On a system of nonlinear Schrödinger equations with quadratic interaction

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Ann. I. H. Poincaré – AN 30 (2013) 661–690

www.elsevier.com/locate/anihpc

On a system of nonlinear Schrödinger equations with quadratic interaction

Nakao Hayashi

^a,^∗

, Tohru Ozawa

^b

, Kazunaga Tanaka

^c

aDepartment of Mathematics, Graduate School of Science, Osaka University, Osaka, Tokyonaka, Japan bDepartment of Applied Physics, Waseda University, Tokyo 169-8555, Japan

cDepartment of Mathematics, Waseda University, Tokyo 169-8555, Japan

Received 13 March 2012; received in revised form 3 September 2012; accepted 22 October 2012 Available online 28 November 2012

Dedicated to the memory of Professor Riichi Iino

Abstract

We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n6. The Cauchy problem is studied inL², inH¹, and in the weightedL² spacex⁻¹L²=F(H¹)under mass resonance condition, where x =(1+ |x|²)^1/2andF is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.

1. Introduction

We study the system of nonlinear Schrödinger equations:

⎧⎪

⎨

⎪⎩

i∂tu+ 1

2mu=λvu, i∂_tv+ 1

2Mv=μu²,

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whereuandv are complex-valued functions of(t, x)∈R×Rⁿ,is the Laplacian inRⁿ,mandM are positive constants, λandμ are complex constants, andu is the complex conjugate ofu. Here the interaction terms in the system (1) are quadratic in(u, v). By the standard scaling arguments on (1), the critical function space isH^n/2⁻², whereH^s=(1−)⁻^s/2L²is the usual Sobolev space of orders(see[3,13,19]). Particularly,L²andH¹are critical spaces forn=4 andn=6, respectively, from the scaling point of view. Those spaces are also important from the point of view of the invariance under group of motion.L²is naturally associated with the conservation of charge,

* Corresponding author.

E-mail address:[email protected](N. Hayashi).

http://dx.doi.org/10.1016/j.anihpc.2012.10.007

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which follows from invariance under Gauge transform.H¹is naturally associated with the conservation of energy, which follows from invariance under time-translation.

The system (1) is regarded as a non-relativistic limit of the system of nonlinear Klein–Gordon equations

⎧⎪

⎪⎨

⎪⎪

⎩ 1

2c²m∂_t²u− 1

2mu+mc²

2 u= −λvu, 1

2c²M∂_t²v− 1

2Mv+Mc²

2 v= −μu²,

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under the mass resonance condition

M=2m (3)

since the modulated wave functions(uc, vc)=(e^{it mc}²u, e^{it Mc}²v)satisfy

⎧⎪

⎨

⎪⎩ 1

2c²m∂_t²u_c−i∂_tu_c− 1

2mu_c= −e^{it c}²^(2m⁻^M)λv_cu_c, 1

2c²M∂_t²v_c−i∂_tv_c− 1

2Mv_c= −e^{it c}²^(M⁻^2m)μu²_c,

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where the phase oscillations on the right hand sides vanish if and only if (3) holds, and under the mass resonance condition (3) the system (4) formally yields (1) as the speed of lightctends to infinity.

The system (2) is closely related to systems studied in[1,7,9]for instance. As regards the non-relativistic limit for the nonlinear Klein–Gordon equations, we refer the reader to[17,18]and references therein. For recent works related to the mass resonance, see[12,24,25].

The Cauchy problem for (1) has been studied from the point of view of small data scattering[10,11]. The purpose of this paper is to study the Cauchy problem for (1) with large data, namely, data which are not necessarily small enough.

The argument in Section 3 is rather standard. We describe it for convenience of readers. Local Cauchy problem is studied inL²and inH¹respectively in Sections3.1and3.2by a contraction argument based on the Strichartz estimates. To extend local solutions we use a priori estimates, which follow from conservation laws of charge and energy.

We show that those conservation laws hold if and only if there existsc∈R\ {0}such thatλ=cμ(Theorems3.3and 3.5below). On the basis of those conservation laws, we prove the existence of unique global solutions inL²and inH¹ regardless of the size of the Cauchy data respectively in Sections3.3and3.4. Local Cauchy problem with the data at t=0 in the weightedL²spacex⁻¹L²=FH¹is discussed in Section3.5under the mass resonance condition, which ensures the invariance of (1) under Galilei transformations. In Section3.6we prove the pseudo-conformal identity and apply it to the proof of the existence of unique global solutions with data att=0 inFH¹.In Section3.7, we derive the virial identity from the energy and pseudo-conformal identities and apply it to the proof of the non-existence of global solutions of negative energy with data inH¹∩FH¹.Section4is devoted to the existence of ground states for (1), which are defined as minimizers of action integrals for standing waves for (1) at frequency(ω,2ω)withω >0.The method of proof depends on Strauss’ compact embedding of the space of radially symmetricH¹functions intoL³: H_r¹⊂L³for 2n5 and on the concentration-compactness argument forn=1. In Section5, we prove that the best constant in a Gagliardo-–Nirenberg type inequality forn=4 is formulated in a variational setting and characterized by ground states at frequency(ω,2ω)=(1,2).In Section6, we prove the existence of threshold on the size of charge of the Cauchy data for which the corresponding solutions to (1) are global in time forn=4.Moreover, the threshold is calculated in terms of the ground states from Section5. This result is regarded as an analogue to Weinstein’s theory in the pseudo-conformal invariant case[26,27]. Under the mass resonance condition (3), we present an explicit repre- sentation formula of blow-up solutions at the threshold by means of the ground states from Section5. In Section7, we study the inverse condition of mass resonance, namely,m=2M, which reduces the problem of the system (1) to the corresponding problem of a single equation. We characterize the structure of ground states for a quadratic scalar field equation, which could clarify how the inverse mass ratio affects the motion of semitrivial standing waves[4,5,14,15].

Existence of stationary solutions to (1) forn=6 is also discussed in this setting. In Section8, we study (1) forn=1 in the framework of Lagrangian systems.

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2. Preliminaries

In this section we collect basic notation and lemmas which will be used subsequent sections. We refer the reader to[2,22,23]for general information. For anypwith 1p∞,L^p=L^p(Rⁿ)denotes the Lebesgue space onRⁿ. The usual scalar product on L² or (L²)ⁿ is denoted by (·,·). For any p with 1p∞ and any non-negative integerm,W_p^m, denotes the usual Sobolev space of ordermbuilt overL^p. Ifp=2,W₂^mis also written asH^m. For any intervalI⊂Rand any Banach spaceX, we denote byC(I;X)the space of strongly continuous functions from I toXand byL^p(I;X)the space of strongly measurable functionsufromI toXsuch that u(·);X ∈L^p(I ). For anypwith 1p∞,pis the dual exponent defined by 1/p+1/p=1.For anya, b∈R,a∨b=max(a, b).The Cauchy problem for (1) with data(u(t₀), v(t₀))=(u₀, v₀)given att=t₀will be treated in the form of the following system of integral equations:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

u(t )=U_m(t−t₀)u₀−i t t₀

U_m t−t

λv t

u t

dt,

v(t )=U_M(t−t₀)v₀−i t t₀

U_M t−t

μu² t

dt,

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whereU_m(t)=exp(i_2m^t )andU_M(t)=exp(i_2M^t )are free propagators with massesmandM, respectively. A pair of indices(q, r)with 2q, r∞is called admissible if 02/q=n/2−n/r1 with the exception(n, q, r)= (2,∞,2).

We use the following Strichartz estimates without particular comments.

Proposition 2.1.Letn1and let(q, r)and(qj, rj)be admissible forj=1,2.Then the following estimates hold U_m(·)φ;L^q

R;L^r C φ;L² and

G_t

0f;L^q²

I;L^r² C f;L^q¹

I;L^r¹ ,

wheret0∈R,I⊂Ris an interval witht0∈I,Gt₀is the integral operator defined as

(Gt₀f )(t)= t t0

U_m t−t

f t

dt, t∈I,

andCis a constant independent oft0,I, andf.

We use Proposition 2.1 to obtain local solutions to (5) by a contraction argument. To be more specific, local solutions to (5) are constructed as a pair of fixed point(u, v)of contraction mapping(u, v)→(Φ(u, v), Ψ (u, v)), where

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

Φ(u, v)

(t)=Um(t−t0)u0−i t t₀

Um

t−t λv

t u

t dt, Ψ (u, v)

(t)=UM(t−t0)v0−i t t₀

UM

t−t μu²

t dt,

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on a suitable complete metric space of functions onI= [t₀−T , t₀+T]for someT >0.

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3. Existence of solutions and non-existence of global solutions

3.1. Local existence ofH¹-solutions

In view of the scaling argument and available results on the Cauchy problem for a single nonlinear Schrödinger equation with power nonlinearities, it is natural to treat (5) inL²space forn4.For anyu₀, v₀∈L²we solve (5) in the spaces

X(I )=

C∩L^∞ I;L²

∩L⁴ I;L^∞

forn=1, X(I )=

C∩L^∞ I;L²

∩L^q⁰ I;L^r⁰

forn=2, where 0<2/q0=1−2/r0<1 withr₀sufficiently large,

X(I )=

C∩L^∞ I;L²

∩L²

I;L^2n/(n⁻²⁾

forn3

on the time intervalI= [t₀−T , t₀+T]withT >0.The associated norms are defined u;X(I ) = u;L^∞

L² ∨ u;L⁴

L^∞ forn=1, u;X(I ) = u;L^∞

L² ∨ u;L^q⁰

L^r⁰ forn=2, u;X(I ) = u;L^∞

L² ∨ u;L²

L^2n/(n⁻²⁾ forn3.

Theorem 3.1.Ifn3, then for anyρ >0there existsT (ρ) >0such that for any(u₀, v₀)∈L²×L²with u₀;L² ∨ v₀;L² ρ,(5)has a unique pair of solutions(u, v)∈X(I )×X(I )withI= [t₀−T (ρ), t₀+T (ρ)]. Ifn=4, then for any(u0, v0)∈L²×L², there existsT (u0, v0) >0such that(5)has a unique pair of solutions(u, v)∈X(I )×X(I ) withI= [t0−T (u0, v0), t0+T (u0, v0)].

Proof. We first consider the casen=1.We estimateΦ(u, v)andΨ (u, v)as Φ(u, v);X(I ) C u₀;L² +C uv;L¹

L² C u₀;L² +CT^3/4 v;L⁴

L^∞ u;L^∞ L² , Ψ (u, v);X(I ) C v₀;L² +CT^3/4 u;L⁴

L^∞ u;L^∞ L² . Similarly,

Φ(u, v)−Φ u, v

;X(I ) CT^3/4 u;L⁴

L^∞ + v;L⁴

L^∞ u−u;L^∞ L² + v−v;L^∞

L² , Ψ (u, v)−Ψ

u, v

;X(I ) CT^3/4 u;L⁴

L^∞ + u;L⁴

L^∞ u−u;L^∞ L² . We next consider the casen=2.We estimateΦ(u, v)andΨ (u, v)as

Φ(u, v);X(I ) C u₀;L² +C uv;L^q⁰ L^r⁰ C u₀;L² +CT^r⁰^/(r⁰⁺²⁾ v;L^r⁰

L^2r⁰^/(r⁰⁻²⁾ u;L^∞ L² , Ψ (u, v);X(I ) C v₀;L² +CT^r⁰^/(r⁰⁺²⁾ u;L^r⁰

L^2r⁰^/(r⁰⁻²⁾ u;L^∞ L² . Similarly,

Φ(u, v)−Φ u, v

;X(I ) CT^r⁰^/(r⁰⁺²⁾ u;L^r⁰

L^2r⁰^/(r⁰⁻²⁾ + v;L^r⁰

L^2r⁰^/(r⁰⁻²⁾

× u−u;L^∞

L² + v−v;L^∞ L² , Ψ (u, v)−Ψ

u, v

;X(I ) CT^r⁰^/(r⁰⁺²⁾ u;L^r⁰

L^2r⁰^/(r⁰⁻²⁾ + u;L^r⁰

L^2r⁰^/(r⁰⁻²⁾ u−u;L^∞ L² .

Note that u;L^r⁰(L^2r⁰^/(r⁰⁻²⁾) u;L^q⁰(L^r⁰) ^2/(r⁰⁻²⁾ u;L^∞(L²) ^(r⁰⁻^4)/(r⁰⁻²⁾.We now consider the casen=3.

We estimateΦ(u, v)andΨ (u, v)as

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Φ(u, v);X(I ) C u₀;L² +C uv;L^4/3 L^3/2 C u0;L² +CT^1/4 v;L²

L⁶ u;L^∞ L² , Ψ (u, v);X(I ) C v₀;L² +CT^1/4 u;L²

L⁶ u;L^∞ L² . Similarly,

Φ(u, v)−Φ u, v

;X(I ) CT^1/4 u;L²

L⁶ + v;L²

L⁶ u−u;L^∞ L² + v−v;L^∞

L² , Ψ (u, v)−Ψ

u, v

;X(I ) CT^1/4 u;L²

L⁶ + u;L²

L⁶ u−u;L^∞ L² . Therefore forn3 we have obtained the following estimates:

Φ(u, v);X(I ) C u₀;L² +CT¹⁻^n/4 u;X(I ) v;X(I ) , Ψ (u, v);X(I ) C v₀;L² +CT¹⁻^n/4 u;X(I ) ²,

Φ(u, v)−Φ u, v

;X(I ) CT¹⁻^n/4 u;X(I ) + v;X(I ) u−u;X(I ) + v−v;X(I ) , Ψ (u, v)−Ψ

u, v

;X(I ) CT¹⁻^n/4 u;X(I ) + u;X(I ) u−u;X(I ) .

Then the standard contraction argument on(u, v)→(Φ(u, v), Ψ (u, v))on a closed ball inX(I )×X(I )goes through by takingT >0 sufficiently small with respect toρ >0 via radius of the ball. This yields the existence and uniqueness of local solutions on[t₀−T , t₀+T]under the size restriction on the radius. The uniqueness of solutions without the size restriction of the radius follows by a similar argument by taking the size of successive time interval sufficiently small.

We finally consider the casen=4.We estimateΦ(u, v)andΨ (u, v)inL²(I;L⁴)as Φ(u, v);L²

L⁴ C Um(·)u0;L²

L⁴ +C uv;L¹ L² C U_m(·)u₀;L²

L⁴ +C v;L²

L⁴ u;L² L⁴ , Ψ (u, v);L²

L⁴ C UM(·)v0;L²

L⁴ +C u;L² L⁴ ². Similarly,

Φ(u, v)−Φ u, v

;L²

L⁴ C u;L²

L⁴ + v;L²

L⁴ u−u;L²

L⁴ + v−v;L² L⁴ , Ψ (u, v)−Ψ

u, v

;L²

L⁴ C u;L²

L⁴ + u;L²

L⁴ u−u;L² L⁴ .

Foru0,v0we know thatUm(·)u0, UM(·)v0∈L²(R;L⁴)and therefore the associated norms may be taken arbitrarily small by takingT >0 sufficiently small. Therefore the contraction argument works on a closed ball inL²(I;L⁴)with center at the origin and radius sufficiently small. Then the solution satisfies the integral equations (5) and then belongs toX(I )by the Strichartz estimates. 2

3.2. Local existence ofH¹-solutions

In view of the scaling argument and available results on the Cauchy problem for a single nonlinear Schrödinger equation with power nonlinearities, it is natural to treat (5) inH¹space forn6.For anyu₀, v₀∈H¹we solve (5) in the spaces

Y (I )=

C∩L^∞ I;H¹

∩L⁴ I;W_∞¹

forn=1, Y (I )=

C∩L^∞ I;H¹

∩L^q⁰ I;W_r¹

0

forn=2, where 0<2/q0=1−2/r0<1 withr0sufficiently large,

Y (I )=

C∩L^∞ I;H¹

∩L²

I;W_2n/(n¹ ₋₂₎

forn3

on the time intervalI= [t₀−T , t₀+T]withT >0.The associated norms are defined by

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u;Y (I ) = u;L^∞

H¹ ∨ u;L⁴

W_∞¹ forn=1, u;Y (I ) = u;L^∞

H¹ ∨ u;L^q⁰ W_r¹

0 forn=2, u;Y (I ) = u;L^∞

H¹ ∨ u;L²

W_2n/(n¹ ₋₂₎ forn3.

Theorem 3.2.Ifn5, then for anyρ >0there existsT (ρ) >0such that for any(u₀, v₀)∈H¹×H¹with u₀;H¹ ∨ v₀;H¹ ρ,(5)has a unique pair of solutions(u, v)∈Y (I )×Y (I )withI = [t₀−T (ρ), t₀+T (ρ)]. Ifn=6, then for any(u₀, v₀)∈H¹×H¹, there existsT (u₀, v₀) >0such that(5)has a unique pair of solutions(u, v)∈Y (I )×Y (I ) withI= [t₀−T (u₀, v₀), t₀+T (u₀, v₀)].

Proof. We first consider the casen3.The contraction argument inY (I )works in the same way as in the proof of Theorem3.1since necessary estimates are those of first derivatives ofΦ(u, v)andΨ (u, v), which depend on(u, v) essentially in a bilinear way. To be specific, we obtain

Φ(u, v);Y (I ) C u0;H¹ +CT¹⁻^n/4 u;Y (I ) v;Y (I ) , Ψ (u, v);Y (I ) C v₀;H¹ +CT¹⁻^n/4 u;Y (I ) ²,

Φ(u, v)−Φ u, v

;Y (I ) CT¹⁻^n/4 u;Y (I ) + v;Y (I ) u−u;Y (I ) + v−v;Y (I ) , Ψ (u, v)−Ψ

u, v

;Y (I ) CT¹⁻^n/4 u;Y (I ) + u;Y (I ) u−u;Y (I ) ,

from which the conclusion follows for n 3. We next consider the case 4 n 6. In this case the pair (4/(n−4), n/2)is admissible and the corresponding dual is given by(4/(8−n), n/(n−2)).As for the estimates on the Duhamel terms, the following bilinear estimate plays an essential role:

uv;L^4/(8⁻ⁿ⁾

I;W_n/(n¹ ₋₂₎ CT^3/2⁻^n/4 u;L^∞

I;H¹ v;L²

I;W_2n/(n¹ ₋₂₎ . Then the conclusion follows in the same way as in the proof of Theorem3.1. 2 3.3. Global existence ofL²-solutions

Letn4 and let(u, v)∈X(I )×X(I )be the unique pair of local solutions of (5) given in Theorem3.1. Then in the same way as in[20], we have

u(t );L² ²= u0;L² ²+2 Im t t₀

λv t

, u² t

dt, v(t );L² ²= v0;L² ²+2 Im

t t₀

μu² t

, v t

dt

for allt ∈I, where the last integrals of the right hand side are understood to be a duality between L^q(I;L^r)and L^q(I;L^r). For the conservation law of total charge it is natural to consider the following condition:

There exists a constantc∈R\{0}such thatλ=cμ. (7)

In fact, we have

Theorem 3.3.Letn4and letλandμsatisfy(7). Then the unique pair of local solutions(u, v)∈X(I )×X(I )of (5)given by Theorem3.1satisfies the following conservation law for allt∈I

u(t );L² ²+c v(t );L² ²= u0;L² ²+c v0;L² ².

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We now state the existence and uniqueness of globalL²solutions on the basis of the function spaceX(R):

X(R)=

C∩L^∞ R;L²

∩L⁴_loc R;L^∞

forn=1, X(R)=

C∩L^∞ R;L²

∩L^q_loc R;L^r

forn=2, where 0<2/q=1−2/r <1 withrsufficiently large,

X(R)=

C∩L^∞ R;L²

∩L²_loc R;L⁶

forn=3.

Theorem 3.4.Letn3and letλandμsatisfy(7)withc >0. Then for any(u₀, v₀)∈L²×L²,(5)has a unique pair of solutions(u, v)∈X(R)×X(R).Moreover,

u(t );L² ²+c v(t );L² ²= u₀;L² ²+c v₀;L² ² for allt∈R.

Proof. The theorem follows from Theorem 3.1and Theorem 3.3by the standard continuation argument of local solutions. 2

3.4. Global existence ofH¹-solutions

Letn6 and let(u, v)∈Y (I )×Y (I )be the unique pair of local solutions of (5) given by Theorem3.2. Then in the same way as in[20], we have

∇u(t );L² ²= ∇u₀;L² ²−2mRe t t0

λv t

, ∂_t u²

t dt, ∇v(t );L² ²= ∇v₀;L² ²−4MRe

t t0

μu² t

, ∂_tv t

dt for allt∈I.Therefore we have

Theorem 3.5.Letn6and letλandμsatisfy(7). Then the unique pair of local solutions(u, v)∈Y (I )×Y (I )of (5)given by Theorem3.2satisfies the following conservation law for allt∈I

1

2m ∇u(t );L² ²+ c

4M ∇v(t );L² ²+Re λ

v(t ), u²(t)

= 1

2m ∇u₀;L² ²+ c

4M ∇v₀;L² ²+Re λ

v₀, u²₀ .

We now state the existence and uniqueness of globalH¹solutions on the basis of the function spaceY (R):

Y (R)=

C∩L^∞ R;H¹

∩L⁴_loc R;W_∞¹

forn=1, Y (R)=

C∩L^∞ R;H¹

∩L^q_loc R;W_r¹

forn=2, where 0<2/q=1−2/r <1 withrsufficiently large,

Y (R)=

C∩L^∞ R;H¹

∩L²_loc

R;W_2n/(n¹ ₋₂₎

forn3.

To obtain an a priori estimate of solutions inH¹×H¹, it is convenient to introduce the following functionals Q(φ, ψ )= φ;L² ²+c ψ;L² ²,

K(φ, ψ )= 1

2m ∇φ;L² ²+ c

4M ∇ψ;L² ², P (φ, ψ )=Re

φ²ψ dx

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and

α₀=inf

J₀(φ, ψ ); (φ, ψ )∈H¹×H¹ , where

J0(φ, ψ )=K(φ, ψ )Q(φ, ψ )^1/2/P

|φ|,|ψ| .

Lemma 3.6.Letn=4and letm, M, c >0.Then there exists a constantC₀>0such that P

|φ|,|ψ|

C₀K

|φ|,|ψ| Q

|φ|,|ψ|1/2

C₀K(φ, ψ )Q(φ, ψ )^1/2 for all(φ, ψ )∈H¹×H¹.

Proof. By the Gagliardo–Nirenberg inequality:

φ;L³ C ∇φ;L² ^2/3 φ;L² ^1/3, we obtain

P

|φ|,|ψ|

C³ ∇|φ|;L² ^4/3 |φ|;L² ^2/3 ∇|ψ|;L² ^2/3 |ψ|;L² ^1/3 C³

2mK

|φ|,|ψ|2/3

Q(φ, ψ )^1/3 4M

c K

|φ|,|ψ|^1/3 1

cQ(φ, ψ ) 1/6

=C³

16m²M1/3

c⁻^1/2K

|φ|,|ψ|

Q(φ, ψ )^1/2. 2

Theorem 3.7.Letn4and letλandμsatisfy(7)withc >0. Ifn3, then for any(u0, v0)∈H¹×H¹,(5)has a unique pair of solutions(u, v)∈Y (R)×Y (R).Ifn=4, then for any(u0, v0)∈H¹×H¹with

|λ|Q(u₀, v₀)^1/2< α₀

(5)has a unique pair of solutions(u, v)∈Y (R)×Y (R).

Proof. By the standard continuation argument, it suffices to obtain a priori estimates onH¹norms ofuandv. By the following Gagliardo–Nirenberg inequality

φ;L³ C ∇φ;L² ^n/6 φ;L² ¹⁻^n/6, we estimate the interaction term in the energy as

λ

u², v|λ|C³ ∇u;L² ^n/3 u;L² ²⁻^n/3 ∇v;L² ^n/6 v;L² ¹⁻^n/6 |λ|C³(2mK)^n/6Q¹⁻^n/6

4M c K

n/12 1 cQ

1/2−n/12

= |λ|C³

16m²Mn/12

c⁻^1/2K(u, v)^n/4Q(u, v)^3/2⁻^n/4

= |λ|C³

16m²Mn/12

c⁻^1/2Q(u₀, v₀)^3/2⁻^n/4K(u, v)^n/4,

where we have used the conservation of charge. Ifn3, thenn/4<1 and the interaction term is dominated by an arbitrarily small constant multiple of the kinetic term of the form

εK(u, v)+CεQ(u0, v0)⁽⁶⁻^n)/(4⁻ⁿ⁾,

which implies the required a priori estimate. Ifn=4, we estimate λ

u², v|λ|P

|u|,|v|

|λ|/α₀

Q(u₀, v₀)^1/2K(u, v)

by Lemma3.6and the required a priori estimate follows if the coefficient toK(u, v)is less than 1. 2

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3.5. Galilei invariance of local solutions under mass resonance

Throughout this section we assume thatM=2mand the mass in the second equation is denoted by 2m. For the free propagatorU_m(t), we introduce the standard generator of Galilei transformations as

J_m=J_m(t)=U_m(t)xU_m(−t )=x+i t

m∇ =M_m(t)i t

m∇M_m(−t ), whereMm(t)=exp(i_2t^m|x|²),t=0.Then we have at least formally

J_m(vu)=M_m(t)i t m∇

M_2m(−t )v·M_m(−t )u

=

M_2m(t)i t

m∇M_2m(−t )v

u−v

M_m(t)i t

m∇M_m(−t )u

=2(J2mv)u−vJ_mu and

J2m

u²

=M2m(t)i t 2m∇

M_m(−t )u2

=uM_m(t)i t

m∇M_m(−t )u=uJ_mu.

For anyu₀, v₀∈L²withJ_m(t₀)u₀, J_2m(t₀)v₀∈L²we solve (5) in the spaceZ_m(I )×Z_2m(I ), where Z_m(I )=

u∈X(I ); J_mu∈X(I )

, I= [t₀−T , t₀+T], T >0 with norm

u;Z_m(I ) = u;X(I ) ∨ J_mu;X(I ) .

Theorem 3.8.Letn6andM=2m. Ifn5, then for anyρ >0there existsT (ρ) >0such that for any(u₀, v₀)∈ L²×L²with(J_m(t₀)u₀, J_2m(t₀)v₀)∈L²×L²and

u₀;L² ∨ J_m(t₀)u₀;L² ∨ v₀;L² ∨ J_2m(t₀)v₀;L² ρ

(5)has a unique pair of solutions(u, v)∈Z_m(I )×Z_2m(I )withI= [t₀−T (ρ), t₀+T (ρ)].Ifn=6, then for any (u0, v0)∈L²with(Jm(t0)u0, J2m(t0)v0)∈L²there existsT (u0, v0) >0such that(5)has a unique pair of solutions (u, v)∈Zm(I )×Z2m(I )withI= [t0−T (u0, v0), t0+T (u0, v0)].

Remark 3.1.Theorem3.8ensures the existence of local solutions of (5) which leave the domain of Galilei generators invariant. In the caset₀=0, the theorem is regarded as a smoothing effect of solutions in terms of Galilei generators.

Proof of Theorem 3.8. Let(u, v)∈Zm(I )×Z2m(I )forI= [t0−T , t0+T]with someT >0. We applyJm(t)and J2m(t)toΦ(u, v)andΨ (u, v), respectively and use

J_m(vu)=2(J2mv)u−vJ_mu, J_2m u²

=uJ_mu.

Then by a similar argument to that of proof of Theorem3.1, we prove Theorem3.8. 2 3.6. Galilei invariance of global solutions under mass resonance

As in Section7, we assume that the mass resonance conditionM=2m. Letn6 and let(u, v)∈Z_m(I )×Z_2m(I ) be the unique pair of local solutions given by Theorem3.8. Then in the same way as in[20], we have

J_m(t )u(t );L² ²= J_m(t₀)u₀;L² ²+2 Im t t₀

λJ_m(s)(vu)(s), J_m(s)u(s) ds, J2m(t)v(t);L² ²= J2m(t0)v0;L² ²+2 Im

t t₀

μJ2m(s) u²

(s), J2m(s)v(s) ds for allt∈I.

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Theorem 3.9.Letn6and letM=2m. Letλandμsatisfy(7). Then the unique pair of solutions(u, v)∈Z_m(I )× Z_2m(I )of (5)given by Theorem3.8satisfies the following identity for allt∈I

J_m(t )u(t );L² ²+c J_2m(t)v(t);L² ²+ 2 mt²Re

λ

v(t ), u²(t)

= J_m(t₀)u₀;L² ²+c J_2m(t₀)v₀;L² ²+ 2 mt₀²Re

λ v₀, u²₀

+4−n m

t t0

sRe λ

v(s), u²(s) ds.

Proof. For simplicity, we give a formal calculation for the proof. Actual proof requires a regularization procedure, see[2,8]. We compute by the condition (7)

Imλ

Jm(vu), Jmu

+cImμ J2m

u² , J2mv

=2 Imλ(J_2mv, uJ_mu)−Imλ

v, (J_mu)²

−cImμ(J_2mv, uJ_mu)

=Imλ(J2mv, uJ_mu)−Imλ

v, (J_mu)²

= t

2mI+ t² 2m²II, where

I=Re

−2λ(xv, u∇u)+λ

∇v, xu²

+4λ(v, ux· ∇u) , II=Im

λ(∇v, u∇u)+2λ

v, (∇u)² . ThenI is written as

I=Re

2λ(xv, u∇u)+λ

∇v, xu²

=Re λ

xv,∇ u²

+λ

x· ∇v, u²

= −nReλ v, u²

, whileIIis written as

II=Im

λ(∇v, u∇u)−2λ(∇v, u∇u)−2λ(v, uu)

=1 2Imλ

v, u²

+2 Imλ(u, uv)

= −2mReλ

∂_tv, u²

−4mReλ(∂tu, uv)= −2md dtReλ

v, u² . Therefore, we obtain

J_m(t )u(t );L² ²+c J_2m(t)v(t);L² ²− J_m(t₀)u₀;L² ²−c J_2m(t₀)v₀;L² ²

= t t₀

s mI+ s²

m²II

ds

= 1 m

t t₀

−d ds

2s²Reλ v, u²

+(4−n)sReλ v, u²

ds, which is the required identity. 2

We now introduce Z_m(R)=

u∈X(R); J_mu∈X(R) .

Theorem 3.10.Letn4and letM=2m.Letλandμsatisfy(7)withc >0. Ifn3, then for any(u₀, v₀)∈L²×L² with(J_m(t₀)u₀, J_2m(t₀)v₀)∈L²×L²,(5)has a unique pair of solutions(u, v)∈Z_m(R)×Z_2m(R).Ifn=4, then for any(u₀, v₀)∈L²×L²with(J_m(t₀)u₀, J_2m(t₀)v₀)∈L²×L²and

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|λ|Q(u₀, v₀)^1/2< α₀

(5)has a unique pair of solutions(u, v)∈Z_m(R)×Z_2m(R).

Proof. By the standard continuation argument, it suffices to obtain a priori estimates on J_mu;L² ∨ J_2mv;L² .If we notice that

J_mu;L² ²+c J_2mv;L² ²= t²

m² ∇M_m⁻¹u;L² ²+c

4 ∇M_2m⁻¹v;L² ²

=2t² mK

M_m⁻¹u, M_2m⁻¹v , v, u²

=

M_2m⁻¹v,

M_m⁻¹u2 ,

then an analogous argument to that of Section6implies the theorem. 2

3.7. Non-existence of global solutions with negative energy under mass resonance

In this section we assume mass resonance conditionM=2m.Letn6 and let(u, v)∈(Zm(I )×Z2m(I ))∩ (Y (I )×Y (I ))be the unique pair of local solutions given by Theorems3.2and3.8with data(u0, v0)∈H¹×H¹ att=t₀satisfying(J_m(t₀)u₀, J_2m(t₀)v₀)∈L²×L², whereI is the intersection of time intervals in Theorems3.2 and3.8. From now on, we taket₀=0 for simplicity. The corresponding pair of local solutions(u, v) satisfies the virial identity:

Theorem 3.11.Let n6and let M=2m.Let λ andμ satisfy (7). Let (u₀, v₀)∈H¹×H¹ satisfy (xu₀, xv₀)∈ L²×L² and let (u, v)∈(Z_m(I )×Z_2m(I ))∩(Y (I )×Y (I )) the corresponding pair of local solutions given by Theorems3.2and3.8witht₀=0.Then

xu(t);L² ²+c xv(t);L² ²= xu₀;L² ²+c xv₀;L² ²+P₀t+ n 2mE₀t² +4−n

m t 0

(t−s) 1

2m ∇u(s);L² ²+ c

8m ∇v(s);L² ² ds for allt∈I, where

P0= 2

mIm(∇u0, xu0)+ c

mIm(∇v0, xv0), E₀= 1

2m ∇u₀;L² ²+ c

8m ∇v₀;L² ²+Reλ v₀, u²₀

.

Proof. For simplicity, we give a formal calculation for the proof. Actual proof requires a regularization procedure, see[2,8]. We compute

d

dt xu;L² ²+c xv;L² ²

=2 Im

i∂_tu,|x|²u

+2cIm

i∂_tv,|x|²v

= 2

mIm(∇u, xu)+ c

mIm(∇v, xv), where the last two terms are rewritten as

2

mIm(∇u, xu)= −1

t J_mu;L² ²− xu;L² ² + t

m² ∇u;L² ², c

mIm(∇v, xv)= −c

t J_2mv;L² ²− xv;L² ² + ct

4m² ∇v;L² ². Therefore we obtain by a direct calculation

d²

dt² xu;L² ²+c xv;L² ²

= n

mE₀+4−n m

1

2m ∇u;L² ²+ c

8m ∇v;L² ² , where we have used Theorems3.5and3.9. This proves the theorem. 2