Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2010, Article ID 504324, 35 pages
doi:10.1155/2010/504324
Research Article
The Initial Value Problem for the Quadratic
Nonlinear Klein-Gordon Equation
Nakao Hayashi1 and Pavel I. Naumkin2
1
Department of Mathematics, Graduate School of Science, Osaka University, Osaka,
Toyonaka 560-0043, Japan
2
Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Mexico
Correspondence should be addressed to Nakao Hayashi, nhayashi@math.sci.osaka-u.ac.jp
Received 18 September 2009; Accepted 23 February 2010
Academic Editor: Dongho Chae
Copyright q 2010 N. Hayashi and P. I. Naumkin. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We study the initial value problem for the quadratic nonlinear Klein-Gordon equation Lu i∂x −1 u2 , t, x ∈ R × R, u0, x u0 x, x ∈ R, where L ∂t ii∂x and i∂x 1 − ∂2x . Using the
Shatah normal forms method, we obtain a sharp asymptotic behavior of small solutions without
the condition of a compact support on the initial data which was assumed in the previous works.
1. Introduction
Let us consider the Cauchy problem for the nonlinear Klein-Gordon equation with a
quadratic nonlinearity in one dimensional case
Lu λi∂x −1 u2 ,
t, x ∈ R × R,
u0, x u0 x,
x ∈ R,
1.1
where λ ∈ C, L ∂t ii∂x , and i∂x 1 − ∂2x .
Our purpose is to obtain the large time asymptotic profile of small solutions to the
Cauchy problem 1.1 without the restriction of a compact support on the initial data which
was assumed in the previous work 1. One of the important tools of paper 1 was based on
the transformation of the equation by virtue of the hyperbolic polar coordinates following to
paper 2. The application of the hyperbolic polar coordinates implies the restriction to the
interior of the light cone, and therefore, requires the compactness of the initial data. Problem
2
Advances in Mathematical Physics
1.1 is related to the Cauchy problem
vtt v − vxx μv2 ,
v0, x v0 x,
t, x ∈ R × R,
vt 0, x v1 x,
x ∈ R,
1.2
where v0 and v1 are the real-valued functions, and μ ∈ R. Indeed we can put u 1/2v ii∂x −1 vt , then u satisfies
Lu i
μi∂x −1 u u2 ,
2
u0, x u0 x,
t, x ∈ R × R,
1.3
x ∈ R,
where u0 1/2v0 ii∂x −1 v1 .
There are a lot of works devoted to the study of the cubic nonlinear Klein-Gordon
equation
vtt v − vxx μv3 ,
v0, x v0 x,
t, x ∈ R × R,
vt 0, x v1 x,
x∈R
1.4
with μ ∈ R. When μ < 0, the global existence of solutions to 1.4 can be easily obtained in
the energy space, which is, however, insufficient for determining the large-time asymptotic
behavior of solutions. The sharp L∞ - time decay estimates of solutions and nonexistence of
the usual scattering states for 1.4 were shown in 3 by using hyperbolic polar coordinates
under the conditions that the initial data are sufficiently regular and have a compact support.
The initial value problem for the nonlinear Klein-Gordon equation with various cubic
nonlinearities depending on v, vt , vx , vxx , vtx and having a suitable nonresonance structure
was studied in 4–6, where small solutions were found in the neighborhood of the free
solutions when the initial data are small and regular and decay rapidly at infinity. Hence
the cubic nonlinearities are not necessarily critical; however the resonant nonlinear term v3
was excluded in these works. In paper 4, the nonresonant nonlinearities were classified into
two types, one of them can be treated by the nonlinear transformation which is different from
the method of normal forms 7 and the other reveals an additional time decay rate via the
operator x∂t t∂x which was used in 2. This nonlinear transformation was refined in 8 and
applied to a system of nonlinear Klein-Gordon equations in one or two space dimensions with
nonresonant nonlinearities. It seems that the method of normal forms is very useful in the
case of a single equation; however it does not work well in the case of a system of nonlinear
Klein-Gordon equations. Some sufficient conditions on quadratic or cubic nonlinearities were
given in 1, which allow us to prove global existence and to find sharp asymptotics of small
solutions to the Cauchy problem including 1.2 with small and regular initial data having
a compact support. Moreover it was proved that the asymptotic profile differs from that of
the linear Klein-Gordon equation. See also 9, 10 in which asymptotic behavior of solutions
to 1.4 was studied as in 1 by using hyperbolic polar coordinates. Compactness condition
on the data was removed in 11 in the case of the cubic nonlinearity v3 and a real-valued
solution. Final value problem with the cubic nonlinearity was studied in 12 for a real-valued
solution. As far as we know the problem of finding the large-time asymptotics is still open
Advances in Mathematical Physics
3
for the case of the cubic nonlinearity v3 and the complex valued initial data. When the initial
data are complex-valued, global existence and L∞ -time decay estimates of small solutions to
the Klein-Gordon equation with cubic nonlinearity |v|2 v were obtained in paper 13 under
the conditions that the initial data are smooth and have a compact support.
The scattering problem and the time decay rates of small solutions to 1.4 with supercritical nonlinearities |v|p−1 v and |v|p with p > 3 were studied in papers of 14, 15. Finally,
we note that the Klein-Gordon equation 1.4 with quadratic nonlinearities in two space
dimensions was studied in 16, where combining the method of the normal forms of 7
and the time decay estimate through the operator x∂t t∂x of 17, it was shown that every
quadratic nonlinearity is nonresonant.
We denote the Lebesgue space by Lp {φ ∈ S ; φLp < ∞}, with the norm φLp 1/p
R |φx|p dx if 1 ≤ p < ∞ and φL∞ supx∈R |φx| if p ∞. The weighted Sobolev space
is
φ ∈ Lp ; xs i∂x m φLp < ∞ ,
Hm,s
p
1.5
√
for m, s ∈ R, 1 ≤ p ≤ ∞, where x 1 x2 . For simplicity we write Hm,s Hm,s
2 . The
index 0 we usually omit if it does not cause a confusion. We denote by Fφ Fx → ξ φ ≡
√ φ 1/ 2π R e−ixξ φxdx the Fourier transform of the function φ. Then the inverse Fourier
√
transformation is F−1 φ Fξ−1→ x φ 1/ 2π R eixξ φξdξ.
Our main result of this paper is the following.
Theorem 1.1. Let u0 ∈ H3,1 and the norm u0 H3,1 ε. Then there exists ε0 > 0 such that for all
0 < ε < ε0 the Cauchy problem 1.1 has a unique global solution
ut ∈ C 0, ∞; H3,1
1.6
≤ Cε1 t−1/2 .
utH1,0
∞
1.7
satisfying the time decay estimate
0,1
∈ H0,1
Furthermore there exists a unique final state W
such that
∞ ∩H
2
|2 log t ut − e−ii∂x t F−1 W
e−2i|λ| Ω|W
ii∂ t
2
|2 log t Fe x ut − W
e−2i|λ| Ω|W
H1,0
H0,1
∞
≤ Cε3/2 tγ−1/4 ,
≤ Cε
1.8
3/2 γ−1/4
t
,
where γ ∈ 0, 1/4, Ωξ ≡ ξ2 /2ξ2ξ 2ξ.
An important tool for obtaining the time decay estimates of solutions to the nonlinear
Klein-Gordon equation is implementation of the operator
J i∂x e−ii∂x t xeii∂x t F−1 ξe−iξt i∂ξ eiξt F i∂x x it∂x ,
1.9
4
Advances in Mathematical Physics
which is analogous to the operator x it∂x e−it/2∂x xeit/2∂x in the case of the nonlinear
Schrödinger equations used in 18. The operator J was used previously in paper of
15 for constructing the scattering operator for nonlinear Klein-Gordon equations with
a supercritical nonlinearity. We have x, i∂x α αi∂x α−2 ∂x ; therefore the commutator
L, J LJ − JL 0, where L ∂t ii∂x is a linear part of 1.1. Since J is not a purely
differential operator, it is apparently difficult to calculate the action of J on the nonlinearity
in 1.1. So, instead we use the first-order differential operator
2
2
P t∂x x∂t
1.10
which is closely related to J by the identity P Lx − iJ and acts easily on the nonlinearity.
Moreover, it almost commutes with L, since L, P −ii∂x −1 ∂x L.
Also we use the method of normal forms of 7 by which we transform the quadratic
nonlinearity into a cubic one with a nonlocal operator. We multiply both sides of equation
1.1 by the free Klein-Gordon evolution group FU−t Feiti∂x eitξ F and put vt, ξ to get
eitξ u
λ
vt t, ξ λeitξ F i∂x −1 u2 √
eitA v t, η − ξ v t, −η dη,
2πξ R
1.11
where Aξ, η ξ ξ − η η. Integrating 1.11 with respect to time, we find
vt, ξ v0, ξ √
t
λ
2πξ
dτ
0
R
eiτA v τ, η − ξ v τ, −η dη.
1.12
Then we integrate by parts with respect to τ, taking into account 1.11,
dη
eitA v t, η − ξ v t, −η √
2πξA ξ, η
R
dη
v0, ξ iλ v 0, η − ξ v 0, −η √
2πξA ξ, η
R
t dη
2
−1
− 2i|λ| dτ √
eiτA−η v τ, η − ξ Fx → η i∂x u2 .
0
R 2πξA ξ, η
vt, ξ iλ
1.13
Returning to the function ut, x UtFξ−1→ x v Fξ−1→ x e−itξ vt, ξ, we obtain the following
equation:
Lu iλGu, u −2i|λ|2 G u, i∂x −1 u2 ,
1.14
with the symmetric bilinear operator
G φ, ψ Fξ−1→ x
R
g ξ − η, η φ ξ − η ψ η dη,
1.15
Advances in Mathematical Physics
5
where
g ζ, η √
1
,
2π ζ η ζ η ζ η
1.16
and ζ ξ − η. Our main point in this paper is to show that the right-hand side of 1.14 can
be decomposed into two terms; one of them is a cubic nonlinearity
1
−2i|λ|2 UtF−1 Ωξ|FU−tut, ξ|2 FU−tut, ξ,
t
1.17
and the other one is a remainder term with an estimate like Ot−5/4 FU−tut3H1,3 .
Remark 1.2. We believe that all quadratic nonlinear terms u2 , u2 , |u|2 of problem 1.3 also
could be considered by this approach. In the same way as in the derivation of 1.14 we get
from 1.3
i
L u μGu, u G1 u, u 2G2 u, u
2
iμ2 G u, i∂x −1 u u2
−1
2
iμ G1 u, i∂x u u
2
1.18
iμ2 G2 u u, i∂x −1 u u2 ,
where
Gj φ, ψ Fξ−1→ x
R
g j ξ − η, η φ ξ − η ψ η dη,
1.19
1
g j ζ, η √
2πAj ζ η, η ζ η
with A1 ξ, η ξ−ξ−η−η, A2 ξ, η ξ−ξ−ηη. Some more regularity conditions
are necessary to treat the bilinear operators Gj . Also we have to show that
G u, i∂x −1 u2 2|u|2 ,
−1
G2 u, i∂x 2
u u
2
,
G1 u, i∂x −1 u2 u2 ,
−1
G2 u, i∂x 2
2
u 2|u|
1.20
are the nonresonant terms i.e., remainders and to remove the resonant terms
G u, i∂x −1 u2 ,
G1 u, i∂x −1 |u|2 ,
G2 u, i∂x −1 |u|2 ,
G2 u, i∂x −1 u2
by an appropriate phase function. We will dedicate a separate paper to this problem.
1.21
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Advances in Mathematical Physics
We prove our main result in Section 3. In the next section we prove several lemmas
used in the proof of the main result.
2. Preliminaries
First we give some estimates for the symmetric bilinear operator
G φ, ψ Fξ−1→ x
R
g ξ − η, η φ ξ − η ψ η dη,
2.1
where
g ζ, η √
1
.
2π ζ η ζ η ζ η
2.2
Denote the kernel as follows:
1
g y, z 4π 2
eiyζizη
dηdζ.
ζ η ζ η ζ η
R2
2.3
Lemma 2.1. The representation is true
G φ, ψ R2
g y, z φ x − y ψx − zdydz,
2.4
where the kernel gy, z obeys the following estimate:
g y, z ≤ C y −3 z−3 ln2 1 1 z − y
2.5
for all y, z ∈ R, y /
z. Moreover the following estimates are valid:
G φ, ψ p ≤ Cφ q ψ r ,
L
L
L
PG φ, ψ p ≤ CPφ q ψ r CPψ q φ r
L
L
L
L
L
C ∂t φ Lq ψ Lr C ∂t ψ Lq φLr
2.6
for 1 ≤ p ≤ ∞, 1 < q ≤ p/α, 1 < r ≤ p/1 − α, α ∈ 0, 1, provided that the right-hand sides are
bounded.
Advances in Mathematical Physics
7
Proof. To prove representation 2.4, we substitute the direct Fourier transforms
1
φζ
√
2π
R
1
ψ η √
2π
e−ix−yζ φ x − y dy,
2.7
R
e
−ix−zη
ψx − zdz
into the definition of the operator G. Then changing ζ ξ − η, we find
G φ, ψ 2π−3/2
R2
φ x − y ψx − z
R2
iyζizη
g ζ, η e
dηdζ dydz
R2
g y, z φ x − y ψx − zdydz,
2.8
where the kernel gy, z is
g y, z 2π−3/2
1
4π 2
R2
g ζ, η eiyζizη dηdζ
eiyζizη
dηdζ.
ζ η ζ η ζ η
R2
2.9
Changing the variables of integration ζ ξ/2 − η and η ξ/2 η the prime we will omit,
we get
1
g y, z 4π 2
eiyζizη
dηdζ
ζ η ζ η ζ η
R2
dξ iyz/2ξ
1
e
B ξ, η eiz−yη dη,
4π 2 R ξ
R
2.10
where Bξ, η 1/ξ ξ/2 − η ξ/2 η. We change s y z/2 and ρ z − y and
denote
1
g s, ρ 4π 2
dξ isξ
e
ξ
R
R
B ξ, η eiρη dη.
2.11
For the case of |ρ| ≤ 1, |s| ≤ 1 we integrate by parts using the identity eiρη Aη∂η ηeiρη ,
where Aη 1/1 iρη. Then we get
1
g s, ρ − 2
4π
dξ isξ
e
ξ
R
R
eiρη η∂η A η B ξ, η dη.
2.12
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Note that
η
η∂η B ξ, η 2
ξ ξ/2 − η ξ/2 η
η − ξ/2
η ξ/2
η − ξ/2
ξ/2 η
.
2.13
Then
−1
ξ η∂η A η B ξ, η ≤
C
.
ξ 1 ρ η ξ η
2.14
Hence we can estimate the kernel g s, ρ as follows:
g s, ρ ≤ C
dη
η
ρ
1
ξ η
R
∞
∞
∞
∞
dη
dη
1
ξ
dξ
dξ
≤C
ln
C
η ξ 1
1 1 ρ η 1 ξ ηξ
1 1 ρ η η
1/|ρ|
∞
ln η
dη
dη
C ∞
dη ≤ C
≤C
ln η
ln η 2
η
η
ρ 1/|ρ|
1 1 ρ η η
1
1
≤ C ln2 1 ρ
dξ
ξ
R
2.15
for the case of |ρ| ≤ 1, |s| ≤ 1. For the case of |ρ| ≤ 1, |s| ≥ 1 we integrate three times by parts
with respect to ξ
g s, ρ C|s|
−3
R
dξe
isξ
R
dη.
eiρη ∂3ξ ξ−1 η∂η A η B ξ, η
2.16
Note that
3
∂ξ ξ−1 η∂η A η B ξ, η
≤
C
.
ξ 1 ρ η ξ η
2.17
Hence we can estimate the kernel gy, z as follows:
g s, ρ ≤ C|s|−3
dξ
R ξ
dη
−3 2
≤ C|s| ln 1 ξ η
R 1 ρ η
1
ρ
2.18
for all |ρ| ≤ 1, |s| ≥ 1. For the case |ρ| ≥ 1, |s| ≤ 1 we integrate by parts three times with respect
to η
−3
g s, ρ Cρ
dξ isξ
e
ξ
R
R
eiρη ∂3η B ξ, η dη.
2.19
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Note that |∂3η Bξ, η| ≤ Cξ η−3 . Hence
g s, ρ ≤ Cρ−3
dξ
R ξ
2
−3
dη
2 ≤ Cρ .
R η
2.20
Finally for the case of |ρ| ≥ 1, |s| ≥ 1 we integrate by parts three times with respect to ξ and η
−3
g s, ρ Csρ
R
dξeisξ
R
eiρη ∂3ξ ∂3η ξ−1 B ξ, η dη.
2.21
Since
3 3
∂ξ ∂η ξ−1 B ξ, η ≤
2
C
2.22
2 ,
ξ ξ η
then we can estimate the kernel g s, ρ as follows:
g s, ρ ≤ Csρ−3
dξ
R ξ
2
−3
dη
2 ≤ Csρ
R η
2.23
for all |ρ| ≥ 1, |s| ≥ 1. Hence estimate 2.5 is true.
By virtue of estimate 2.5 applying the Hölder inequality with 1/p 1/p1 1/p2 and
the Young inequality with 1/p1 1 1/q 1/q1 and 1/p2 1 1/r 1/r1 , we find
dy G φ, ψ p ≤ C φ x − y L
3
R
y dz 1
2
ψx−z
ln 1 3
z
−
y
z
p1
R
≤ CφLq ψ Lr ,
p
Lx2 L∞
y
Lx
2.24
where 1 ≤ p ≤ ∞, 1 < q ≤ p/α, 1 < r ≤ p/1 − α, α ∈ 0, 1.
We now estimate the operator P t∂x x∂t as follows:
PG φ, ψ G Pφ, ψ G φ, Pψ R2
yg y, z ψx − z∂t φ x − y dydz
R2
zg y, z φ x − y ∂t ψx − zdydz.
2.25
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Advances in Mathematical Physics
Then by virtue of estimate 2.4 applying the Hölder and Young inequalities, we get
PG φ, ψ p ≤ G Pφ, ψ p G φ, Pψ p
L
L
L
yg y, z ψx − z∂t φ x − y dydz
R2
Lp
R
Lp
zg
y,
z
φ
x
−
y
∂
ψx
−
zdydz
t
2
2.26
≤ CPφLq ψ Lr CPψ Lq φLr C∂t φLq ψ Lr
C∂t ψ Lq φLr .
Lemma 2.1 is proved.
We now decompose the free Klein-Gordon evolution group Ut e−ii∂x t F−1 EtF,
where Et e−itξ similarly to the factorization of the free Schrödinger evolution group. We
denote the dilation operator by
1 x
,
Dω φ √ φ
iω ω
Dω −1 iD1/ω .
2.27
Define the multiplication factor Mt e−itixθx , where θx 1 for |x| < 1 and θx 0 for
|x| ≥ 1. We introduce the operator
x
.
φ
Bφ ix
ix3/2
θx
2.28
The inverse operator B−1 acts on the functions φx defined on −1, 1 as follows:
B−1 φ 1
ξ
φ
3/2
ξ
ξ
2.29
for all ξ ∈ R, since ξ x/ix ∈ R and x ξ/ξ ∈ −1, 1. We now introduce the operators
Vt B−1 MtDt−1 F−1 e−itξ ,
Wt Mt1 − θDt−1 F−1 e−itξ ,
2.30
so that we have the representation for the free Klein-Gordon evolution group
UtF−1 e−iti∂x F−1 F−1 e−itξ Dt MtBVt Wt
Dt MtB Dt MtBVt − 1 Dt MtWt.
2.31
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11
The first term Dt MtBφ of the right-hand side of 2.31 describes inside the light cone
the well-known leading term of the large-time asymptotics of solutions of the linear KleinGordon equation Lu 0 with initial data φ. The second term of the right-hand side of
2.31 is a remainder inside of the light cone, whereas the last term represents the large time
asymptotics outside of the light cone which decays more rapidly in time. We also have
FU−t Feiti∂x eitξ F V −1 tB−1 MtDt−1 W−1 tDt−1
B−1 MtDt−1 V −1 t − 1 B−1 MtDt−1 W−1 tDt−1 ,
2.32
where the right-inverse operators are
V −1 t EtFDt MtB,
2.33
W−1 t EtFDt 1 − θ,
where Et e−itξ .
In the next lemma we state the estimates of the operators Vt B−1 MtDt−1 F−1 e−itξ .
Lemma 2.2. The estimates hold as follows:
Vtφ 2 ≤ Cφ 2 ,
L
L
ρ
α−1 ξ ∂ξ Vtφ 2 ≤ Cξα ∂ξ φ 2 C
φ
ξ
2,
L
L
2.34
L
where ρ < α, α ∈ 1, 2, and
ρ
ξ ∂ξ V −1 tφ
L2
≤ Cξ1/4α ∂ξ φ 2 Cξα−3/4 φ 2 ,
L
L
2.35
where ρ < α, α ∈ 0, 1, provided the right-hand sides are finite.
Proof. Changing the variable of integration x ξξ−1 , we see that
−1 2
φ
B 2
L
1
2
φ ξ dξ φx2 dx φ2 2
.
L −1,1
ξ
ξ3
−1
R
2.36
Hence VtφL2 φL2 .
Consider the estimate for the derivative ∂ξ Vtφ. Define Sξ, η η − ξη 1/ξ.
Note that ∂S/∂η η/η − ξ/ξ 1 ηξ − ηξ/ξηη ξη − ξ and ∂S/∂ξ ξ − η/ξ3 . Hence integrating by parts one time yields
∂ξ
R
e
−itS
φ η dη −itξ−3
ξ
−3
R
R
e
e−itS φ η ξ − η dη
−itS
φ η g ξ, η φ η gη ξ, η dη,
2.37
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where gξ, η ξ − η/ξ/ξ − η/η ξηη ξ/1 ηξ − ηξ. Also we have
−1
ξ η
≤ C 1 η ξ − ηξ
2 .
η ξ
2.38
2
ξ2 η
ξ η η ξ
≤ C .
g ξ, η 1 η ξ − ηξ
η ξ
2.39
Hence the estimate is true
Since ∂ξ ξa aξξa−2 , we get
ρ
ξ ∂ξ Vtφ ξ
it
∂ξ ξ−3/2
2π
ρ
R
e−itS φ η dη
3
it
ρ−2
ρ−9/2
− ξξ Vtφ ξ
e−itS φ η g ξ, η dη
2
2π R
it
e−itS φ η gη ξ, η dη.
ξρ−9/2
2π R
2.40
Consider the L2 -estimate of the integral
√
tξρ−9/2
R
eitξη/ξ ψ η g ξ, η dη.
2.41
We have
2
√
t eitξη/ξ ψ η g ξ, η ξρ−9/2 dη
2
R
t
L
R
R
dηψ η
dηψ η
R
R
dζψζ eitξ/ξη−ζ g ξ, η gξ, ζξ2ρ−9 dξ
2.42
R
dζψζG η, ζ ,
where the kernel
G η, ζ t
R
eitξ/ξη−ζ g ξ, η gξ, ζξ2ρ−9 dξ
η ξ
ζ ξ
ξ2ρ−7 dξ.
1 η ξ − ηξ 1 ζξ − ζξ
t η ζ eitξ/ξη−ζ R
2.43
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After a change ξ x/ix, we find
η 1/ix
ζ 1/ix
ix4−2ρ dx.
− ζx/ix
1
ζ/ix
1 η /ix − ηx/ix
2.44
1
G η, ζ t η ζ
eitxη−ζ −1
We now change the contour z x iy ∈ Γ of integration in G as y signη − ζix2 ; then
η 1/iz
ζ 1/iz
iz4−2ρ dz.
1 η /iz − ηz/iz 1 ζ/iz − ζz/iz
2.45
G η, ζ t η ζ eitzη−ζ Γ
2
1/4
Since |iz| ix2 y2 4x2 y2 ∼ ix, using 2.38 and the inequality η ξ ≥
ηα−1 ξ2−α for α ∈ 1, 2, we get |η 1/iz/1 η/iz − ηz/iz| ≤ Cηα−1 ixα−2 ;
also we have
2
Cix−2δ
itzη−ζ e−ix t|η−ζ| ≤ e
δ
1 tη − ζ
2.46
for δ > 1. Therefore we obtain the estimate
α
α
Ct η ζα 1
Ct η ζα
2α−ρ−δ
G η, ζ ≤ dx ≤ ix
δ
δ ,
−1
1 tη − ζ
1 tη − ζ
2.47
if we choose ρ < α, α ∈ 1, 2 and 1 < δ < 1 α − ρ. Thus we get
2
√
t eitξη/ξ ψ η g ξ, η ξρ−9/2 dη dηψ η
dζψζG η, ζ
2
R
L
R
α t
α
≤ C η ψ η L2 dζψζζ δ
R
1t η−ζ
2
≤ Ct·α ψ L2
since δ > 1.
R
L2
R
α 2
dη
δ ≤ C· ψ L2
1 tη
2.48
14
Advances in Mathematical Physics
In the same manner we consider the estimate of the integral
2
√
t eitξη/ξ ψ η gη ξ, η ξρ−9/2 dη
2
R
t
L
R
R
dηψ η
dηψ η
R
R
dζψζ eitξ/ξη−ζ gη ξ, η gζ ξ, ζξ2ρ−9 dξ
2.49
R
η, ζ ,
dζψζG
where the kernel
η, ζ t eitξ/ξη−ζ gη ξ, η gζ ξ, ζξ2ρ−9 dξ
G
t
R
R
e
itξ/ξη−ζ
η ξ η/ η ξ ζ ξ ζ/ζξ
ξ2ρ−7 dξ,
1 ζξ − ζξ
1 η ξ − ηξ
2.50
since by a direct calculation
η η ξ
η ξ η/ η ξ
∂η .
1 η ξ − ηξ
1 η ξ − ηξ
2.51
In the same way as in 2.47 we have
Ctηα−1 ζα−1
G η, ζ ≤ δ
1 tη − ζ
2.52
if we choose ρ < α, α ∈ 1, 2, and 1 < δ < 1 α − ρ. Therefore we get
2
√
α−1 2
t eitξη/ξ ψ η gη ξ, η ξρ−9/2 dη ≤ C
ψ η 2 η
2
L
R
L
R
tdη
α−1 2
δ ≤ C· ψ L2 .
1 tη
2.53
Hence ξρ ∂ξ VtφL2 ≤ Cξα ∂ξ φL2 Cξα−1 φL2 . Thus we have the second estimate of
the lemma.
Consider the estimate for the derivative ∂ξ V −1 tφ. Note that ∂Sη, ξ/∂ξ ξ/ξ −
η/η 1 ηξ − ηξ/ξηη ξξ − η and ∂Sη, ξ/∂η η − ξ/η3 . Hence
integrating by parts one time yields
√
ρ
−1
ξ ∂ξ V tφ − √
t
ρ−1
1 η ξ − ηξ 1/2 itSη,ξ
φ η de
η
η ξ
R
ξ
2iπ
√
t
ρ−1
√
eitSη,ξ φ η g1 ξ, η φ η g1η ξ, η dη,
ξ
2iπ
R
2.54
Advances in Mathematical Physics
15
where Sη, ξ ξ − ξη 1/η and g1 ξ, η 1 ηξ − ηξ/η ξη1/2 . The
estimate is true
3/2
1 η ξ − ηξ 1/2
η
ξ
g1 ξ, η ≤ C η
.
η ξ
η ξ
2.55
Consider the L2 -estimate of the integral
√
ξρ−1 t
R
e−itξη/η ψ η g1 ξ, η dη.
2.56
We have, changing η/η y and ζ/ζ z,
ρ−1 √
2
−itξη/η
ξ
t
e
ψ
η
g
ξ,
η
dη
1
2
R
t
R
dηψ η
1
t
−1
L
R
3
dy iy ψ
dζψζ e−itξη/η−ζ/ζ g1 ξ, η g1 ξ, ζξ2ρ−2 dξ
R
y
iy
1
−1
dziz3 ψ
z
iz
2.57
y
z
× e
g1 ξ, g1 ξ,
ξ2ρ−2 dξ
iz
iy
R
1
1
3
y
z
G1 y, z ,
dy iy ψ dziz3 ψ
iz
iy
−1
−1
−itξy−z
where the kernel
G1 y, z
z
g1 ξ,
g1
t e
ξ2ρ−2 dξ
iz
R
1 ξ/ iy − yξ/ iy 1 ξ/iz − zξ/iz
−1/2
−1/2
−itξy−z
t iy
e
ξ2ρ−2 dξ.
iz
1/iz ξ
1/ iy ξ
R
2.58
−itξy−z
y
ξ, iy
We now change the contour ξ x iy ∈ Γ of integration in G as y signy − zx
2 ; then
G1 y, z
−1/2
t iy
iz−1/2
Γ
e
−itξy−z
1 ξ/ iy − yξ/ iy 1 ξ/iz − zξ/iz
ξ2ρ−2 dξ.
1/iz ξ
1/ iy ξ
2.59
16
Advances in Mathematical Physics
1/2
2
and by 2.55 |1 ξ/iy − yξ/iy/1/iy Since |ξ| x
2 − y2 4x2 y2 ∼ x
1−α
ξ| ≤ Cξ iy−α for α ∈ 0, 1, and also
−itξy−z e
≤ C
C
2 ≤
δ δ
1 tx
y−z
tx
y − z y − z
2.60
for δ < 1, we obtain the estimate
−1/2−α
C iy
iz−1/2−α
G1 y, z ≤
−2α2ρ−δ dx
x
y − zδ y − z
R
2.61
−1/2−α
C iy
iz−1/2−α
≤
,
y − zδ y − z
if we choose ρ < α, and 1 > δ > 1 − 2α 2ρ. Therefore we get
2
√
t eitξη/ξ ψ η g ξ, η ξρ−9/2 dη
2
R
1
−1
5/2−α
dy iy
ψ
y
iy
L
1
−1
5/2−α
dziz
z
C
ψ
iz y − zδ y − z
2
5/2−2α y
≤C
dy iy
ψ iy
−1
1
≤C
1
−1
2.62
2
2
dξξ1/22α ψξ ≤ C·1/4α ψ 2 .
L
In the same manner we consider the estimate of the integral with φηg1η ξ, η. Hence
ρ
ξ ∂ξ V −1 tφ
L2
≤ Cξ1/4α ∂ξ φ 2 Cξα−3/4 φ 2 ,
L
L
2.63
where α > ρ. Thus we have the second estimate of the lemma. Lemma 2.2 is proved.
∞
2
In the next lemma we prove an auxiliary asymptotics for the integral 0 e−itz Φξ, zdz.
Lemma 2.3. The estimate holds as follows:
∞
π α
−itz2
e
Φξ, zdz − Φξ, 0
ξ
≤ Ct−3/4 ξα Φz ξ, zL∞ L2z
ξ
4it 0
provided the right-hand side is finite, where α ∈ R.
2.64
Advances in Mathematical Physics
17
Proof. We represent the integral
∞
e
−itz2
Φξ, zdz Φξ, 0
∞
e
0
−itz2
dz R Φξ, 0
0
π
R,
4it
2.65
where the remainder term is
R
∞
e−itz Φξ, z − Φξ, 0dz.
2
2.66
0
In the remainder term we integrate by parts via identity e−itz 1/1 − 2itz2 ∂z ze−itz 2
∞
α ξ R ∞ ≤ Cξα
L
ξ
0
2
1
z
−
Φξ,
0dz
Φξ,
∞
2
1 tz
L
ξ
α ∞ z
C
Φ
zdz
ξ,
ξ
z
2
1 tz
L∞
ξ
0
−1 √
2
≤ Cξα Φz ξ, zL∞ L2z z
1
tz
ξ
2.67
L1z
−1 2
z 1 tz
L∞ L2z Cξα Φz ξ, z
ξ
L2z
≤ Ct−3/4 ξα Φz ξ, zL∞ L2z ,
ξ
since
|Φξ, z − Φξ, 0| ≤
z
0
|Φz ξ, z|dz ≤ C |z|Φz ξ, zL∞ L2z .
2.68
ξ
Thus we have the estimate for the remainder in the asymptotic formula. Lemma 2.3 is proved.
In the next lemma we obtain the asymptotics for the operator Vt
B−1 MtDt−1 F−1 e−itξ and the right-inverse operator V −1 t EtFDt MtB.
Lemma 2.4. The estimates hold as follows:
β
ξ Vt − 1φ
L∞
≤ Ct−1/4 ξβ3/4 ∂ξ φ 2 ξβ−1/4 φ 2 ,
β−3/4 −1
V t − 1 φ
ξ
L
L∞
≤ Ct
−1/4
L
3/2β
∂ξ φ 2 ξ1/2β φ 2 ,
ξ
L
for all t ≥ 1, where 0 ≤ β ≤ 3/2, provided the right-hand sides are finite.
L
2.69
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Advances in Mathematical Physics
Proof. We have the identities
1 η ξ − ηξ 2
ξ S ξ, η η − ξ −
η − ξ η−ξ ,
2
ξ
η ξ ξ
1 η ξ − ηξ η
ξ
Sη ξ, η −
η−ξ ,
ξ ξ η η ξ
η
2.70
and Sηη ξ, η η−3 . We now change z Sξ, η. We denote The inverse functions by
ηj z, so that η1 z : 0, ∞ → ξ, ∞ and η2 z : 0, ∞ → −∞, ξ.
Thus the stationary point z 0 transforms into η ξ. Hence we can write the
representation
Vtφ ξ
it
2π
−3/2
2ξ−3/2
R
e−itSξ,η φ η dη
2 ∞
S ξ, ηj z
it
−itz2
e
φ ηj z dz.
Sη ξ, ηj z 2π j1 0
2.71
By a direct calculation we find
1 η ξ − ηξ
dz Sη ξ, η .
1/2
dη
η ξ
S ξ, η
2.72
Therefore we obtain
Vtφ ∞
e−itz Φξ, zdz,
2
2.73
0
where
Φξ, z 2ξ−1
2
it
2π j1
η φ η
1 ξ η − ξη .
2.74
ηηj z
Application of Lemma 2.3 yields
Vtφ φξ O t−3/4 Φz ξ, zL∞ L2z .
ξ
2.75
Advances in Mathematical Physics
19
By a direct calculation we have
dz
dη
Φz ξ, z
3/2 2
2 η
φ η
it 3/4
3/4
2π j1 ξ
1 η ξ − ηξ
2.76
2 η ηξ ξ η / η ξ
it φ η .
1/2
3/4
2π j1 ξ3/4 η
1 η ξ − ηξ
By 2.38 we get the estimate
β
ξ Φz ξ, z
L∞
L2
ξ z
dz β
ξ Φz ξ, z
dη L∞
L2
ξ η
β 9/4
√
ξ η
≤ C t φ η η ξ3/2
L∞
L2
ξ η
β 5/4
√
ξ η
C t φ η η ξ3/2
√
√
3/4β β−1/4 ≤ C t η
φ 2 C t η
φ
L
L2
L∞
L2
ξ η
2.77
for 0 ≤ β ≤ 3/2 since η ξ ≥ Cξ2/3β η1−2/3β . This yields the first estimate of the
lemma.
We now prove the second estimate. We have
√ 1
−3/2
y
t
itξ−ξy−iy
e
φ dy,
V tφ √
iy
iy
2iπ −1
−1
2.78
then changing y ηη−1 , we find
√
−1
V tφx √
t
2iπ
−3/2
eitSη,ξ φ η η
dη,
2.79
R
where
1 η ξ − ηξ 2
ξη 1
S η, ξ ξ − 2 η − ξ .
η
η ξ η
2.80
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Advances in Mathematical Physics
As above we change z Sη, ξ and represent
√
−1
V tφx √
t
2iπ
R
−3/2
eitSη,ξ φ η η
dη
√
2 ∞
S ηj z, ξ
2t 2 −3/2
itz
e φ ηj z ηj z
√
dz.
Sη ηj z, ξ iπ j1 0
2.81
By a direct calculation we find Sη η, ξ η − ξ/η3 and
Sη η, ξ η ξ
dz
.
5/2
dη 2 S η, ξ
2 η
1 η ξ − ηξ
2.82
Therefore we obtain
−1
V tφx ∞
2
eitz Φξ, zdz,
2.83
0
where
√ 2 2 2t η 1 η ξ − ηξ Φξ, z √
φ η η ξ
iπ j1
2.84
.
ηηj z
Application of Lemma 2.3 yields
V −1 tφx φξ O t−3/4 Φz ξ, zL∞ L2z .
2.85
ξ
By a direct calculation we have
Φz ξ, z
3/4
√ 2 1 η ξ − ηξ
9/4 dz 2 2t √
φ η
η
3/2
dη
iπ j1
η ξ
⎞
⎛
1/4 √ 2 5/4 1
η
−
ηξ
η
ξ
η
1 η ξ − ηξ
2 2t
⎟ ⎜
√
⎠ φ η ,
⎝
η ξ
iπ j1
η ξ
η
2.86
Advances in Mathematical Physics
21
Since
⎛
⎞ ⎜ η 1 η ξ − ηξ ⎟ η 5/4 1 η ξ − ηξ 1/4
⎝
⎠ η ξ
η ξ
η
5/4 ξηη1/4 1 ηξ − ηξ3/4
ηξ − ξ η
η
5/2
3/2
1/4
η ξ
1 η ξ − ηξ
2 η ξ
2.87
2
ξ3/4 η
≤ C
3/2 ,
ξ η
therefore we get the estimate
β−3/4
Φz ξ, z
ξ
L∞
L2
ξ z
dz β−3/4
ξ
Φz ξ, z
dη L∞
L2
ξ η
⎞
⎛
β 3
ξβ η2
√ ⎜
η
ξ
⎟
≤ C t⎝ φ η φ η ⎠
∞ 2
∞ 2 ξ η3/2
η ξ3/2
L Lη
L Lη
ξ
ξ
√ 1/2β 3/2β ≤C t η
φ 2 η
φ 2
L
L
2.88
for 0 ≤ β ≤ 2/3, since η ξ ≥ Cξ2/3β η1−2/3β . This yields the second estimate of the
lemma. Lemma 2.4 is proved.
In the next lemma we find the estimates for the operator Wt 1−θMtDt−1 F−1 e−itξ .
Lemma 2.5. The estimates hold as follows:
Wtφ r ≤ Ct−1/21−δ1−1/r ξ12δ1−1/r ∂ξ φ 1 ξ12δ1−1/r−1 φ 1
L
L
for all t ≥ 1, where 1 < r ≤ ∞, δ ∈ 0, 1, provided the right-hand side is finite.
L
2.89
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Advances in Mathematical Physics
Proof. Note that Wtφ 0 for |x| ≤ 1. To prove the estimate, we integrate by parts via the
identity eitxξ−ξ 1/itx − ξ/ξ ∂ξ eitxξ−ξ √
Wtφ √
it
2π
R
1
−√
2πit
−√
eitxξ−ξ φξdξ
1
2πit
R
eitxξ−ξ
1
φ ξdξ
x − ξ/ξ
R
eitxξ−ξ
ξ−3
x − ξ/ξ2
2.90
φξdξ
for all |x| ≥ 2. Since 1/|x − ξ/ξ| ≤ Cx−1 for all |x| ≥ 2, then we get
Wtφ ≤ Ct−1/2 x−1
φ ξ ξ−3 φξ dξ
R
≤ Ct−1/2 x−1 ∂ξ φL1 Cξ−3 φ 1 .
2.91
L
For the case of 1 < |x| ≤ 2 we integrate by parts via the identity eitxξ−ξ 1/ξ/ξ itξx − ξ/ξ∂ξ ξeitxξ−ξ √ it
ξ
φ ξdξ
Wtφ − √
eitxξ−ξ
ξ/ξ itξx − ξ/ξ
2π R
√ it
1/ξ2 itξξx/ξ − 1
eitxξ−ξ
φξdξ.
√
2π R
ξ/ξ itξx − ξ/ξ2
2.92
Since |x−ξ/ξ| ≥ ||x|−1||1−|ξ|/ξ| and |ξx/ξ−1| ≤ |ξ|/ξ||x|−1||1−|ξ|/ξ| ≤ |x−ξ/ξ|
for 1 < |x|, therefore
√
Wtφ ≤ C t
√
ξφ ξ φξ
dξ
R |ξ|/ξ tξ||x| − 1| |1 − |ξ|/ξ|
ξφ ξ φξ
≤C t
R1
tξ−2 tξ||x| − 1|
dξ
2.93
Advances in Mathematical Physics
23
for 1 < |x| ≤ 2. Hence we get
Wtφ r ≤ Ct−1/2 x−1 L
Lr |x|≥2
∂ξ φ 1 C
ξ−3 φ 1
L
L
√
ξφ ξ φξ 1
C t dξ
1 tξ/1 tξ − 2 ||x| − 1| r
−2
1 tξ
R
L 1<|x|≤2
≤ Ct−1/2 ∂ξ φL1 Cξ−3 φ 1
Ct1/2−1/r
L
R
1/r−1
dξ
ξ1−1/r φ ξ ξ−1/r φξ 1 tξ−2
≤ Ct−1/21−δ1−1/r ξ12δ1−1/r ∂ξ φ
L1
ξ12δ1−1/r−1 φ 1
L
2.94
for δ ∈ 0, 1. This yields the estimate of the lemma. Lemma 2.5 is proved.
We next prove the time decay estimate in terms of the operator J.
Lemma 2.6. The estimate is valid
1/2
1/2
uL∞ ≤ Ct−1/2 u1/2
Ju
u
3/2
3/2
1/2
H
H
H
2.95
for all t ≥ 0, provided that the right-hand side is finite.
Proof. Since uL1 ≤ Cxu1/2
u1/2
, by applying the L∞ -L1 time decay estimate of the free
L2
L2
evolution group e−ii∂x t see paper of 19, Lemma 1, we get
uL∞ e−ii∂x t eii∂x t u
L∞
≤ Ct−1/2 i∂x 3/2 eii∂x t u
L1
1/2 1/2
≤ Ct−1/2 xi∂x 3/2 eii∂x t u 2 i∂x 3/2 eii∂x t u 2
L
L
2.96
≤ Ct−1/2 Ju1/2
u1/2
H1/2
H3/2
for all t > 0. Then by the Sobolev inequality we have uL∞ ≤ CuH1 . Thus the desired
estimate follows. Lemma 2.6 is proved.
Next we obtain the asymptotics for the integral
R
eitAξ,y φ ξ, y dy,
where Aξ, y ξ ζ − 2η, and ζ ξ 2y, η ξ y.
2.97
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Advances in Mathematical Physics
Lemma 2.7. The following asymptotics is true:
R
eitAξ,y φ ξ, y dy ⎛
3/2 3/2
η
iπ
⎜ −3/4 ζ
3/2
φ ξ, y ξ φξ, 0 O⎝t
ζ η3/4 y
t
L∞
L2
ξ y
1/2 1/4 φ ξ, y O t−3/4 ζ1/2 η
ζ η
⎞
⎟
⎠
2.98
L∞
L2
ξ y
for t ≥ 1 uniformly with respect to ξ ∈ R, provided the right-hand side is finite.
Proof. Denote ζ ξ 2y, η ξ y, so that y ζ − η η − ξ. We have the identities
1 η ζ − ηζ ξζ − ξζ ξ η − ξη 2
A ξ, y 2
y ,
ξ ζ ξ η ζ η
1 η ζ − ηζ
η
ζ
− 2
Ay ξ, y 2
y,
ζ
η
ζ η ζ η
2.99
and Ayy ξ, y 22/ζ3 − 1/η3 . We now change z Aξ, y. We denote the inverse
functions by yj z, so that y1 z : 0, ∞ → 0, ∞ and y2 z : 0, ∞ → −∞, 0. Thus the
stationary point z 0 transforms into y 0. Hence we can write the representation
2 ∞
A ξ, yj z
2 eitAξ,y φ ξ, y dy 2
e−itz φ ξ, yj z dz.
Ay ξ, yj z R
j1 0
2.100
By a direct calculation we find
Ay ξ, y 1
dz
,
dy 2 A ξ, y
Q ξ, ζ, η
2.101
where
√
2ζ
η
η 1 η ζ − ηζ ξζ − ξζ ξ η − ξη
ζ
Q ξ, ζ, η ≡
.
1 η ζ − ηζ ξ ζ ξ η
2.102
Therefore we obtain
R
e
itAξ,y
φ ξ, y dy ∞
0
e−itz Φξ, zdz,
2
2.103
Advances in Mathematical Physics
25
where
Φξ, z 2
√ 2 φ ξ, y Q ξ, ζ, η yyj z .
2.104
j1
Since Qξ, ξ, ξ ξ3/2 , then the application of Lemma 2.3 yields
R
e
itAξ,y
φ ξ, y dy iπ
ξ3/2 φξ, 0 O t−3/4 Φz ξ, zL∞ L2z .
ξ
t
2.105
By a direct calculation we have
Φz ξ, z
2
3/2 dz φy ξ, y Q ξ, ζ, η
yyj z
dy j1
2.106
2
φ ξ, y Q ξ, ζ, η 2Qζ ξ, ζ, η Qη ξ, ζ, η yyj z
j1
.
By 2.38
ζ η
Q ξ, ζ, η ≤ C 1/2 ,
ζ η
η − ξ
C η
|ζ − ξ|
Qζ ξ, ζ, η ≤ Cζ−1 Q ξ, ζ, η 1 ≤
1/2 ,
η ξ ζ ξ
ζ η
2.107
Qη ξ, ζ, η ≤ Cζ
1/2 .
ζ η
Therefore we get the estimate
Φz ξ, zL∞ L2z
ξ
dz Φz ξ, z
dy L∞
L2
ξ y
ζ3/2 η3/2
≤ C φ ξ, y ζ η3/4 y
1/2 1/4 Cζ1/2 η
φ ξ, y ζ η
L∞
L2
ξ y
L∞
L2
ξ y
2.108
.
This yields the estimate of the lemma. Lemma 2.7 is proved.
In the next lemma we obtain the asymptotics for the nonlinear term FU−tGu,
i∂x −1 u2 in 1.14.
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Advances in Mathematical Physics
Lemma 2.8. Let v ∈ H1,3 . Then the asymptotic formula holds as follows:
ξ FU−tG u, i∂x −1 u2 − 1 Ωξ|vt, ξ|2 vt, ξ ≤ Ct−5/4 v3 1,3
H
t
2.109
t, ξ, Ωξ ≡ ξ2 /2ξ2ξ2ξ.
for t ≥ 1 uniformly with respect to ξ ∈ R, where vt, ξ eitξ u
Proof. We first find the representation for FU−ti∂x −1 u2 t. We introduce the operator
Qt BVt Wt MDt−1 F−1 e−itξ , so that the representation for the free evolution group
is true UtF−1 e−iti∂x F−1 F−1 e−itξ Dt MQt. We also have Feiti∂x Q−1 tMDt−1 ,
where the right-inverse
Q−1 t V −1 tB−1 W−1 t eitξ FDt M.
√ operator
−1
Since Dωt iDω Dt , Dω iD1/ω , and FD1/ω Dω F, we find
√
Q−1 tMω−1 eitξ FDt Mω ieitξ Dω FDωt Mω
√
ieitξ Dω e−iωtξ Q−1 ωt
2.110
with ω > 0. Applying this formula with ω 2 and putting ut UtF−1 v Dt MQtv, we
get
FU−t i∂x −1 u2 t ξ−1 Q−1 tMDt−1 Dt MQtv2
1
eitξ e−2itξ −1
Q 2tQtv2 .
√ Q−1 tMQtv2 √ D2
2ξ
t
ξ it
2.111
Since FU−t Feii∂x t eitξ F, we can write
1
e−2itη
Fx → η i∂x −1 u2 t √ D2 Q−1 2tQtv2
2η
t
η
1
,
√ e−2itη/2 Φ t,
2
2it
2.112
where Φt, η 2η−1 Q−1 2tQtv2 . Hence we obtain
FU−tG u, i∂x −1 u2 eitξ FG u, i∂x −1 u2
−1 2
u
dη
t,
η
−
ξ
F
u
t
i∂
x
→
η
x
e
√
.
ξ ξ − η η
2πξ R
itξ
2.113
Advances in Mathematical Physics
27
t, η − ξ eitη−ξ vt, η − ξ, we get
Since u
−1 2
FU−tG u, i∂x u
eitA v t, η − ξ Φ t, η/2
√
dη
2 πitξ R ξ ξ − η η
1
2.114
ξ η − ξ − 2η/2.
with A
Since Qt BVt for |x| ≤ 1 and Qt Wt for |x| ≥ 1, we then have
Qtv2 BVtv2 Wtv2
2.115
for all x ∈ R. In the same manner applying the identity B−1 Bv2 ξ3/2 v2 ξ, we obtain
−1
−1
Φ t, η 2η V −1 2tB−1 BVtv2 2η W−1 2tWtv2
−1
−1
2η V −1 2tξ3/2 Vtv2 2η W−1 2tWtv2
Φ1 t, η Φ2 t, η .
2.116
After a change η 2ξ 2y, we find
−1 2
FU−tG u, i∂x u
R
e
itAξ,y
φ1 ξ, y dy R
eitAξ,y φ2 ξ, y dy,
2.117
where
v t, ξ 2y Φ1 t, ξ y
φ1 ξ, y ≡ √
,
πitξ ξ ξ 2y 2 ξ y
v t, ξ 2y Φ2 t, ξ y
φ2 ξ, y ≡ √
,
πitξ ξ ξ 2y 2 ξ y
2.118
and Aξ, y ξ ξ 2y − 2ξ y. Note that
−1
W 2tφ
L2
E2tFD2t 1 − θφ
L2
D2t 1 − θφL2
1 − θφL2 ≤ φL2 .
2.119
Hence by Lemma 2.5 with δ 5/6, r 4
Wtφ2 4 ≤ Ct−3/4 ξ2 ∂ξ φ 1 ξφL1
L
L
2.120
for all t ≥ 1, and
−1
W 2tWtv2 L2
2
≤ Wtv2L4 ≤ Ct−3/4 ξ2 ∂ξ v 1 ξvL1 .
L
2.121
28
Advances in Mathematical Physics
Thus
ξ eitAξ,y φ2 ξ, y dy
R
L∞
≤ Ct−1/2 vtL2 ξ−2 W−1 2tWtv2 L2
≤ Ct−5/4 v3H1,3 .
2.122
We apply Lemma 2.7 to find
ξFU−tG u, i∂x −1 u2 − t−1 ξ−1/2 Ωξvt, ξV −1 2tξ3/2 Vtv2 v
t,
ξ
2y
Φ
t,
ξ
y
1
1/2
1/2
1/4
ξ
2y
≤ Ct−5/4 ξ
y
ξ
2y
ξ
y
ξ 2y 2 ξ y
L∞
L2
ξ y
3/2 3/2
ξ
2y
v
t,
ξ
2y
Φ
ξ
y
t,
ξ
y
1
−5/4 Ct
3/4 ∂y
ξ ξ 2y 2 ξ y ξ 2y ξ y
Ct−5/4 v3H1,3
L∞
L2
ξ y
≤ Ct−5/4 vL∞ Φ1 H0,1/4 Φ1 L∞ vH0,1/4
vH0,1
v3H1,3
Φ1 H1,1/4 vH1,1 Φ1 H0,1/4
∞
∞
≤ Ct−5/4 vH1,1 Φ1 H1,1/4 v3H1,3
2.123
since
v t, ξ 2y Φ1 t, ξ y
ξφ1 ξ, y ≡ √ .
πit ξ ξ 2y 2 ξ y
2.124
By Lemma 2.2 we have
Φ1 tH1,1/4 ≤ Cξ−3/4 ∂ξ V −1 2tξ3/2 Vtv2 L2
≤ Cξ3/21/4γ ∂ξ Vtv2 2 Cξ3/2−3/4γ Vtv2 L
L2
≤ Cv2H1,2 .
2.125
We now write the representation for Ψt, ξ as
Ψt, ξ ξ3/2
vt, ξV −1 2tξ3/2 Vtv2
2ξ2ξ 2ξ
2
ξΩξ|v| v R,
2.126
Advances in Mathematical Physics
29
where the remainder is
R ξ−1/2 2ξΩξvt, ξ2ξ−1
× V −1 2t − 1 ξ3/2 Vtv2 ξ3/2 Vtv2 − v2 .
2.127
By the second estimate of Lemma 2.4 with β 5/4 and the first estimate of Lemma 2.4 with
β 3/4, we obtain
2ξ−1 V −1 2t − 1 ξ3/2 Vtv2 L∞
≤ Ct−1/4 ξ3/4 Vtv ∞ ξ2 ∂ξ Vtv 2 ξVtv2L4
L
2.128
L
≤ Ct−1/4 ξ3/4 Vt − 1φ ∞ vH1,2 Ct−1/4 v2H1,2 ≤ Ct−1/4 v2H1,2
L
for all t ≥ 1. Also
1/2
Vtv2 − v2 ξ
L∞
≤ C ξ1/4 Vt − 1v
L∞
ξ1/4 v ∞ ξ1/4 Vt − 1v
L∞
L
≤ Ct−1/4 v2H1,2 .
2.129
Thus we find the estimate for the remainder Rt as
RtL∞ ≤ Cξ−1/2 2ξΩξv ∞ 2ξ−1 V −1 2t − 1 ξ3/2 Vtv2 L∞
L
Cξ−1/2 2ξΩξv ∞ ξ1/2 Vtv2 − v2 L
L∞
≤ Ct
−1/4
2.130
v3H1,2 .
Hence
Ψt, ξ ξΩξ|v|2 v O t−1/4 v3H1,2 .
Therefore the asymptotics of the lemma is true. Lemma 2.8 is proved.
2.131
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Advances in Mathematical Physics
3. Proof of Theorem 1.1
We introduce the function space
!
XT φ ∈ C 0, T ; L2 ; φXT < ∞ ,
3.1
where
φ sup t−γ φt 3 t−γ Jφt 1 t−3γ Jφt 2 t1/2 φt 1 ,
XT
H
H
H
H∞
t∈0,T 3.2
and γ > 0 is small.
The local existence in the function space XT can be proved by a standard contraction
mapping principle. We state it without a proof.
Theorem 3.1. Let u0 ∈ H3,1 and the norm u0 H3,1 ε. Then there exist ε0 > 0 and T > 1 such that
for all 0 < ε < ε0 the initial value problem 1.1 has a unique local solution u ∈ C0, T ; H3,1 with
√
the estimate uXT < ε.
Let us prove that the existence time T can be extended to infinity which then yields
the result of Theorem 1.1. By contradiction, we assume that there exists a minimal time T > 0
√
√
such that the a priori estimate uXT < ε does not hold; namely, we have uXT ≤ ε.
We apply the energy method to 1.14 Lu iλGu, u −2i|λ|2 Gu, i∂x −1 u2 , i.e.,
multiplying both sides of the above equation by i∂x 6 u iλGu, u, taking the real part,
and integrating over the space to obtain
d
u iλGu, uH3 ≤ CG u, i∂x −1 u2 3
H
dt
≤
Cu2H1∞ uH3
≤ Cε
3/2
3.3
γ−1
t
since by Lemma 2.1
G u, i∂x −1 u2 H3
≤ CG ∂3x u, u2 2 CG u, ∂2x u2 2 CG ∂2x u, u2 L
CG ∂x u, u2 L
L2
≤ Cu2H1∞ uH3
L2
3.4
√
√
√
and uH1∞ ≤ εt−1/2 , uH3 ≤ εtγ by the estimate uXT < ε. Hence by Theorem 3.1
γ
we have u iλGu, uH3 ≤ Cεt and
uH3 ≤ u iGu, uH3 Gu, uH3
≤ Cεtγ CuH1∞ uH3 ≤ Cεtγ .
3.5
Advances in Mathematical Physics
31
Next we use the commutator relations x, i∂x α αi∂x α−2 ∂x , Pi∂x α αi∂x ∂x ∂t , ∂t u −ii∂x u ii∂x −1 u2 , and LP P − ii∂x −1 ∂x L, to get Lu iGu, u −2i|λ|2 Gu, i∂x −1 u2 . Hence
α−2
LPu iλGu, u −2i|λ|2 PG u, i∂x −1 u2 2|λ|2 i∂x −1 ∂x G u, i∂x −1 u2 .
3.6
Then by the energy method i.e., multiplying both sides of the above equation by
i∂x 4 Pu − Gu, u, taking the real part, and integrating over the space, by Lemma 2.1,
using ∂t u −ii∂x u ii∂x −1 u2 , we obtain
PG u, i∂x −1 u2 H2
≤ CPuH2 ∂t uH2 u2H1∞
≤ CPuH2 3.7
uH3 u2H1∞ .
Hence
d
Pu iλGu, uH2 ≤ CPG u, i∂x −1 u2 2 CG u, i∂x −1 u2 2
H
H
dt
≤ CPuH2 uH3 u2H1∞
−1
3.8
γ−1
≤ Cεt PuH2 Cε t
2
.
Therefore by Theorem 3.1 it follows that
PuH2 ≤ Cεtγ .
3.9
The energy estimate and the identity Lx xL − ii∂x −1 ∂x imply that
d
xu iλGu, uH2 ≤ CxG u, i∂x −1 u2 2 u iGu, uH2
H
dt
≤ Cu2H1∞ xuH2 Cu2H1∞ uH2 Cεtγ
3.10
≤ Cεt−1 xuH2 Cεtγ ,
which yields
xuH2 ≤ Cεtγ1 .
3.11
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Advances in Mathematical Physics
Then by the identity J iP − ixL − i∂x −1 ∂x we obtain
Ju iλGu, uH2 ≤ Pu iλGu, uH2 xLu iλGu, uH2 u iλGu, uH2
≤ Cεtγ CxG u, i∂x −1 u2 2
H
γ
≤ Cεt Cu2H1∞ xuH2
Cu2H1∞ uH2 ≤ Cεtγ .
3.12
We use Lemma 2.8 to obtain for the new variable vt eitξ u
t, ξ
1
ξFU−tG u, i∂x −1 u2 ξΩξ|vt, ξ|2 vt, ξ O t−5/4 v3H1,3
t
3.13
for t ≥ 1 uniformly with respect to ξ ∈ R, where Ωξ ≡ ξ2 /2ξ2ξ 2ξ. Hence for the
new function φ eitξ Fu iλGu, u we get
2
1
∂t ξφ −2i|λ|2 ξΩξφt, ξ φt, ξ O t−5/4 ε3 t3γ
t
3.14
in Lr , 2 ≤ r ≤ ∞. Then integrating, we obtain supt≥1 ξφtL∞ ≤ Cε. Hence
supvtH0,1
≤ Cε.
∞
3.15
t≥1
By the decomposition of the free evolutions group we have the identity
i∂x ut Dt MtBVt Dt WtξFeii∂x t ut
Dt MtBξFeii∂x t ut
Dt MtBξ−3/2 ξ3/2 Vt − 1 Dt Wt ξFeii∂x t ut.
3.16
By Lemmas 2.4 and 2.5 we find that the last term of the right-hand side of 3.16 is a
remainder. Indeed we have the estimate
utH1∞ ≤ CDt MtBξFeii∂x t ut
L∞
Cεt−5/4 ξFeii∂x t ut
≤ Ct−1/2 supvtH0,1
Ct−5/4 JutH2 ,
∞
H1,2
3.17
t≥1
which yields
supt1/2 utH1∞ ≤ Cε.
t≥1
3.18
Advances in Mathematical Physics
33
From 3.15 and 3.18 it follows that
uXT ≤ Cε <
√
3.19
ε,
which implies the desired contradiction. Thus there exists a unique global solution u ∈
C0, ∞; H3,1 to 1.1 with the time decay estimate.
We now prove the asymptotics of solutions. By 3.14 as in the proof of Lemma 2.7 we
have
ψt − ψs 0,1 ψt − ψs 0,1 ≤ Cε3/2
H∞
H
t
τ γ−5/4 dτ ≤ Cε3/2 sγ−1/4
3.20
s
with γ ∈ 0, 1/4, where
2
ψt ve−2i|λ| Ω|v|
2
log t
2
Feii∂x t ute−2i|λ| Ω|Fe
ii∂x t
2
ut| log t
.
3.21
0,1
such that
Thus we see that there exists a unique final state ψ ∈ H0,1
∞ ∩H
ψt − ψ 0,1 ψt − ψ 0,1 ≤ Cε3/2 tγ−1/4 .
H∞
H
3.22
We consider the asymptotics of the phase function
Φt iΩ
t |vτ|2 − |vt|2
1
dτ
τ
iΩ
t 1
ψτ2 − ψt2 dτ .
τ
3.23
By a direct calculation we have
Φt − Φs iΩ
t
s
2 dτ 2 2 2 ψτ − ψt
ψt − ψs log s ,
τ
3.24
where 1 < s < τ < t. Hence
t
ψτ − ψt 0,1 ψτ ∞ ψt ∞ dτ
H∞
L
L
τ
s
ψs ∞ ψt ∞ log s
Cψs − ψtH0,1
L
L
∞
≤C
Φt − ΦsH0,1
∞
≤ Cε5/2
t
3.25
τ γ−5/4 dτ Cε5/2 sγ−1/4 log s,
s
from which it follows that there exists a unique real valued function Φ such that
≤ Cε3/2 tγ−1/4 .
Φt − iΦ H0,1
∞
3.26
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Advances in Mathematical Physics
Similarly, we find Φt − iΦ H0,1 ≤ Cε3/2 tγ−1/4 . Therefore we have the asymptotics of the
phase function
t
2 dτ
iΩ
iΩ ψτ
τ
1
t 2 dτ
ii∂x t
uτ
Fe
τ
1
2 2
2
iΦ iΩψ log t Φt − iΦ iΩ Feii∂x t ut − ψ log t.
3.27
We also find
2
Feii∂x t ut − ψ e2|λ| iΦ iΩ|ψ |
2
2
log t
t
2
Feii∂x t ut − ψ e2|λ| iΩ 1 |ψτ| dτ/τ
t
2
2
2
2
2
ψ e2|λ| iΩ 1 |ψτ| dτ/τ − e2|λ| iΦ 2|λ| iΩ|ψ | log t
2
t
2
t
2
3.28
2
ψte2|λ| iΩ 1 |ψt| dτ/τ − ψ e2|λ| iΩ 1 |ψτ| dτ/τ
t
2
2
2
2
2
ψ e2|λ| iΩ 1 |ψτ| dτ/τ − e2|λ| iΦ 2|λ| iΩ|ψ | log t .
Collecting these estimates, we obtain
2
2
2
ii∂x t
ut − ψ e2|λ| iΦ 2|λ| iΩ|ψ | log t Fe
H0,1
≤ Cψt − ψ H0,1 Cψ H0,1 Φt − iΦ L2
ψ H0,1 ψt − ψ L∞ ψ L2 ψtL2 log t
3.29
≤ Cε3/2 tγ−1/4 ,
and similarly
2
2
2
ii∂x t
ut − ψ e2|λ| iΦ 2|λ| iΩ|ψ | log t Fe
H0,1
∞
≤ Cε3/2 tγ−1/4 .
3.30
Therefore we have
ii∂ t
|2 log t Fe x ut − W
eiΩ|W
≤ Cε3/2 tγ−1/4 ,
ii∂ t
|2 log t Fe x ut − W
eiΩ|W
≤ Cε3/2 tγ−1/4 ,
H0,1
H0,1
∞
3.31
Advances in Mathematical Physics
35
ψ exp2|λ|2 iΦ . Estimate 3.31 means that
where W
2
|2 log t ut − e−ii∂x t F−1 W
e2|λ| iΩ|W
H1,0
≤ Cε3/2 tγ−1/4 .
3.32
Theorem 1.1 is proved.
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