Electronic Journal of Differential Equations, Vol. 2001(2001), No. 54, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
WITH A QUADRATIC NONLINEARITY IN ONE-SPACE
DIMENSION
NAKAO HAYASHI & PAVEL I. NAUMKIN
Abstract. We consider the Cauchy problem for the Schrödinger equation
with a quadratic nonlinearity in one space dimension
1
iut + uxx = t−α |ux |2 , u(0, x) = u0 (x),
2
where α ∈ (0, 1). From the heuristic point of view, solutions to this problem
should have a quasilinear character when α ∈ (1/2, 1). We show in this paper
that the solutions do not have a quasilinear character for all α ∈ (0, 1) due to
the special structure of the nonlinear term. We also prove that for α ∈ [1/2, 1)
if the initial data u0 ∈ H 3,0 ∩ H 2,2 are small, then the solution has a slow
time decay such as t−α/2 . For α ∈ (0, 1/2), if we assume that the initial data
u0 are analytic and small, then the same time decay occurs.
1. Introduction
In this paper we consider the Schrödinger equation, with a quadratic derivative
term,
Lu = t−α |ux |2 , t, x ∈ R
(1.1)
u(0, x) = u0 (x), x ∈ R,
where L = i∂t + 21 ∂x2 , and α ∈ (0, 1). The Cauchy problem for Schrödinger equations
with a cubic derivative term was studied in [9]. There the authors considered
Lu = t1−δ F (u, ux ),
u(0, x) = u0 (x),
t, x ∈ R
x ∈ R,
(1.2)
where 0 < δ < 1, is a sufficiently small constant, and the nonlinear interaction
term F consists of cubic nonlinearities.
F (u, ux ) = λ1 |u|2 u + iλ2 |u|2 ux + iλ3 u2 ūx + λ4 |ux |2 u + λ5 ūu2x + iλ6 |ux |2 ux ,
where the coefficients λ1 , λ6 ∈ R, λ2 , λ3 , λ4 , λ5 ∈ C, λ2 − λ3 ∈ R, λ4 − λ5 ∈ R. In
[9], the authors found a time decay estimate for the solutions of this problem,
ku(t)k∞ ≤ C|t|−1/2 .
(1.3)
2000 Mathematics Subject Classification. 35Q55, 74G10, 74G25.
Key words and phrases. Schrödinger equation, large time behaviour, quadratic nonlinearity .
c
2001
Southwest Texas State University.
Submitted May 22, 2001. Published July 25, 2001.
1
2
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
The same result is also true for the case δ > 1. From the heuristic point of view
problem (1.1) corresponds to problem (1.2), when δ = α + 21 . Therefore it is
natural to make a conjecture that the solutions of (1.1) also have the decay property
(1.3). However, as we will show in the present paper, due to the special oscillating
structure of the nonlinear term, for α ∈ (0, 1) the asymptotic behavior of solutions
to (1.1) do not obey the estimate (1.3). Our result stated below depends on the
structure of nonlinearity which appears in the identity
Z
2
1/2
(FU(−t)|ux | )(t, ξ) = (2π)
e−itξη (FU(−t)ux (t, η))(FU(−t)ux )(t, ξ + η)dη.
In the cases of u2x and ū2x we have
(FU(−t)u2x )(t, ξ)
=
i
(2π)1/2 e 4 tξ
2
Z
2
e−ity (FU(−t)ux )(t,
ξ
ξ
− y)(FU(−t)ux )(t, + y)dy
2
2
and
(FU(−t)ū2 )(t, ξ)
=
3i
(2π)1/2 e 4 tξ
2
Z
2
eity (FU(−t)ux )(t,
ξ
ξ
− y)(FU(−t)ux )(t, + y)dy,
2
2
where U(t) is the linear Schrödinger evolution group
Z
2
i
it 2
1
U(t)φ = √
e 2t (x−y) φ(y)dy = F −1 e− 2 ξ Fφ,
2πit
R
Fφ ≡ φ̂ = √12π e−ixξ φ(x)dx denotes the Fourier transform of the function φ. The
2
oscillating function e±ity yields an additional time decay term through integration
by parts. However, the oscillating function e±itξy does not give an additional time
decay uniformly with respect to ξ. This is the main reason why we do not have
estimate (1.3) for solutions of (1.1). In [6] we proved (1.3) for solutions of the
Cauchy problem
Lu = λ(ux )2 + µu2x , with λ, µ ∈ C.
However, the nonlinearity |ux |2 was out of our scope. In the present paper we
intend to fill up this gap studying the case of quadratic nonlinearity t−α |ux |2 .
The methods developed for the nonlinear Schrödinger equations with quadratic
nonlinearities u2x , |ux |2 and u2x can be applied also to the study of the large time
asymptotic behavior for other quadratic nonlinear equations, such as BenjaminOno and Korteweg-de Vries equations (in paper [8], mBO equation was reduced
to the cubic nonlinear Schrödinger equation). In paper [2], Cohn used the method
of normal forms of Shatah [11] to study the nonlinear Schrödinger equations with
quadratic nonlinearity u2x and showed that the solution exists on [0, T ) with T
bounded from below by Cε−6 , where ε is the size of the data in some Sobolev norm.
In paper [10] the nonlinearity u2x was studied by the Hopf-Cole transformation. The
L2 -estimate of solutions involving the operator J = x + it∂x plays a crucial role
in the large time asymptotic behavior of solutions. However the nonlinearity N (u)
under consideration does not posses a self-conjugate structure eiω N (u) = N (eiω u)
for all ω ∈ R, therefore we can not use the operator J = x + it∂x directly in
(1.1). To overcome these obstacles we use the method developed in [7] and apply
systematically the operator I = x∂x + 2t∂t .
EJDE–2001/54
ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
3
We now state our strategy for the proof. If we put v = ux . Then the problem is
written as
Lv = t−α ∂x |v|2 , t, x ∈ R.
By the identity
∂x J |v|2 = ∂x (vJ v + itv x v) = vJ ∂x v + 2v x J v − vJ ∂x v
we have
LJ v = t−α J ∂x |v|2 = t−α (−|v|2 + vJ ∂x v + 2v x J v − vJ ∂x v).
Therefore, the operator J acts on this problem also. Thus global existence in time
of small solutions to the problem can be proved for α ∈ (1/2, 1) and the derivative
ux should have the same asymptotic behaviour as the solutions to the corresponding
linear problem (along with time-decay estimate (1.3)). Combining this fact and the
identity (1) we prove the time decay of solutions. Roughly speaking, we show there
exists a constant c and a positive constant γ such that
√
|u(t, t) − ct−α/2 | ≤ Ct−(α/2)−γ .
In the case of α ∈ (0, 1/2) we use the fact that
1
(uJ u − uJ u)
it
which implies that usual derivative yields an additional time decay, in particular, the
fractional derivative |∂x |β gives us an additional time decay like t−β (see Lemma 2.4
below). However we have the derivative loss on the nonlinear term which requires
us to use some analytic function space.
To state our results we need some
R notation. We denote the inverse Fourier
transformation by F −1 φ = φ̌ = √12π eixξ φ(ξ)dξ. We essentially use the estimates
∂x |u|2 =
of the operators J = x + it∂x = U(t)xU(−t) = itM (t)∂x M (t) and I = x∂x + 2t∂t ,
2
M = e(ix )/(2t) . Note that the relation J ∂x = I +2itL is valid, where L = i∂t + 21 ∂x2
and√U(t) = M (t)D(t)FM (t), D(t) is the dilation operator defined by (D(t)ψ)(x) =
(1/ it)ψ(x/t). Then since D−1 (t) = iD(1/t) we have U(−t) = M F −1 D−1 (t)M =
iM F −1 D(1/t)M .
We denote Rthe usual Lebesgue space Lp = {φ ∈ S0 ; kφkp < ∞}, where the
norm kφkp = ( R |φ(x)|p dx)1/p if 1 ≤ p < ∞ and kφk∞ = ess.sup {|φ(x)|; x ∈ R} if
p = ∞. For simplicity we write k · k = k · k2 . Weighted Sobolev space is
Hpm,k = φ ∈ S0 : kφkm,k,p ≡ hxik hi∂x im φp < ∞ ,
√
m, k ∈ R, 1 ≤ p ≤ ∞, hxi = 1 + x2 . The fractional derivative |∂x |α , α ∈ (0, 1) is
equal to
Z
dz
|∂x |α φ = F −1 |ξ|α Fφ = C (φ(x + z) − φ(x)) 1+α .
|z|
R
We denote also for simplicity H m,k = H2m,k and the norm kφkm,k = kφkm,k,2 .
Different positive constants are denoted by the same letter C. Denote Φ(x) =
R − i (ξ−x)2 α−1
e 2
|ξ|
dξ.
Now we state the main results of this paper.
Theorem 1.1. Let α ∈ [1/2, 1). We assume that the initial data u0 ∈ H 3,0 ∩ H 2,2
and the norm ku0 k3,0 + ku0 k2,2 is sufficiently small. Then there exists a unique
global solution u of the Cauchy problem (1.1) such that u ∈ C(R; H 3,0 ). Moreover
4
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
there exist unique constant B and functions P, Q such that |ξ|1−α P (ξ) ∈ L∞ (R),
|ξ|1−α Q(ξ) ∈ L∞ (R) and the following asymptotic statement is valid
u(t, x) = Be
ix2
2t
α
α
x
x
x
t− 2 Φ( √ ) + O(t− 2 −γ (h √ iα−1 + h √ i−α ))
t
t
t
(1.4)
for all t ≥ 1, uniformly in |x| ≤ t1−ρ , and
ix2 1
1
x
x
x
u(t, x) = t−α P ( ) + e 2t √ Q( ) + O(t−α−γ + t− 2 −γ h i−α )
t
t
t
t
(1.5)
for all t ≥ 1, uniformly in |x| ≥ t1−ρ , where ρ, γ > 0 are small.
In the case α ∈ (0, 1/2) we have to assume that the initial data are analytic.
Denote
∞
X
1
1
A0 = φ ∈ L2 : kφkA0 ≡
k|∂x | 2 −α (x∂x )n φk1,0 < ∞ .
n!
n=0
Theorem 1.2. Let α ∈ (0, 1/2). We assume that the initial data u0 ∈ A and the
norm ku0 kA0 is sufficiently small. Then there exists a unique global solution u of
the Cauchy problem (1.1) such that u ∈ C(R; H 1,0 ). Moreover there exist unique
constant B and functions P, Q such that asymptotics (1.4) and (1.5) are valid.
Remark 1.1. In the region |x| = t1−ρ asymptotics (1.4) coincides with (1.5).
In Section 2 we prove some preliminary estimates. In Section 3 we prove Theorem
1.1. Section 4 is devoted to the proof of Theorem 1.2.
2. Preliminaries
First we prove some time decay estimates.
Lemma 2.1. We have the estimate
kux k∞ ≤ Ct−1/2 kFU(−t)ux k∞ + Ct−
1+β−γ
2
1
(kux k + k|∂x | 2 −β J ∂x uk),
for all t > 0, where β ∈ (0, 21 ], γ ∈ (0, β).
Proof. Denote w = U(−t)ux . Then since U(t) = M DFM , where M = e
Dφ = √1it φ( xt ) is the dilation operator, J = x + it∂x = U(t)xU(−t), we get
ix2
2t
,
ux = U(t)w = M DFw + M DF(M − 1)w
and by virtue of the Hölder inequality and Sobolev embedding theorem kφkp ≤
1
1
Ck|∂x | 2 − p φk if 2 ≤ p < ∞, we have
kM DF(M − 1)wk∞
≤ Ct−1/2 kF(M − 1)wk∞ ≤ Ct−1/2 k(M − 1)wk1 ≤ Ct−
≤ Ct−
1+β−γ
2
(kwk + kxwk β1 ) ≤ Ct−
1+β−γ
2
≤ Ct−
1+β−γ
2
(kux k + k|∂x | 2 −β xU(−t)ux k)
≤ Ct−
1+β−γ
2
(kux k + k|∂x | 2 −β J ∂x uk),
1+β−γ
2
k|x|β−γ wk1
1
(kwk + k|∂x | 2 −β xwk)
1
1
therefore the result of the lemma follows. Lemma 2.1 is proved.
EJDE–2001/54
ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
5
Denote
1
kφkY = sup tα hti1−2γ k∂t φk0,1,∞ + sup t−γ k|ξ| 2 −β ∂ξ φk + sup kφk0,1,∞ ,
t>0
t>0
t>0
where β ∈ (0, 21 ], γ > 0 is small. In the next lemma we obtain the asymptotic
representation as ξ → 0 for the integral
Z t
Z
I=
τ −α dτ e−iτ ξη φ1 (τ, ξ + η)φ2 (τ, η)dη
0
which corresponds to the identity (1).
Lemma 2.2. If φl ∈ Y, l = 1, 2, then we have
Z
πα
I = Γ(1 − α)|ξ|α−1 (sin( ) φ1 (t, η)φ2 (t, η)|ξ|α−1 dη
2
Z
πα
+i sign ξ cos( ) φ1 (t, η)φ2 (t, η)|η|α−1 sign η dη)
2
+O(t−γ |ξ|α−1 kφ1 kY kφ2 kY ).
for all |ξ| ≤ t−µ , t ≥ 1, where µ =
Proof. We write I =
P4
l=1 Il ,
I1 =
Z
I2 =
I3 =
γ > 0 is small.
where
tν /|ξ|
Z
0
t
tν /|ξ|
Z
3γ
α2 ,
ν
t /|ξ|
τ −α dτ
Z
τ −α dτ e−iτ ξη φ1 (t, η)φ2 (t, η)dη,
Z
−α
τ dτ e−iτ ξη φ1 (τ, ξ + η)φ2 (τ, η)dη,
Z
e−iτ ξη (φ1 (τ, η)φ2 (τ, η) − φ1 (t, η)φ2 (t, η))dη
0
I4 =
Z
tν /|ξ|
τ
−α
dτ
Z
e−iτ ξη (φ1 (τ, ξ + η) − φ1 (τ, η))φ2 (τ, η)dη,
0
where ν = 2γ/α. If τ |ξ| ≥ 1, we integrate by parts with respect to η to obtain
Z
| e−iτ ξη φ1 (t, x + η)φ2 (t, η)dη|
Z
−1
≤ hτ ξi | e−iτ ξη ∂η (φ1 (t, x + η)φ2 (t, η))dη|
≤ Chτ ξi−1 tγ
2
X
1
kφ3−l k∞ sup t−γ k|ξ| 2 −γ ∂ξ φl k ≤ Chτ ξi−1 tγ kφ1 kY kφ2 kY ,
t>0
l=1
hence changing τ |ξ| = z we obtain
Z ∞
Z
−α
|
τ dτ e−iτ ξη φ1 (t, x + η)φ2 (t, η)dη|
tν /|ξ|
≤ Ctγ kφ1 kY kφ2 kY
Z
∞
hτ ξi−1 τ −α dτ ≤ Ctγ |ξ|α−1 kφ1 kY kφ2 kY
tν /|ξ|
≤ C|ξ|α−1 tγ−αν kφ1 kY kφ2 kY ≤ Ct−γ |ξ|α−1 kφ1 kY kφ2 kY .
Z
∞
tν
z −α−1 dz
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NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
Since
Z
∞
τ −α eiτ ξη dτ
∞
Z
=
0
τ −α cos(τ ξη)dτ + i
0
∞
τ −α sin(τ ξη)dτ
0
πα
)|ξη|α−1
2
πα
+iΓ(1 − α) cos( )|ξη|α−1 sign(ξη)
2
Z
e−iτ ξη φ1 (t, η)φ2 (t, η)dη
Z ∞
Z
−
τ −α dτ e−iτ ξη φ1 (t, η)φ2 (t, η)dη
tν /|ξ|
Z
πα
= Γ(1 − α) sin( )|ξ|α−1 φ1 (t, η)φ2 (t, η)|η|α−1 dη
2
Z
πα
+iΓ(1 − α) cos( )|ξ|α−1 sign(ξη)φ1 (t, η)φ2 (t, η)|η|α−1 dη
2
+O(t−γ |ξ|α−1 kφ1 kY kφ2 kY ).
τ
−α
∞
Γ(1 − α) sin(
=
(see [1]), we find
Z
I1 =
Z
dτ
0
In the same manner we obtain
Z t
Z
|
τ −α dτ e−iτ ξη φ1 (τ, x + η)φ2 (τ, η)dη|
tν /|ξ|
γ
≤ Ct kφ1 kY kφ2 kY
t
Z
hτ ξi−1 τ −α dτ ≤ Ct−γ |ξ|α−1 kφ1 kY kφ2 kY ,
tν /|ξ|
hence
|I2 | ≤ Ct−γ |ξ|α−1 kφ1 kY kφ2 kY .
To estimate I3 we note that
kφl (t, ξ) − φl (τ, ξ)k0,1,∞ = k
Z
t
∂τ φl (τ, ξ)dτ k0,1,∞ = O(τ 2γ−α kφl kY )
τ
which implies
Z
tν /|ξ|
τ −α dτ
Z
≤ Ckφ1 kY kφ2 kY |
Z
|I3 | = |
e−iτ ξη (φ1 (τ, η)φ2 (τ, η) − φ1 (t, η)φ2 (t, η))dη|
0
tν /|ξ|
τ 2γ−2α dτ | ≤ Ct−γ |ξ|α−1 kφ1 kY kφ2 kY
0
since µα ≥ γ + ν and |ξ| ≤ t−µ . Now using the estimate
Z ξ
−1
−1
khηi (φ(t, ξ + η) − φ(t, η))k1 = khηi
∂y φ(t, y + η)dyk1
0
1
≤ C|ξ|k|ξ| 2 −β ∂ξ φk ≤ Ctγ |ξ|kφkY
for all |ξ| ≤ 1, we get
|I4 | ≤ Ckφ1 kY kφ2 kY |ξ|
Z
tν /|ξ|
τ γ−α dτ ≤ Ct−γ |ξ|α−1 kφ1 kY kφ2 kY
0
since µ(1 − γ) ≥ γ + ν and |ξ| ≤ t−µ . Lemma 2.2 is proved.
EJDE–2001/54
ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
7
In the next lemma we consider the asymptotic behaviour of the integral
Z
it
x 2
I(t, x) = e− 2 (ξ− t ) f (t, ξ)dξ
as t → ∞ uniformly with respect to x ∈ R. Define Φ(x) =
Note that
Φ(x) = O(hxi−α + hxiα−1 )
R
i
2
e− 2 (ξ−x) |ξ|α−1 dξ.
as |x| → ∞. Let γ be a small positive number and
β = min(1/2, α) − γ, µ = 3γ/α2 ,
5γ
6γ
ρ= 2
, θ= 2
, δ = θ + γ.
α (1 − α)
α (1 − α)2
Lemma 2.3. Let ∂ξ f (t, ξ) = O(|ξ|α−2 ) and f (t, ξ) = t1−α Ψ(tξ) + O(t1−α−δ ) for
all |ξ| ≤ tθ−1 , ∂ξ f (t, ξ) = (α−1)B|ξ|α−1 ξ −1 +O(t−γ |ξ|α−2 ) for all tθ−1 ≤ |ξ| ≤ t−µ
1
and k|ξ| 2 −β ξ∂ξ f (t, ξ)k ≤ Ctγ , then we have the asymptotic formula
α
1
α
1
1
I(t, x) = Bt− 2 Φ(xt− 2 ) + O(t− 2 −γ (hxt− 2 i−α + hxt− 2 iα−1 ))
for all t ≥ 1 uniformly in |x| ≤ t1−ρ and
√
√
1
ix2
x
π
x
I(t, x) = 2πt−α e− 2t Ψ̌( ) + √ f (t, ) + O(t−α−γ + t− 2 −γ hxt−1 i−α )
t
t
it
for all t ≥ 1 uniformly in |x| ≥ t1−ρ .
Proof. For x > 0, we have
f (t, ξ)
= f (t, 1) +
Z
ξ
∂η f (t, η)dη +
t−µ
+O(k|ξ|
Z
ξ
|η|α−2 dη + O(t−γ
t−µ
Z
ξ∂ξ f (t, ξ)k(
= B|ξ|α−1 + O(1 + t−γ |ξ|
= B|ξ|
+ O(t
−γ
α−1
|ξ|
Z
ξ
t−µ
|η|α−2 dη)
t−µ
1
α−1
α−1
∂η f (t, η)dη
1
= f (t, 1) + (α − 1)B
1
2 −β
t−µ
Z
|ξ|2β−3 dξ)1/2 )
+ tµ(1−β)+γ )
)
for all tµ−1 ≤ |ξ| ≤ 2t−ρ since µ(1 − β) + 2γ ≤ ρ(1 − α). We make a change of
variable of integration ξ = zt−1/2 , then we have
Z
2
i
I(t, x) = t−1/2 e− 2 (z−b) f (t, zt−1/2 )dz,
√
√
1
where b = a t = x/ t. First consider the case |x| ≤ t1−ρ , i.e. b ≤ t 2 −ρ . We
represent
α
I = Bt− 2 Φ(b) + R1 + R2 ,
where the remainder terms are
Z
1−α
2
i
−1/2
Rj = t
e− 2 (z−b) (f (t, zt−1/2 ) − Bt 2 |z|α−1 )ϕj (z)dz,
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NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
the function ϕ1 (z) ∈ C1 (R) : ϕ1 (z) = 1 if z < b/3 and ϕ1 (z) = 0 if z > 2b/3,
ϕ2 (z) = 1 − ϕ1 (z). In the remainder term R1 we integrate by parts via the identity
2
i
e− 2 (z−b) =
2
i
1
d
(ze− 2 (z−b) )
1 − iz(z − b) dz
(2.1)
to get
Z
α
≤ Ct− 2 −γ |z|α−1 hzbi−1 (|ϕ1 | + |zϕ01 |)dz
Z
−α
2
+Ct
|z|α−1 hzbi−1 dz
1
|R1 |
|z|≤tµ− 2
≤ Ct
−α
2 −γ
√
α
hbi−α ≤ Ct− 2 −γ ha ti−α .
(2.2)
In the remainder term R2 we use the identity
i
2
i
1
d
((z − b)e− 2 (z−b) )
2
1 − i(z − b) dz
2
e− 2 (z−b) =
(2.3)
to find
|R2 |
≤ Ct
−α
2 −γ
α
+Ct− 2
Z
|z|α−1 hz − bi−2 (|ϕ2 | + |zϕ02 |)dz
Z
1
|z|≤tµ− 2
+Ct−1/2
Z
|z|α−1 hz − bi−2 dz
1
|z|>2t 2 −ρ
hz − bi−2 |zt−1/2 ||f 0 (t, zt−1/2 )|dz
√
α
α
= O(t− 2 −γ hbiα−1 ) = O(t− 2 −γ ha tiα−1 ),
(2.4)
since
Z
hz − bi−2 |zt−1/2 ||f 0 (t, zt−1/2 )|dz
Z
1
≤ Ck|ξ| 2 −β ξf 0 (t, ξ)k(
|zt−1/2 |2β−1 hz − bi−4 dz)1/2
1
|z|>2t 2 −ρ
Z
1−2β
1
≤ Ct 4 hbi−1 k|ξ| 2 −β ξf 0 (t, ξ)k(
z 2β−3 dz)1/2
1
1
|z|>2t 2 −ρ
|z|>2t 2 −ρ
≤ Ct
− 14 +ρ(1−β)
−1
hbi
k|ξ|
1
2 −β
0
ξf (t, ξ)k ≤ Ct−γ hbi−1 .
1
We consider now the case |x| > t1−ρ , i.e. b > t 2 −ρ . Then we represent I in the
form
r
Z
π
−1/2
− 2i (z−b)2
−1/2
I=t
e
f (t, zt
)dz +
f (t, a) + R3 + R4 ,
θ− 1
it
|z|≤t 2
where the remainder terms are
Z
2
i
−1/2
R3 = t
e− 2 (z−b) f (t, zt−1/2 )ϕ1 (z)dz
θ− 1
|z|>t 2
Z
2
i
R4 = t−1/2 e− 2 (z−b) (f (t, zt−1/2 ) − f (t, a))ϕ2 (z)dz.
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ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
9
Consider the integral
Z
Z
2
i
−1/2
− 2i (z−b)2
−1/2
t
e
f (t, zt
)dz =
e− 2 t(ξ−a) f (t, ξ)dξ
θ− 1
|z|≤t 2
|ξ|≤tθ−1
Z
2
i
= t1−α
e− 2 t(ξ−a) Ψ(tξ)dξ + O(t−α−γ )
|ξ|≤tθ−1
Z
ix2
eiya Ψ(y)dy + O(t−α−γ )
= t−α e− 2t
√
=
|y|≤tθ
2
− ix
2t
2πt−α e
b
Ψ(a)
+ O(t−α−γ ).
In the remainder term R3 above we integrate by parts via identity (2.1) to get
Z
α
|z|α−1 hzbi−1 (|ϕ1 | + |zϕ01 |)dz
|R3 | ≤ Ct− 2
θ− 1
|z|≥t 2
Z
−1/2
hzbi−1 |zt−1/2 ||f 0 (t, zt−1/2 )|dz
(2.5)
+Ct
1
|z|>2t 2 −ρ
≤ Ct−α+ρ−θ(1−α) + Ct−α−γ ≤ Ct−α−γ
since θ(1 − α) − ρ ≥ γ. In the remainder term R4 we integrate by parts via (2.3)
to find
Z ∞
−1/2
|R4 | ≤ Ct
|f (t, zt−1/2 ) − f (t, a)|hz − bi−4 dz
+t−1
Z
b/3
∞
1
|f 0 (t, zt− 2 )|hz − bi−1 dz
(2.6)
b/3
≤ C|a|−1 tγ−
1+β
2
1
≤ Chai−1 t− 2 −γ ,
since
−1/2
|f (t, zt
) − f (t, a)| = |
Z
a
∂ξ f (t, ξ)dξ| ≤
zt−1/2
Z
1
≤ Ck|ξ| 2 −β ξ∂ξ f (t, ξ)k(
Z
a
3
zt−1/2
3
|ξ|β− 2 |ξ| 2 −β |∂ξ f (t, ξ)|dξ
a
zt−1/2
|ξ|2β−3 dξ)1/2
β
≤ C|a|−1 tγ− 2 |z − b|β .
Collecting estimates (2.2), (2.4)-(2.6) we get the asymptotic statement needed and
Lemma 2.3 is proved.
In the next lemma we obtain time-decay estimate via additional derivative for
the nonlinear term. We will use this estimate in the proof of Theorem 1.2.
Lemma 2.4. We have the estimate
1
k|∂x | 2 −β (ux vx )k1,0
1
1
≤ Ctβ−1 k|∂x | 2 −β uk1,0 k|∂x | 2 −β vk1,0
1
1
1
1
+Ct−1 (tβ kFU(−t)ux k∞ + k|∂x | 2 −β uk1,0 )k|∂x | 2 −β J ∂x vk1,0
+Ct−1 (tβ kFU(−t)vx k∞ + k|∂x | 2 −β vk1,0 )k|∂x | 2 −β J ∂x uk1,0
for all t > 0, where β ∈ (0, 1/2].
10
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
Proof. Application of the Fourier transformation yields
Z
1
û(t, ξ + η)v̂(t, η)(ξ + η)ηdη,
F(ux vx ) = √
2π
it
2
2
it
then changing iηû(t, η) = e− 2 η φ(t, η) and iηv̂(t, η) = e− 2 η ψ(t, η) we obtain
Z
1
e−itξη φ(t, ξ + η)ψ(t, η)dη,
(2.7)
FU(−t)(ux vx ) = √
2π
whence integrating by parts with respect to η we get
1
1
k|∂x | 2 −β (ux vx )k = Ck|ξ| 2 −β FU(−t)(ux vx )k
Z
1
= Ck|ξ| 2 −β e−itξη φ(t, ξ + η)ψ(t, η)dηk
Z
1
−β
−1
2
≤ Ckhtξi |ξ|
k(k e−itξη φ(t, ξ + η)ψ(t, η)dηk∞
Z
+k e−itξη φξ (t, ξ + η)ψ(t, η)dηk∞
Z
+k e−itξη φ(t, ξ + η)ψη (t, η)dηk∞ )
1
1
≤ Ctβ−1 kφkkψk + Ctβ−1 kφk∞ k|ξ| 2 −β ∂ξ ψk + Ctβ−1 kψk∞ k|ξ| 2 −β ∂ξ φk
1
≤ Ctβ−1 kux kkvx k + Ctβ−1 kFU(−t)ux k∞ k|∂x | 2 −β J ∂x vk
1
+Ctβ−1 kFU(−t)vx k∞ k|∂x | 2 −β J ∂x uk
and
3
3
k|∂x | 2 −β (ux vx )k = Ck|ξ| 2 −β FU(−t)(ux vx )k
Z
3
= Ck|ξ| 2 −β e−itξη φ(t, ξ + η)ψ(t, η)dηk
Z
1
−1
−β
2
≤ Ct (k|ξ|
e−itξη φξ (t, ξ + η)ψ(t, η)dηk
Z
1
+Ck|ξ| 2 −β e−itξη φ(t, ξ + η)ψη (t, η)dηk)
1
1
≤ Ct−1 khξi 2 −β φkk∂ξ ψk1 + Ct−1 kφkk|ξ| 2 −β ∂ξ ψk1
1
1
+Ct−1 khξi 2 −β ψkk∂ξ φk1 + Ct−1 kψkk|ξ| 2 −β ∂ξ ψk1
1
1
≤ Ct−1 k|∂x | 2 −β uk1,0 k|∂x | 2 −β J ∂x vk1,0
1
1
+Ct−1 k|∂x | 2 −β vk1,0 k|∂x | 2 −β J ∂x uk1,0 .
Lemma 2.4 is proved.
3. Proof of Theorem 1.1
By virtue of the method in [4], [5] (see also the proof of a-priori estimates below
in Lemma 3.2) we easily obtain the local existence of solutions in the functional
space
XT = φ ∈ C((−T, T ); L2 (R)) : sup kφ(t)kX < ∞ ,
t∈(−T,T )
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ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
11
where the norm in X is
kukX
= hti−γ kuk3,0 + hti−γ kIuk1,0 + hti−3γ kI 2 uk
+tα hti1−2γ k∂t FU(−t)ux (t)k0,1,∞ ,
with I = x∂x + 2t∂t .
Theorem 3.1. Let the initial data u0 ∈ H 3,0 ∩ H 2,2 . Then for some time T > 0
there exists a unique solution u ∈ XT of the Cauchy problem (1.1). If we assume
in addition that the norm of the initial data ku0 k3,0 + ku0 k2,2 = ε2 is sufficiently
small, then there exists a unique solution u ∈ XT of (1.1) for some time T > 1,
such that the following estimate supt∈[0,T ] kukX < ε is valid.
In the next lemma we obtain the estimates of global solutions in the norm X.
Lemma 3.2. Let α ∈ [1/2, 1). We assume that the initial data u0 ∈ H 3,0 ∩ H 2,2
and the norm ku0 k3,0 + ku0 k2,2 = ε2 is sufficiently small. Then there exists a
unique global solution of the Cauchy problem (1.1) such that u ∈ C(R; H 3,0 ) and
the following estimate is valid
sup kukX < ε.
(3.1)
t>0
Proof. Applying the result of Theorem 3.1 and using a standard continuation argument we can find a maximal time T > 1 such that the inequality
kukX ≤ ε
(3.2)
is true for all t ∈ [0, T ]. If we prove (3.1) on the whole time interval [0, T ], then
by the contradiction argument we obtain the desired result of the lemma. In view
of the local existence Theorem 3.1 it is sufficient to consider the estimates of the
solution on the time interval t ≥ 1 only.
As a consequence of (3.2) we have
Z t
kFU(−t)ux (t)k0,1,∞ ≤ Cε +
k∂τ FU(−τ )ux (τ )k0,1,∞ dτ
0
Z t
≤ Cε + Cε
hτ iγ−1 τ −α dτ ≤ Cε.
0
Note that J ∂x = I + 2itL, where J = x + it∂x . Hence
kJ ∂x uk1,0 ≤ kIuk1,0 + CtkLuk1,0 ≤ kIuk1,0 + Ct1/2 kux k∞ kuk2,0
and
kJ ∂x Iuk ≤ kI 2 uk + CtkLIuk ≤ kI 2 uk + Ct1/2 kux k∞ (kux k + kIux k).
Then by Lemma 2.1 with β = 12 , using estimate (3.4) we find
kux k1,0,∞
γ
3
≤ Ct−1/2 kFU(−t)ux k0,1,∞ + Ct 2 − 4 (kuk2,0 + kJ ∂x uk1,0 )
1
≤ Cεt−1/2 + Cεt3γ− 4 kux k∞ ,
whence
kux k1,0,∞ ≤ Cεt−1/2 .
(3.3)
Therefore by virtue of (3.2) we have also the estimates
t−γ kJ ∂x uk1,0 + t−3γ kJ ∂x Iuk ≤ Cε.
(3.4)
12
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
Let us estimate norms kuk3,0 , kIuk1,0 and kI 2 uk. Differentiating three times
equation (1.1) we get for h0 = (1 + ∂x3 )u
Lh0 = t−α (ux ∂x h0 + ux ∂x h0 ) + R0
where
1
L = i∂t + ∂x2 , R0 = t−α (−|ux |2 + 3uxx uxxx + 3uxxx uxx ).
2
Via (3.2), (3.3) we have the estimate
kR0 k ≤ Ct−α kux k1,0,∞ kh0 k ≤ Cε2 tγ−1 .
Applying the operator I to both sides of equation (1.1) and using the commutator
relations LI = (I + 2)L and [I, t−α ] = −2αt−α , we find
Lhk = t−α (ux ∂x hk + ux ∂x hk ) + Rk ,
(3.5)
2
where k = 1, 2, h1 = (1 + ∂x )Iu, h2 = I u,
R1 = t−α (uxx Iux + uxx Iux + 2(1 − α)(1 + ∂x )|ux |2 ),
and
R2 = 2t−α (|Iux |2 + (2 − α)I|ux |2 + 2(1 − α)2 |ux |2 ).
By (3.2) and (3.4) we have
1
3
1
kIux Iux k ≤ Ct− 2 kIux k 2 kJ Iux k1/2 ≤ Cε2 t3γ− 2 ,
then by virtue of (3.2), (3.3) we estimate the remainder terms
kR1 k ≤ Ct−1/2 kux k1,0,∞ (kuk1,0 + kIuk1,0 ) ≤ Cε2 tγ−1
and
kR2 k ≤ Ct−1/2 kux k∞ (kuk1,0 + kIuk1,0 ) + Ct−1/2 kIux Iux k ≤ Cε2 t3γ−1 .
−α
To cancel the higher-order derivative t−α ūx ∂x hk , we multiply (3.5) by E ≡ e−t ū .
The other higher-order derivative t−α ux ∂x hk will be eliminated via integration by
parts. Since E(L − t−α ux ∂x ) = (L − g)E, where g = −t−α uxx + 12 t−2α (ux )2 −
t−2α |ux |2 , from equation (3.5) we obtain
LEhk = t−α ux E∂x hk + ERk + gEhk .
−1
(3.6)
Note that kEk1,0,∞ ≤ C and kgk∞ ≤ Cεt by virtue of (3.2), (3.3). Applying the
energy method to (3.6) we obtain
Z
d
kEhk k2 ≤ Ct−α | ux E∂x (hk )2 dx| + C(kERk k + kgEhk k)kEhk k,
dt
whence integration by parts yields
d
kEhk k ≤ Cεt−1 kEhk k + CkRk k,
(3.7)
dt
where k = 0, 1, 2. Integrating (3.7) with respect to time t ∈ [1, T ] we obtain the
estimate
ε
hti−γ kuk3,0 + hti−γ kIuk1,0 + hti−3γ kI 2 uk < .
(3.8)
2
for all t ∈ [0, T ]. We now estimate k∂t FU(−t)ux (t)k0,1,∞ . We apply the Fourier
transformation to equation (1.1), then changing the dependent variable Fux =
it 2
e− 2 ξ w, in view of (2.7) we obtain
Z
iξt−α
e−itξη w(t, ξ + η)w(t, η)dη,
(3.9)
iwt (t, ξ) = − √
2π
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ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
13
where w(t, ξ) = FU(−t)ux . When t ∈ (0, 1) we get
Z
kξ e−itξη w(t, ξ + η)w(t, η)dηk0,1,∞ ≤ Ckwk20,2 ≤ Ckuk23,0 ≤ Cε2
and if t ≥ 1, we integrate by parts with respect to η,
Z
kξ e−itξη w(t, ξ + η)w(t, η)dηk0,1,∞
Z
−1
≤ Chti k e−itξη ∂η (w(t, ξ + η)w(t, η))dηk0,1,∞
≤ Chti−1 k∂η wk0,1 kwk0,1
≤ Chti−1 kJ ∂x uk1,0 kuk2,0 ≤ Cε2 hti2γ−1 ;
therefore,
ε
.
(3.10)
2
By (3.8) and (3.10) we see that estimate (3.1) is true for all t ∈ [0, T ]. The contradiction obtained proves (3.1) for all t > 0.
tα hti1−2γ k∂t FU(−t)ux (t)k0,1,∞ <
To complete the proof of Theorem 1.1 we evaluate the large time asymptotic
estimate of the solution u. Note that by Lemma 2.1 derivative ux has a quasi linear
asymptotic formula
3
3
ux = M Dw + O(tγ− 4 kJ ∂x uk) = M Dw + O(t2γ− 4 ),
where M = e
ix2
2t
, Dφ =
√1 φ( x ).
t
it
For the solution u(t, x) we have
Z
it
it 2
x 2
M
e− 2 (ξ− t ) v(t, ξ)dξ,
u(t, x) = F −1 e− 2 ξ v = √
2π
where v = FU(−t)u. In the same way as in the proof of (3.10) we have the estimate
k∂t FU(−t)Iux (t)k0,1,∞ ≤ Ct5γ−α−1 .
To apply Lemma 2.3 we need to prove the representation
∂ξ v(t, ξ) = (α − 1)B|ξ|α−1 ξ −1 + O(t−γ |ξ|α−2 )
for all tθ−1 ≤ |ξ| ≤ t−µ , ∂ξ v(t, ξ) = O(|ξ|α−2 ) and v(t, ξ) = t1−α Ψ(tξ) + O(t1−α−δ )
for all |ξ| ≤ tθ−1 , with δ ≥ θ + γ. From (3.4) we get kξ∂ξ vk ≤ k∂x J uk ≤ Ctγ . We
have
b
∂ξ v(t, ξ) = ξ −1 (2t∂t − v − Iv),
−1
where Ib = FIF = −∂ξ ξ + 2t∂t .
Similarly to (3.5) we get
LIu = t−α (ux Iux + ux Iux ) + 2(1 − α)t−α |ux |2 ,
hence
b
Iv(t,
ξ)
Z t
Z
i
b
b
= Iv(0,
ξ) + √
τ −α dτ e−iτ ξη Iw(τ,
ξ + η)w(τ, η)dη
2π 0
Z t
Z
i
b
τ −α dτ e−iτ ξη w(τ, ξ + η)Iw(τ,
η)dη
+√
2π 0
Z
Z
2i(1 − α) t −α
+ √
τ dτ e−iτ ξη w(τ, ξ + η)w(τ, η)dη.
2π
0
14
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
Since t∂t v = O(t1−α htξi−1 ) = O(t−γ |ξ|α−1 ) for tθ−1 ≤ |ξ| and t∂t v = O(|ξ|α−1 ) for
|ξ| ≤ tθ−1 , applying Lemma 2.2 we get
vξ (t, ξ)
b + v) + O(t−γ |ξ|α−2 )
= −ξ −1 (Iv
Z t
Z
i
−α
b
= −√
τ dτ e−iτ ξη Iw(τ,
ξ + η)w(τ, η)dη
2π 0
Z t
Z
i
b
η)dη
−√
τ −α dτ e−iτ ξη w(τ, ξ + η)Iw(τ,
2π 0
Z t
Z
2iα
τ −α dτ e−iτ ξη w(τ, ξ + η)w(τ, η)dη + O(t−γ |ξ|α−2 )
+√
2π 0
= G(t)|ξ|α−1 ξ −1 + O(t−γ |ξ|α−2 )
for all tθ−1 ≤ |ξ| ≤ t−µ , and ∂ξ v(t, ξ) = O(|ξ|α−2 ) for all |ξ| ≤ tθ−1 , where
G(t)
Z
2i
πα
b
= √ Γ(1 − α) sin( )< Iw(t,
η)w(t, η)|η|α−1 dη
2
2π
Z
πα
2iα
+ √ Γ(1 − α) sin( ) |w(t, η)|2 |η|α−1 dη,
2
2π
b
since |w(t, η)|2 signη and <(Iw(t,
η)w(t, η))signη are odd functions. We have by
(3.10)
kw(t, η) − w(τ, η)k0,1,∞ ≤ k
Z
t
∂s w(s, η)dsk0,1,∞ ≤ C
Z
τ
t
s2γ−α−1 ds ≤ Cτ 2γ−α
τ
0,1
for all 1 ≤ τ ≤ t. Therefore there exists a limit W = limt→∞ w(t) in H∞
(R) such
that
kw(t, η) − W k0,1,∞ ≤ Cτ 2γ−α .
Similarly to (3.10) we get by (3.4)
b 0,1,∞ ≤ Cεt5γ−α−1
k∂t Iwk
(3.11)
0,1
b
for all t ≥ 1. Hence there exists a limit K = limt→∞ Iw(t)
in H∞
(R) such that
b
kIw(t,
η) − Kk0,1,∞ ≤ Cτ 5γ−α .
Thus
∂ξ v(t, ξ) = B1 |ξ|α−1 ξ −1 + O(t−γ |ξ|α−2 ),
where
B1
=
Z
2i
πα
√ Γ(1 − α) sin( )< K(η)W (η)|η|α−1 dη
2
2π
Z
πα
2iα
+ √ Γ(1 − α) sin( ) |W (η)|2 |η|α−1 dη.
2
2π
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ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
15
Also we have
v(t, ξ)
=
Z
t
dτ
Z
e−iτ ξη w(τ, ξ + η)w(τ, η)dη
τ −α dτ
Z
e−iτ ξη |W (η)|2 dη + O(t−γ min(|ξ|α−1 , t1−α ))
τ
−α
0
=
Z
t
0
= t1−α
Z
1
|z|−α dz
Z
e−izηtξ |W (η)|2 dη + O(t1−α−δ )
0
= t1−α Ψ(tξ) + O(t1−α−δ )
R
R1
for |ξ| ≤ tθ−1 , where δ = α − 2γ, Ψ(x) = 0 |z|−α dz e−izηx |W (η)|2 dη. Now
application of Lemma 2.3 yields asymptotics (1.4) for the solution
√ u(t, x). Using
Lemma 3.2 we get the result of Theorem 1.1 with B = B1 / 2π, P = √12π Ψ̌,
Q = √12π V . Theorem 1.1 is proved.
4. Proof of Theorem 1.2
By the method in [3] (see also the proof of a-priori estimates below in Lemma
4.2), we easily obtain the local existence of solutions in the analytic functional space
AT = φ ∈ C([−T, T ]; L2 (R)) : sup kφ(t)kAt < ∞ ,
t∈[−T,T ]
where the norm At is defined as
kukAt
1
= hti−γ k|∂x | 2 −β uk3,0 + tα hti1−γ k∂t FU(−t)ux (t)k0,1,∞
∞
X
1
hti−nγ
+
k|∂x | 2 −β I n uk1,0 .
n!
n=1
Denote
kukZ =
∞ −nγ
X
1
t
k|∂x | 2 −β I n uk1,0 .
n!
n=0
Theorem 4.1. Let α ∈ (0, 1). We assume that the initial data u0 ∈ A0 . Then
for some time T > 0 there exists a unique solution u ∈ AT of the Cauchy problem
(1.1). If we assume in addition that the norm of the initial data ku0 kA0 = ε2 is
sufficiently small, then there exists a unique solution u ∈ AT of (1.1) for some time
T > 1, such that the following estimate supt∈[0,T ] kukA < ε is valid.
In the next lemma we obtain the estimates of global solutions in the norm At .
Lemma 4.2. Let the initial data u0 ∈ A0 are such that the norm ku0 kA0 = ε2 is
sufficiently small. Then there exists a unique global solution of the Cauchy problem
(1.1) such that u ∈ A∞ . Moreover the following estimate is valid
kukAt < ε
(4.1)
for all t > 0.
Proof. As in Lemma 3.2 we argue by contradiction and find a maximal time T > 1
such that the estimate
kukAt ≤ ε
(4.2)
16
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
is valid for all t ∈ [0, T ]. Via Theorem 4.1 it is sufficient to consider t ≥ 1.
1
As above in Lemma 3.2 we can estimate the norms kux k1,0,∞ , k|∂x | 2 −β J ∂x uk1,0 ,
1
k|∂x | 2 −β J ∂x Iuk1,0 via the norm kukZ . Indeed we have the estimate
1
k|∂x | 2 −β J ∂x uk1,0
1
1
≤ k|∂x | 2 −β Iuk1,0 + 2tk|∂x | 2 −β Luk1,0
≤ CkukZ + Ckuk2Z ≤ Cε
and
1
k|∂x | 2 −β J ∂x Iuk1,0
1
1
k|∂x | 2 −β I 2 uk1,0 + 2tk|∂x | 2 −β LIuk1,0
≤
≤ CkukZ + Ckuk2Z ≤ Cε.
We first estimate k∂t FU(−t)ux (t)k0,1,∞ . We apply the Fourier transformation
it 2
to equation (1.1), then changing the dependent variable Fux = e− 2 ξ w, in view of
(2.7) we obtain
iξt−α
iwt (t, ξ) = √
2π
Z
e−itξη w(t, ξ + η)w(t, η)dη,
(4.3)
where w(t, ξ) = FU(−t)ux . For t ≥ 1, we integrate by parts with respect to η
kξ
Z
e−itξη w(t, ξ + η)w(t, η)dηk0,1,∞
Z
≤ Chti−1 k e−itξη ∂η (w(t, ξ + η)w(t, η))dηk0,1,∞ ≤ Chti−1 k∂η wk0,1 kwk0,1
≤ Chti−1 kJ ∂x uk1,0 kuk2,0 ≤ Cε2 hti2γ−1 ;
therefore,
tα+1−2γ k∂t FU(−t)ux (t)k0,1,∞ < ε
(4.4)
for all t ≥ 1. In the same manner
tα+1−5γ k∂t FU(−t)Iux (t)k0,1,∞ < ε.
We estimate the norm kukZ =
P∞
n=0
1
t−nγ
2 −β I n uk
1,0
n! k|∂x |
for all t ≥ 1. Note that
∞ −γn
X
t
k(I + a)n uk1,0
n!
n=0
≤
∞ −γn X
n
∞ n−j
X
X
X |a|n−j t−γn
t
Cnj |a|n−j kI j uk1,0 =
kI j uk1,0
n!
(n
−
j)!
j!
n=0 j=0
n=0
j=0
=
∞ −γj
X
t
j=0
j!
kI j uk1,0
∞
∞ −γj
X
−γ X t
|at|k−j
= e|a|t
kI j uk1,0 ≤ CkukZ
(k − j)!
j!
j=0
k=j
(4.5)
EJDE–2001/54
ASYMPTOTIC BEHAVIOUR FOR SCHRÖDINGER EQUATIONS
We have by Lemma 2.4 denoting Cnm =
kLukZ
= kt−α |ux |2 kZ =
= t−α
= t
=
−α
17
n!
m!(n−m!)
∞ −nγ
X
1
t
k|∂x | 2 −β I n t−α |ux |2 k1,0
n!
n=0
∞ −nγ
X
1
t
k|∂x | 2 −β (I − 2α)n |ux |2 k1,0
n!
n=0
∞ −nγ X
n
X
1
t
Cnj k|∂x | 2 −β (I − 2α)n−j ux (I − 2α)j ux k1,0
n!
n=0
j=0
∞ X
n
X
1
t−α−3nγ
k|∂x | 2 −β ((I − 1 − 2α)n−j u)x ((I − 1 − 2α)j u)x k1,0
(n
−
j)!j!
n=0 j=0
≤ C
∞ X
n
X
t−α−3nγ+β−1
(n − j)!j!
n=0 j=0
(k|∂x |
1
2 −β
1
k|∂x | 2 −β (I − 1 − 2α)n−j uk1,0
1
(I − 1 − 2α)j uk1,0 + k|∂x | 2 −β (I − 1 − 2α)j J ∂x uk1,0 )
≤ Ct−α+β−1 kukZ (kukZ + kIukZ + tkLukZ ),
hence we get
kLukZ
≤ Ct−α+β−1 kuk2Z + Ct−α+β−1 kukZ kIukZ
≤ Ct−γ−1 (kukZ + kIukZ ).
In particular we have
kJ ∂x ukZ ≤ CkukZ + CkIukZ .
Using the commutation relations I n ∂x = ∂x (I + 1)n , LI n = (I + 2)n L, I n t−α =
t−α (I − 2α)n applying operator I to equation (1.1) we get
LI n u = t−α (I + 2(1 − α))n |ux |2 .
Via (4.3) we obtain
∞
X
1
t−α−γn (n!)−1 k|∂x | 2 −β (I + 2 − 2α)n |ux |2 k1,0
n=0
≤ C
∞
X
t
×
m
X
j
Cm
k|∂x | 2 −β ((I + 1)j u)x ((I + 1)m−j u)x k1,0
−α−γn
−1
(n!)
n=0
n
X
Cnm (2 − 2α)n−m
m=0
1
j=0
≤ Ctβ−α−1 kuk2Z + Ctβ−α−1 kukZ kJ ∂x ukZ
≤ Cεt−γ−1 (kukZ + kIukZ ) .
Using Lemma 2.4 we obtain
1
k|∂x | 2 −β ((I + 1)j u)x ((I + 1)m−j u)x k1,0
1
1
≤ Ctβ−1 k|∂x | 2 −β (I + 1)j uk1,0 k|∂x | 2 −β (I + 1)m−j uk1,0
1
1
+Ctβ−1 k|∂x | 2 −β (I + 1)j uk1,0 k|∂x | 2 −β (I + 1)m−j J ∂x uk1,0
1
1
+Ctβ−1 k|∂x | 2 −β (I + 1)j J ∂x uk1,0 k|∂x | 2 −β (I + 1)m−j uk1,0 .
(4.6)
18
NAKAO HAYASHI & PAVEL I. NAUMKIN
EJDE–2001/54
Hence by the energy method, in view of (4.2) we find
d
kukZ ≤ −γt−1−γ kIukZ + Cεt−γ−1 kukZ + Cεt−γ−1 kIukZ ≤ Cεtβ−α−1 kukZ .
dt
(4.7)
Integration of (4.7) with respect to time t ≥ 1 yields kukZ < 2ε for all t ≥ 1. The
1
norm k|∂x | 2 −β uk3,0 is estimated in the same manner as in the proof of Lemma 3.2.
Therefore, Lemma 4.2 is proved.
Now we complete the proof of Theorem 1.2 by applying Lemmas 2.2 and 2.3 as
in the previous section.
Acknowledgments. The second author was partially supported by a grant from
CONACYT.
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Nakao Hayashi
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka,
Osaka 560-0043, Japan
E-mail address: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin
Instituto de Fı́sica y Matemáticas, Universidad Michoacana, AP 2-82, Morelia, CP
58040, Michoacán, Mexico
E-mail address: pavelni@zeus.ccu.umich.mx