We prove the existence of the modified wave operators to those equations for small final data

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

MODIFIED WAVE OPERATORS FOR NONLINEAR

SCHR ¨ODINGER EQUATIONS IN ONE AND TWO DIMENSIONS

NAKAO HAYASHI, PAVEL I. NAUMKIN, AKIHIRO SHIMOMURA, & SATOSHI TONEGAWA

Dedicated to Professor ShigeToshi Kuroda on his 70th birthday and to Professor Masaru Yamaguchi on his 60th birthday

Abstract. We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schr¨odinger equations with cubic and qua- dratic nonlinearities in one or two space dimensions. The nonlinearities are summation of gauge invariant term and non-gauge invariant terms. The scat- tering problem of these equations belongs to the long range case. We prove the existence of the modified wave operators to those equations for small final data. Our result is an improvement of the previous work [13].

1. Introduction

In this paper, we study the existence global solutions and scattering theory for the nonlinear Schr¨odinger equations

Lu=N_n(u) +G_n(u), (t, x)∈R×Rⁿ, (1.1) in one or two space dimensionsn= 1 and 2, whereL=i∂t+¹₂∆ and

N1(u) =λ₁u³+λ₂u²u+λ₃u³, N2(u) =λ₁u²+λ₂u²,

Gn(u) =λ₀|u|ⁿ²u

withλ0∈Randλj ∈C,j= 1,2,3. We construct a modified wave operator inL² to equation (1.1) for small final dataφ∈H^0,2∩H˙^−δ with ⁿ₂ < δ <2, where the weighted Sobolev space is defined by

H^m,s={u∈ S⁰;kuk_Hm,s =k hi∇i^mhxi^suk_L2<∞}, wherehxi=p

1 +|x|² and the homogeneous Sobolev space is H˙^m=

u∈ S⁰;kukH˙^m=k(−∆)^m2uk_L2 <∞ .

We intend to weaken the assumptionφ∈H˙⁻⁴ from the previous work [13].

2000Mathematics Subject Classification. 35Q55, 35B40, 35B38.

Key words and phrases. Modified wave operators, nonlinear Schr¨odinger equations.

c

2004 Texas State University - San Marcos.

Submitted March 10, 2004. Published April 21, 2004.

1

Many works have been devoted to the global existence and asymptotic behavior of solutions for the nonlinear Schr¨odinger equations. We remind the definition of the wave operators in the scattering theory for the linear Schr¨odinger equation.

Assume that for a solutionu_f(t, x) of the free Schr¨odinger equationLuf = 0 with given initial data u_f(0, x) = φ(x), there exists a unique global in time solution u(t, x) of the perturbed Schr¨odinger equation such that u(t, x) behaves like free solutionu_f(t, x) as t → ∞ (this case is called by the short range case, otherwise it is called by the long range case). Then we define a wave operator W₊ by the mapping from φto u|t=0. In the long range case, ordinary wave operators do not exist and we have to construct modified wave operators including a suitable phase correction in their definition. Analogously we can define the wave operators and introduce the modified wave operators for the nonlinear Schr¨odinger equation.

We first recall several known results concerning the scattering problem for the nonlinear Schr¨odinger equation

Lu=λ|u|^p−1u, (t, x)∈R×Rⁿ (1.2) withλ∈Randp >1. We consider the existence of wave operatorsW±for equation (1.2). The wave operatorW+ is defined for equation (1.2) as follows. Let Σ beL² or a dense subset of it. Letφ∈Σ, and define the free solution

u_f(t) =U(t)φ, where

U(t)≡e^it2∆.

Note thatuf is the solution to the Cauchy problem of the free Schr¨odinger equation Lu= 0, (t, x)∈R×Rⁿ,

u(0, x) =φ(x), x∈Rⁿ.

If there exists a unique global solutionuof equation (1.2) such that ku(t)−uf(t)kL² →0,

ast→+∞, then a mapping

W+:φ7→u(0)

is well-defined on Σ. We call the mappingW+ by the wave operator. The function φ is called by a final state, final data, a scattered state or scattered data. It is known that, when p > 1 + _n² and n ≤3, there exist the wave operators W_± on a suitable weighted Sobolev space (see [3]). In the case of n ≥ 4, the existence of wave operators is proved if p > ¹₄(√

n²+ 4n+ 36−n+ 2) in [3] and if p =

1 4(√

n²+ 4n+ 36−n+ 2) in [11]. (Note that 1 +_n² <¹₄(√

n²+ 4n+ 36−n+ 2) if n≥4, so for the casen≥4 and 1 +_n² < p < ¹₄(√

n²+ 4n+ 36−n+ 2) the problem is open). On the other hand, when 1≤p≤1 +_n², non-trivial solutions of equation (1.2) does not have a free profile inL², that is, we cannot define the wave operators onL² (see, e.g., [1]). Intuitive meaning of these facts is as follows. Recalling the well-known time decay estimates kuf(t)k_L2 = kφk_L2 = O(1), and kuf(t)kL^∞ = O(t⁻ⁿ²), we see that k|u_f(t)|^pk_L2 =O(t^−n2(p−1)). Roughly speaking, according to the linear scattering theory (the Cook-Kuroda method), wave operators exist if and only ifk|uf(t)|^pk_L2 is integrable with respect to tover the interval [1,∞), that is, p >1 + _n².

There are several results concerning the long range scattering for equation (1.2) in the critical case p= 1 +_n². In the long range case, as we already mentioned, the usual wave operators do not exist, so we introduce the modified wave operators Wf+ as follows. We construct a suitable modified free profileA+(t), and consider a unique solutionu(t) of equation (1.2) which approachesA+(t) in L² ast→ ∞:

ku(t)−A+(t)kL² →0, t→ ∞.

Then the mapping

Wf+:A+(0)7→u(0)

is called the modified wave operator. Ozawa [12] and Ginibre and Ozawa [2] proved the existence of modified wave operators for small final data in one space dimen- sion and in two and three space dimensions, respectively, by the phase correction method. More precisely, they put a modified free profile of the form A₊(t) = U(t)e^{−iS(t,−i∇)}φ, whereφis a final state, and chose the phase functionSsuch that kLA₊(t)− |A(t)|ⁿ²A(t)k_L2 decays faster than k|U(t)φ|²ⁿU(t)φk_L2 = O(t⁻¹). Re- cently, Ginibre and Velo [4] have partially extended above results removing the size restrictions of the final data in the case of the nonlinearity a(t)|u|²u. where a(t) has a suitable growth rate with respect tot. The large time asymptotic behavior of solutions to the initial value problem for equation (1.2) with 1≤n≤3 was studied and the asymptotic completeness of the wave operator was partially shown in [6].

The phase correction method is applicable only for the gauge invariant nonlineari- ties, likeλ|u|^p−1u, whereλ∈R, because we can regard|u|^p−1as a time dependent long range potential. We cannot apply the phase correction method to non-gauge invariant nonlinearities of the formu^p or|u|^p−1u+u^p, because we should consider the non-gauge invariant nonlinearity as a time dependent external force.

There are some results on the scattering theory for equation (1.1) in one or two space dimensions. In [10] it was shown the existence of the wave operator for equation (1.1) with Gn(u) = 0 by using the method by H¨ormander [8], where he studied the life span of solutions of nonlinear Klein-Gordon equations and in [13] it was constructed the modified wave operator for equation (1.1) by combining the methods in [8] and [12]. More precisely, the following two propositions were obtained in [13]:

Proposition 1.1. Letn= 1,φ∈H^0,3∩H˙⁻⁴ andkφkH^0,3+kφkH˙⁻⁴ be sufficiently small. Then there exists a unique global solutionuof (1.1) such thatu∈C(R⁺;L²),

sup

t≥1

t^bku(t)−up(t)k_L2+ sup

t≥1

t^bZ ∞ t

ku(τ)−up(τ)k⁴_L∞dτ1/4

<∞, where ¹₂ < b <1, and

up(t) = 1 (it)ⁿ²e^ix

2 2t φ(b x

t) exp −iλ0|φ(b x

t)|²ⁿlogt .

Proposition 1.2. Let n= 2,φ∈H^0,4∩H˙⁻⁴,xφ∈H˙⁻² andkφkH^0,4+kφkH˙⁻⁴+ kxφkH˙⁻² be sufficiently small. Then there exists a unique global solutionuof equa- tion (1.1) such thatu∈C(R⁺;L²),

sup

t≥1

t^bku(t)−u_p(t)kL²+ sup

t≥1

t^bZ ∞ t

ku(τ)−u_p(τ)k⁴_L4dτ^1/4

<∞, where ¹₂ < b <1.

Throughout this paper, we denote the norm of a Banach spaceZbyk · kZ. Our purpose in this paper is to improve the condition on a final data φ ∈ H˙⁻⁴. In order to explain the reason why the previous proof by [10] and [8] requires such a condition, we give briefly the idea of paper [13] on the example of the Cauchy problem

Lu=u², (t, x)∈R×R². (1.3) If a solution u of (1.3) behaves like a free solutionU(t)φ as t −→ ∞ for a given φ, then. u0(t, x) = ¹

(it)ⁿ²e^ix2t2φ(b ^x_t) can be considered as an approximate solution of (1.3) since

U(t)φ= 1 ite^ix

2 2t φb x

t

+O t^−1−α

|x|^2αφ _L1

.

By a direct calculation we find thatL(u−u0) =u²−_2it¹3e^ix2t2| · |[²φ(η). withη=^x_t. The last term of the right-hand side of the above equation is a remainder term which we denote byR. Hence the problem becomes

L(u−u0) =u²−u²₀+u²₀+R. (1.4) We find a solution in the neighborhood of u₀. howeveru²₀ can not be considered as a remainder term since ku²₀k_L2 = t⁻¹kφb²k_L2. In order to cancel u²₀ we try to find ur such that Lur−u²₀ is a remainder term. We put ur = t^−bP(^x_t)e^iax2t2 to getLur=t^{−b a(1−a)}₂ ^x_t2²P ^x_t

e^iax

2

2t +R1 which implies that we should takeP(η) =

2 a(a−1)

1

η²φ(η)b ² and a= b = 2 to cancel u²₀ in the right-hand side of (1.4) and we note thatR1 contains a term liket⁻⁴e^ix

2 t 1

η⁴φ(η)b ². Thus we get L(u−u0−ur) =u²−u²₀+R+R1.

This is the reason why we require a vanishing condition ofφ(η) at the origin.b Our main result in the present paper is the following.

Theorem 1.3. Let φ ∈ H^0,2∩H˙^−δ and kφk_H0,2 +kφkH˙^−δ be sufficiently small, where ⁿ₂ < δ < 2. Then there exists a unique global solution u of (1.1) such that u∈C(R⁺;L²),

sup

t≥1

t^δ2ku(t)−u_p(t)kL²+ sup

t≥1

t^δ2Z ∞ t

ku(τ)−u_p(τ)k⁴_Xndτ^1/4

<∞ whereX1=L^∞, X2=L⁴,

up(t) = 1 (it)ⁿ²e^ix

2 2t φb x

t

exp −iλ0

bφ x

t

2 nlogt

. Furthermore the modified wave operator

Wf+:φ7→u(0) is well-defined.

Similar result holds for the negative time.

Remark 1.4. If we consider the asymptotic behavior of solutions to the Cauchy problem for equation (1.1) with initial data u(0, x) = φ₀(x), x ∈ Rⁿ, then we see from Theorem 1.3 that for any initial data φ0 belonging to the range of the modified wave operatorWf+, there exists a unique global solutionu∈C(R⁺;L²) of the Cauchy problem for equation (1.1) which has a modified free profileu_p. More

precisely,usatisfies the asymptotic formula of Theorem 1.3. However it is not clear how to describe the initial data beloging to the range of the operatorWf+.

Remark 1.5. Ifφ∈H^0,2 and φ(0) = 0, thenb φ∈H^0,2∩H˙^−α for 0≤α <1 +ⁿ₂ with n= 1,2. This follows from the fact that ˙H⁰ =L² ⊃H^0,2 and the following inequalities:

(a) k| · |^−αfkL² ≤Ck| · |^−α+1∇fkL² forα > ⁿ⁺¹₂ , provided thatf(0) = 0, (b) k| · |^−α+1fkL² ≤CkfkH^1,0 for 1< α <1 +ⁿ₂ withn= 1,2.

Note that this implies thatR

φ(x)dx= 0 andφ∈H^0,2, thenφ∈H^0,2∩H˙^−α. Proof of (a): From the equality

f(ξ) =f(ξ)−f(0) = Z 1

0

d

dtf(tξ)dt= Z 1

0

ξ· ∇f(tξ)dt.

and Schwarz’ inequality, it follows that

|f(ξ)|²≤ |ξ|² Z 1

0

|∇f(tξ)|²dt.

Therefore, we have k| · |^−αfk²_L2 =

Z 1

|ξ|^2α|f(ξ)|²dξ≤ Z 1

|ξ|^2α−2 Z 1

0

|∇f(tξ)|²dtdξ

= Z 1

0

Z 1

|ξ|^2α−2|∇f(tξ)|²dξdt= Z 1

0

Z t^2α−2

|η|^2α−2|∇f(η)|²dη tⁿdt

= 1

2α−1−nk| · |^−α+1∇fk²_L2

forα > ⁿ⁺¹₂ .

Proof of (b) : We split the norm on the left hand side as follows:

k| · |^−α+1fkL² ≤ k| · |^−α+1fkL²(|·|≥1)+k| · |^−α+1fkL²(|·|<1)=I1+I2. Sinceα≥1, it is easily seen thatI1≤ kfk_L2. By the H¨older inequality, we have

I₂≤ k| · |^−α+1kL^p(|·|<1)kfkL^q(|·|<1),

where 2 ≤ p, q ≤ ∞ and ¹_p +¹_q = ¹₂. Here, we put (p, q) = (2,∞) for n = 1 and (p, q) = _α−1^α ,_2−α^2α

for n = 2 so that we have k| · |^−α+1kL^p(|·|<1) < ∞ and kfk_Lq(|·|<1)≤ kfkL^q ≤Ckfk_H1,0 by the Sobolev embedding.

Remark 1.6. In the previous paper [7], we considered the Cauchy problem for the cubic nonlinear Schr¨odinger equation

iut+1

2uxx=N(u), x∈R, t >1 u(1, x) =u1(x), x∈R,

where N(u) =λ1u³+λ2u²u+λ3u³. λj ∈C. j= 1,2,3. It was shown that there exists a global small solution u ∈ C([1,∞), L^∞), if the initial data u₁ belong to some analytic function space and are sufficiently small. For the coefficients λ_j it

was assumed that there existsθ0>0 such that Re λ1

√3e^2ir−iλ₂e^−2ir+ λ3

√3e^−4ir

≥C >0, Im λ₁

√

3e^2ir−iλ2e^−2ir+ λ₃

√ 3e^−4ir

r≥Cr²,

for all|r|< θ0. and also it was assumed that the initial datau1(x) are such that

arge^−i2ξ2cu1(ξ)

< θ0, inf

|ξ|≤1|uc1(ξ)| ≥Cε,

where εis a small positive constant depending on the size of the initial data in a suitable norm. Moreover it was shown that there exist unique final statesW₊, r₊∈ L^∞ and 0< γ <1/20 such that the asymptotic statement

u(t, x) = (it)⁻¹²W+(^x_t)e^ix2t2 q

1 +χ ^x_t

|W+ x t

|²log_t+x^t22

+O

t⁻¹² 1 + log t² t+x²

−¹₂−γ

is valid fort→ ∞uniformly with respect tox∈R, whereγ >0 and χ(ξ) is given by

χ(ξ) = Reλ1

√3exp(2ir₊(ξ))−iλ₂exp(−2ir₊(ξ)) + λ3

√3exp(−4ir₊(ξ)) . This asymptotic formula shows that, in the short range region|x|<√

t. the solution has an additional logarithmic time decay comparing with the corresponding linear case. Thus we can see that the vanishing condition at the origin on the Fourier transform of the final data seems to be essential for our result in the present paper.

For the convenience of the reader we now state the strategy of the proof. We consider the linearized version of equation (1.1)

Lu=N_n(v) +G_n(v), (t, x)∈R×Rⁿ. We take

u0(t, x) = 1 (it)ⁿ²e^ix

2 2t φb x

t

exp −iλ0|φb x t

|ⁿ² logt

as the first approximation for solutions to (1.1). By a direct calculation we get Lu0=Gn(u0) +R1(t),

whereR1(t) is a remainder term. Hence

L(u−u₀) =N_n(v) +G_n(v)− G_n(u₀) +R₁. We define the second approximationu₁ for solutions of (1.1) as

u₁(t) =−i Z t

∞

U(t−τ)N_n(u₀)dτ which implies that

Lu1=Nn(u0) and

u(t)−u0(t) =−i Z t

∞

U(t−τ)(Nn(v)− Nn(u0) +Gn(v)− Gn(u0))dτ

−i Z t

∞

U(t−τ)R1(τ)dτ +u1(t).

We define the function space X =

f ∈C([T,∞);L²);kfkX<∞ kfkX = sup

t∈[T ,∞)

t^bkf(t)−u0(t)k_L2+ sup

t∈[T ,∞)

t^b( Z ∞

t

kf(t)−u0(t)k⁴_Xndt)^1/4, where

X1=L^∞, X2=L⁴, b > n 4.

In order to get the result we need to prove the following estimate foru1(t), ku1(t)k+ (

Z ∞ t

ku1(τ)k⁴_Xndτ)^1/4≤C(k| · |^−eδφkb +kφk_H0,2)¹⁺ⁿ²t^−eδ/2, forn/2<eδ <2. which is the main estimate of the present paper. Note that the choice ofu1 differs from that used in the previous papers.

2. Preliminaries

Lemma 2.1. We have for ω6= 1. f, g∈L¹∩L² andh∈C², Z t

∞

h(iτ)U(t−τ)∆(e^iωx

2

2τ e^{ig(xτ) logτ}f(x τ))dτ

=− 2iω

1−ωh(it)e^iωx

2 2t e^{ig xt}

logtf x t

− 2ω (1−ω)²

Z t

∞

X

(F,k)

F(iτ)e^iωx

2

2τ e^{ig(xτ) logτ}k(x τ)

−iωU(t−τ) Z τ

∞

X

(F,k)

F⁰(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds

−iωU(t−τ) Z τ

∞

X

(F,k)

F(is)e^iωx

2

2s e^{ig(xs) logs}1

sk(g−in 2 )(x

s)ds

dτ +R(t),

where the summation is taken over(F, k) = (h⁰, f),(hτ⁻¹, f(g−in/2)), R(t) =− iω

(1−ω)² Z t

∞

U(t−τ) Z τ

∞

X

(F,k)

F(is)R_0,k(s)ds dτ

+ 1

1−ω Z t

∞

h(iτ)U(t−τ)R0,f(τ)dτ, and

R0,k(t) =e^iωx

2 2t k(x

t)∆e^{ig xt} logt

+ 2i1 t²

X∂jg x t

∂jk x t

e^iωx

2

2t e^{ig(xt) logt}logt + 1

t²(∆k) x t

e^iωx

2 2t e^{ig xt}

logt

. Proof. By a direct computation we find that

(2iω∂_t+ ∆)e^iωx

2 2t e^{ig xt}

logt

f x t

=−2ω1

tf(g−id 2)(x

t)e^iωx

2 2t e^{ig xt}

logt

+R_0,f(t),

where

R_0,f(t) =e^iωx

2 2t f(x

t)∆e^{ig xt}

logt+ 2i1 t²

X(∂_jg·∂_jf) x t

e^iωx

2

2t e^{ig(xt) logt}logt + 1

t²(∆f) x t

e^iωx

2 2t e^{ig xt}

logt.

Therefore,

U(−t)∆(e^iωx2t2e^{ig xt} logt

f x t

)

=−∂t(U(−t)2iω(e^iωx2t2e^{ig xt} logt

f(x

t))) +ωU(−t)∆(e^iωx2t2e^{ig xt} logt

f x t

) +U(−t)(−2ω1

tf(g−in 2 ) x

t e^iωx

2 2t e^{ig xt}

logt

+R0,f(t)) from which it follows that

U(−t)∆(e^iωx2t2e^{ig xt} logt

f x t

)

=− 2iω

1−ω∂_t(U(−t)e^iωx2t2e^{ig xt} logt

f x t

)

− 2ω

1−ωU(−t)1

tf(g−in 2 )(x

t)e^iωx

2 2t e^{ig xt}

logt

+ 1

1−ωU(−t)R0,f(t).

(2.1)

Hence Z t

∞

h(iτ)U(t−τ)∆(e^iωx

2

2τ e^{ig(xτ) logτ}f(x τ))dτ

=− 2iω 1−ωU(t)

Z t

∞

h(iτ)∂τ(U(−τ)e^iωx

2

2τ e^{ig(xτ) logτ}f(x τ))dτ

− 2ω 1−ω

Z t

∞

h(iτ)U(t−τ)1

τf(g−in 2 )(x

τ)e^iωx

2

2τ e^{ig(xτ) logτ}dτ +R1,f(t)

=− 2iω

1−ωh(it)e^iωx

2 2t e^{ig xt}

logt

f x t

− 2ω 1−ω

Z t

∞

h⁰(iτ)U(t−τ)e^iωx

2

2τ e^{ig(xτ) logτ}f(x τ)dτ

− 2ω 1−ω

Z t

∞

h(iτ)U(t−τ)1

τf(g−in 2 )(x

τ)e^iωx

2

2τ e^{ig(xτ) logτ}dτ +R1,f(t),

(2.2)

where

R1,f(t) = 1 1−ω

Z t

∞

h(iτ)U(t−τ)R0,f(τ)dτ.

We write

F(iτ)U(−τ)e^iωx

2

2τ e^{ig(xτ) logτ}k(x τ)

=∂τ(U(−τ) Z τ

∞

F(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds) + i

2U(−τ) Z τ

∞

F(is)∆(e^iωx

2

2s e^{ig(xs) logs}k(x s))ds

=∂_τ(U(−τ) Z τ

∞

F(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds) +ωF(iτ)U(−τ)e^iωx

2

2τ e^{ig(xτ) logτ}k(x τ)

−ωU(−τ) Z τ

∞

iF⁰(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds

−iωU(−τ) Z τ

∞

F(is)e^iωx

2

2s e^{ig(xs) logs}1

sk(g−in 2 )(x

s)ds + i

2U(−τ) Z τ

∞

F(is)R0,k(s)ds hence

(1−ω)F(iτ)U(−τ)e^iωx2τ2e^{ig(xτ) logτ}k(x τ)

=∂τ(U(−τ) Z τ

∞

F(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds)

−ωU(−τ) Z τ

∞

iF⁰(is)e^iωx

2

2s e^{ig(xs) logs}k(x s)ds

−iωU(−τ) Z τ

∞

F(is)e^iωx

2

2s e^{ig(xs) logs}1

sk(g−in 2 )(x

s)ds + i

2U(−τ) Z τ

∞

F(is)R0,k(s)ds.

(2.3)

We apply (2.3) with (F, k) = (h⁰, f) or (F, k) = (hτ⁻¹, f(g−in/2)) to the right-hand

side of (2.1) to get the desired result.

In the next lemma we state the Strichartz estimate forRt

sU(t−τ)f(τ)dτ obtained by Yajima [14].

Lemma 2.2. For any pairs(q, r) and (q⁰, r⁰) such that 0 ≤ ²_q = ⁿ₂ − ⁿ_r <1 and 0≤_q²0 = ⁿ₂−_rⁿ0 <1. for any (possibly unbounded) intervalI and for anys∈I the Strichartz estimate

( Z

I

Z t s

U(t−τ)f(τ)dτ

q

L^rdt)^1q ≤C(

Z

I

kf(t)k^q_L⁰_r0dt)^q10,

is true with a constantC independent ofI ands, where ¹_r+¹_r = 1and ¹_q +¹_q = 1.

Denote

Re1(t) = Z t

∞

U(t−τ) Z τ

∞

F(is)R0,k(s)ds dτ Re2(t) =

Z t

∞

U(t−τ)h(iτ)R0,k(τ)dτ,

where

R_0,k(t) =e^iωx

2 2t k(x

t)∆e^{ig xt}

logt+ 2i1 t²

X∂_jg x t

∂_jk x t

e^iωx

2

2t e^{ig(xt) logt}logt + 1

t²(∆k) x t

e^iωx

2 2t e^{ig xt}

logt.

Lemma 2.3. Let

|F(it)| ≤C|t|⁻²⁻ⁿ², |h(it)| ≤C|t|⁻¹⁻ⁿ². Then

kRej(t)k_L2+ ( Z ∞

t

kRej(t)k⁴_Xndt)^1/4

≤Ct⁻²(k∆kk_L2+k∇k· ∇gk_L2logt+kk∆gk_L2logt+kk∇g· ∇gk_L2(logt)²), whereX1=L^∞, X2=L⁴.

Proof. We have by the Strichartz estimate (see Lemma 2.2) kRej(t)k_L2+Z ∞

t

Rej(t)

4 Xn

dt1/4

≤C Z ∞

t

Z ∞ τ

|s|⁻²⁻ⁿ²kR0,k(s)k_L2 ds+|τ|⁻¹⁻ⁿ² kR0,k(τ)k_L2

dτ.

It is easy to see that kR_0,k(t)k_L2

≤Ct⁻²⁺²ⁿ(k∆kk_L2+k∇k· ∇gk_L2logt+kk∆gk_L2logt+kk∇g· ∇gk_L2(logt)²).

Therefore, we have the result of the lemma.

Lemma 2.4. Assume that|G(it)|+|t||G⁰(it)| ≤C|t|^−q−n2, then

Z t

∞

G(iτ)e^iωx

2

2τ e^{ig(xτ) logs}k(x τ)dτ

_Lp

≤



























Ct^{−δ2−q+1−n2(1−2p)}k| · |^−δkkL^p

+Ct^{−δ2e−q+1−n2(1−2p)}(k| · |^1−eδ∇kkL^p+k| · |^1−eδk∇gkL^plogt), for0< δ,eδ <2, 1≤p <∞,

Ct^{−δ2−q+1−n2(1−1p)}k| · |^−δkk_L^∞

+Ct^{−δ2e−q+1−n2(1−1p)}(k| · |^1−eδ∇kkL^∞+k| · |^1−eδk∇gkL^∞logt), for0< δ,eδ <2−ⁿ_p, 1≤p <∞.

Proof. Using the identity

1

1−^iωx_2τ²∂_tτ e^iωx

2 2τ =e^iωx

2 2τ

we have

Z t

∞

G(iτ)e^iωx

2

2τ e^{ig(xτ) logτ}k(x τ)dτ

= Z t

∞

G(iτ)e^{ig(xτ) logτ}k(x

τ) 1

1−^iωx_2τ²∂ττ e^iωx

2 2τ

dτ

=G(it)k x t

e^{ig(xt) logt} 1

1−^iωx_2t²te^iωx

2 2t

− Z t

∞

τ e^iωx

2 2τ ∂τ

G(iτ)k(x τ) 1

1−^iωx_2τ²e^{ig(xτ) logτ} dτ.

We also obtain G(it)k x

t

e^{ig(xt) logt} 1

1−^iωx_2t²te^iωx

2 2t )

_Lp

≤Ct^{−δ2−q+1−n2}Z _t1/2^x

δ

1 + _t1/2^x

2

x

t

−δ

k x t

p

dx^1/p

≤

(Ct^{−δ2−q+1−n2(1−2p)}k| · |^−δkk_Lp, 0< δ <2,1≤p <∞ Ct^{−δ2−q+1−n2(1−1p)}k| · |^−δkkL^∞, 0< δ <2−ⁿ_p,1≤p <∞ and in the same way we get

te^iωx

2 2t ∂_t

G(it)k x t

1

1−^iωx_2t²e^{ig xt} logt

_Lp

≤































Ct^{−δ2−q−n2(1−2p)} |·|^−δk

_Lp +Ct^{−δ2e−q−n2(1−2p)}(

|·|^1−eδ∇k _Lp+

|·|^1−eδk∇g

_Lplogt), for 0< δ,eδ <2, 1≤p <∞,

Ct^{−δ2−q−n2(1−1p)} |·|^−δk

_L∞ +Ct^{−δ2e−q−n2(1−1p)}(

|·|^1−eδ∇k _L∞+

|·|^1−eδk∇g

_L∞logt), for 0< δ,eδ <2−ⁿ₂, 1≤p <∞.

Hence we have the result of the lemma.

3. Proof of Theorem 1.3 We consider the linearized version of equation (1.1)

Lu=Nn(v) +Gn(v), (t, x)∈R×Rⁿ. (3.1) We take

u0(t, x) = 1 (it)ⁿ²e^ix

2 2t φb x

t exp

−iλ0|φb x t

|ⁿ² logt

as the first approximation for solutions of (3.1). By a direct calculation we get Lu0=Gn(u₀) +R₁,

where

R₁(t) = 1 (it)ⁿ²e^ix

2 2t φb x

t 1

2∆ exp(−iλ₀|φb x t

|ⁿ² logt)

−2 nλ0

1 t²

1 (it)ⁿ²e^ix

2 2t ∇φb x

t

exp(−iλ0|φb x t

|ⁿ² logt)

×2 Re∇φb x t

φb x

t |φ(b x

t)|ⁿ²⁻²logt +1

2 1 (it)ⁿ²e^ix

2

2t t⁻²∆φb x t

exp(−iλ0|φb x t

|ⁿ² logt).

Hence

L(u−u₀) =N_n(v) +G_n(v)− G_n(u₀) +R₁. By Lemma 2.2 we obtain

Z ∞ t

U(t−τ)R₁(τ)dτ

_L2+Z ∞ t

Z ∞ t

U(t−τ)R₁(τ)dτ

4 X_n

dt^1/4

≤C Z ∞

t

kR₁(τ)k_L2dτ ≤Ct⁻¹(logt)²kφk¹⁺_H0,2²ⁿ

(3.2)

since by the H¨older inequality we have kR1(t)kL²

≤Ct⁻²k∆φkb _L2+Ct⁻²(logt)²kφkb _Lⁿ²∞⁻¹k∇bφk²_L4+Ct⁻²(logt)kφkb _L²ⁿ∞k∆φkb _L2

≤Ct⁻²(logt)²kφk¹⁺_H0,2²ⁿ. We now defineu1as

u₁(t) =−i Z t

∞

U(t−τ)N_n(u₀)dτ which impliesLu1=Nn(u0) and

u(t)−u0(t) =−i Z t

∞

U(t−τ)(Nn(v)− Nn(u0) +Gn(v)− Gn(u0))dτ

−i Z t

∞

U(t−τ)R₁(τ)dτ +u₁(t).

(3.3)

Note that

i∂tu1(t) =Nn(u0) + i 2

Z t

∞

U(t−τ)∆Nn(u0)dτ. (3.4) Now, we define the function space

X =

f ∈C([T,∞);L²);kfkX<∞ , where kfkX= sup

t∈[T ,∞)

t^bkf(t)−u₀(t)kL²+ sup

t∈[T ,∞)

t^bZ ∞ t

kf(t)−u₀(t)k⁴_Xndt^1/4 ,

and

X₁=L^∞, X₂=L⁴, b > n 4.

LetXρ be a closed ball inX with a radiusρand a center u0. Letv ∈Xρ. From (3.4) and Lemma 2.1 it follows that

i∂tu1(t) =Nn(u0) + i 2

X

(ω,h,g,f)

− 2iω

1−ωh(it)e^iωx

2

2t e^{ig(xt) logt}f(x t)

− 2ω (1−ω)²

Z t

∞

X

(F,k)

F(iτ)e^iωx

2

2τ e^{ig(xτ) logτ}k(x τ)

−iωU(t−τ) Z τ

∞

X

(F,k)

F⁰(is)e^iωx

2

2s e^{(xs) logs}k(x s)ds

−iωU(t−τ) Z τ

∞

X

(F,k)

F(is)e^iωx

2

2s e^{ig(xs) logs}1

sk(g−in 2 )(x

s)ds

dτ+R(t),

where the summation with respect to (ω, h, g, f) is taken over (ω, h, g, f) =

3,(it)^−3/2, λ0|φˆ x t

|², λ1φˆ x t

3 , −1,(−i)^−1/2t^−3/2, λ₀|φ(ˆ x

t)|², λ₂φˆ x t

φˆ x t

2 , −3,(−it)^−3/2, λ0|φ(ˆ x

t)|², λ3φ(ˆ x t)

3 ,

whenn= 1, and (ω, h, g, f) =

2,(it)⁻¹, λ0|φˆ x t

|, λ1φˆ x t

2 ,

−2,(−it)⁻¹, λ0|φ(ˆ x

t)|, λ2φˆ x t

2 ,

when n = 2, and the summation with respect to (F, k) is taken over (F, k) = (h⁰, f),(hτ⁻¹, f(g−in/2)). We have

Gn(v)− Gn(u0)

=λ0|v|ⁿ²v−λ0|u0|ⁿ²u0

=λ0(|v|ⁿ² − |u0|ⁿ²)(v−u0) +λ0(|v|ⁿ² − |u0|ⁿ²)u0+λ0|u0|ⁿ²(v−u0). Therefore, by the Strichartz estimate we get

Z ∞ t

U(t−τ)(Gn(v)− Gn(u0))dτ _L2 +Z ∞

t

Z ∞ t

U(t−τ)(Gn(v)− Gn(u0))dτ

4 L⁴

dt1/4

≤CZ ∞ t

kv(τ)−u0(τ)k²_L2dτ¹₂Z ∞ t

kv(τ)−u0(τ)k⁴_L4dτ1/4

+C Z ∞

t

kv(τ)−u0(τ)kL²ku0(τ)kL^∞dτ

≤Cρ²t^−2b+12 +Ct^−bρkφk_L1,

(3.5)

forn= 2. Also

Z ∞ t

U(t−τ)(Gn(v)− Gn(u0))dτ _L2 +Z ∞

t

Z ∞ t

U(t−τ)(Gn(v)− Gn(u0))dτ

4 X1

dt1/4

≤CZ ∞ t

k|v(τ)−u0(τ)|³k_L⁴³1dτ3/4

+C Z ∞

t

k|v(τ)−u0(τ)||u0(τ)|²k_L2dτ

≤CZ ∞ t

kv(τ)−u0(τ)k_L⁴³∞kv(τ)−u0(τ)k_L⁸³2dτ^3/4 +C

Z ∞ t

kv(τ)−u₀(τ)kL²ku0(τ)k²_L∞dτ

≤CZ ∞ t

kv(τ)−u₀(τ)k⁴_L∞dτ¹₄Z ∞ t

kv(τ)−u₀(τ)k⁴_L2dτ^1/2 +C

Z ∞ t

kv(τ)−u0(τ)k_L2ku0(τ)k²_L∞dτ

≤Cρt^−bZ ∞ t

ρ⁴τ^−4bdτ1/2

+Cρkφk²_L1

Z ∞ t

τ^−b−1dτ

≤Cρ³t^−3b+12 +Ct^−bρkφk²_L1,

(3.6)

forn= 1, where we have used the facts thatb > n/4 and

|Gn(v)− Gn(u₀)| ≤C(|v−u₀|ⁿ² +|u0|ⁿ²)|v−u₀|.

Similarly, we see that the above estimate holds valid withGnreplaced byNn. Thus by (3.2), (3.3), (3.5) and (3.6)

ku(t)−u₀(t)kL²+Z ∞ t

ku(τ)−u₀(τ)k⁴_Xndτ^1/4

≤Cρ¹⁺ⁿ²t^{−(1+n2)b+12} +Ct^−bρkφk_Lⁿ²1+Ct⁻¹(logt)²kφk¹⁺_H0,2ⁿ²

+ku1(t)k_L2+Z ∞ t

ku1(τ)k⁴_Xndτ1/4

.

(3.7)

To get the result we now estimate u₁(t). By Lemma 2.1, Lemma 2.3 and Lemma 2.4 we get

ku1(t)k_L2+Z ∞ t

ku1(τ)k⁴_Xndτ^1/4

≤C(k| · |^−eδφkb _L2+kφk_H0,2)¹⁺ⁿ²t^−δ2e, (3.8)