ON THE CAUCHY PROBLEM FOR A NONLOCAL NONLINEAR SCHRÖDINGER EQUATION

Vol. 27, No. 12, December 2022, pp. 7185–7206 doi:10.3934/dcdsb.2022039

ON THE CAUCHY PROBLEM FOR A NONLOCAL NONLINEAR SCHR ¨ODINGER EQUATION

Hongwei Wang

School of Mathematics and Statistics, Anyang Normal University Anyang 455000, China

Amin Esfahani^∗

Department of Mathematics, Nazarbayev University Nur-Sultan 010000, Kazakhstan

(Communicated by Shouhong Wang)

Abstract. This paper considers the one-dimensional Schr¨odinger equation with nonlocal nonlinearity that describes the interactions of nonlinear disper- sive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the con- ditions under which the solutions are uniformly bounded in the energy space.

1. Introduction. In this paper, we study the following Schr¨odinger-type equation iut−uxx=ςD^β(|u|²u), x, t∈R, (1) introduced in [4,18] as a model for assessing the validity of weak turbulence theory for random waves, whereς =±1,β ∈Rand D^β withβ ∈Ris the usual Fourier multiplier operator with the symbol|ξ|^β. The parameterβcontrols the nonlinearity, in particular small value ofβmakes the nonlinearity weaker because of a smoothing effect inx. Equation (1) was originally derived as

iut−uxx=ς(|D^βu|²D^βu),

through a heuristic approach, later it was proved that it can be rigorously obtained as an approximation of the fully nonlinear wave system equations [24]. Whenβ = 0, (1) turns into the classical Schr¨odinger equation

iu_t−u_xx=ς|u|²u. (2)

Gr¨unrock studies the local well-posdesness for (2) in [11] in alternative function spaces belowL²(R), but only in settings where the local Lipschitz dependence on the initial data still holds. In the focusing case, (2) admits soliton and multisoliton solutions. Moreover, it is globally well posed inL²thanks to the conservation of the L²-norm. The scaling-critical Sobolev index associated with (2) is−1/2. Christ,

2020Mathematics Subject Classification. Primary: 35A15, 35Q55; Secondary: 35A01.

Key words and phrases. Schr¨odinger equation, Well-posedness, Standing wave.

H. W. was supported by Key Project of Natural Science Foundation of Educational Committee of Henan Province(No. 20A110007). A. E. was supported by the Social Policy Grant (SPG) funded by Nazarbayev University, Kazakhstan.

∗Corresponding author: Amin Esfahani.

7185

Colliander and Tao [6] showed that the data-to-solution map is unbounded from H^s(R) to H^s(R) for s < −1/2. Recently, Harrop-Griffiths et al. in [12] proved that (2) is globally well-posed for all initial data in H^s(R) with s > −1/2 in the sense that the solution map extends uniquely from the Schwartz space to a jointly continuous mapR×H^s(R)→H^s(R).

Equation (1) mathematically is similar to the derivative nonlinear Schr¨odinger equation

iut−uxx=ςi(|u|²u)x. (3) It is known from [20] that (3) is localy well-posed in H^s(R) for s ≥ 1/2. His method of proof combined the gauge transform with Bourgain’s Fourier restriction norm method. This result was shown by Biagioni and Linares [2], and Takaoka [21]

to be sharp in the sense that the flow map fails to be uniformlyC⁰ fors <1/2.

Analogous to (2), the − sign in (1) corresponds to the defocusing (repulsive) problem, while the + sign corresponds to the focusing (attractive) problem. Equa- tion (1) similar to (3) and contrary to (2) is not invariant by the pseudoconformal transformation and no explicit blow-up solution is known for both (2) and (1). An- other main difference between (1) and (2) is that the former one is not invariant by a Galilean transform. An interesting property of (3) is that all these types of derivative nonlinear Schr¨odinger equations are related via gauge transformations, any result on one of the forms can a priori be transferred to the other forms, but it seems that (1) has no such a property. Interestingly, given a solutionuof (1), then

uλ(x, t) =λ^1−β2u(λx, λ²t)

is also a solution of (1). As a consequence, the scale-invariant Sobolev spaces for (1) are ˙H^sc(R) with s_c = −¹₂(1−β). One of the aims of the current paper is to address the question for local well-posedness of (1). We should also note that the functionals

E(u) = Z

R

|D^−β/2ux|²+ς1 2|u|⁴

dx F(u) =

Z

R

|D^−β/2u|²dx are formally motion invariants of (1).

Theorem 1.1. Let β ∈ (0,1). Then the Cauchy problem associated with (1) is locally well-posed inH^s(R)ifs > β/2.

The local well-posedness of (1) can be extended to the case β < 0. Indeed, whenβ∈(−1,0), then the nonlinear termD^β(|u|²u) can be rewritten by the Riesz potential by

I_−β∗ |u|²u,

whereI_−β=Cβ|x|^−1−β. By using the Hardy-Littlewood-Sobolev lemma, the frac- tional chain rule and the Strichartz estimates for the Sch¨odinger equation (see [16]), one can consider a suitable ball in the space

L^∞_t (R;H^s(Rx))∩L^q_t(R;H_r^s(Rx))

to show the local-wellposedness by a contraction argument (see [16]), whereH_r^s= (1−∂_x²)^−s/2L^r. So we omit the detilas and focus on the caseβ ≥0. Forb, s,∈R, we define the (inhomogeneous)X^s,b spaces with respect to the norm

X^s,b ={u∈S⁰, kuk_Xs,b:=khξi^shτ−ξ²i^bu(ξ, τˆ )kL²(R²)<+∞},

where ˆuin this definition is the Fourier transform of uwith respect to both time and space andh·i=a+| · |. As (1) with an initial datau(0) =u0 is related to the integral form

u(t) =U(t)u₀−iς Z t

0

U(t−τ)D^β(|u|²u)(τ)dτ,

where U is the unitary Schr¨odinger group of (1), then the main difficulty, by a standard iteration argument, is to show a trilinear estimate related to this integral form inX^s,b spaces.

It is worth pointing out there is an important relation between the mixed Lebesgue spaces and the aboveX^s,b spaces, based on the Strichartz estimates for the group U. We have that

kU(t)uk_L^q

t(R)L^px(R).kukL²(R²)

for all pairs (q, p) satisfying ²_q +_p¹ =¹₂ with 2≤p, q≤ ∞. This leads us to kuk_L^q

t(R)L^p_x(R).kuk

X^0,12 +.

This estimate can be deduced by adapting suitably the dyadic method introduced in [22, 23] in which multilinear estimates in weightedL²(R) spaces were generally studied.

In order to prove the trilinear estimates, related to Theorem1.1and the integral for inX^s,b, we prove a bilinear estimate for (2) with the resonance functionh(ξ) = 2ξ1ξ2.

Next aim in this paper is to study the ill-posedness of (1). We use the ideas of [6]. More precisely, the strategy is to approximate the solution of the Cauchy problem associated with (1) by the solutions of (2), and use the ill-posedness result [6] of (2) inH^s(R). The main difficulty here lies in small dispersion analysis due to the non-local nonlinearity.

Theorem 1.2. Let _2(3−3β)^3β−2 < s < ^β₂, 0 < β < ²₃. Then the solution map of the following initial value problem

(iu_t−u_xx−D^β |u|²u

= 0, u(0) =ϕ

fails to be locally uniformly continuous. More precisely, for 0 < δ ε 1 and T >0 arbitrary, there are two solutionsu1, u2 with initial data φ1, φ2 such that

kφ1kH^s,kφ2kH^s .ε, (4)

kφ₁−φ₂k_Hs .δ, (5)

sup

0≤t≤T

ku1(t)−u₂(t)kH^s&ε. (6) Finally, we study the existence of standing waves of (1). By a standing wave we mean a solution of (1) of the formu(x, t) = e^−iωtϕ(x), whereω >0 is the standing wave frequency. Substituting this form into (1), it transpires that ϕmust satisfy

ωD^−βϕ+D^2−βϕ=±|ϕ|²ϕ. (7) Thus one sees that the natural space to study (1) is

X = ˙H^−β2(R)∩H˙^1−β2(R), equipped with the norm

kfk_X =kfk_˙

H^−β2(R)+kfk_˙

H^1−β2(R).

By multiplying (7) by ϕand integrating overR, one can easily see that it does not have any nontrivial solution in the defocusing (repulsive) case. So we study the following case

ωD^−βϕ+D^2−βϕ=|ϕ|²ϕ. (8) We use the concentration-compactness principle [17] to show the existence of the ground states in X. We recall that the solutionϕof (8) is called a ground state, ifϕminimizes the action

S =E+ωF (9)

among all non-trivial solutions of (8). The essential tools in this way is the embed- dingX into L⁴(R) by showing the following Gagliardo-Nirenberg type inequality:

kgkL⁴(R)≤C D^−β2g

θ₁ L²(R)

D^−β2gx

θ₂ L²(R)

for some θ₁, θ₂ ≥ 0 (see (35) below). We also obtain the best constant for this inequality. Indeed, we show a profile decomposition by mimicking the proof lines of [10,13]. This also enables us to give another approach to study the existence of so- lutions of (8). As an application of aforementioned Gagliardo-Nirenberg inequality, we show the uniform bound of solutions of (1) inX.

The paper is organized as follows. In section2, we use some useful known esti- mates which we need in the subsequent section to show the well-posedness result.

In section3, we prove the ill-posedness result. Finally, the existence of the standing waves and their applications are proved in section4.

We end this section by presenting the following convention. We use A .B to denote the statement that A ≤CB for some positive harmless constant C which may vary from line to line and depend on various parameters. We use A ∼B to denote the statement thatA.B andB.A.

2. Well-posedness. In this section, we prove our well-posedness result.

We use the following result [19, Proposition 3] to obtain the bilinear estimates.

So we need to recall the definitions and notation accordingly.

For any integerk≥2, we denote Γk(Z) as the hyperplane Γk(Z) =

ξ= (ξ1, . . . , ξk)∈Z^k, ξ1+. . .+ξk= 0 with the measure

Z

Γ_k(Z)

f :=

Z

Z^k−1

f(ξ) dξ₁. . .dξ_k−1,

where Z can be any abelian additive group with an invariant measure dξ. By following the concepts introduced in [22], the [k;Z]-multiplier is a function η : Γ_k(Z)→Cand the multiplier normkηk_[k;Z] is the best constant such that

Z

Γ_k(Z)

η(ξ)

k

Y

j=1

fi(ξi)

≤ kηk_[k;Z]

k

Y

j=1

kfik_L2(Z)

holds for all test functionsfi onZ.

We should review some of Tao’s notations in [22]. Any summations over capital- ized variables such asNi,LiandHare presumed to be dyadic. It will be convenient for N1, N2, N3 > 0 to define the quantitiesNmax ≥Nmed ≥ Nmin which are the maximum, median, and minimum of N1, N2, N3, respectively. Likewise, we have L_max ≥L_med ≥L_min if L₁, L₂, L₃ >0. We also adopt the following summation

convention. Any summation of the formLmax∼. . .is a sum over the three dyadic variablesL1, L2, L3&1. Hence, we have for instance that

X

Lmax∼H

= X

L1,L2,L3&1:Lmax∼H

.

Analogously, any summation of the form Nmax ∼ . . . sum over the three dyadic variablesN1, N2, N3>0. For example, we have

X

N_max∼N_med∼N

:= X

N₁,N₂,N₃>0:N_max∼N_med∼N

.

If τ, ξ and φ(ξ) are given with τ1+τ2+τ3 = 0, then we write λ :=τ−φ(ξ).

Similarly we have λi := τi−φ(ξi). We refer to h: Γ3(Z) → R as the resonance function, which is defined by

h(ξ) :=φ(ξ1) +φ(ξ2) +φ(ξ3) =−λ1−λ2−λ3. (10) By the dyadic decomposition of each variableξ_iorλ_i, as well as the functionh(ξ), we are led to considerkX_{N1,N2,N3;H;L1,L2,L3}k_[3;

R×R],whereX_{N1,N2,N3;H;L1,L2,L3} is the multiplier

XN₁,N₂,N₃;H;L₁,L₂,L₃(ξ, τ) =χ_{{|h(ξ)|∼H}}

3

Y

j=1

χ_{|ξj|∼N_j}χ_{|λj|∼L_j}, (11) whereχ is the characteristic function. From the identities

ξ₁+ξ₂+ξ₃= 0 and

λ1+λ2+λ3+h(ξ) = 0 (12)

on the support of the multiplier, we see thatX_{N1,N2,N3;H;L1,L2,L3} vanishes unless

Nmax∼Nmed (13)

and

L_max∼max(H, L_med). (14)

The following estimates can be found in [22] (see also [19, Proposition 3]) Lemma 2.1. Let H, N1, N2, N3, L1, L2, L3 >0 satisfy Nmax ∼Nmed, H ∼N2N3

and

Lmax∼max(H, Lmed). Then we have the following estimates.

(1)

kXN₁,N₂,N₃;H;L₁,L₂,L₃k_[3;

R×R].L

1 2

minN

1 2

min. (15)

(2)

kXN1,N2,N3;H;L1,L2,L3k_[3;

R×R] .min L₁L₂

N3

,L₁L₃ N2

¹₂

. (16)

(3) IfL1=Lmax andN1∼N2∼N3 holds, then kXN₁,N₂,N₃;H;L₁,L₂,L₃k_[3;R×R].L

1 2

minL

1 4

med, (17)

If L1=Lmax andN1∼N2∼N3 does not hold, then kXN₁,N₂,N₃;H;L₁,L₂,L₃k_[3;

R×R]. L_min¹² L_med¹² N

1

max2

. (18)

Lemma 2.2. If s > ^β₂,0< β <1 then for allu, v onR×R, we have ku¯vk_L2(R×R).kuk

X^−β2,1 2−εk¯vk

X^s,12+ε, (19) ku¯vk_L2(R×R).kuk

X^s,12+εk¯vk

X^−β2,1

2−ε. (20)

Proof. We first dispense with the estimate (19). Indeed, it suffices to show by the Plancherel theorem that

hξ₁i^β2

hξ2i^shτ1−ξ²₁i^12−εhτ2+ξ₂²i^12+ε _[3;

R×R]

.1.

The comparison principle and orthogonality reduce our estimate to show that X

N_max∼Nmed∼N

X

L₁,L₂,L₃&1

N

β 2

1 N₂^−s L₁^12−εL₂^12+ε

× kX_{N1,N2,N3;Lmax;L1,L2,L3}k_[3;R×R].1 (21) and

X

Nmax∼Nmed∼N

X

Lmax∼Lmed

X

LmaxH

N

β 2

1 N₂^−s L

1 2−ε

1 L

1 2+ε 2

× kXN₁,N₂,N₃;H;L₁,L₂,L₃k_[3;

R×R].1.

(22)

Fix N, we first prove (22). If we have Lmax ∼ Lmed N2N3, we apply (15).

Hence, the sum in (22) is estimated by X

N_max∼N_med∼N

X

L_max∼L_med

X

L_maxH

N

β 2

1 N₂^−s L

1 2−ε

1 L

1 2+ε 2

L

1 2

minN

1 2

min

. X

Nmax∼Nmed∼N

N

β 2

1 N₂^−sN

1 2

min

(N2N3)^12−ε .

(23)

IfNmin=N1, then we have X

N_max∼N_med∼N

N^{−s+β2−12+2ε}.1, provideds > ^β₂ −¹₂.

IfN_min=N₂, then X

N_max∼Nmed∼N

N

β 2

1 N₂^−sN_min¹²

(N2N3)^12−ε . X

N_max∼Nmed∼N

N_min^−s+εN^β2−12+ε,

ifs >0 then the estimate is certainly true for β <1, if s≤0, then the estimate is similar to the caseNmin=N1.

If Nmin = N3, then N1 ∼ N2 ∼ N, we can obtain the desired estimate for s > ^β₂ −¹₂.

Now we show (21). We may assumeL_max∼N₂N₃.

IfLmax=L2, we apply (15), the estimate is similar to (23), we omit it.

IfLmax=L3, we apply (16), performing theLsummations, then the summation in (21) is estimated by

X

Nmax∼N_med∼N

X

L₁,L₂,L₃&1

N

β 2

1 N₂^−s L

1 2−ε

1 L

1 2+ε 2

L

1 2

1L

1 2

2

N

1 2

3

. X

Nmax∼N_med∼N

N

β 2

1 N₂^−s(N2N3)^ε N

1 2

3

,

ifNmin =N3, then (21) is true fors > ^β₂, if Nmin 6=N3, then we can get (21) for s > ^β₂ −¹₂.

IfL_max=L₁andN₁∼N₂∼N₃∼N, then we can estimate the sum in (21) via (17),

X

Nmax∼Nmed∼N

X

L1,L2,L3&1

N

β 2

1 N₂^−s L

1 2−ε

1 L

1 2+ε 2

L

1 2

minL

1 4

med. X

Nmax∼Nmed∼N

N

β 2

1 N₂^−s(N2N3)^−14+ε .1

fors > ^β₂ −¹₂. Otherwise, (17) can be used to estimate the sum in (21) X

Nmax∼Nmed∼N

X

L1,L2,L3&1

N

β 2

1 N₂^−s L

1 2−ε

1 L

1 2+ε 2

L_min¹² L_med¹⁴ N

1

max2

. X

Nmax∼Nmed∼N

N

β 2

1 N₂^−s L

1 4−ε

1 N

1

max2

.1 fors > ^β₂ −¹₂.

The proof of (20) is similar to (19), we omit it.

Theorem 2.3. Let s > ^β₂, for anyu, v, w, then D^β(u¯vw)

X^s,−12+ε .kuk

X^s,12+εk¯vk

X^s,12+εkwk

X^s,12+ε. (24) Proof. By duality and the Plancherel theorem, it suffices to show that

|ξ1+ξ2+ξ3|^βhξ4i^s

hτ₄+ξ₄²i^12−εhξ₁i^shτ₁−ξ₁²i^12+εhξ₂i^shτ₂+ξ₂²i^12+εhξ₃i^shτ₃−ξ₃²i^12+ε _[4;

R×R]

.1.

(25) We estimate|ξ1+ξ2+ξ3|byhξ4i, then apply the inequality

hξ4i^s+β.hξ4i^β2

3

X

j=1

hξji^s+β2. If max

j=1,2,3|ξ_j|=|ξ₂|, we may minorize hτ₂+ξ²₂i^12+ε byhτ₂+ξ²₂i^12−ε, and reduce to showing that

hξ2i^β2

hξ₁i^shτ₁−ξ₁²i^12+εhτ₂+ξ²₂i^12−ε· hξ4i^β2

hξ₃i^shτ₃−ξ₃²i^12+εhτ₄+ξ₄²i^12−ε _[4;

R×R]

.1, (26) thenT T^∗ identity reduce our estimate to show that

hξ2i^β2

hξ1i^shτ1−ξ₁²i^12+εhτ2+ξ₂²i^12−ε [3;R×R]

.1,

hξ4i^β2

hξ3i^shτ3−ξ₃²i^12+εhτ4+ξ₄²i^12−ε _[3;

R×R]

.1.

These estimates follows from (20).

If max

j=1,2,3|ξj| = |ξ1|, we may monorize hτ1−ξ²₁i^12+ε by hτ1−ξ₁²i^12−ε, reduce to showing that

hξ₁i^β2

hξ2i^shτ1−ξ₁²i^12−εhτ2+ξ₂²i^12+ε· hξ₄i^β2

hξ3i^shτ3−ξ₃²i^12+εhτ4+ξ₄²i^12−ε _[4;

R×R]

.1, (27)

thenT T^∗ identity reduce our estimate to show that

hξ1i^β2

hξ2i^shτ1−ξ₁²i^12−εhτ2+ξ₂²i^12+ε _[3;

R×R]

.1,

hξ₄i^β2

hξ3i^shτ3−ξ₃²i^12+εhτ4+ξ₄²i^12−ε _[3;

R×R]

.1.

The first estimate is valid by (19), and the second estimate is valid by (20).

By symmetry, the estimate of max

j=1,2,3|ξj|=|ξ3|is the same as the case max

j=1,2,3|ξj|=

|ξ1|.

With Theorem 2.3 in our hands, the completeness of proof of Theorem 1.1 is obtained by the standard approach and we omit it (see [6]).

3. Ill-posedness. In this section, we prove our ill-posedness result in Theorem1.2.

Our idea is to approximate the solutions of (1) with ones of (2). For convenience we the caseς = +1:

(iut−uxx−D^β |u|²u

= 0,

u(0) =ϕ (28)

We also consider (2) correspondingly. To prove the ill-posedness result, we recall the following result of [6, Lemma 2.1] (see also [5]).

Lemma 3.1. Let s >−¹₂ and w∈H^σ(R)with σ >0. For M >1, τ >0, x₀∈R andA >0 let

v(x) =Ae^{iM x}u

x−x0

τ

. (1) Ifs≥0, then

kvkH^s≤C1|A|τ¹²M^skukH^s

for all u, A, x0 wheneverM ·τ ≥1, where C1 is a positive constant depending only ons.

(2) Ifs <0 andσ≥ |s|, then

kvkH^s ≤C₁|A|τ¹²M^skukH^σ

for allu, A, x0wheneverM^1+sσ·τ ≥1, whereC1is a positive constant depending only ons andσ.

(3) There existsc1>0 such that for eachuthere existsCu<∞such that kvkH^s ≥c1|A|τ¹²M^skuk_L2

wheneverτ·M ≥Cu.

Now we recall the ill-posedness result of [6] for (2).

Lemma 3.2. Let s <0. The solution map of the initial value problem associated with (2) fails to be uniformly continuous. More precisely, for0 < δ ε1 and T >0 arbitrary, there are two solutions v1, v2 to (2) with initial dataφ1, φ2 such that

kφ₁k_Hs,kφ₂k_Hs .ε, kφ1−φ₂kH^s .δ, sup

0≤t≤T

kv1(t)−v₂(t)kH^s &ε.

Moreover we can find solutions to satisfy sup

0≤t≤∞

kvj(t)k_H5 .ε, j= 1,2. (29) Suppose thatN 1 is a large parameter that will be chosen later. Let v(s, y) be a solution of the cubic NLS equation (2) and

(s, y) := (t, x+ 2N t).

We are going to construct an approximate solution given by

V(t, x) = e^{iN x}e^iN2tv(s, y). (30) Sincev(s, y) is a solution of (2), we have

i∂tV −∂_x²V −D^β |V|²V

=E, where

E= e^{iN x}e^iN2t|v|²v−D^β |V|²V

=E2−E1.

Lemma 3.3. Let ube a smooth solution to (28)andV be a smooth solution to the equation

i∂_tV −∂_x²V −D^β |V|²V

=E

for some error function E. Let e be the solution to the inhomogeneous problem i∂_te−∂_x²e=E withe(0) = 0and suppose thatζ(t)is a compactly supported smooth time cut-off function such that ζ = 1 on I = [0,1]. If ku(0)k

H^β2+ε, kV(0)k

H^β2+ε, kζ(t)ek

X^β2+ε,1

2+ε.εandε is sufficiently small, then we have ku−Vk

X^β2+ε,1

2+ε .ku(0)−V(0)k

H^β2+ε+kζ(t)ek

X^β2+ε,1 2+ε. In particular, we have

sup

0≤t≤1

ku(t)−V(t)k

H^β2 .ku(0)−V(0)k

H^β2 +kζ(t)ek

X^β2,1 2+ε. Proof. If we consider in the integral form forV, then we have

V(t)−e(t) =U(t)V(0)−i Z t

0

U(t−t⁰)(D^β(|V|²V))(t⁰) dt⁰. We get by takingX^β2,12+ε(I) norm on both sides that

kVk

X^β2+ε,1

2+ε .kV(0)k

H^β2+ε+kζ(t)ek

X^β2+ε,1 2+ε+

D^β(|V|²V)

X^β2+ε,−1 2+ε(I)

.kV(0)k

H^β2+ε+kζ(t)ek

X^β2+ε,1

2+ε+kVk³

X^β2+ε,1 2+ε(I). Now it reveals by the continuity argument with sufficiently smallεthatkVk

X β 2+ε,1

2+ε (I). ε.

Letw:=u−V. Thenwsatisfies the equation i∂tw−∂_x²w=D^β |w|²w+ 2|w|²V + 2w|V|²+w²V¯+ ¯wV²

−E, w(0) =u(0)−V(0).

which is written in integral form as w(t) =U(t)w(0)−e(t)

−i Z t

0

U(t−t⁰) D^β |w|²w+ 2|w|²V + 2w|V|²+w²V¯ + ¯wV² (t⁰) dt⁰.

We have by taking again X^β2+ε,12+ε(I) norm on both sides of the above equation that

kwkX^β2+ε,1

2+ε.ku(0)−V(0)k

H^β2+ε+kζ(t)ek

X^β2+ε,1 2+ε

+

|w|²w+ 2|w|²V + 2w|V|²+w²V¯ + ¯wV²

X^{β2+ε,−12+ε}(I)

.ku(0)−V(0)k

H^β2+ε+kζ(t)ek

X^β2+ε,1 2+ε

+kwk

X^β2+ε,1 2+ε(I)

kwkX^β2+ε,1

2+ε(I)+kVk

X^β2+ε,1 2+ε(I)

² . Finally the desired bound is deduced by the continuity argument with respect to time providedεis sufficiently small.

Lemma 3.4. If e is a solution of equation i∂_te−∂²_xe =E with e(0) = 0, and ζ is the smooth time cut-off function given above, the V(t, x) in E satisfy (30) and (29), then

kζ(t)ek

X^β2+ε,1

2+ε.εN^−1+3β2+ε. Proof. Using the Plancherel theorem we have

kζ(t)e1k

X^β2+ε,1

2+ε.kζ(t)E1k

X^β2+ε,−1 2+ε

=

hτ−ξ²i^−12+εhξi^β2+εζ(t)E\1

_L2

τL²_ξ

≤

hξi^β2+εζ(t)E\1

_L2

τL²_ξ

=

ζ(t)hξi^β2+εEc₁(t, ξ) _L2

tL²_ξ

≤

hξi^β2+εEc1(t, ξ) _L∞

t L²_x([0,1]×R)

. A direct computation leads to that

ζ(t)E\₁(τ, ξ) =|ξ|^βζ|v|\²v(τ+N²−2N ξ, ξ−N).

LetP_λ,µbe the Littlewood-Paley projection with dyadic numbersλ, µ. The fact thatζ(t) is compactly supported yields

Pλ,µ\ζ|v|²v(τ, ξ) _L2

τL²_ξ . ε hλi^Khµi^K by choosingK large enough, and so

Pλ,µ\ζ|v|²v(τ+N²−2N ξ, ξ−3N) _L2

τL²_ξ. ε

hλ+N²−2N ξi^Khµ−Ni^K. Rewritingkζ(t)E1k

X^{β2+ε,−12+ε} by dyadic decomposition, kζ(t)E1k²

X^β2+ε,−1

2+ε. X

λ,µ≥1,dyadic

hλ−µ²i^−1+2εhµi^3β

×

Pλ,µ\ζ|v|²v(τ+N²−2N ξ, ξ−3N) _L2

τL²_ξ

. X

λ,µ≥1,dyadic

hλ−µ²i^−1+2εhµi^3β ε²

hλ+N²−2N ξi^2Khµ−Ni^2K .ε²N^−2+3β+2ε.

The estimate ofkζ(t)E2k²

X^β2,−1

2+ε is similar tokζ(t)E1k²

X^β2,−1

2+ε. We omit it.

Now we are ready to prove our ill-posedness result.

Proof of Theorem 1.2. Let 0< δ ε1 and T >0 be given. From Lemma 3.2 we have two global solutionv1, v2 with initial dataφ1, φ2, respectively, such that

kφ1kH^s,kφ2kH^s .ε, kφ1−φ₂kH^s .δ, sup

0≤t≤T

kv1(t)−v2(t)kH^s&ε. (31) sup

0≤t≤∞

kv1(t)k_H5,kv2(t)k_H5 .ε.

DefineV1, V2 by

Vj(t, x) =e^{iN x}e^iN2tvj(s, y), j= 1,2.

And let u₁, u₂ be smooth solutions of (28) with initial data V₁(0, x), V₂(0, x), re- spectively. Let λ 1 be a large parameter to be chosen later. For j = 1,2, set

u^λ_j :=λu_j(λ²t, λx), V_j^λ:=λV_j(λ²t, λx).

Thus we have

u^λ_j(0, x) =e^{iN λx}V_j^λ(0, x) =λe^{iN λx}v_j(0, λx).

Lemma3.1withM =N λ,τ=λ⁻¹ implies that ifs≥0, u^λ_j(0)

_Hs.λ^s+12N^skvj(0)k_Hs, while fors <0, for sufficiently largekwe obtain

u^λ_j(0)

_Hs .λ^s+12N^skvj(0)k_Hk. We chooseλ=N

−s s+ 12, then

u^λ_j(0)

H^s .ε,

u^λ₁(0)−u^λ₂(0) _Hs.δ.

Rescaling gives

u^λ_j(t)−V_j^λ(t)

_Hs .λ^max(s,0)+12

uj(λ²t)−Vj(λ²t) _Hs .λ^max(s,0)+12

u_j(λ²t)−V_j(λ²t) H^β2+ε. Induction argument on time interval up to ^log_λ2^N yields

u_j(λ²t)−V_j(λ²t)

H^β2+ε.εN^−1+3β2+ε, whenever 0< t^log_λ2^N. Hence we have

u^λ_j(t)−V_j^λ(t)

H^s .λ^max(s,0)+12εN^−1+3β2+ε. From the hypothesiss >_2(3−3β)^3β−2 , it follows that

u^λ_j(t)−V_j^λ(t)

_Hs .ε. (32) Applying Lemma3.1, we have

u^λ_j(t) H^s≤

u^λ_j(t)−V_j^λ(t) H^s+

V_j^λ(t)

H^s.ε+

vj(λ²t)

_Hs .ε. (33)

By the ill-posedness result (31), we can find a time t0 > 0 such that kv1(t0)− v2(t0)kL² &ε. Using Lemma3.1we obtain

V1

t0

λ²

−V2

t0

λ²

_Hs &λ^s+12εN^s∼ε. (34) Combining (32), (33) and (34), a triangle inequality shows

u1

t₀ λ²

−u2

t₀ λ²

_Hs&ε,

fort0 logN. Choosingλlarge enough that _λ^t02 < T, we get (6). This completes the proof.

4. Standing wave. In this section we study the existence of standing waves of (1).

The following result is direct consequence of the Gagliardo-Nirenberg inequality.

Lemma 4.1. Let −¹₂ < β < ³₂ and0≤q≤^q₂^∗ −1, where q^∗=

( ₂

β−1, β >1,

∞⁻, β≤1,

where ∞⁻ is any number q1 <∞. Then there is a constant C > 0 such that for any g∈X,

kgk_L2q+2(R)≤C D^−β2g

1−¹₂(β+_q+1^q )

L²(R)

D^−β2gx

β 2+_2(q+1)^q

L²(R) . (35) As a consequence, it follows thatX is continuously embedded in L^2q+2(R).

Proof. If β ≥0, then (35) is obtained from the following Gagliardo-Nirenberg in- equality

kD^β2gk_L2q+2(R)≤Ckgk¹⁻

1 2(β+_q+1^q ) L²(R) kgxk

β 2+_2(q+1)^q

L²(R) . (36)

If −¹₂ < β ≤0, then we have again from the Gagliardo-Nirenberg inequality with p >1 that

kgk_L2q+2(R)≤Ckgk^1−θ_Lp(R)

D^1−β2g

θ L²(R), whereθ= ²⁻

p q+1

2+(1−β)p ∈(0,1). Finally inequality (35) is followed from kgkL^p(R)≤CkD^−β2gkL²(R),

withp=_β+1² .

Lemma 4.2. Assume that ω > 0. Equation (8) possesses no nontrivial solution ϕ∈X ∩L⁴(R)if β≥ ³₂ orβ ≤ −¹₂ holds.

Proof. Letϕbe a nontrivial solution of (8). First we multiply (8) byϕand integrate overRto get

ωkD^−β2ϕk²_L2(R)+kD^1−β2ϕk²_L2(R)=kϕk⁴_L4(R). (37) Next, we multiply (8) byxϕ_xand integrate overR, and use the identity (see [14, Lemma 3] and [15, Theorem 4.1])

Z

R

xϕxD^αϕdx= α−1

2 kD^α2ϕk²_L2(R)

to deduce that 1

2kϕk⁴_L4(R)+ (1−β)kD^1−β2ϕk²_L2(R)=ω(β+ 1)kD^−β2ϕk²_L2(R). (38) Combining (37) and (38), we derive

kD^1−β2ϕk²_L2(R)=2β+ 1

4 kϕk⁴_L4(R)

ωkD^−β2ϕk²_L2(R)= 3−2β

4 kϕk⁴_L4(R).

(39) And the proof is deduced form (39).

Theorem 4.3. Let −¹₂ < β < ³₂. Ifϕ∈X be a nontrivial solution of (8), then ϕ∈L¹(R)∩H^∞(R)and

|ϕ(x)|=O(e⁻

√ω|x|), (40)

at infinity ifβ ≥0and|ϕ(x)|=O(|x|^−1−β)ifβ ≤0.

Proof. One can observe that any nontrivial solutionϕ∈X satisfies

ϕ=K∗ϕ³, (41)

where

K(ξ) =ˆ |ξ|^β ω+|ξ|².

We note thatK=D^βK₀, where K₀(x) = (ω+D²)⁻¹. It is known that K₀(x)∼= 1

√ωe⁻

√ω|x|.

It is easy to see thatkK0k_Lq(R)=ω^−12(1+1q). It is known from [7] that K(x) =π

2 csc(π

2(1 +β)) cosh(√ ωx) +1

2Γ(1 +β) cos(π

2(1 +β))

× e⁻

√ω|x|−βπiγ(−β,−√

ω|x|)−e

√ω|x|γ(−β,√ ω|x|), whereγ(·,·) is the lower incomplete gamma function. Furthermore,

D^βK0∈L^q(R) (42)

for all 1 ≤q≤ ∞ andβ ≥0 with β−1/q <1 by an interpolation. To show the regularity of the solutions of (8), we claim thatϕ∈L^∞(R). Actually, it is obvious from the Sobolev inequality for th caseβ ≤1. In the caseβ >1, we can use (41) and apply the Young inequality and (42) to getϕ∈L^r(R) when ²_r ≥5β−7; so that ϕ∈L^∞(R) ifβ < ⁷₅. If we repeat this process once again, we concludeϕ∈L^r(R) when ²_r ≥7β−11; so thatϕ∈L^∞(R) for anyβ < ³₂.

Now, (42) shows thatϕ∈H^2(1−β2)(R), so ifβ <1, thenϕ∈H¹(R). We proceed to find from (41) that

kD^3−βϕkL²(R).kϕ²ϕxkL²(R).kϕxkL²(R)<+∞.

Soϕ∈H^3−β(R). In the caseβ >1, let L >0 be an integer such that 1/(L+ 1)≤ 1−^β₂ <1/L <1. Then by an iteration argument as done for the caseβ <1, we can deduce that ϕ∈H^3−β(R). Repeating the above argument show thatϕ∈ H^k(R) for anyk≥1.