CUBO A Mathematical Journal

Vol.11, No¯ 04, (15–28). September 2009

Small Data Global Existence and Scattering for the
Mass-Critical Nonlinear Schrödinger Equation with Power

Convolution in R3

George Venkov

Department of Differential Equations,
Faculty of Applied Mathematics and Informatics,

Technical University of Sofia,
8 "Kliment Ohridski" Str., 1756 Sofia, Bulgaria.

email: gvenkov@tu-sofia.bg

ABSTRACT

The main purpose of the present paper is to consider the well-posedness of the L2-

critical nonlinear Schrödinger equation of a Hartree type

i∂tψ + △ψ = (|x|−1 ∗ |ψ|
8
3 )ψ, (t,x) ∈ R+ × R3.

More precisely, we shall establish the local existence of solutions for initial data ψ0 in

L
2
(R

3
), as well as the existence of global solutions for small initial data. Moreover, we

shall prove the existence of scattering operator.

RESUMEN

El principal objetivo del artículo es considerar si la ecuación de Schrödinger no lineal

L
2- crítica del tipo Hartree

i∂tψ + △ψ = (|x|−1 ∗ |ψ|
8
3 )ψ, (t,x) ∈ R+ × R3.

está bien puesta o no. En efecto, estableceremos la existencia local de soluciones para

datos iniciales ψ0 en L
2
(R

3
), así como la existencia de soluciones globales para datos

iniciales pequeños. Más aún, probaremos la existencia del operador de scattering.

Key words and phrases: Nonlinear Schrödinger equation, power convolution, Hartree equation,
local and global existence.

Math. Subj. Class.: 35A05, 35Q55.



16 George Venkov CUBO
11, 4 (2009)

1 Introduction

In this paper we consider the Cauchy problem for the defocussing mass-critical nonlinear Schrödinger

equation of a Hartree type

i∂tψ + △ψ = (|x|−1 ∗ |ψ|
8
3 )ψ, (t,x) ∈ R+ × R3, (1.1)

ψ(0,x) = ψ0(x), (1.2)

where ∗ denotes the usual convolution operator in R3. Here ψ = ψ(t,x) is a complex valued
function and the initial value ψ0 : R

3 7→ C is given.
Equation (1.1) can be written in terms of the wave function ψ and the potential V as the

Schrödinger-Poisson system of the form

i∂tψ + △ψ = V ψ, (1.3)

△V = −4π|ψ| 83 , (1.4)

where the (−) sign in the Poisson equation (1.4) corresponds to the repulsive type interaction.
Equation (1.1) is known as the Schrödinger equation with nonlocal power nonlinearity of a Hartree

type.

The main purpose of the present work is to obtain local and global existence, well-posedness

and scattering of solutions to (1.1)–(1.2). When studying the problem of local and global exis-

tence of solution to the nonlinear Schrödinger equation, one is primarily interested in the scaling

symmetry of the equation under the transformation

ψλ(t,x) =
1

λa
ψ(

t

λ2
,
x

λ
), λ > 0, (1.5)

with some constant a depending on the equation. On the other hand, after the above scaling the

L
p-norm has dimension, namely

‖ψλ‖Lp(R3) =
1

λ
a− 3

p

‖ψ‖Lp(R3). (1.6)

A simple calculation shows that for a = 3
2

the scaling transformation (1.5) leaves equation

(1.1) unperturbed and, in addition, preserves the L2-norm.

The above scaling symmetry of (1.1) is closely related to the so-called pseudoconformal sym-

metry,

P[ψ](τ,y) = φ(τ,y) = 1
τ

3
2

e
i

y2

4τ ψ

(
1

τ
,
y

τ

)
. (1.7)

This is a symmetry in the sense that, if ψ(t,x) is a solution to (1.1) on (t,x) ∈ [t1, t2] × R3,
then φ(τ,y) is a solution to the same equation on (τ,y) ∈ [ 1

t2
,

1

t1
] × R3. In general, the scaling and

the pseudoconformal symmetries (1.5) and (1.7) relate (1.1) to a wide class of equations, referred to



CUBO
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Small Data Global Existence and Scattering for the ... 17

as the mass-critical (L2-critical or pseudoconformal) nonlinear Schrödinger equations. The name

comes from the fact that the transforms (1.5) and (1.7) leave both the equation and the mass (the

L
2-norm) invariant. Mass is one of the basic structures used in physics and is defined by

M(ψ(t)) =

∫

R3

|ψ(t,x)|2dx. (1.8)

For (1.1), we shall prove (see Corollary 2.4 bellow) that the mass is a conserved quantity, i.e.

M(ψ(t)) = ‖ψ(t)‖2L2(R3) = ‖ψ0‖2L2(R3) = M(ψ0). (1.9)

As in the papers [1,4,11,13,18–20], our results make use of mixed spaces of the type Lq([0,T ],Lr(R3))

for admissible q and r. Thus, we make the following definition.

Definition 1. We say that the pair (q,r) of exponents is Schrödinger–admissible if q and r satisfy

2

q
= 3

(
1

2
− 1
r

)
, 2 ≤ q ≤ ∞. (1.10)

In the frame of the mass-critical NLS, equation (1.1) is similar to the Schrödinger equation

with local (pure power) nonlinearity, which in an arbitrary spatial dimension n ≥ 1 has the form

i∂tψ + △ψ = |ψ|
4
n ψ, (t,x) ∈ R+ × Rn. (1.11)

The problems of global existence and well-posedness for solutions to (1.11) have been inten-

sively studied, see for example [1–4, 18–21]. The local theory for (1.11) is due to Cazenave and

Weissler, who in [4] constructed local-in-time solutions for arbitrary initial data in L2(Rn) and

also constructed global solutions for small initial data. Their results can be summarized as fol-

lows: Given ψ0 ∈ L2(Rn), there exists a unique local solution ψ to (1.11) with ψ(0,x) = ψ0(x).
The solution ψ has a conserved mass M(ψ(t)) = M(ψ0). Moreover, if M(ψ0) is sufficiently small

depending on n, then ψ is a global solution and
∫

R+

∫

R3

|ψ(t,x)|
2(n+2)

n dxdt ≤ M(ψ0). (1.12)

The condition that ψ ∈ L2(n+2)/nt,x is natural for (1.11). This balanced space appears in
the original Strichartz inequality [17] and it is necessary in order to ensure local existence and

uniqueness of solutions to (1.11).

Following the strategy developed for (1.11), we aim to establish the local well-posedness theory

for (1.1) and to construct global solutions for sufficiently small L2-initial data. More precisely we

shall use the following

Definition 2. A function ψ : [0,T∗) × R3 7→ C, 0 < T∗ ≤ ∞ is a L2(R3) solution to (1.1)
if ψ ∈ C0([0,T ],L2(R3)) ∩ L14/3([0,T ],L14/5(R3)) for 0 < T < T∗, and we have the Duhamel’s
integral representation

ψ(t) = U(t)ψ0 − i
∫ t

0

U(t − s)(|x|−1 ∗ |ψ(s)| 83 )ψ(s)ds, (1.13)



18 George Venkov CUBO
11, 4 (2009)

for any t ∈ [0,T ]. Here U(t) = eit∆ is the free Schrödinger evolution group defined via the Fourier
transform

f̂(ξ) =
1

(2π)3/2

∫

R3

e
−ix·ξ

f(x)dx, (1.14)

by

êit∆f(ξ) = e
−it|ξ|2

f(ξ). (1.15)

We say that ψ is a global solution to (1.1) if T∗ = ∞.

The first main result of the present paper is the following

Theorem 1.1. For every initial data ψ0 ∈ L2(R3) there exists a unique maximal solution ψ ∈
C

0
([0,T

∗
),L

2
(R

3
)) ∩ L143 ([0,T∗),L145 (R3)) of (1.1). Furthermore:

(i) ψ ∈ Lq([0,T ],Lr(R3)), for 0 < T < T∗ and every admissible pair (q,r);

(ii) the mass is conserved, i.e. M(ψ(t)) = M(ψ0) for t ∈ [0,T∗);

(iii) there exists a constant ε > 0 sufficiently small, such that if ‖ψ0‖L2(R3) < ε , then T∗ = ∞
and ψ ∈ Lq(R+,Lr(R3)) for every admissible pair (q,r);

(iv) if T∗ < ∞, then ‖ψ‖Lq ([0,T∗),Lr (R3)) = ∞ for every r > 14/5;

(v) ψ depends continuously on the initial data ψ0 ∈ L2(R3) in the space ψ ∈ C0([0,T∗),L2(R3)) ∩
L

14
3 ([0,T

∗
),L

14
5 (R

3
)).

There exists an extensive literature on the scattering theory for the Schrödinger equation with

convolution nonlinearity [8, 14, 15] and for the Hartree equation [5–7, 9, 10], of which the existence

of a wave operator is the question of crucial importance. Let v(t) = U(t)ψ+ be a solution to the

free Schrödinger equation

i∂tv + △v = 0, (1.16)

with initial data ψ+ ∈ X (called the asymptotic state), where X = Xψ0 is a suitable Banach
space, depending on the initial data. That question can be formulated as follows. Does there

exist a solution of (1.1)–(1.2), which behaves asymptotically as v when t → ∞ in a suitable sense,
depending on the choice of the space X? If that is the case, then the map Ω+ : X 7→ X is called
the wave operator for positive times. In other words, a global strong X-solution ψ to the nonlinear

equation (1.1) with an initial data ψ0 scatters in X to a solution v(t) = U(t)ψ+ if we have

lim
t→∞

‖ψ(t) − U(t)ψ+‖X = 0, (1.17)

or equivalently (by using the unitarity of U(t))

lim
t→∞

‖U(−t)ψ(t) − ψ+‖X = 0. (1.18)



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Small Data Global Existence and Scattering for the ... 19

Suppose that for every asymptotic state ψ+ ∈ X, there exists a unique initial data ψ0 ∈ X,
whose corresponding X-wellposed solution is global and scatters to v(t) as t → ∞. Then, we can
define the wave operator Ω+ : X 7→ X in the sense of the space X by

Ω+ψ+ = ψ0. (1.19)

The problem of the existence of ψ for given ψ+ is referred to as the problem of existence of

the wave operator. When the wave operator Ω+ is injective, we say that the Cauchy problem

(1.1)–(1.2) is asymptotically complete in X. The same problem can be constructed for negative

times, but for definiteness, hereinafter, we shall restrict our attention only to positive time.

A standard way to construct the wave operator Ω+ consists in solving the Cauchy problem

for (1.1) with initial data ψ+ at t = ∞ in the form of the integral equation

ψ(t) = U(t)ψ0 + i

∫ ∞

t

U(t − s)(|x|−1 ∗ |ψ(s)| 83 )ψ(s)ds. (1.20)

One usually solves (1.20) by a contraction method in a neighborhood of infinity in time (or

in the time interval [T,∞) for T sufficiently large) and then continues that solution to all times.
Thus, the problem is an immediate consequence of the global well-posedness and uses the results

of Theorem 1.1.

With our second main result, we shall construct scattering theory in L2(R3) with small initial

data. In fact, we shall prove the following

Theorem 1.2. Let ε > 0 be sufficiently small and consider the ball Bε = {ψ ∈ L2(R3); ‖ψ‖L2 < ε}.
Let ψ ∈ C0([0,T∗),L2(R3)) ∩ L143 ([0,T∗),L145 (R3)) be the unique maximal solution of (1.1),

given by part(iii) of Theorem 1.1. Then we have:

(i) for any ψ± ∈ Bε, there exists a unique ψ0 ∈ Bε, such that

lim
t→±∞

‖U(−t)ψ(t) − ψ±‖L2 = 0; (1.21)

(ii) for any ψ0 ∈ Bε, there exists unique ψ± ∈ Bε, such that (1.21) is satisfied;

(iii) the wave operators Ω± : ψ± 7→ φ0 and the scattering operator S = Ω−1+ ◦ Ω− are homeomor-
phisms from Bε onto itself and isometric in the L2(R3) norm.

The paper is organized as follows. In Section 2 we state some useful results and prove Theorem

1.1. In Section 3 we prove Theorem 1.2 and give some generalization notes about the scattering

problems for (1.1).

We shall conclude this section by giving some of the notations, used in the paper. As usual,

L
r
(R

n
) = {ϕ ∈ S′; ‖ϕ‖Lr < ∞}, where ‖ϕ‖Lr = (

∫
|ϕ(x)|rdx) 1r if 1 ≤ r < ∞ and ‖ϕ‖L∞ =

ess.sup{|ϕ(x)|; x ∈ Rn} if r = ∞. We use r′ for denoting the exponent dual to r and defined by
1/r + 1/r

′
= 1. Given Lebesgue exponents q, r and a function f(t,x) in Lq(R,Lr(R3)), we write

‖f‖Lq (R,Lr (R3)) = (
∫
‖f(t)‖q

Lr (R3)
dt)

1
q .



20 George Venkov CUBO
11, 4 (2009)

2 The local existence result

We shall start this section by collecting some preliminaries and useful results.

As we shall see, the L
14
3
t L

14
5
x norm in space-time plays a fundamental role. This is better

understood if we recall some of the estimates available for the corresponding linear problem. We

begin by recalling the following properties of the free Schrödinger evolution group U(t) = eit△ (see

for instance [12, 13, 22]).

Lemma 2.1. Let (q,r) be an admissible pair. Then, for every ϕ ∈ L2(R3) the following estimate
holds

‖U(t)ϕ‖Lq (R,Lr (R3)) ≤ C0‖ϕ‖L2(R3). (2.1)

Moreover, for every admissible pair (θ,ρ) and f ∈ Lθ′ ([0,T ],Lρ′ (R3)) we have
∥∥∥∥
∫ ·

0

U(· − s)f(s)ds
∥∥∥∥
Lq ([0,T],Lr (R3))

≤ C‖f‖Lθ′ ([0,T],Lρ′ (R3)), (2.2)

for 0 < T ≤ ∞. Here the constants C0,C > 0 and depend only on the spatial exponents r and ρ.

The classical Strichartz estimates for the Schrödinger equation [17] are one of the main tools

in the study of local and global existence, time decay and scattering both for the linear and the

nonlinear equation, due to the fact that they fit the assumptions required by the contraction

argument (see for instance [5, 13, 17, 18, 22]).

The Strichartz type estimates for the inhomogeneous Schrödinger equation

i∂tv(t,x) + △v(t,x) = F(t,x), v(0,x) = f(x), (2.3)

in R+ × R3 are given, up to the end-point, namely the pair (q,r) = (2, 6) by the following result
due to Keel and Tao [13].

Lemma 2.2. If (q,r) and (q̃, r̃) satisfy (1.10), then the solution to the Cauchy problem (2.3)
satisfies the estimate

‖v‖Lq ([0,T∗),Lr (R3)) + ‖v‖L∞([0,T∗),L2(R3))
≤ C(‖F‖

Lq̃
′
([0,T∗),Lr̃

′(R3))
+ ‖f‖L2(R3)). (2.4)

Very important tool in our functional analysis background is the following lemma.

Lemma 2.3. (Hardy-Littlewood-Sobolev Inequality) For 0 < α < 3 consider the Riesz potential

Iα(g)(x) =

∫

R3

g(y)

|x − y|3−αdy. (2.5)

Then for any 1 < θ < r < ∞ and g ∈ Lr(R3), we have

‖Iα(g)‖Lθ ≤ C‖g‖Lr, (2.6)

where 1
θ

=
1

r
− α

3
.



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Small Data Global Existence and Scattering for the ... 21

For the proof of Lemma 2.3, see equation (31), Chapter VIII.4.2 in Stein [16].

The first important result in the present study of (1.1) is the following conservation law.

Lemma 2.4. Let ψ ∈ C0([0,T∗),L2(R3)) be a solution to (1.1) with initial data ψ(0) = ψ0 ∈
L

2
(R

3
). Then

M(ψ(t)) = M(ψ0), (2.7)

for any 0 ≤ t < T∗.

Proof. We multiply (1.1) by ψ to get

iψ∂tψ + ψ△ψ = (|x|−1 ∗ |ψ|
8
3 )|ψ|2. (2.8)

We conjugate (2.8) and subtract the result from the above expression to obtain

i∂t|ψ|2 = (ψ△ψ − ψ△ψ) = ∇ · (ψ∇ψ − ψ∇ψ). (2.9)

By integration over R3 we obtain the desired result

d

dt
M(ψ(t)) =

d

dt
‖ψ(t)‖2L2(R3) = 0 (2.10)

and the proof of the Lemma is completed.

We should note here that the specific power of 8/3 in the Hartree-type nonlinearities of (1.1)

does not allow the establishment of the energy conservation law (due mainly to the presence of

the convolution). The latter would cause difficulties in proving, for example, global existence for

solution of (1.1) at the H1-level.

The arguments of Theorem 1.1 rely primarily on the Strichartz estimate (2.4) in Lemma 2.2

and on the Hölder inequality.

Let us denote by

N(ψ) = (|x|−1 ∗ |ψ| 83 )ψ (2.11)

the nonlinear term in the Hartree equation (1.1). Consider the form

V (ψ1,ψ2)(t,x) = |x|−1 ∗ |ψ1(t,x)|α|ψ2(t,x)|
8
3
−α
, 0 ≤ α ≤ 8

3
. (2.12)

Then

N(ψ) = V (ψ,ψ)ψ

and we shall need the following estimates.

Lemma 2.5. For any 3 < p < ∞ and r1,r2 satisfying 1p =
α
r1

+
8/3−α
r2

− 2
3
, 0 ≤ α ≤ 8

3
we have

‖V (ψ1,ψ2)(t, ·)‖Lp(R3) ≤ C‖ψ1(t, ·)‖αLr1 (R3)‖ψ2(t, ·)‖
8
3
−α

Lr2 (R3)
. (2.13)



22 George Venkov CUBO
11, 4 (2009)

Proof. It is sufficient to apply the Hardy-Littlewood-Sobolev inequality Lemma 2.3 and Hölder
inequality to get

‖| · |−1 ∗ |ψ1(t,x)|α|ψ2(t,x)|
8
3
−α‖Lp ≤ C‖ψ1(t, ·)‖αLr1 ‖ψ2(t, ·)‖

8
3
−α

Lr2 ,

provided

3 < p < ∞, 1
p

=
α

r1
+

8/3 − α
r2

− 2
3
.

Lemma 2.6. For any r̃ satisfying 2 ≤ r̃ ≤ 6 and for any r1,r2,r3, 2 ≤ rj ≤ 6,j = 1, 2, 3, satisfying

1

r̃′
=

α

r1
+

8/3 − α
r2

+
1

r3
− 2

3
, (2.14)

we have

‖V (ψ1,ψ2)(t, ·)ψ3(t, ·)‖Lr̃′ (R3)
≤ C‖ψ1(t, ·)‖αLr1 (R3)‖ψ2(t, ·)‖

8
3
−α

Lr2 (R3)
‖ψ3(t, ·)‖Lr3 (R3). (2.15)

Proof. The Hölder inequality implies

‖V (ψ1,ψ2)(t, ·)ψ3(t, ·)‖Lr̃′ ≤ C‖V (ψ1,ψ2)(t, ·)‖Lp ‖ψ3(t, ·)‖Lr3 ,

where
1

r̃′
=

1

p
+

1

r3
.

Applying further the estimate of Lemma 2.5, we complete the proof of the Lemma.

The corresponding space - time estimate follows easily from the above estimate.

Lemma 2.7. Let 0 < T ≤ ∞ and consider the Schrödinger-admissible pair (q,r) = (14/3, 14/5).
Then, we have the estimate

‖N(ψ)‖
L

14
11 ([0,T],L

14
9 (R3)

≤ C‖ψ‖
11
3

L
14
3 ([0,T],L

14
5 (R3)

. (2.16)

Proof. Applying Hölder inequality in time to (2.16) in Lemma 2.6 we obtain

‖V (ψ1,ψ2)(t, ·)ψ3(t, ·)‖Lq̃′ ([0,T],Lr̃′ )
≤ C‖ψ1(t, ·)‖αLq1 ([0,T],Lr1 )‖ψ2(t, ·)‖

8
3
−α

Lq2 ([0,T],Lr2 )
‖ψ3(t, ·)‖Lq3 ([0,T],Lr3 ),

where 0 ≤ α ≤ 8/3 and
1

r̃′
=

α

r1
+

8/3 − α
r2

+
1

r3
− 2

3
, (2.17)

1

q̃′
=
α

q1
+

8/3 − α
q2

+
1

q3
. (2.18)



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Small Data Global Existence and Scattering for the ... 23

Now we shall choose the couples (qj,rj) so that

1

qj
=

3

2

(
1

2
− 1
rj

)
, j = 1, 2, 3 (2.19)

and the relations (2.17), (2.18) are satisfied. Indeed, the simplest choice is

r = r̃ = r1 = r2 = r3, q = q̃ = q1 = q2 = q3.

Then (2.17) reads as
1

r′
=

11

3r
− 2

3
, (2.20)

while (2.18) becomes
1

q′
=

11

3q
(2.21)

which together give the couple (q,r) = (14/3, 14/5) and the proof is completed.

Lemma 2.8. Let 0 < T ≤ ∞ and let (q,r) be a Schrödinger-admissible pair. Then there exists a
constant C > 0, independent of T such that

∥∥∥∥
∫ ·

0

U(· − s)[N(ψ)(s) − N(χ)(s)]ds
∥∥∥∥
Lq ([0,T],Lr )

(2.22)

≤ C
(
‖ψ‖

8
3

L
14
3 ([0,T],L

14
5 )

+ ‖χ‖
8
3

L
14
3 ([0,T],L

14
5 )

)
‖ψ − χ‖

L
14
3 ([0,T],L

14
5 )
,

for every ψ,χ ∈ L143 ([0,T ],L145 (R3)).

Proof. To prove the Lemma, we shall use the estimates in Lemma 2.1. Then the estimate (2.22)
follows directly from (2.15), (2.16), (2.2) and Hölder inequality.

Proof of Theorem 1.1 . The proof of (ii) follows from Lemma 2.4.

We shall prove the existence of solution to (1.1) by a fix point argument. Let ψ0 ∈ L2(R3)
with ‖ψ0‖L2(R3) < ε, where ε > 0 is sufficiently small. Let R > 0 and consider the ball

B2R(T) = {ψ ∈ C0([0,T ],L2)) ∩ L
14
3 ([0,T ],L

14
5 ); ‖ψ‖

L
14
3 ([0,T],L

14
5 )

≤ 2R},

endowed with the metric

d(ψ,χ) = ‖ψ − χ‖
L

14
3 ([0,T],L

14
5 )
.

Since the space L
14
3 ([0,T ],L

14
5 ) is reflexive, the ball B2R is weakly compact, implying B2R is a

complete metric space.

Consider the map Φ[ψ](t), defined by the right-hand side of the Duhamel’s integral represen-

tation (1.13). Then, for ψ ∈ B2R, using (2.1), (2.2) and (2.16), we can write

‖ψ‖
L

14
3 ([0,T],L

14
5 )

≤ ‖U(·)ψ0‖
L

14
3 ([0,T],L

14
5 )

+ C1‖ψ‖
11
3

L
14
3 ([0,T],L

14
5 )

≤ C0ε + C1‖ψ‖
11
3

L
14
3 ([0,T],L

14
5 )

. (2.23)



24 George Venkov CUBO
11, 4 (2009)

Now, write R = C0ε and choose ε > 0 so small that there exists a positive number y, satisfying

C1y
11
3 − y + R > 0, 0 < y ≤ 2R. For that purpose, it is sufficient to take

C1(2R)
8
3 <

1

2
,

or equivalently

ε <
1

2C0(2C1)
3
8

,

implying that Φ[ψ] ∈ C0([0,T ],L2)) ∩ L143 ([0,T ],L145 ).
On the other hand, from (2.22) it follows that

‖N(ψ) − N(χ)‖
L

14
11 ([0,T],L

14
9

≤ C2
(
‖ψ‖

8
3

L
14
3 ([0,T],L

14
5 )

+ ‖χ‖
8
3

L
14
3 ([0,T],L

14
5 )

)
‖ψ − χ‖

L
14
3 ([0,T],L

14
5 )

≤ C22(2R)
8
3 ‖ψ − χ‖

L
14
3 ([0,T],L

14
5 )
, (2.24)

for every ψ,χ ∈ B2R. If we choose R such that

C22(2R)
8
3 ≤ 1

2
,

or equivalently

ε ≤ 1
2C0(4C2)

3
8

,

we finally obtain that the map Φ[ψ](t) is a strict contraction on the ball B2R. Thus Φ[ψ] has a

fixed point ψ, which is the unique solution of (1.1) in B2R. So far, we have proved the statement

of Theorem 1.1, as well as the part (i).

Notice, that the Strichartz estimate (2.4) implies a similar inequality, namely

‖ψ‖
L

14
3 ([0,T],L

14
5 )

≤ C0‖ψ0‖L2 + C1‖ψ‖
11
3

L
14
3 ([0,T],L

14
5 )

(2.25)

where we can take both the constants in (2.23) and (2.25) to be the same.

The second comment on the above proof is that from (2.23) and (2.25), the size of the ball

B2R depends directly on the size of the norm ‖U(t)ψ0‖
L

14
3 ([0,T],L

14
5 (R3))

and it can be done small

by taking either ‖ψ0‖L2(R3) small or the interval [0,T ] small.
Let us denote by T∗ the supremum of all T > 0 for which there exists a solution of (1.1)

in C0([0,T ],L2(R3)) ∩ L143 ([0,T ],L145 (R3)). To prove (iii), observe that if ψ0 is sufficiently small,
then (2.23) holds regardless of the value of T . Thus we may accomplish the fixed point procedure

in the ball B2R(∞), providing T∗ = ∞.
Further, we claim that if T∗ < ∞, then ‖ψ‖Lq ([0,T∗),Lr (R3)) = ∞ for every r > 14/5. Indeed,

on the contrary, let us assume that T∗ < ∞ and ‖ψ‖
L

14
3 ([0,T∗),L

14
5 )

< ∞. For any t ∈ [0,T∗) let



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Small Data Global Existence and Scattering for the ... 25

τ ∈ [0,T∗ − t). Using Duhamel’s formula (1.13), we can write

ψ(t + τ) = U(t + τ)ψ0 − i
∫ t+τ

0

U(t + τ − s)N(ψ)(s)ds

= U(τ)U(t)ψ0 − i
∫ t+τ

0

U(t + τ − s)N(ψ)(s)ds

= U(τ)ψ(t) + iU(τ)

∫ t

0

U(t − s)N(ψ)(s)ds

−i
∫ t+τ

0

U(t + τ − s)N(ψ)(s)ds

= U(τ)ψ(t) − i
∫ t+τ

t

U(t + τ − s)N(ψ)(s)ds. (2.26)

From (2.26) and the estimate (2.22) in Lemma 2.8 we obtain

‖U(·)ψ(t)‖
L

14
3 ([0,T∗−t),L

14
5 )

≤ C
(
‖ψ(t)‖

L
14
3 ([t,T∗),L

14
5 )

+ ‖ψ‖
11
3

L
14
3 ([t,T∗),L

14
5 )

)
. (2.27)

Observing now that ‖U(·)ψ‖
L

14
3 ([0,T],L

14
5 )

→ 0 as T → 0 and taking t close enough to T∗, it
follows that ‖U(·)ψ(t)‖

L
14
3 ([0,T∗−t),L

14
5 )

can be made small enough and the assumptions in (iii) are

fulfilled. Therefore, ψ can be extended after T∗, which contradicts the maximality. Let (q,r) be a

Schrödinger-admissible pair with r ≥ 14
5

. Then, from Hölder inequality for T < T∗, we can write

‖ψ‖
L

14
3 ([0,T],L

14
5 )

≤ ‖ψ‖1−α
L∞([0,T],L2)

‖ψ‖αLq ([0,T],Lr ), α ∈ (0, 1). (2.28)

Letting T → T∗, we obtain that ‖ψ‖Lq ([0,T],Lr ) = ∞, which proves the statement (iv).
To prove (v), consider a sequence ψk

0
∈ L2(R3), such that ψk

0
→ ψ0 ∈ L2(R3) as k → ∞.

Thus, for k large enough, ‖ψk
0
‖L2(R3) < ε. We can use the Duhamel’s formula (1.13) to construct a

sequence of solutions ψk ∈ L143 ([0,T ],L145 (R3)) to (1.1) with initial datum ψk
0
. Applying the proof

of (iii), we obtain that ψk → ψ in C0([0,T ],L2(R3)) ∩ L14/3([0,T ],L14/5(R3)) as k → ∞, and in
fact in every Lq([0,T ],Lr(R3))) for (q,r) be an admissible pair. Thus, the proof of the Theorem

is completed.

3 Small data scattering theory in L2(R3)

In this section we shall prove Theorem 1.2. The arguments for proving the existence of the wave

operator are standard and follows the exposition in [12,15]. We shall prove only the (+) case since

the (−) case can be proved similarly. Let ψ0 ∈ L2(R3) with ‖ψ0‖L2 < ε and ψ ∈ B2R, where the
ball B2R is defined in the previous section. Then, for t > t0, using Duhamel’s integral formula

(1.13) we have

U(−t)ψ(t) = U(−t0)ψ(t0) − i
∫ t

t0

U(−s)N(ψ)(s)ds. (3.1)



26 George Venkov CUBO
11, 4 (2009)

Therefore, using the estimate (2.16), we have

‖U(−t)ψ(t) − U(−t0)ψ(t0)‖L2 =
∥∥∥∥
∫ t

t0

U(−s)N(ψ)(s)ds
∥∥∥∥
L2

≤ C ‖ψ‖
11
3

L
14
3 ([t0,t],L

14
5 )

→ 0, (3.2)

as t0 → ∞. Since U(−t0)ψ(t0) ∈ L2, then the proof of part (ii) is completed.
To prove (i), assume that ψ+ ∈ L2(R3), ψ ∈ B2R and consider the map

Φ+[ψ](t) = U(t)ψ+ + i

∫ ∞

t

U(t − s)N(ψ)(s)ds, t > T, (3.3)

for some T = T(ψ+) large enough. Then, using the same arguments as in the proof of part

(iii) of Theorem 1.1, we find that Φ+ is a contraction on B2R and has a unique fixed point if

‖ψ+‖L2 < ε. Using the global well-posedness result established in Theorem 1.1 for small data, one
can then extend this solution uniquely for any t ∈ [0,∞), and in particular ψ will take some value
ψ0 = ψ(0) ∈ L2 at time t = 0. This gives existence of the wave operator Ω+, defined by

ψ0 = Ω+φ+ = ψ+ + i

∫ ∞

0

U(−s)N(ψ)(s)ds. (3.4)

This proves part (i) of the Theorem 1.2.

To prove (iii), we shall use the following observations. Since, the wave operators Ω± are

isometric in the space Bε, it is clear that the scattering operator S : φ− 7→ φ+ is well defined as a
map from Bε onto itself and isometric in the L

2 norm, i.e. ‖Sψ‖L2 = ‖ψ‖L2 . This completes the
proof of the Theorem 1.2.

It is clear that the finiteness of the L
14/3
t L

14/5
x -norm is sufficient to yield global well-posedness

and scattering results for small data in L2. As the L
2(n+2)/n
t,x -norm for the Schrödinger equation

with pure power nonlinearity (1.11), this is the best possible choice of admissible indices for (1.1),

which ensures the space-time integrability via the Strichartz estimates and the contraction property

of the nonlinear map. Moreover, the above results does not depend on the repulsive character of

the problem and can be proved in a similar way for the focussing analogue of (1.1). Finally, we note

that the global existence, well-posedness and scattering problems for (1.1) with an arbitrary initial

data require spaces, strictly smaller than the L2 one. For example, in another work, following the

approach of Hayashi, Naumkin and Ozawa [9], we shall consider the above problems for data in

the pseudoconformal space Σs = {ψ0 ∈ Hs; |x|sψ0 ∈ L2} for some 0 < s < 1, depending on the
behavior of the leading term of the solution ψ for large time.

Received: March 2008. Revised: July 2008.



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Small Data Global Existence and Scattering for the ... 27

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