 0 )dη 0 dζ 0 ,
=
e
A(η , ζ )ψ(t, ξ 0 − η 0 )φ(t,
R2
where
η0 ζ 0
e 0 , ζ 0 ) = t−1 e a2 i(η0 )2 + 2b i(ζ 0 )2 h(t, √
A(η
, √ ),
t
t
0
0
ξ
ξ
e ξ 0 ) = φ(t, √ ), ψ(t,
e ξ 0 ) = ψ(t, √ ),
φ(t,
t
t
Also we denote
Z
e
Q(ξ)
=
R2
0
0
e 0 , ζ 0 )dη 0 dζ 0 .
e−iξ(aη +bζ ) A(η
14
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
Now application of Lemma
√ 3.3 yields
√ the first two
√ estimates of the lemma. As
above we make a change tξ = ξ 0 , tη = η 0 and tζ = ζ 0
Z
0 2
i
e ξ 0 − η0 )
e 0 , ζ 0 )e 2i a(ξ0 −η0 )2 φ(t,
Ha,b (φ, ψ) = e 2 (a+b)(ξ )
A(η
R2
× ∂ζ 0 (e
0
0 2
i
2 b(ξ −ζ )
e ξ 0 − η 0 ))dη 0 dζ 0 ,
ψ(t,
then applying Lemma 3.4 we find the estimates of the lemma. The proof is complete.
4. Inverse transformation
We consider the transformation
3
I(v) = v − E G−1,−1 (v, v),
it
2
where E(t) = e 2 ξ and
Z
a+b
Ga,b (φ, ψ) = E
a
2
2
b
h(t, η, ζ)e 2 it(ξ−η) φ(t, ξ − η)e 2 it(ξ−ζ) ψ(t, ξ − ζ)dηdζ.
R2
We first give estimates of the operator Ga,b in the norm
3
1
kφkXα,λ ≡ sup (kBα φ(t)kL∞ + t− 4 kBλ ∂ξ φ(t)kL2 + t− 4 k∂ξ2 φ(t)kL2 ).
T
1≤t≤T
Lemma 4.1. Let condition (3.2) be fulfilled. Then the estimate
kE q (Ga,b (φ, ψ) − Qa,b φψ)kXθ,σ ≤ CkφkXα,λ kψkXβ,δ
T
is true, where θ =
small,
min( 23
+ α + δ,
3
2
T
T
+ β + λ, 2 + λ + δ − ), α, β, λ, δ ∈ R, > 0 is
3
+ λ + δ)
2
with an additional condition in the case of q 6= 0 that α, β, λ, δ are such that
1
σ < min(α + δ, β + λ, + λ + δ),
2
1
min(α + δ, β + λ, + λ + δ) > 1.
2
Proof. By the first estimate of Lemma 3.5 with p = ∞ we have
σ < min(2 + α, 2 + β, 2 + λ, 2 + δ, 1 + α + δ, 1 + β + λ,
kBθ (Ga,b (φ, ψ) − Qa,b φψ)kL∞ ≤ CkφkXα,λ kψkXβ,δ .
T
T
We now estimate the derivative
∂ξ (E q (Ga,b (φ, ψ) − Qa,b φψ))
= E q (Ga,b (∂ξ φ, ψ) − Qa,b ψ∂ξ φ) + E q (Ga,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψ)
√
q
e
+ tE q (Gg
a,b (φ, ψ) − Qa,b φψ) + itqξE (Ga,b (φ, ψ) − Qa,b φψ),
√
e
e
where Gg
a,b and Qa,b are defined by the kernel h(t, η, ζ) = −i t(aη + bζ)h(t, η, ζ).
Then by the first estimate of Lemma 3.5 with p = 2 we find the estimate
kBσ ∂ξ (E q (Ga,b (φ, ψ) − Qa,b φψ))kL2
≤ CkBσ (Ga,b (∂ξ φ, ψ) − Qa,b ψ∂ξ φ)kL2
+ CkBσ (Ga,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψ)kL2
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
15
√
e
+ C tkB− 12 − kL2 kBσ+ 21 + (Gg
a,b (φ, ψ) − Qa,b φψ)kL∞
√
+ Cq tkB− 12 − kL2 kBσ+ 23 + (Ga,b (φ, ψ) − Qa,b φψ)kL∞
≤ Ct1/4 kφkXα,λ kψkXβ,δ
T
T
with the additional condition
1
+λ+δ
2
for the case of q =
6 0. Finally we estimate the second derivative
σ < min 2 + α, 2 + β, 2 + λ, 2 + δ, α + δ, β + λ,
∂ξ2 (E q (Ga,b (φ, ψ) − Qa,b φψ))
= E q (Ga,b (∂ξ2 φ, ψ) − Qa,b ψ∂ξ2 φ)
+ 2E q (Ga,b (∂ξ φ, ∂ξ ψ) − Qa,b ∂ξ φ∂ξ ψ) + E q (Ga,b (φ, ∂ξ2 ψ) − Qa,b φ∂ξ2 ψ)
√ q
√
e
e
tE (Gg
+ tE q (Gg
a,b (∂ξ φ, ψ) − Qa,b ∂ξ φψ) +
a,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψ)
g
ee
2
q
+ tE q (Gg
a,b (φ, ψ) − Qa,b φψ) + iqt(1 + iqtξ )E (Ga,b (φ, ψ) − Qa,b φψ)
+ itqξE q (Ga,b (∂ξ φ, ψ) − Qa,b ψ∂ξ φ) + itqξE q (Ga,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψ)
3
e
+ 2iqt 2 ξE q (Gg
a,b (φ, ψ) − Qa,b φψ),
g
ee
where Gg
a,b and Qa,b are defined by the kernel
√
e
e
h(t, η, ζ).
h(t, η, ζ) = −i t(aη + bζ)e
Then by Lemma 3.5 we find the estimates
k∂ξ2 (E q (Ga,b (φ, ψ) − Qa,b φψ))kL2
≤ CkGa,b (∂ξ2 φ, ψ) − Qa,b ψ∂ξ2 φkL2
+ CkGa,b (∂ξ φ, ∂ξ ψ) − Qa,b ∂ξ φ∂ξ ψkL2 + CkGa,b (φ, ∂ξ2 ψ) − Qa,b φ∂ξ2 ψkL2
√
√
e
g
e
+ C tkGg
a,b (∂ξ φ, ψ) − Qa,b ψ∂ξ φkL2 + C tkGa,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψkL2
g
ee
+ CtkB− 21 − kL2 kB 12 + (Gg
a,b (φ, ψ) − Qa,b φψ)kL∞
√
+ Cq tkB1 (Ga,b (∂ξ φ, ψ) − Qa,b ψ∂ξ φ)kL2
√
+ Cq tkB1 (Ga,b (φ, ∂ξ ψ) − Qa,b φ∂ξ ψ)kL2
e
+ CqtkB− 21 − kL2 kB 23 + (Gg
a,b (φ, ψ) − Qa,b φψ)kL∞
+ CqtkB− 21 − kL2 kB 25 + (Ga,b (φ, ψ) − Qa,b φψ)kL∞
≤ Ct3/4 kφkXα,λ kψkXβ,δ
T
T
with the additional condition
1
+λ+δ >1
2
for the case of q =
6 0. Lemma 4.1 is proved.
min α + δ, β + λ,
Now let us find the inverse transformation I −1 . We consider the equation
φ = I(v).
(4.1)
16
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
3
We look for the solution of (4.1) in the form v = φ + ψ1 + E ψ2 and substitute it
into (4.1), then we find
3
3
ψ1 + E ψ2 = E G−1,−1 (φ + ψ1 , φ + ψ1 ) + 2G−1,2 (φ + ψ1 , ψ2 ) + E 3 G2,2 (ψ2 , ψ2 ). (4.2)
3
Comparing in (4.2) the terms with the same oscillating exponents like E we find
a system of equations
ψ1 = G−1,2 (φ + ψ1 , ψ2 ) + E 3 G2,2 (ψ2 , ψ2 ),
(4.3)
ψ2 = G−1,−1 (φ + ψ1 , φ + ψ1 ).
In the next lemma we solve this system in the space
Zε = (ψ1 , ψ2 ) ∈ (C([1, T ]; C2 (R)))2 : k(ψ1 , ψ2 )kZ ≤ Cε2
with the norm
k(ψ1 , ψ2 )kZ ≡ kψ1 k
2,1+
XT
γ
3
+ kψ2 k
1,1+
XT
γ
2
,
where γ ∈ (0, 12 ).
Lemma 4.2. Suppose that φ ∈ C([1, T ]; C2 (R)) and the estimate is true
kφkX0,γ ≤ ε,
T
where ε > 0 is sufficiently small. Then there exists unique solutions ψ1 , ψ2 ∈
C([1, T ]; C2 (R)) of a system (4.3) such that k(ψ1 , ψ2 )kZ ≤ Cε2 .
Proof. We solve equations (4.3) by the contraction mapping principle in the set Zε .
Define the transformation M(ψ1 , ψ2 ) = (M1 , M2 ), where
M1 = G−1,2 (φ + ψ1 , ψ2 ) + E 3 G2,2 (ψ2 , ψ2 ),
(4.4)
M2 = G−1,−1 (φ + ψ1 , φ + ψ1 )
for (ψ1 , ψ2 ) ∈ Zε . Applying Lemma 4.1, in view of the fact that kφkX0,γ ≤ ε and
T
k(ψ1 , ψ2 )kZ ≤ Cε2 , we obtain, by (4.4),
kM1 k
2,1+
XT
γ
3
≤ kG−1,2 (φ + ψ1 , ψ2 ) − Q−1,2 (φ + ψ1 )ψ2 k
2,1+
XT
γ
3
2
+ kE 3 (G2,2 (ψ2 , ψ2 ) − Q2,2 ψ2 )k
2,1+
XT
+ kQ−1,−1 (φ + ψ1 )ψ2 k
≤ Ckψ2 k
1,1+
XT
γ
2
γ
3
2
2,1+
XT
+ kE 3 Q2,2 ψ2 k
γ
3
(kφkX0,γ + kψ1 kX0,γ + kψ2 k
T
T
1,1+
XT
2,1+
XT
γ
3
) ≤ Cε3 .
γ
2
In the same manner we have
kM2 k
1,1+
XT
γ
2
≤ kG−1,−1 (φ + ψ1 , φ + ψ1 ) − Q−1,−1 (φ + ψ1 )2 k
1,1+
XT
+ kQ−1,−1 (φ + ψ1 )2 k
1,1+
XT
γ
2
γ
2
≤ C(kφkX0,γ + kψ1 kX0,γ )2 ≤ Cε2 .
T
T
Thus the mapping M(ψ1 , ψ2 ) transforms the set Zε into itself. In the same manner
we find
f1 , ψ
f2 )kZ ≤ 1 k(ψ1 , ψ2 ) − (ψ
f1 , ψ
f2 )kZ .
kM(ψ1 , ψ2 ) − M(ψ
2
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
17
Therefore (M1 , M2 ) is a contraction mapping in Zε . Hence there exist a unique
solution (ψ1 , ψ2 ) ∈ Zε of a system of integral equations (4.3). The proof is complete.
5. Estimates for derivatives
We now make a change φ = I(v) in equation (2.5)
Lφ = t−1 P,
(5.1)
φ(1, ξ) = φ0 (ξ),
where L = i∂t +
1
2
2t2 ∂ξ ,
i
2
E = e 2 tξ ,
3
P = E Ge−1,−1 (v, v) − H−1,2 (v, v 2 )
with v = I −1 (φ) and the operator
Ha,b (φ, ψ) = t−1/2 E
a+b
Z
2
a
h(t, η, ζ)e 2 it(ξ−η) φ(t, ξ − η)
R2
× ∂ζ e
2
b
2 it(ξ−ζ)
ψ(t, ξ − ζ) dηdζ.
Define the norms
kφkVTα ≡ sup kBα φ(t)kL∞ ,
1≤t≤T
kφkWTγ ≡ sup (t
− 14
3
kBγ φ(t)kL2 + t− 4 k∂ξ φ(t)kL2 ).
1≤t≤T
First we state the local existence result for equation (5.1). Denote Y = {φ ∈
L∞ , φ0 ∈ H1,1 }.
Theorem 5.1. Assume that the initial data φ0 ∈ Y. Then for some time T > 1
there exists a unique solution φ ∈ C([1, T ]; Y) of the Cauchy problem (5.1).
In the next lemma we give a representation for the derivatives of the operator
Ha,b .
Lemma 5.2. Let condition (3.2) be fulfilled. Then the estimate is true
3
k∂ξ (E q Ha,b (φ, ψ)) − iqbt 2 ξ 2 E q Qa,b φψkWTρ ≤ CkφkXα,λ kψkXβ,δ ,
T
T
where
1
+ α + β − , λ + δ − , 1 + δ − , 1 + λ − 2
α, β, λ, δ ∈ R, > 0 is small. Also in the case of q 6= 0 we assume that α, β ∈ R,
λ, δ > 0 are such that ρ ≥ 1.
ρ = min α + δ, β + λ, 1 + α, 1 + β,
Proof. We note that
√
(1)
Ha,b (φ, ψ) = ib tξGa,b (φ, ψ) + t−1/2 Ga,b (φ, ∂ξ ψ) + Ga,b (φ, ψ),
√
(1)
where Ga,b and Q(1) are defined by the kernel h(1) (t, η, ζ) = tbζh(t, η, ζ). Then
we obtain by Lemma 3.5
√
kBρ ξtE q (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
√
√
≤ tkBρ+1 (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
√
≤ C tkB2+ρ (Ga,b (φ, ψ) − Qa,b φψ)kL2
18
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
√
(1)
+ CkBρ+1 Ga,b (φ, ∂ξ ψ)kL2 + C tkBρ+1 Ga,b (φ, ψ)kL2
≤ Ct1/4 kφkXα,λ kψkXβ,δ .
T
T
We now estimate by Lemma 3.5 the derivative,
√
kBρ ∂ξ (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
√
≤ C tkBρ ∂ξ ξ(Ga,b (φ, ψ) − Qa,b φψ)kL2
(1)
+ Ct−1/2 kBρ ∂ξ Ga,b (φ, ∂ξ ψ)kL2 + CkBσ ∂ξ Ga,b (φ, ψ)kL2
≤ Ct1/4 kφkXα,λ kψkXβ,δ .
T
Hence
T
√
k∂ξ ξtE q (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
√
≤ CtkHa,b (φ, ψ) − ib tξQa,b φψkL2
√
+ CqtkB2 (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
√
√
+ C tkB1 ∂ξ (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2
3
≤ Ct 4 kφkXα,λ kψkXβ,δ
T
T
if ρ ≥ 1 for the case q 6= 0. Also by Lemma 3.5 we have
kBρ E q ∂ξ Ha,b (φ, ψ)kL2
√
√
≤ CkBρ ∂ξ (Ha,b (φ, ψ) − ib tξQa,b φψ)kL2 + C tkBρ ∂ξ (ξQa,b φψ)kL2
≤ Ct1/4 kφkXα,λ kψkXβ,δ
T
T
and
k∂ξ E q ∂ξ Ha,b (φ, ψ)kL2
√
≤ k∂ξ2 Ha,b (φ, ψ)kL2 + Cq tkB1 E q ∂ξ Ha,b (φ, ψ)kL2
≤ Ct3/4 kφkXα,λ kψkXβ,δ
T
T
if ρ ≥ 1 in the case of q 6= 0. The lemma is proved.
3
We now substitute the inverse transformation v = I −1 (φ) = φ1 + E ψ2 with
φ1 = φ + ψ1 into the operator P to get
3
P = E H2,2 (ψ2 , φ21 ) + 2E 3 H−1,−1 (φ1 , φ1 ψ2 ) + E 6 H−1,−4 (φ1 , ψ22 )
3
+ E Ge−1,−1 (φ1 , φ1 ) + H−1,2 (φ1 , φ21 ) + 2Ge−1,2 (φ1 , ψ2 )
+ E 3 Ge2,2 (ψ2 , ψ2 ) + 2H2,−1 (ψ2 , φ1 ψ2 ) + E 3 H2,−4 (ψ2 , ψ22 ).
Denote ε = kφkX0,γ . Since ψ2 = G−1,−1 (φ + ψ1 , φ + ψ1 ) then by Lemma 3.5 we
T
have
2
ktξ(ψ2 − Q−1,−1 φ )kW1+γ ≤ Cε2
T
Then by virtue of Lemma 5.2 with ρ = 1 we have
3
3
3
k∂ξ E H2,2 (ψ2 , φ21 ) + 6it 2 ξ 2 E Q2,2 Q−1,−1 φ4 kWT1
3
3
3
≤ k∂ξ (E H2,2 (ψ2 , φ21 )) + 6it 2 ξ 2 E Q2,2 ψ2 φ21 kWT1
3
3
3
3
+ Ckt 2 ξ 2 E Q2,2 ψ2 φ21 − t 2 ξ 2 E Q2,2 Q−1,−1 φ4 kWT1
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
≤ Ckψ2 k
1,1+
XT
γ
2
19
kφ21 kX0,γ ≤ Cε2 .
T
In the same manner
3
3
k∂ξ E 3 H−1,−1 (φ1 , φ1 ψ2 ) + 3it 2 ξ 2 E 3 Q2−1,−1 φφ kWT1 ≤ Cε2 ,
5
3
k∂ξ E 6 H−1,−4 (φ1 , ψ22 ) + 24it 2 ξ 2 E 6 Q2−1,−1 Q−1,−4 φ kWT1 ≤ Cε2 .
All the other terms in the derivative ∂ξ P can be estimated in the norm WT1 , the
worst term is H−1,2 (φ1 , φ21 ) which yields the restriction γ < 21 − . Therefore we
can represent ∂ξ P in the form
∂ξ P = t1/2
3
X
E ωj Ωj Nj + R,
(5.2)
j=1
3
5
where N1 = φ4 , N2 = φφ , N3 = φ , ω1 = −3, ω2 = 3, ω3 = 6,
Ω1 = −6itξ 2 Q−1,−1 Q2,2 ,
Ω2 = −3itξ 2 Q2−1,−1 ,
Ω3 = −24itξ 2 Q2−1,−1 Q−1,−4
with the estimate of the remainder kRkWTγ ≤ Cε2 and
√
k
t− 2 +l |∂ξk ∂tl Ωj (t, ξ)| ≤ Chξ ti−l−k
for all t ≥ 1, ξ ∈ R, j = 1, 2, 3, k, l = 0, 1, 2.
Lemma 5.3. Let the initial data φ0 ∈ Y and kφ0 kY ≤ ε, where ε > 0 is sufficiently
small. Assume that representation (5.2) is valid with the estimate of the remainder
kRkWTγ ≤ Cε2 .
Suppose that
kφkVT0 ≤ ε.
(5.3)
Then the solutions φ ∈ C([1, T ]; Y) of (5.1) satisfy the estimate
k∂ξ φkWTγ < 10ε.
(5.4)
Proof. We prove estimate (5.4) by contradiction. By the continuity of φ we can
find a maximal time Te ∈ (1, T ] such that
k∂ξ φkWγe ≤ 10ε.
(5.5)
T
Thus we have kφkX0,γ ≤ 10ε. By a direct calculation we have
e
T
L(E ωj χj ) = t−1 E ωj
iωj
χj + iωj ξ∂ξ χj + tLχj
2Aj
(5.6)
with Aj = (1 + (1 + ωj )itξ 2 )−1 . This identity is useful in the case of ω 6= −1. Then
we get, from (5.1) and (5.2),
L φξ +
3
X
E ωj χj
j=1
= t−1
3
X
j=1
E ωj t1/2 Ωj Nj +
3
X
iωj χj + t−1
E ωj (iωj ξ∂ξ χj + tLχj ) + t−1 R.
2Aj
j=1
(5.7)
20
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
To eliminate the first summand in the right-hand side of (5.7) we choose χj =
P3
2i 1/2
Ωj Aj Nj and denote Φ = φξ + j=1 E ωj χj . By the identities
ωj t
1
uξ vξ + uLv,
t2
1
Lφ = −Lφ + 2 φξξ ,
t
L(uv) = vLu +
LNj = Njφ Lφ − Njφ Lφ +
1
1
(Njφφ φ2ξ + 2Njφφ |φξ |2 + Njφφ (φξ )2 ) + 2 Njφ φξξ ,
2t2
t
in view of (5.1), we obtain
2i
Nj tL(t1/2 Ωj Aj ) + t−1/2 ∂ξ (Ωj Aj )∂ξ Nj
tLχj =
ωj
i −1/2
+
t
Ωj Aj (Njφφ φ2ξ + 2Njφφ |φξ |2 + Njφφ (φξ )2 )
ωj
2i 1/2
+
t Ωj Aj (Njφ P − Njφ P)
ωj
3
X
1
1
2i −1/2
ωj
+
t
Ωj Aj Njφ Φξ + 2it 2
∂ξ (E Ωj Aj Nj )
ωj
ω
j=1 j
Therefore, from (5.7),
LΦ = 2it−3/2
3
X
1 ωj
E Ωj Aj Njφ Φξ + t−1 R1 ,
ω
j=1 j
(5.8)
where in view of (5.1) we have
R1 = R − 2
3
X
E ωj ξ∂ξ (Ωj Aj Nj ) −
j=1
+
3
X
i
Nj tL(t1/2 Ωj Aj )
ωj
E ωj t−1/2 ∂ξ (Ωj Aj )∂ξ Nj
j=1
+
3
X
E ωj
i −1/2
2
t
Ωj Aj (Njφφ φ2ξ + 2Njφφ |φξ |2 + Njφφ φξ )
ωj
E ωj
2i 1/2
t Ωj Aj Nj (Njφ P − Njφ P)
ωj
j=1
+
3
X
j=1
−
3
X
E ωj
j,l=1
4
ωl
Ωj Aj Njφ ∂ξ (E Ωl Al Nl ).
ωj ωl
Since
|Ωj Aj | + |ξ∂ξ (Ωj Aj )| + |tL(Ωj Aj )| ≤ CB−2 , t−1/2 |∂ξ (Ωj Aj )| ≤ CB−3 ,
it follows that R1 satisfies
kR1 kWγe ≤ Cε2 .
T
2 γ/2
We multiply (5.8) by (M + tξ )
2 γ/2
L((M + tξ )
2 γ/2
Φ) = (M + tξ )
and use the commutator
γ
LΦ + ΦL(M + tξ 2 )γ/2 + 2γt−1 ξ(M + tξ 2 ) 2 −1 ∂ξ Φ
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
21
then we get L((M + tξ 2 )γ/2 Φ) = R2 , where
γ
R2 = ΦL(M + tξ 2 )γ/2 + 2γt−1 ξ(M + tξ 2 ) 2 −1 Φξ
+ 2it−3/2
3
X
1 ωj
E Bγ Ωj Aj Njφ Φξ + t−1 (M + tξ 2 )γ/2 R1 .
ω
j
j=1
Since
γ
1 1+
(M + tξ 2 )γ/2 ,
2t
M
by (5.5) choosing M sufficiently large we have
3
5
γ
kR2 kL2 ≤ k(M + tξ 2 )γ/2 ΦkL2 + Cε 4 t− 4 .
2t
Then we apply the energy method to estimate the L2 -norm of Bγ Φ,
|L(M + tξ 2 )γ/2 | ≤
5
3
d
γ
k(M + tξ 2 )γ/2 ΦkL2 ≤ k(M + tξ 2 )γ/2 ΦkL2 + Cε 4 t− 4 .
dt
2t
Hence integration with respect to time yields
5
k(M + tξ 2 )γ/2 ΦkL2 ≤ ε + Cε 4 t1/4
5
for all t ∈ [1, Te] if 0 < γ < 12 . Therefore, kBγ φξ kL2 ≤ ε + Cε 4 t1/4 for all t ∈ [1, Te].
We now differentiate (5.8) with respect to ξ to get
LΦξ = 2it−3/2
where
R3 = 2it−3/2
3
X
1 ωj
E Ωj Aj Njφ Φξξ + R3 ,
ω
j=1 j
(5.9)
3
X
1
Φξ ∂ξ (E ωj Ωj Aj Njφ ) + t−1 ∂ξ R1 .
ω
j
j=1
1
By (5.5) we see that kR3 kL2 ≤ Cε2 t− 4 . Then to estimate the L2 -norm of Φξ we
apply the energy method to (5.9)
Z
3
X
1
d
kΦξ k2L2 ≤ Ct−3/2
| Ωj Aj E ωj Njφ Φξξ Φξ dξ| + Cε3 t 2 .
dt
R
j=1
Hence integrating by parts with respect to ξ we avoid the derivative loss and obtain
3 3
kΦξ kL2 ≤ ε + Cε 2 t 4 for all t ∈ [1, Te]. Therefore
3
3
k∂ξ2 φkL2 < ε + Cε 2 t 4
for all t ∈ [1, Te]. Thus we have k∂ξ φkWTγ < 10ε. The contradiction proves estimate
(5.4). Lemma 5.3 is proved.
6. Estimates in the uniform norm
We now estimate φ in the norm kφkVT0 = sup1≤t≤T kφ(t)kL∞ .
Lemma 6.1. Let the initial data φ0 ∈ Y and kφ0 kY ≤ ε, where ε > 0 is sufficiently
small. Suppose that
k∂ξ φkWTγ < 10ε.
(6.1)
Then the solutions φ ∈ C([1, T ]; Y) of (5.1) satisfy the estimate
kφkVT0 < 10ε.
(6.2)
22
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
Proof. We prove estimate (6.2) by the contradiction. By the continuity of φ we can
find a maximal time Te ∈ (1, T ] such that
kφkV0e ≤ 10ε.
(6.3)
T
Now (6.3) along with (6.1) imply that
kφkX0,γ ≤ Cε.
(6.4)
e
T
Denote w(t) = V(−t)φ(t), where
V(−t) = FM F −1 =
√ Z
2
it
t dηe− 2 (ξ−η) .
R
Applying operator V(−t) to equation (5.1), we have
iwt = t−1 V(−t)P,
w(1) = V(−1)φ0 ,
(6.5)
2
3
i
where P = E Ge−1,−1 (v, v) − H−1,2 (v, v 2 ), and E = e 2 tξ . Note that P has the
form
P = E∂ξ (G0 (v, v) + G1 (v, v 2 )),
where
Z
2
2
1
1
e
G0 (v, v) =
h1 (t, η, ζ)e− 2 it(ξ−η) v(ξ − η)e− 2 it(ξ−ζ) v(ξ − ζ)dηdζ
2
R
Z
2
2
1
2
G1 (v, v ) =
h1 (t, η, ζ)e− 2 it(ξ−η) v(ξ − η)eit(ξ−ζ) v 2 (ξ − ζ)dηdζ
R2
with kernels
r
√
t
4t 2
e
(η − ηζ + ζ 2 ) ,
h1 (t, η, ζ) = − √ (2 + t∂t )t∂t K0
3i
π 3
r
√
t
4t 2
h1 (t, η, ζ) = √ (1 + t∂t )∂ζ K0
(η − ηζ + ζ 2 ) .
3i
π 3
Then by the identities
V(−t)E∂ξ = FM F −1 (∂ξ + itξ)E
= FM (−ix + t∂x )F −1 E
= tF∂x M F −1 E = itξV(t)E,
we find
V(−t)P = itξV(−t)E(G0 (v, v) + G1 (v, v 2 )).
Integration by parts yields
|(Bγ V(−t) − V(−t)Bγ )φ|
√ Z − it (ξ−η)2 √ γ
√
= t
e 2
(hξ ti − hη tiγ )φ(η)dη
R
√
√
Z
hξ tiγ − hη tiγ
−1/2
− it
(ξ−η)2
2
=t
(ξ − η)∂η (
φ(η))dη
e
1 + it(ξ − η)2
R
≤ CkφkL∞ + Ct−1/4 k∂ξ φkL2
≤ CkφkX0,γ .
e
T
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
23
Hence
kBγ (w − φ)kL∞ ≤ k(Bγ V(−t) − V(−t)Bγ )φkL∞ + k(V(−t) − 1)Bγ φkL∞
1
≤ CkφkX0,γ + Ct− 4 k∂ξ Bγ φkL2 ≤ CkφkX0,γ .
e
T
e
T
Thus by the estimates of Section 4 we see that
v = φ + O(εB−1 ) = w + O(εB−γ ).
Then by Lemma 3.5 we get the representation for the operator P,
t−1 V(−t)P = iξV(−t)E(G0 (v, v) + G1 (v, v 2 ))
e 2 w + O(ε2 t−1/2 ξB−1−γ ),
= −it−1/2 ξ Q|w|
where
Z
e=
Q
eitξ(η−2ζ) h1 (t, η, ζ)dηdζ
R2
√ Z
√
√
ζ − η2
t
−hη ti
= √
dηdζ + O(hξ ti−2 )
e
eitξ(η−2ζ) 2
2
η − ηζ + ζ
π 3 R2
√
2i
√
=√
+ O(hξ ti−2 ).
3(1 + ξ 3t)
Thus we can write (6.5) in the form
1
wt = iΘ|w|2 w + O(ε2 t− 2 ξB−1−γ ).
(6.6)
where
Θ(t, ξ) = − √
2ξ
√ .
3t(1 + ξ 3t)
The first term in the right-hand side of (6.5) is divergent, so we eliminate it by the
change w(t, ξ) = ϕ(t, ξ)Ew , where
Z t
Ew = exp i
Θ(τ, ξ)|w(τ, ξ)|2 dτ .
1
Then we get from (6.6)
ϕt = O(ε2 t−1/2 ξB−1−γ ).
(6.7)
Integrating in time
|ϕ| ≤ ε + Cε2
Z
1
≤ ε + Cε
2
Z
t
√
|ξ|dτ
√
τ (1 + |ξ| τ )1+γ
√
|ξ| t
|ξ|
dz
≤ ε + Cε2 .
(1 + z)1+γ
Hence kφkV0e < 10ε. This contradiction proves (6.2)and completes the proof.
T
24
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
7. Proof of Theorem 1.1
By Lemma 5.3 we see that the a priori estimate of kφkVT0 implies the a priori
estimate of k∂ξ φkWTγ . Vice versa by Lemma 6.1 the a priori estimate of k∂ξ φkWTγ
yields the a priori estimate of kφkVT0 . Therefore the global existence of solution
v = I −1 (φ) ∈ C([1, ∞); Y) of the Cauchy problem (2.5) satisfying a priori estimate
kφkX0,γ
≤ Cε
∞
follows by a standard continuation argument from Lemma 5.3, Lemma 6.1 and the
local existence Theorem 5.1. This yields the solution of the Cauchy problem (1.1).
Theorem 1.1 is proved.
8. Proof of Theorem 1.2
Existence of a self-similar solution of (2.1) of the form
Appendix 10 since
1
√
M S( √xt )
t
follows from
u(t) = U(t)F −1 w(t) = Dt E(t)v(t), w(t) = V(−t)v(t)
√
and w(t) has the form w(t, ξ) = M S(ξ t). We now prove the stability of solutions
in the neighborhood of a self-similar solution of the equation (2.1). We consider the
difference r(t, ξ) = φ1 (t, ξ) − φ2 (t, ξ) and may assume that V(−t)(φ1 (t) − φ2 (t)) = 0
at ξ = 0. Define the norm
1
µ
3
krkWTµ,ν = sup t 2 t− 4 kBν r(t)kL2 + t− 4 k∂ξ r(t)kL2 .
1≤t≤T
Lemma 8.1. Let the initial data φj ∈ Y and kφj kY ≤ ε, where ε > 0 is sufficiently
small. Suppose that V(−t)(φ1 (t) − φ2 (t)) = 0 at ξ = 0. Assume that representation
(5.2) is valid. Suppose that
kφj kX0,γ ≤ ε.
T
Then the solutions φj ∈ C([1, T ]; Y) of (5.1) satisfy the estimate
k∂ξ (φ1 − φ2 )kWTµ,ν < Cε
for 0 < µ < γ and
1
2
−γ ≤ν <
1
2
− µ.
Proof. Denote r(t) = φ1 (t) − φ2 (t), then by (5.1) we have
Lr = t−1 (P(φ1 ) − P(φ2 )).
(8.1)
We prove the estimate of the lemma by contradiction. By the continuity of r(t) we
can find a maximal time Te ∈ (1, T ] such that
k∂ξ (φ1 − φ2 )kWµ,ν
≤ Cε
e
(8.2)
T
Since V(−t)r(t) = w1 (t, ξ) − w2 (t, ξ) vanishes at ξ = 0 for all t ≥ 1, we can estimate
the norm
Z ξ
|V(−t)r(t)| =
B−ν Bν ∂ξ V(−t)r(t)dξ
0
p
≤ C |ξ|B−ν kBν ∂ξ V(−t)r(t)kL2
√ 1
µ
1
≤ |ξ|µ h tξi 2 −ν−µ t 2 − 4 kBν ∂ξ r(t)kL2
√ 1
≤ Cε|ξ|µ h tξi 2 −ν−µ .
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
Also we have
25
1
|(1 − V(−t))r(t)| ≤ t− 4 k∂ξ r(t)kL2 ≤ Cεt−µ/2 .
Hence
|r(t)| ≤ |(1 − V(−t))r(t)| + |V(−t)r(t)|
√ 1
≤ Cεt−µ/2 + Cε|ξ|µ h tξi 2 −ν−µ .
Then we get from (8.1) in view of (5.6),
L rξ +
3
X
3
X
iωj χj
E ωj χj = t−1
E ωj t1/2 Ωj Mj +
2Aj
j=1
j=1
+ t−1
3
X
(8.3)
E ωj (iωj ξ∂ξ χj + tLχj ) + t−1 (R1 − R2 ),
j=1
where we denote Mj = Nj (φ1 ) − Nj (φ2 ). To eliminate the first summand in the
right-hand side of (8.3) we choose χj = ω2ij t1/2 Ωj Aj Mj and denote Φ = rξ +
P3
ωj
j=1 E χj . Therefore, from (8.3) we obtain
LΦ = 2it−3/2
3
X
1 ωj
E Ωj Aj Mjr Φξ + t−1 R3 ,
ω
j
j=1
where in view of (8.1), we have
R3 = R1 − R2 − 2
3
X
E ωj (ξ∂ξ (Ωj Aj Mj ) −
j=1
+
3
X
i
Mj tL(t1/2 Ωj Aj ))
ωj
E ωj t−1/2 ∂ξ (Ωj Aj )∂ξ Mj
j=1
+
3
X
E ωj
i −1/2
t
Ωj Aj (Mjrr rξ2 + 2Mjrr |rξ |2 + Mjrr rξ 2 )
ωj
E ωj
2i 1/2
t Ωj Aj Mj (Mjr (P1 − P2 ) − Mjr (P1 − P2 ))
ωj
j=1
+
3
X
j=1
−
3
X
j,l=1
E ωj
4
ωl
Ωj Aj Mjr ∂ξ (E Ωl Al Ml ).
ωj ωl
Since
|Ωj Aj | + |ξ∂ξ (Ωj Aj )| + |tL(Ωj Aj )| ≤ CB−2 , t−1/2 |∂ξ (Ωj Aj )| ≤ CB−3 ,
then R3 can be estimated as
kR3 kWµ,ν
≤ Cε2
e
T
1
2
if γ ≥ − ν. Then as in the proof of Lemma 5.3 we multiply (5.8) by (M + tξ 2 )ν/2
and use the commutator
ν
L((M + tξ 2 )ν/2 Φ) = (M + tξ 2 )ν/2 LΦ + ΦL(M + tξ 2 )ν/2 + 2νt−1 ξ(M + tξ 2 ) 2 −1 ∂ξ Φ
then we get
L((M + tξ 2 )ν/2 Φ) = R4 ,
26
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
where
ν
R4 = ΦL(M + tξ 2 )ν/2 + 2νt−1 ξ(M + tξ 2 ) 2 −1 Φξ
−3/2
+ 2it
3
X
1 ωj
E Bν Ωj Aj Njφ Φξ + t−1 (M + tξ 2 )ν/2 R3 .
ω
j=1 j
Since
ν
1
(1 +
)(M + tξ 2 )ν/2 ,
2t
M
by (5.5) choosing M sufficiently large we have
5 µ
3
ν
kR4 kL2 ≤ k(M + tξ 2 )ν/2 ΦkL2 + Cε 4 t 2 − 4 .
2t
Then we apply the energy method to estimate the L2 - norm of Bγ Φ
|L(M + tξ 2 )ν/2 | ≤
ν
5 µ
3
d
ν
k(M + tξ 2 )ν/2 ΦkL2 ≤ k(M + tξ 2 ) 2 ΦkL2 + Cε 4 t 2 − 4 .
dt
2t
Hence the integration with respect to time yields
5
1
µ
k(M + tξ 2 )ν/2 ΦkL2 ≤ ε + Cε 4 t 4 − 2
µ
5 1
for all t ∈ [1, Te] if 0 < ν < 21 − µ. Therefore kBν rξ kL2 ≤ ε + Cε 4 t 4 − 2 for all
t ∈ [1, Te].
We now differentiate (8.3) with respect to ξ to get
LΦξ = 2it−3/2
where
R5 = 2it−3/2
3
X
1 ωj
E Ωj Aj Mjr Φξξ + R5 ,
ω
j=1 j
3
X
1
Φξ ∂ξ (E ωj Ωj Aj Mjr ) + t−1 ∂ξ R3 .
ω
j
j=1
ν
1
By (8.2) we see that kR5 kL2 ≤ Cε2 t− 2 − 4 . Then to estimate the L2 - norm of Φξ
we apply the energy method to (5.9)
Z
3
X
1
d
2
−3/2
kΦξ kL2 ≤ Ct
| Ωj Aj E ωj Mjr Φξξ Φξ dξ| + Cε3 t 2 −ν .
dt
R
j=1
Hence integrating by parts with respect to ξ we avoid the derivative loss and obtain
ν
3 3
kΦξ kL2 ≤ ε + Cε 2 t 4 − 2 for all t ∈ [1, Te]. Therefore
3
3
ν
k∂ξ2 rkL2 < ε + Cε 2 t 4 − 2
for all t ∈ [1, Te]. Thus we have k∂ξ rkWTµ,ν < Cε. This contradiction proves estimate
of the lemma. Lemma 8.1 is proved.
To study the asymptotic behavior we make a change as in the proof of Lemma
6.1 w = Vφ = ϕEϕ and s = VS = gEs , where
Z t
Es = exp i
Θ(τ, ξ)|s(τ, ξ)|2 dτ .
1
Then from (6.5) we get for the difference f = ϕ − g
ift = Eϕ t−1 VP(φ) − iΘ|ϕ|2 ϕ − Eg t−1 VP(S) + iΘ|g|2 g
= (Eϕ − Eg )(t−1 VP(Vs) − iΘ|s|2 s) + t−1 Eϕ (V − 1)(P(φ) − P(S))
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
27
− iΘEϕ ((V − 1)φ)(Vφ)2 − ((V − 1)S)(VS)2
+ φ((V − 1)φ)(V + 1)φ − S((V − 1)S)(V + 1)S
+ Eϕ (t−1 P(φ) − iΘ|φ|2 φ − t−1 P(S) + iΘ|S|2 S)
µ
= O(ε2 t− 2 −1 ).
Then we get
|f (t) − f (t1 )| ≤ Cε2
Z
t
µ
τ − 2 −1 dτ ≤ ε2 t1 −µ/2 .
t1
Also by (6.7),
|ϕ(t) − ϕ(t1 )| ≤ Cε
2
Z
t
√
τ −1/2 ξhξ τ i−1−γ dτ ≤
t1
Cε2
√
hξ t1 iγ
for all t > t1 > 1. Therefore the limits exist
F (ξ) = lim f (t, ξ), Φ(ξ) = lim ϕ(t, ξ) and G(ξ) = lim g(t, ξ)
t→∞
t→∞
t→∞
with the estimates |F (ξ)| + |Φ(ξ)| + |G(ξ)| ≤ Cε. Then using the estimates
|f (t, ξ) − F (ξ)| ≤ ε2 t−µ/2 ,
|g(t, ξ) − G(ξ)| ≤
Cε2
√ ,
hξ tiγ
Cε2
√ ,
hξ tiγ
√
|1 − Eg Eϕ | ≤ Cε2 t−µ/2 hξ tiγ ,
|ϕ(t, ξ) − Φ(ξ)| ≤
we obtain
φ = S + Eϕ (f + (1 − Eg Eϕ )g) + (1 − V)r
µ
= S + Eϕ (F + (1 − Eg Eϕ )G) + O(ε2 t− 2 ).
Note that
√
|ϕ(t, ξ)|2 = |Φ(ξ)|2 + O(ε2 hξ ti−γ ).
We now denote
Z
t
Ψ1 (t) = −i
(|ϕ(τ, ξ)|2 − |ϕ(t, ξ)|2 )Θdτ,
1
Z t
Ψ2 (t) = −i
(|g(τ, ξ)|2 − |ϕ(τ, ξ)|2 − |g(t, ξ)|2 + |ϕ(t, ξ)|2 )Θdτ.
1
We then get
Z
t
Ψ1 (t)−Ψ1 (t1 ) = −i
(|ϕ(τ, ξ)|2 −|ϕ(t, ξ)|2 )Θdτ +i(|ϕ(t, ξ)|2 −|ϕ(t1 , ξ)|2 )
Z
t1
t1
Θdτ
1
and
Z
t
Ψ2 (t) − Ψ2 (t1 ) = −i
(|g(τ, ξ)|2 − |ϕ(τ, ξ)|2 − |g(t, ξ)|2 + |ϕ(t, ξ)|2 )Θdτ
t1
+ i(|g(t, ξ)|2 − |ϕ(t, ξ)|2 − |g(t1 , ξ)|2 + |ϕ(t1 , ξ)|2 )
Z
Θdτ
1
for all 1 < t1 < τ < t. Using these estimates we get
√
|Ψj (t) − Ψj (t1 )| ≤ Cε2 hξ t1 i−γ ,
j = 1, 2.
t1
28
N. HAYASHI, P. I. NAUMKIN
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Therefore, there exist unique functions Φj+2 ∈ L∞ , such that
iΦj+2 = lim Ψj (t)
t→∞
and
√
|iΦj+2 − Ψj (t)| ≤ Cε2 hξ ti−γ , j = 1, 2.
Then we find
Z
t
−i
√
|ϕ(τ )|2 Θdτ = i|Φ|2 log(ξ t) + iΦ3 + O(ε2 t−µ/2 )
1
and
t
Z
−i
√
(|g(τ )|2 − |ϕ(τ )|2 )Θdτ = i(|G|2 − |Φ|2 ) log(ξ t) + iΦ4 + O(ε2 t−µ/2 )
1
with some functions Φ3 , Φ4 ∈ L∞ . Hence
√
1 + |ξ| t
2
) − iΦ3 + O(ε2 t−µ/2 )
Eϕ = exp − i|Φ| log(
1 + |ξ|
and
√
1 + |ξ| t
Eg Eϕ = exp − i(|G|2 − |Φ|2 ) log(
) − iΦ4 + O(ε2 t−µ/2 ) .
1 + |ξ|
Thus
φ = S + e−i|Φ|
2
log(
√
1+|ξ| t
)−iΦ3
1+|ξ|
F + G 1 − e−i(|G|
2
−|Φ|2 ) log(
√
1+|ξ| t
)−iΦ4
1+|ξ|
+ O(ε3 t−γ )
=S+
2
X
Hj eiBj log(
√
1+|ξ| t
)
1+|ξ|
+ O(ε2 t−µ/2 ).
j=1
The asymptotic formula follows now from the inverse transformation [11] of u(t, x) =
it 2
t−1/2 e 2 ξ v(t, ξ) and ξ = x/t,
u(t, x) = t−1/2 Ev(t) = t
− 21
ES + t−1/2 E
2
X
iBj log(
Hj e
|x|
1+ √
t
|x|
1+
t
)
1
µ
+ O(t− 2 − 2 ).
j=1
Theorem 1.2 is proved.
9. Appendix 1, Hope-Cole transformation
We consider the quadratic nonlinear Schrödinger equation
1
iut + uxx = ∂x (u2 ), x ∈ R, t ∈ R,
2
u(0, x) = u0 (x), x ∈ R.
(9.1)
There exists a self-similar solution of the form s(t, x) = √1t ϕ(ξ), with ξ = √xt .
Indeed if we substitute √1t ϕ(ξ) into (9.1) we get the ordinary differential equation
for the function ϕ(ξ),
i
1
− (ξϕ)0 + ϕ00 = (ϕ2 )0 ,
2
2
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
29
hence integrating with respect to ξ and choosing the integration constant as zero,
we obtain the Bernoulli equation ϕ0 = iξϕ + 2ϕ2 , which has a solution
r
√
i 2
i −1
ξ
ϕ(ξ) = e 2
C − 2πi erf(ξ − )
,
2
Rx
2
with the error function erf(x) = √2π 0 e−y dy. We suppose that the constant C
have a sufficiently large absolute value. Now we represent
a solution
R
R u(t, x) =
s(t,
x)
+
w(t,
x)
and
choose
a
constant
C
such
that
s(t,
x)dx
=
ϕ(ξ)dξ =
R
u(t, x)dx, then
Z
w(t, x)dx = 0,
so we can consider w as a full derivative: w = vx . Thus we have
1
ivt + vxx = 2svx + (vx )2 .
2
Now by the Hopf-Cole anzatz v = − 21 log(1 + φ) we linearize the above equation
1
iφt + φxx = 2sφx .
2
We can eliminate the right-hand side of the
R xabove equation by virtue of the change
of the dependent variable φ(t, x) = exp(2 0 s(t, y)dy) ψ(t, x),
1
iψt + ψxx = 0, x ∈ R, t ∈ R,
2
ψ(0, x) = ψ0 (x), x ∈ R,
Rx
Rx
where ψ0 (x) = (exp(−2 0 (u0 (y) − s(0, y))dy) − 1) exp(−2 0 s(0, y)dy). Thus the
solution of ( 9.1) have a form
Z x
1
s(t, y)dy)).
u(t, x) = s(t, x) − ∂x log(1 + ψ(t, x) exp(2
2
0
10. Appendix 2, Existence of a self-similar solution
√
Here we study a self-similar solution of the form v(t, ξ) = w(ξ t)
Z
vt (t, ξ) = ξ eitS v(t, η − ξ)v(t, −η)dη,
R
where
1 2
(ξ + (ξ − η)2 + η 2 )
2
and the solution u(t, x) of (1.1) can be obtained from v(t, ξ) through the formula
S=
u(t) = U(t)F −1 v(t).
√
For w(ξ t) we get
w0 (ξ) = 2
Z
eiS w(η − ξ)w(−η)dη.
R
We use the identity eiS = (∂ξ ξ + ∂η η)(KeiS ) + F eiS , where
K=
2 + iS
,
2(1 + iS)2
F =
1 − iS
.
(1 + iS)3
30
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
Then we get
Z
d
(w(ξ) + 2 ξKeiS w(η − ξ)w(−η)dη)
dξ
R
Z
Z
d
iS
= 2 F e w(η − ξ)w(−η)dη − KηeiS (w(η − ξ)w(−η))dη.
dη
R
R
Denote
Z
φ(ξ) = w(ξ) + 2
ξK(ξ, η)eiS w(η − ξ)w(−η)dη,
(10.1)
R
then in view of the symmetry η − ζ ↔ ζ we have
Z
Z
Z
dφ
iS
iS
= 2 F e w(η − ξ)w(−η)dη − 4 dηe Kηw(η − ξ) e−iQ w(η − y)w(y)dy
dξ
R
R
R
(10.2)
with
1
Q = (η 2 + (η − y)2 + y 2 ).
2
We will see later that the right-hand side of (10.2) has the asymptotic
16iπ
|φ(ξ)|2 φ(ξ) + O(hξiγ−2 ).
3hξi
To exclude the first divergent term we make a change φ = ϕE, where
16 Z ξ
dξ E = exp
|ϕ(ξ)|2
i
,
3 0
hξi
then we get from (10.2),
Z
dϕ
16iπ
= 2E
|ϕ(ξ)|2 ϕ(ξ)
F eiS w(η − ξ)w(−η)dη −
dξ
3hξi
R
Z
Z
iS
− 4E
dηe Kηw(η − ξ) e−iQ w(η − y)w(y)dy.
R
(10.3)
R
Integrating the above expression, we obtain
Z
Z ξ
F eiS w(η − ξ)w(−η)dηdξ
ϕ(ξ) = ϕ(0) +
2E
0
ξ
Z
−
R
16iπ
3hξi
0
Z
|ϕ(ξ)|2 ϕ(ξ)
dηeiS Kηw(η − ξ)
+ 4E
R
(10.4)
Z
e−iQ w(η − y)w(y)dy dξ.
R
We solve integral equation (10.4) by the contraction mapping principle. Define the
transformation
Z
Z ξ
A(ϕ)(ξ) = ϕ(0) +
2E
F eiS w(η − ξ)w(−η)dηdξ
0
Z
ξ
−
16iπ
3hξi
0
Z
R
|ϕ(ξ)|2 ϕ(ξ)
iS
(10.5)
Z
dηe Kηw(η − ξ)
+ 4E
R
R
e−iQ w(η − y)w(y)dy dξ.
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
31
We now find the inverse transformation to (10.1). Changing the variable of integration η = η 0 + 2ξ we rewrite (10.1) as (the prime we will omit)
Z
2
2
3
ξ
ξ
w(ξ) = φ(ξ) − e 4 iξ
eiη K1 (ξ, η)w(η − )w(−η − )dη
(10.6)
2
2
R
where
3
ξ
ξ(2 + iS1 )
, S1 = ξ 2 + η 2 .
K1 (ξ, η) = ξK(ξ, η + ) =
2
(1 + iS1 )2
4
We substitute the representation
3
2
w(ξ) = φ(ξ) + ψ1 (ξ) − e 4 iξ ψ2 (ξ)
(10.7)
into (10.6). In view of the symmetry η → −η, changing in the second integral
η = 23 ξ − η 0 we find
Z
2
2
2
3
3
ξ
ξ
eiη K1 (ξ, η)ψ3 (η − )ψ3 (−η − )dη
ψ1 (ξ) − e 4 iξ ψ2 (ξ) = −e 4 iξ
2
2
R
Z
2
1
+ 2 e 4 iη K2 (ξ, η)ψ3 (ξ − η)ψ2 (η − 2ξ)dη
R
−e
2
3
8 iξ
Z
2
1
ξ
ξ
e− 2 iη K1 (ξ, η)ψ2 (−η − )ψ2 (−η − )dη,
2
2
R
where ψ3 (ξ) = φ(ξ) + ψ1 (ξ) and K2 (ξ, η) = K1 (ξ, 23 ξ − η). Thus we obtain the
system of integral equations for the functions ψ1 (ξ) and ψ2 (ξ)
Z
2
1
ψ1 (ξ) = 2 e 4 iη K2 (ξ, η)ψ2 (η − 2ξ)ψ3 (ξ − η)dη
R
,
Z
2
2
3
1
ξ
ξ
+ e− 8 iξ
e− 2 iη K1 (ξ, η)ψ2 (η − )ψ2 (−η − )dη
(10.8)
2
2
R
Z
2
ξ
ξ
ψ2 (ξ) =
eiη K1 (ξ, η)ψ3 (η − )ψ3 (−η − )dη.
2
2
R
Denote the norm
kψkZ ≡ sup(hξi1−γ |ψ(ξ)| + hξi1−2γ |ψ 0 (ξ)|).
ξ∈R
Lemma 10.1. Let φ ∈ C1 (R) satisfy the estimate
sup(|φ(ξ)| + hξi1−γ |φ0 (ξ)|) ≤ ε,
(10.9)
ξ∈R
where ε > 0 is sufficiently small and γ ∈ (0, 1). Then there exist unique solutions
ψ1 , ψ2 ∈ C1 (R) of a system of integral equations (10.8) such that
kψ1 kZ + kψ2 kZ ≤ Cε2 .
(10.10)
Proof. We solve equations (10.8) by the contraction mapping principle in the set
Zε = (ψ1 , ψ2 ) ∈ (C1 (R))2 : kψ1 kZ + kψ2 kZ ≤ Cε2 .
Define the transformation
Z
1
2
e 4 iη K2 (ξ, η)ψ2 (η − 2ξ)ψ3 (ξ − η)dη
M1 (ψ1 , ψ2 )(ξ) = 2
R
− 83 iξ 2
Z
+e
− 21 iη 2
e
R
ξ
ξ
K1 (ξ, η)ψ2 (η − )ψ2 (−η − )dη
2
2
,
32
N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
Z
M2 (ψ1 , ψ2 )(ξ) =
2
ξ
ξ
eiη K1 (ξ, η)ψ3 (η − )ψ3 (−η − )dη
2
2
R
for (ψ1 , ψ2 ) ∈ Zε . Via ( 10.9) and by the fact that kψ1 kZ + kψ2 kZ ≤ Cε2 we obtain
the estimate
|M1 (ψ1 , ψ2 )(ξ)|
Z
ξ
ξ
3
≤ Cε
(1 + |ξ| + |η|)−1 hη − 2ξiγ−1 + hη − iγ−1 hη + iγ−1 dη
2
2
R
≤ Cε3 hξiγ−1 .
2
2
Integrating by parts with respect to η via the identity eiη = A∂η (ηeiη ) with
A = (1 + 2iη 2 )−1 we get
Z
2
ξ
ξ
|M2 (ψ1 , ψ2 )(ξ)| =
eiη ψ3 (η − )ψ3 (−η − )η∂η (AK1 )dη
2
2
R
Z
2
ξ
ξ
+ 2 eiη ηAK1 ψ3 (η − )ψ30 (−η − )dη
2
2
ZR 2
≤ Cε
hξi−1 hηi−2 + (1 + |ξ| + |η|)−2 hηi−1
R
+ (1 + |ξ| + |η|)−1 hηi−1 h2η + ξi2γ−1 dη
≤ Cε2 hξi−1 .
We now estimate the derivatives
d
M1 (ψ1 , ψ2 )(ξ)
dξ
Z
2
1
= 2 e 4 iη ψ2 (η − 2ξ)ψ3 (ξ − η)∂ξ K2 dη
R
Z
2
1
+ 2 e 4 iη (ψ2 (η − 2ξ)ψ30 (ξ − η) − 2ψ20 (η − 2ξ)ψ3 (ξ − η))K2 dη
R
Z
2
1
ξ
ξ
e− 2 iη K1 ψ20 (η − )ψ2 (−η − )dη
2
2
R
Z
2
2
3
1
ξ
ξ
3
+ e− 8 iξ
e− 2 iη ψ2 (η − )ψ2 (−η − )(∂ξ K1 − iξK1 )dη.
2
2
4
R
− 83 iξ 2
−e
and
d
M2 (ψ1 , ψ2 )(ξ) = −
dξ
Z
2
ξ
ξ
eiη K1 ψ30 (η − )ψ3 (−η − )dη
2
2
R
Z
+
2
ξ
ξ
eiη ψ3 (η − )ψ3 (−η − )∂ξ K1 dη.
2
2
R
Then we find the estimates
Z
d
| M1 (ψ1 , ψ2 )(ξ)| ≤ Cε3 (hη − 2ξi2γ−1 + hξ − ηi2γ−1 )(1 + |ξ| + |η|)−1 dη
dξ
R
Z
ξ
ξ
4
+ Cε
hη − iγ−1 hη + iγ−1 dη ≤ Cε3 hξi2γ−1
2
2
R
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
33
and
|
d
M2 (ψ1 , ψ2 )(ξ)| ≤ Cε2
dξ
Z
ξ
(1 + |ξ| + |η|)−1 hη − i2γ−1 dη
2
R
Z
+ Cε2 (1 + |ξ| + |η|)−2 dη ≤ Cε2 hξi2γ−1 .
R
Thus the mapping (M1 , M2 ) transforms the set Z into itself. In the same manner
we find
f1 kZ + 1 kψ2 − ψ
f2 kZ .
f1 , ψ
f2 )kZ ≤ 1 kψ1 − ψ
kMj (ψ1 , ψ2 ) − Mj (ψ
2
2
Therefore, (M1 , M2 ) is a contraction mapping in Z. Hence there exist unique
solutions ψ1 , ψ2 ∈ Z of a system of integral equations (10.8), which satisfy estimate
(10.10). Lemma 10.1 is proved.
We now evaluate the asymptotic form of the integral
Z
2
I≡
eαi(η−µξ) A(ξ, η)φ1 (η − ξ)Φdη,
R
where
Z
2
eβiy φ2 (a1 η − b1 y)φ3 (a2 η − b2 y)dy,
Φ=
R
with α, β, µ, a1 , a2 , b1 , b2 ∈ R\{0}. Also we assume that a1 b2 − a2 b1 6= 0.
Lemma 10.2. Suppose that
kφj kY ≡ sup(|φj (ξ)| + hξi1−2γ |φ0j (ξ)|) ≤ C,
ξ∈R
|A(ξ, η)| ≤ C(1 + |ξ| + |η|)−1 ,
|∂η A(ξ, η)| ≤ C(1 + |ξ| + |η|)−2 .
Then the asymptotic form is true
iπ
I = √ A(ξ, µξ)φ1 ((µ − 1)ξ)φ2 (a1 µξ)φ3 (a2 µξ) + O(hξiγ−2 ).
αβ
2
(10.11)
2
Proof. We first integrate by parts with respect to y via identity eβiy = B∂y (yeβiy )
with B = (1 + 2iβy 2 )−1 we get Φ = Φ1 + Φ2 + Φ3 , where
Z
2
Φ1 = 2 eβiy φ2 (a1 η − b1 y)φ3 (a2 η − b2 y)B(B − 1)dy,
R
Z
2
Φ2 = b1 eβiy φ02 (a1 η − b1 y)φ3 (a2 η − b2 y)yB dy,
ZR
2
Φ3 = b2 eβiy φ2 (a1 η − b1 y)φ03 (a2 η − b2 y)yB dy,
R
Now in the integral
Z
I=
2
eαi(η−µξ) A(ξ, η)φ1 (η − ξ)Φ1 (η)dη
R
we integrate by parts with respect to η via identity
2
2
eαi(η−µξ) = H∂η ((η − µξ)eαi(η−µξ) )
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N. HAYASHI, P. I. NAUMKIN
EJDE-2008/15
with H = (1 + 2iα(η − µξ)2 )−1 to obtain I = I1 + · · · + I5 , where
Z
2
I1 = 2φ1 ((µ − 1)ξ)Φ1 (µξ)A(ξ, µξ) eαi(η−µξ) H(H − 1)dη,
R
Z
αi(η−µξ)2
I2 =
e
φ1 (η − ξ)Φ1 (η)(η − µξ)H∂η Adη
ZR
2
I3 =
eαi(η−µξ) φ1 (η − ξ)Φ1 (η)A(ξ, η)
R
− φ1 ((µ − 1)ξ)Φ1 (µξ)A(ξ, µξ) H(H − 1)dη
Z
2
I4 =
eαi(η−µξ) φ1 (η − ξ)Φ01 (η)A(η − µξ)Hdη
ZR
2
I5 =
eαi(η−µξ) φ01 (η − ξ)Φ1 (η)A(η − µξ)Hdη
R
We prove that the integral I1 is the main term. Since
√
Z
iπ
βiy 2
2 e
B(B − 1)dy = √ ,
β
R
and hyi|φj (ai µξ − bi y) − φj (ai µξ)| ≤ Chξiγ−1 hyi−γ , we have
Z
2
Φ1 (µξ) = 2φ2 (a1 µξ)φ3 (a2 µξ) eβiy B(B − 1)dy
R
Z
βiy 2
+2 e
φ2 (a1 µξ − b1 y)φ3 (a2 µξ − b2 y)
R
− φ2 (a1 µξ)φ3 (a2 µξ) B(B − 1)dy
√
Z
iπ
γ−1
= √ φ2 (a1 µξ)φ3 (a2 µξ) + O hξi
hyi−1−γ dy .
β
R
By a direct calculation,
√
Z
Z
iπ
1
iη 2
αi(η−µξ)2
e dη = √ .
2 e
H(H − 1)dη = √
α R
α
R
Therefore,
iπ
I1 = √ A(ξ, µξ)φ1 ((µ − 1)ξ)φ2 (a1 µξ)φ3 (a2 µξ) + O hξiγ−2 .
αβ
We have
Z
|Φ1 (η)| ≤ C
|Φ01 (η)|
ha1 η − b1 yi−γ1 ha2 η − b2 yi−γ2 hyi−2 dy ≤ Chηi−γ1 −γ2 ,
ZR
ha1 η − b1 yiγ−1 ha2 η − b2 yi−γ2 hyi−2 dy
Z
+ C ha1 η − b1 yi−γ1 ha2 η − b2 yiγ−1 hyi−2 dy ≤ Chηiγ−1
≤C
R
R
Hence
γ−2
Z
|I2 | ≤ Chξi
hη − µξi−1 hηi−γ dη ≤ Chξiγ−2 .
R
Since
hη − µξi−1 |φ(η) − φ(µξ)| ≤ Chξiγ−1 hηi−γ
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
35
and
hη − µξi−1 |A(ξ, η) − A(ξ, µξ)| ≤ Chξiγ−2 hηi−γ
we find
|I3 | ≤ Chξiγ−2
Z
hηi−γ hη − µξi−1 dη ≤ Chξiγ−2 .
R
We also have
−1
Z
|I4 | + |I5 | ≤ Chξi
(hηiγ−1 + hη − ξiγ−1 )hη − µξi−1 dη ≤ Chξiγ−2 .
R
In the integral Φ2 we change a1 η − b1 y = y 0 (the prime we will omit) then with
1
b2
(a2 b1 − a1 b2 ) 6= 0, be2 = ,
b1
b1
e1 = (1 + 2β i(a1 η − y)2 )−1 , β1 = β
B
b21
b21
ae2 =
we have
Z
2
e
eiβ1 (a1 η−y) φ02 (y)φ3 (ae2 η − be2 y)(a1 η − y)Bdy.
Φ2 = C
R
Then with Q = β1 (a1 η − y)2 + α(η − µξ)2 . we define
Z
Z
e .
I6 = C
dyφ02 (y) A(ξ, η)φ1 (η − ξ)eiQ φ3 (ae2 η − be2 y)(a1 η − y)Bdη
R
R
Then we integrate by parts with respect to η via identity eiQ = H∂η (e
η eiQ ) with
−1
H = (1 + ie
η Qη ) , where
ηe = η − eby − e
aξ,
eb =
a1 β1
,
α + β1 a21
e
a=
αµ
,
α + β1 a21
if α + β1 a21 6= 0 and ηe = η if α + β1 a21 = 0 we get
Z
Z
e
I6 = C
dyφ02 (y) eiQ ηe∂η HA(ξ, η)φ1 (η − ξ)φ3 (ae2 η − be2 y)(a1 η − y)Bdη
R
−1
since he
η Qη i
R
−2
≤ Che
ηi
if α + β1 a21 6= 0 and
he
η Qη i−1 ≤ C|e
η |γ−1 |a1 β1 y + αµξ|γ−1
if α + β1 a21 = 0 we obtain
−1
Z
|I6 | ≤ Chξi
−1
Z
hη −
dyhyi
ZR
≤ Chξi
R
γ−1
dyhyi
R
if α +
γ−1
y −1
i hη − eby − e
aξi−1 dη
a1
ha3 y − e
aξiγ−1 ≤ Chξi2γ−2
1b
6= 0, since a1 , e
a, a3 ≡ 1−a
6= 0, and
a1
Z
Z
|I6 | ≤ Chξiγ−1 dyhyiγ−1 |a1 β1 y + αµξ|γ−1 ha1 η − yi−1 |η|γ−1
R
R
γ−1
γ−1
e
+ hη − ξi
+ hae2 η − b2 yi
+ ha1 η − yi−1 dη ≤ Chξi2γ−2
β1 a21
e
if α + β1 a21 = 0. In the same manner we estimate the integral with Φ3 . Thus we
have the asymptotic form (10.11). Lemma 10.2 is proved.
36
N. HAYASHI, P. I. NAUMKIN
3
EJDE-2008/15
2
Now we substitute (10.7) w(ξ) = ψ3 (ξ) − e 4 iξ ψ2 (ξ) into the second summand in
the right-hand side of (10.2) as above changing the variables of integration y = y 0 + η2
or y = 2η − y 0 we find
Z
Z
2
2
3
η
η
e−iQ w(η − y)w(y)dy = e− 4 iη
e−iy ψ3 ( − y)ψ3 ( + y)dy
2
2
R
Z R
2
1
− 2 e− 4 iy ψ3 (y − η)ψ2 (2η − y)dy
R
Z
2
3
1
η
η
− 8 iη 2
+e
e 2 iy ψ2 ( − y)ψ2 ( + y)dy.
2
2
R
So we have
Z
Z
dηeiS Kηw(η − ξ) e−iQ w(η − y)w(y)dy
R
R
Z
2
1
i(η−2ξ)
Kηψ3 (η − ξ)Ψ1 (η)dη
=
e4
R
Z
2
2
3
1
+ 2e 4 iξ
ei(η− 2 ξ) Kηψ3 (η − ξ)Ψ2 (η)dη
Z R
2
2
3
5
4
iξ
+ e5
e 8 i(η− 5 ξ) Kηψ3 (η − ξ)Ψ3 (η)dη
ZR
2
2
3
1
1
iξ
+ e8
e− 2 i(η− 2 ξ) Kηψ2 (η − ξ)Ψ1 (η)dη
R
Z
2
1
+ 2 e 4 i(η+ξ) Kηψ2 (η − ξ)Ψ2 (η)dη
R
Z
2
2
3
1
iξ
+ e4
e− 8 i(η−2ξ) Kηψ2 (η − ξ)Ψ3 (η)dη
R
where
Z
2
η
η
Ψ1 (η) =
e−iy ψ3 ( − y)ψ3 ( + y)dy,
2
2
ZR
2
1
Ψ2 (η) =
e− 4 iy ψ3 (y − η)ψ2 (2η − y)dy,
R
Z
2
1
η
η
Ψ3 (η) =
e 2 iy ψ2 ( − y)ψ2 ( + y)dy,
2
2
R
Applying Lemma 10.2 we obtain the asymptotic form
Z
Z
dηeiS Kηw(η − ξ) e−iQ w(η − y)w(y)dy
R
R
Z
Z
2
2
1
η
η
i(η−2ξ)
=
e4
Kηψ3 (η − ξ) e−iy ψ3 ( − y)ψ3 ( + y)dydη + O(hξiγ−2 )
2
2
R
R
4iπ
=−
ψ3 (ξ)ψ3 (ξ)ψ3 (ξ) + O(hξiγ−2 )
3hξi
4iπ
=−
|φ(ξ)|2 φ(ξ) + O(hξiγ−2 ).
3hξi
Thus we can estimate the right-hand side of (10.5)
Z
d
16iπ
A(ϕ)(ξ) = 2E
F eiS w(η − ξ)w(−η)dηdξ −
|ϕ(ξ)|2 ϕ(ξ)
dξ
3hξi
R
EJDE-2008/15
ASYMPTOTIC BEHAVIOR
Z
− 4E
dηeiS Kηw(η − ξ)
R
γ−2
= O(ε3 hξi
Z
37
e−iQ w(η − y)w(y)dy
R
).
We solve equation (10.4) by the contraction mapping principle in the set
Xε = {ϕ ∈ C1 (R) : kϕkX ≤ Cε2 }
where the norm
kϕkX ≡ sup |ϕ(ξ)| + hξi2−2γ |ϕ0 (ξ)| .
ξ∈R
When ϕ ∈ Xε , then kφkY ≡ supξ∈R (|φ(ξ)| + hξi1−2γ |φ0 (ξ)|) ≤ Cε and by Lemma
10.1
kψ1 kZ + kψ2 kZ + kψ3 kZ ≤ Cε.
Then by Lemma 10.2
d
| A(ϕ)(ξ)| ≤ Cεhξi2γ−2 .
dξ
And integrating we have
Z ξ
|A(ϕ)(ξ)| ≤ Cε
hξi2γ−2 dξ + |ϕ(0)| ≤ Cε.
0
In the same manner we can estimate the difference
|A(ϕ1 ) − A(ϕ2 )| ≤ Cεkϕ1 − ϕ2 kX .
Therefore, A is a contraction mapping in X. Hence there exists a unique solution
ϕ ∈ X of integral equation (10.4).
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Nakao Hayashi
Department of Mathematics, Graduate School of Science, Osaka University, Osaka,
Toyonaka, 560-0043, Japan
E-mail address: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia,
AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico
E-mail address: pavelni@matmor.unam.mx