We show global well-posedness and scattering in the energy space

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 96, pp. 1–24.

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

DEFOCUSING FOURTH-ORDER COUPLED NONLINEAR SCHR ¨ODINGER EQUATIONS

RADHIA GHANMI, TAREK SAANOUNI

Abstract. We study the initial value problem for some defocusing coupled nonlinear fourth-order Schr¨odinger equations. We show global well-posedness and scattering in the energy space.

1. Introduction

This manuscript is concerned with the initial value problem for some defocusing fourth-order Schr¨odinger system with power-type nonlinearities

iu˙_j+ ∆²u_j+X^m

k=1

a_jk|uk|^p

|uj|^p−2u_j= 0;

uj(0,·) =ψj,

(1.1)

whereuj:R×R^N →Cforj∈[1, m] andajk=akj are positive real numbers.

Fourth-order Schr¨odinger equations have been introduced by Karpman [9] and Karpman-Shagalov [10] to take into account the role of small fourth-order disper- sion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity.

The m-component classical coupled nonlinear Schr¨odinger system with power- type nonlinearity

iu˙j+ ∆uj =±

m

X

k=1

ajk|uk|^p|uj|^p−2uj, (1.2) arises in many physical problems. This models physical systems in which the field has more than one component. For example, in optical fibers and waveguides, the propagating electric field has two components that are transverse to the direction of propagation. Readers are referred to various works [8, 27] for the derivation and applications of this system. For mathematical point of view, well-posedness issues of (CN LS)_p were investigated by many authors. Indeed, global existence of solutions and scattering hold [19, 3, 26, 24, 25].

2010Mathematics Subject Classification. 35Q55.

Key words and phrases. Nonlinear fourth-order Schr¨odinger system; global well-posedness;

scattering.

c

2016 Texas State University.

Submitted February 29, 2016. Published April 12, 2016.

1

System (1.1) is a mixture of the two previous problems. A solution u :=

(u1, . . . , um) to (1.1) formally satisfies respectively conservation of the mass and the energy

M(uj) :=

Z

R^N

|uj(t, x)|²dx=M(ψj);

E(u(t)) :=1 2

m

X

j=1

Z

R^N

|∆uj|²dx+ 1 2p

m

X

j,k=1

a_jk Z

R^N

|uj(t, x)|^p|uk(t, x)|^pdx

=E(u(0)).

To use the conservation laws, it is natural to study (1.1) inH², called energy space.

Problem (1.1) is a natural extension of the classical one component fourth-order Schr¨odinger equation which was first studied in [5], where various properties of the equation in the subcritical regime were described. Related references [2] gave sharp dispersive estimates for the biharmonic Schr¨odinger operator which lead to the Strichartz estimates. The model case given by a pure power nonlinearity is of particular interest. Indeed, the question of well-posedness in the energy space H² was widely investigated. We denote forp >1 the fourth-order Schr¨odinger problem iu˙ + ∆²u±u|u|^p−1= 0, u:R×R^N →C. (1.3) This equation satisfies a scaling invariance. Indeed, ifuis a solution to (1.3) with datau0, thenuλ:=λ^p−14 u(λ⁴·, λ·) is a solution to (1.3) with dataλ^p−14 u0(λ·). For sc:= ^N₂ −_p−1⁴ , the space ˙H^sc whose norm is invariant under the dilatationu7→uλ

is relevant in this theory. Whensc= 2 which is the energy critical case, the critical power ispc :=^N_N⁺⁴₋₄,N ≥5. Pausader [20] established global well-posedness in the defocusing subcritical case, namely 1 < p < pc. Moreover, he established global well-posedness and scattering for radial data in the defocusing critical case, namely p=pc. The same result without radial condition was obtained by Miao, Xu and Zhao [17], for N ≥ 9. See also [15], for similar results in the more general case sc ≥1. The focusing case was treated by the last authors in [16]. They obtained results similar to one proved by Kenig and Merle [12, 11] in the classical Schr¨odinger case. See [23] in the case of exponential nonlinearity.

In this note, which seems to be one of the first papers studying a system of nonlinear coupled fourth-order Schr¨odinger equations, we combine in some meaning the two problems (1.3) and (CN LS)_p. Thus, we have to overcome two difficulties.

The first one is the presence of a bilaplacian in Schr¨odinger operator and the second one is the coupled nonlinearities. It is the purpose of this manuscript to obtaining global well-posedness in the energy space and scattering of (1.1) via Morawetz estimate.

The rest of the paper is organized as follows. The next section contains the main results and some technical tools needed in the sequel. The third and fourth sections are devoted to prove well-posedness of (1.1). In section five, scattering is established. In appendix, we give a proof of Morawetz estimate and a blow-up criterion.

We close this section with some notations. Define the product space H :=H²(R^N)× · · · ×H²(R^N) = [H²(R^N)]^m,

whereH²(R^N) is the usual Sobolev space endowed with the complete norm kuk_H2(R^N):=

kuk²_L2(R^N)+k∆uk²_L2(R^N)

^1/2 . Let us denote the real number

p^∗:=

( _N

N−4 ifN >4;

∞ if 1≤N≤4.

We mention thatCwill denote a constant which may vary from line to line and ifA andB are non negative real numbers,A.B means thatA≤CB. For 1≤r≤ ∞ and (s, T)∈[1,∞)×(0,∞), we denote the Lebesgue spaceL^r:=L^r(R^N) with the usual normk · kr:=k · kL^r,k · k:=k · k2 and

kukL^s_T(L^r):=Z T 0

ku(t)k^s_rdt^1/s

, kukL^s(L^r):=Z +∞

0

ku(t)k^s_rdt^1/s .

For simplicity, we denote the usual Sobolev space W^s,p := W^s,p(R^N) and H^s :=

W^s,2. IfX is an abstract spaceC_T(X) :=C([0, T], X) stands for the set of contin- uous functions valued in X and X_rd is the set of radial elements in X, moreover for an eventual solution to (1.1), we denoteT^∗>0 its lifespan.

2. Background and main results

In what follows, we give the main results and collect some estimates needed in the sequel.

2.1. Main results. First, local well-posedness of the fourth-order Schr¨odinger problem (1.1) is claimed.

Theorem 2.1. Let 1≤N ≤8,2≤p≤p^∗ andΨ∈H. Then, there existT^∗>0 and a unique maximal solution to (1.1),u∈C([0, T^∗), H). Moreover,

(1) u∈ L

8p N(p−1)

loc ([0, T^∗], W^2,2p)^(m)

;

(2) usatisfies conservation of the energy and the mass;

(3) T^∗=∞in the subcritical case (2≤p < p^∗).

Remark 2.2. The artificial conditionp≥2, which is due to some technical diffi- culty, requires the restrictionN ≤8.

Second, system (1.1) scatters in the energy space. Indeed, every global solution of (1.1) is asymptotic, ast→ ±∞, to a solution of the associated linear fourth-order Schr¨odinger system.

Theorem 2.3. Let 4 < N < 8 and 2 ≤p < p^∗. Take u ∈ C(R, H) be a global solution to (1.1). Then

u∈ L^N(p−1)8p (R, W^2,2p)(m)

and there exists(ψ^±₁, . . . , ψ_m^±)∈H such that

t→±∞lim ku(t)−(e^it∆2ψ^±₁, . . . , e^it∆2ψ_m^±)k_H2 = 0.

Remark 2.4. When proving scattering, the intermediate result Proposition 4.2 gives a decay of global solutions to (1.1), in some Lebesgue norms.

Finally, in the critical case, global existence and scattering for small data hold in the energy space.

Theorem 2.5. Let 4 < N ≤ 8 and p = p^∗. There exists 0 > 0 such that if Ψ := (ψ1, . . . , ψm) ∈ H satisfies ξ(Ψ) := Pm

j=1

R

R^N|∆ψj|²dx ≤0, system (1.1) possesses a unique global solutionu∈C(R, H), which scatters.

In the next subsection, we give some standard estimates needed in the paper.

2.2. Tools. We start with some properties of the free fourth-order Schr¨odinger kernel.

Proposition 2.6. Denoting the free operator associated to the fourth-order frac- tional Schr¨odinger equation

e^it∆2u0:=F⁻¹(e^it|y|4)∗u0, yield

(1) e^it∆2u₀ is the solution to the linear problem associated to (1.3);

(2) e^it∆2u₀∓iRt

0e^i(t−s)∆2u|u|^p−1ds is the solution to the problem (1.3);

(3) (e^it∆2)^∗=e^−it∆2;

(4) e^it∆2 is an isometry of L².

Now, we give the so-called Strichartz estimate [20].

Definition 2.7. A pair (q, r) of positive real numbers is said to be admissible if 2≤q, r≤ ∞, (q, r, N)6= (2,∞,4) and 4

q =N1 2−1

r .

Proposition 2.8. Let two admissible pairs (q, r), (a, b). There exists a positive real number C:=C_q,a such that for anyT >0,

kuk_L^q

T(W^2,r)≤C

ku₀k_H2+kiu˙ + ∆²uk_La0 T(W^2,b0)

; (2.1)

k∆uk_L^q

T(L^r)≤C

k∆u0k_L2+kiu˙+ ∆²uk

L²_T( ˙W^1,N+22N )

. (2.2)

The following Morawetz estimate which is essential in proving scattering, is proved in the appendix, in the spirit of [17, 18].

Proposition 2.9. Let 5≤N ≤8,2≤p≤p^∗ andu∈C(I, H)be the solution to (1.1). Then,

(1) ifN >5,

m

X

j=1

Z

I

Z

R^N×R^N

|uj(t, x)|²|uj(t, y)|²

|x−y|⁵ dx dy dt.u1; (2.3) (2) ifN = 5,

m

X

j=1

Z

I

Z

R⁵

|uj(t, x)|⁴dxdt.u1. (2.4) Let us gather some useful Sobolev embeddings [1].

Proposition 2.10. The continuous injections hold

(1) W^s,p(R^N),→L^q(R^N)whenever1< p < q <∞,s >0and ¹_p ≤ ¹_q +_N^s;

(2) W^s,p1(R^N),→W^{s−N(p11−p12),p2}(R^N)if 1≤p1≤p2<∞.

Now, we give some fractional Gagliardo-Nirenberg inequality [7].

Lemma 2.11. Let 1< p, p₁, p₂<∞,s, s₁∈Randµ∈[0,1]. Then, the fractional inequality

kukH˙^s,p .kuk^1−µ_Lp0kuk^µ_˙

H^s1,p1, holds whenever

N

p −s= (1−µ)N p0

+µ(N p1

−s1) and s≤µs1. We close this subsection with some absorption result.

Lemma 2.12. Let T >0 andX ∈C([0, T],R+)such that X≤a+bX^θ on [0, T], wherea, b >0,θ >1,a <(1−¹_θ) ¹

(θb)¹θ

andX(0)≤ ¹

(θb)^θ−11

. Then

X ≤ θ

θ−1a on [0, T].

Proof. The functionf(x) :=bx^θ−x+ais decreasing on [0,(bθ)^1−θ1 ] and increasing on [(bθ)^1−θ1 ,∞). The assumptions imply thatf((bθ)^1−θ1 )<0 andf(_θ−1^θ a)≤0. As f(X(t))≥0,f(0)>0 andX(0)≤(bθ)^1−θ1 , we conclude the proof by a continuity

argument.

3. Local well-posedness

This section is devoted to prove Theorem 2.1. The proof contains two steps.

First we prove existence of a unique local solution to (1.1), second we establish global existence in the subcritical case.

3.1. Local existence and uniqueness. We use a standard fixed point argument.

(1)Subcritical case 2≤p < p^∗. ForT, R >0, we denote the space ET ,R:=n

u∈(CT(H²)∩L

8p N(p−1)

T (W^2,2p))^(m): kuk

(L^∞_T(L²)∩L

8p N(p−1)

T (L^2p))^(m)

+k∆uk

(L^∞_T(L²)∩L

8p N(p−1)

T (L^2p))^(m)

≤Ro

endowed with the distance d(u,v) :=

m

X

j=1

kuj−v_jkL^∞_T(L²)+kuj−v_jk

L

8p N(p−1) T (L^2p)

.

Define the function φ(u)(t) :=T(t)Ψ−i

m

X

k=1

a_jk Z t

0

T(t−s)

|uk|^p|u1|^p−2u₁, . . . ,|uk|^p|um|^p−2u_m ds,

where T(t)Ψ := (e^it∆2ψ₁, . . . , e^it∆2ψ_m). We prove the existence of some small T, R >0 such thatφis a contraction ofE_{T ,R}.

• First step 3< N ≤8. Take u,v∈ET ,R, applying the Strichartz estimate (2.1), we obtain

d(φ(u), φ(v)).

m

X

j,k=1

k|uk|^p|uj|^p−2uj− |vk|^p|vj|^p−2vjk

L

8p p(8−N)+N(L

2p 2p−1)

.

To derive the contraction, consider the function

f_j,k:C^m→C,(u₁, . . . , u_m)7→ |u_k|^p|u_j|^p−2u_j. With the mean value Theorem,

|fj,k(u)−fj,k(v)|

.max{|uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2,|vk|^p|vj|^p−2+|vk|^p−1|vj|^p−1}|u−v|. (3.1) Using H¨older’s inequality, Sobolev embedding and denoting the quantity

(I) :=kf_j,k(u)−f_j,k(v)k

L

8p p(8−N)+N

T (L

2p 2p−1)

,

we compute via a symmetry argument (I).

|u_k|^p−1|u_j|^p−1+|u_k|^p|u_j|^p−2

|u−v|

L

8p p(8−N)+N

T (L

2p 2p−1)

.T

8p−2N(p−1)

8p ku−vk

L

8p N(p−1) T (L^2p)

|uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2 L^∞_T(L

p p−1)

.T^{4p−N(p−1)4p} ku−vk

L

8p N(p−1) T (L^2p)

kukk^p−1_L∞

T(L^2p)kujk^p−1_L∞ T(L^2p)

+kukk^p_L∞

T(L^2p)kujk^p−2_L∞ T(L^2p)

.T^{4p−N(p−1)4p} ku−vk

L

8p N(p−1) T (L^2p)

kukk^p−1_L∞

T(H²)kujk^p−1_L∞ T(H²)

+kukk^p_L∞

T(H²)kujk^p−2_L∞ T(H²)

. Thus kfj,k(u)−fj,k(v)k

L

8p p(8−N)+N

T (L

2p 2p−1)

.T^{4p−N(p−1)4p} kuk^2(p−1)_L∞

T(H)ku−vk

L

8p N(p−1) T (L^2p)

. (3.2)

Then

d(φ(u), φ(v)).T^{4p−N(p−1)4p} R^2(p−1)d(u,v).

Moreover, takingv= 0 in the previous inequality, yields kφ(u)k

(L^∞_T(L²)∩L

8p N(p−1)

T (L^2p))^(m) .kΨk+T^{4p−N(p−1)4p} R^2p−1. It remains to estimate the quantity

(A) :=k∆(f_j,k(u))k

L

8p p(8−N)+N

T (L

2p 2p−1)

.kD(fj,k)(u)∆uk

L

8p p(8−N)+N

T (L

2p 2p−1)

+k|∇u|²D²(fj,k)(u)k

L

8p p(8−N)+N

T (L

2p 2p−1)

.(I1) + (I2).

Via H¨older inequality and Sobolev embedding, we obtain (I1).k∆uk

L

8p N(p−1) T (L^2p)

|uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2

L

8p 8p−2N(p−1)

T (L

p p−1)

.T^{4p−N(p−1)4p} k∆uk

L

8p N(p−1) T (L^2p)

ku_kk^p−1_L∞

T(L^2p)ku_jk^p−1_L∞ T(L^2p)

+ku_kk^p_L∞

T(L^2p)ku_jk^p−2_L∞ T(L^2p)

.T^{4p−N(p−1)4p} k∆uk

L

8p N(p−1) T (L^2p)

kukk^p−1_L∞

T(H²)kujk^p−1_L∞ T(H²)

+kukk^p_L∞

T(H²)kujk^p−2_L∞ T(H²)

.T

4p−N(p−1)

4p k∆uk

L

8p N(p−1) T (L^2p)

kuk^2(p−1)_L∞ T(H).

Using the interpolation inequalityk∇ · k²_2p.k · k2pk∆· k2p, we obtain k|∇u|²(fj,k)ii(u)k

L

2p 2p−1 x

.k|∇u|²(|uk|^p−2|uj|^p−1+|uk|^p|uj|^p−3)k

L

2p 2p−1 x

.k∇uk²_L2p x

ku_kk^p−2

L^2p_x ku_jk^p−1

L^2p_x +ku_kk^p

L^2p_x ku_jk^p−3

L^2p_x

.k∆uk_L2p

x kuk^2p−2_H . This implies that

(I2).kk∆uk_L2pkuk^2p−2_H k

L

8p p(8−N)+N T

.T^{4p−N(p−1)4p} k∆uk

L

8p N(p−1) T (L^2p)

kuk^2p−2_L∞ T(H). Then

kφ(u)k

(L^∞_T(L²)∩L

8p N(p−1)

T (L^2p))^(m)

+k∆(φ(u))k

(L

8p N(p−1)

T (L^2p)∩L^∞_T(L²))^(m)

≤C

kΨkH+T^{4p−N(p−1)4p} R^2p−1 .

ChoosingR > CkΨk_H andT >0 sufficiently small via the fact that 2≤p < p^∗,φ is a contraction ofE_{T ,R}.

• Second step 1≤N ≤3. In this case, we use the Sobolev embeddingH²,→L^∞. Applying Strichartz estimate tou,v∈ET ,Ryields

d(φ(u), φ(v)).

m

X

j,k=1

k|u_k|^p|u_j|^p−2u_j− |v_k|^p|v_j|^p−2v_jk_L1 T(L²)

.

m

X

j,k=1

k|uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2kL^∞_T(L^∞)ku−vk_L¹

T(L²)

.Tkuk^2(p−1)_L∞

T(H²)ku−vkL^∞_T(L²)

.T R^2(p−1)d(u,v).

(3.3)

It remains to estimate

(B) :=k∆(fj,k(u))k_L¹

T(L²)

.kD(f_j,k)(u)∆uk_L1

T(L²)+k|∇u|²D²(f_j,k)(u)k_L1 T(L²). Thanks H¨older and Sobolev inequalities, we obtain

kD(fj,k)(u)∆uk_L¹

T(L²).k∆uk_L¹

T(L²)k|uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2kL^∞_T(L^∞)

.Tk∆uk_L^∞

T(L²)kuk^2(p−1)_L∞ T(H²)

.T R^2p−1.

Using the Sobolev injectionH²,→W^1,4, we obtain k|∇u|²(f_j,k)_ii(u)k_L¹

T(L²).k|∇u|²(|uk|^p−2|uj|^p−1+|uk|^p|uj|^p−3)k_L¹

T(L²)

.Tk∇uk²_L∞

T(L⁴)kuk^2p−3_L∞ T(H²)

.T R^2p−1. This implies

kφ(u)k

(L

8p N(p−1)

T (L^2p)∩L^∞_T(L²))^(m)

+k∆(φ(u))k

(L

8p N(p−1)

T (L^2p)∩L^∞_T(L²))^(m)

≤C

kΨkH+T R^2p−1 .

ChoosingR > CkΨk_H andT >0 sufficiently small,φis a contraction ofE_{T ,R}. Finally, thanks to a classical fixed point Theorem, We deduce the existence of a fixed pointu∈BT(R), which is a solution to (1.1). Moreover, uniqueness follows thanks to (3.2) and (3.3).

(2)Critical case 4< N ≤8 andp=p^∗. The proof follows by arguing as in the subcritical case, where we take rather thanET ,R, the complete space

FT ,ρ:=n u∈(L

8p N(p−1)

T (W^2,2p))^(m):kuk

(L

8p N(p−1)

T (W^2,2p))^(m)

≤ρo

endowed with the distance

d(u,v) =ku−vk

(L

8p N(p−1)

T (L^2p))^(m)

,

via the fact that limT→0kT(t)Ψk

(L

8p N(p−1)

T (L^2p))^(m)

= 0 and the next result.

Lemma 3.1. Let Ψ∈H and suppose that u∈(L

8p N(p−1)

T (W^2,2p))^(m) is a solution to (1.1). Then, there exists0< T⁰≤T such that u∈CT⁰(H).

Proof. Using the previous computation via Duhamel formula (second point in Proposition 2.6), yields

kuk_L^∞

T(H).kΨkH+kuk^2(p−1)_L∞ T(H)kuk

(L

8p N(p−1)

T (W^2,2p))^(m)

.

The proof is complete thanks to Lemma 2.12.

3.2. Existence of global solutions. We prove that the maximal solution of (1.1) is global in the subcritical case. The global existence is a consequence of energy conservation and previous calculations. Letu∈C([0, T^∗), H) be the unique max- imal solution of (1.1). We prove thatuis global. By contradiction, suppose that T^∗<∞. For 0< s < T^∗ consider the problem

iv˙_j+ ∆²v_j=X^m

k=1

|vk|^p

|vj|^p−2v_j; vj(s,·) =uj(s,·).

(3.4) By the same arguments used in the local existence, we can find a real numberτ >0 and a solutionv= (v₁, . . . , v_m) to (3.4) onC [s, s+τ], H). Using the conservation

of energy we see thatτ does not depend ons. Thus, if we letsbe close toT^∗such thatT^∗< s+τ, this fact contradicts the maximality ofT^∗.

4. Scattering

This section we establish Theorem 2.3 about the scattering of (1.1). For any time slabI, take the Strichartz space

S(I) :=C(I, H²)∩L^N(p−1)8p (I, W^2,2p) endowed the norm

kukS(I):=kukL^∞(I,H²)+kuk

L

8p

N(p−1)(I,W^2,2p). The first intermediate result is as follows.

Lemma 4.1. For any time slabI, we have ku(t)−e^it∆2Ψk_(S(I))(m) .kuk

2pN(p−1)−8p N(p−1)

L^∞(I,L^2p)^(m)kuk

8p−N(p−1) N(p−1)

L

8p

N(p−1)(I,W^2,2p)(m), wheree^it∆2(Ψ₁, . . . ,Ψ_m) := (e^it∆2Ψ₁, . . . , e^it∆2Ψ_m).

Proof. Using Strichartz estimate, we have ku(t)−e^it∆2Ψk_(S(I))(m) .

m

X

j,k=1

kf_j,k(u)k

L

8p

p(8−N)+N(I,W^2,

2p 2p−1)

.

Thanks to H¨older inequality, we obtain kfj,k(u)k

L

2p 2p−1 x

.

|uk|^p|uj|^p−1

L

2p 2p−1 x

.kukk^p

L^2p_x kujk^p−1

L^2p_x . Lettingθ:= ^8p−N_2N(p−1)^(p−1), we obtain the inequality

1

2 ≤θ≤p−1 2.

The left part of the inequality follows fromp≤p^∗. DenotingX:=p−1, the right part of the claim is equivalent to

T(X) :=N X²+ (N−4)X−4≥0.

T has two roots X1 = −1 < 0 < X2 = _N⁴, since 2 ≤ p ≤ p^∗, it follows that X =p−1≥X2. This proves of the inequality.

Now, using an interpolation argument, write kf_j,k(u)k

L

8p p(8−N)+N(I,L

2p 2p−1)

.

kukk^p_L2pkujk^p−1_L2p

L

8p p(8−N)+N(I)

.kukk^p−_L∞¹²(I,L^−θ2p)kujk^p−_L∞¹²(I,L^−θ2p)

kukk^θ+_L2p¹²kujk^θ−_L2p¹²

L

8p p(8−N)+N(I)

.kukk^p−_L∞¹²(I,L^−θ2p)kujk^p−_L∞¹²(I,L^−θ2p)kukk^θ+12

L

8p

N(p−1)(I,L^2p)

kujk^θ−12

L

8p

N(p−1)(I,L^2p)

.

Then

kfj,k(u)k

L

8p p(8−N)+N(I,L

2p 2p−1)

.kukk^p−_L∞¹²(I,L^−θ2p)kujk^p−_L∞¹²(I,L^−θ2p)kukk^θ+12

L

8p

N(p−1)(I,L^2p)

kujk^θ−12

L

8p

N(p−1)(I,L^2p)

.ku_kk^p−_L∞¹²(I,L^−θ2p)ku_kk^θ+12

L

8p

N(p−1)(I,L^2p)

ku_jk^p−_L∞¹²(I,L^−θ2p)ku_jk^θ−12

L

8p

N(p−1)(I,L^2p)

.kuk

2pN(p−1)−8p N(p−1)

L^∞(I,L^2p)(m)kuk

8p−N(p−1) N(p−1)

L

8p

N(p−1)(I,L^2p)(m).

(4.1)

It remains to estimate the quantity (I) := k∆(f_j,k(u))k

L

8p p(8−N)+N(I,L

2p 2p−1)

. We write

(I).

m

X

i=1

k∆u(fj,k)i(u) k

L

8p p(8−N)+N(I,L

2p 2p−1)

+k|∇u|²(fj,k)ii(u)k

L

8p p(8−N)+N(I,L

2p 2p−1)

.(I1) + (I2).

Using H¨older inequality, we obtain k∆u(fj,k)_i(u)k

L

2p 2p−1 x

.

∆u |uk|^p−1|uj|^p−1+|uk|^p|uj|^p−2

L

2p 2p−1 x

.k∆uk_(L2p x)^m

kukk^p−1

L^2p_x kujk^p−1

L^2p_x +kukk^p

L^2p_x kujk^p−2

L^2p_x

.

Lettingµ:=^{4p−N(p−1)}_N(p−1) , we obtain the inequality µ≤ p

2 ≤p−1.

Note that

p≥2µ⇔p≥8p−2N(p−1) N(p−1)

⇔N p(p−1)≥8p−2N(p−1)

⇔T(X:=p−1) :=N X²+ (3N−8)X−8≥0.

T has two rootsp₁<0< p₂. Since T(1) = 4(N−4)≥0 andX =p−1≥1. The inequality is proved.

Now, using H¨older inequality (I₁).

k∆uk_(L2p)^m

ku_kk^p−1_L2pku_jk^p−1_L2p +ku_kk^p_L2pku_jk^p−2_L2p

_Lp(8−N)+N^8p .k∆uk

L

8p

N(p−1)(I,L^2p)^(m)

×

kukk^p−1−µ_L∞(I,L^2p)kujk^p−1−µ_L∞(I,L^2p)

kukk^µ_L2pkujk^µ_L2p

L

8p 8p−2N(p−1)

+kukk^p−2µ_L∞(I,L^2p)kujk^p−2_L∞(I,L^2p)kkukk^2µ_L2pk

L

8p 8p−2N(p−1)

.k∆uk

L

8p

N(p−1)(I,L^2p)^(m)

×

kukk^p−1−µ_L∞(I,L^2p)kujk^p−1−µ_L∞(I,L^2p)kukk^µ

L

8p

N(p−1)(I,L^2p)

kujk^µ

L

8p

N(p−1)(I,L^2p)

+kukk^p−2µ_L∞(I,L^2p)kujk^p−2_L∞(I,L^2p)kukk^2µ

L

8p

N(p−1)(I,L^2p)

.

Then,A:=Pm

i,j,k=1k∆u(f_j,k)_i(u)k

L

8p p(8−N)+N(I,L

2p 2p−1)

satisfies A.k∆uk

L

8p

N(p−1)(I,L^2p)(m)

X^m

k=1

kukk^p−1−µ_L∞(I,L^2p)kukk^µ

L

8p

N(p−1)(I,L^2p)

×

m

X

j=1

kujk^p−1−µ_L∞(I,L^2p)kujk^µ

L

8p

N(p−1)(I,L^2p)

+

m

X

k=1

kukk^p−2µ_L∞(I,L^2p)kukk^2µ

L

8p

N(p−1)(I,L^2p) m

X

j=1

kujk^p−2_L∞(I,L^2p)

.

So,

A.kuk

L

8p

N(p−1)(I,W^2,2p)(m)kuk

2pN(p−1)−8p N(p−1)

L^∞(I,L^2p)(m)kuk

8p−2N(p−1) N(p−1)

L

8p

N(p−1)(I,L^2p)^(m). (4.2) Similarly

B:=

m

X

j,k=1

k|∇u|²(fj,k)ii(u)k

L

8p p(8−N)+N(I,L

2p 2p−1)

.

m

X

j,k=1

|∇u|²

|u_k|^p−2|u_j|^p−1+|u_k|^p|u_j|^p−3

L

8p p(8−N)+N(I,L

2p 2p−1)

.

Using the interpolation inequalityk∇ · k²_2p.k · k2pk∆· k2p, we obtain k|∇u|²(|uk|^p−2|uj|^p−1+|uk|^p|uj|^p−3)k

L

2p 2p−1 x

.k∇uk²_L2p x

kukk^p−2

L^2p_x kujk^p−1

L^2p_x +kukk^p

L^2p_x kujk^p−3

L^2p_x

.k∆uk_L^2p

x kuk_L^2p

x kuk^2p−3

L^2p_x

.k∆uk_L2p

x kuk^2p−2_H . Thus, arguing as previously, we obtain

B.kuk

L

8p

N(p−1)(I,W^2,2p)(m)kuk

2pN(p−1)−8p N(p−1)

L^∞(I,L^2p)(m)kuk

8p−2N(p−1) N(p−1)

L

8p

N(p−1)(I,L^2p)(m). (4.3) Finally, thanks to (4.1)–(4.3), it follows that

ku(t)−e^it∆2Ψk_(S(I))(m) .kuk

2pN(p−1)−8p N(p−1)

L^∞(I,L^2p)(m)kuk

8p−N(p−1) N(p−1)

L

8p

N(p−1)(I,W^2,2p)(m). The next auxiliary result is about the decay of solutions.

Proposition 4.2. For any 2< r < _N^2N₋₄, we have

t→∞lim ku(t)k_(Lr)^(m) = 0.

Proof. Letχ∈C₀^∞(R^N) be a cut-off function andϕn := (ϕⁿ₁, . . . , ϕⁿ_m) be a sequence inH satisfying sup_nkϕnkH<∞and

ϕn * ϕ:= (ϕ1, . . . , ϕm)∈H.

Letun:= (uⁿ₁, . . . , uⁿ_m) (respectivelyu:= (u1, . . . , um)) be the solution inC(R, H) to (1.1) with initial dataϕn respectivelyϕ). In what follows, we prove a claim.

Claim. For every >0, there exist T>0 andn∈Nsuch that kχ(un−u)k_(L^∞

T(L²))^(m) < for alln > n. (4.4) Indeed, denoting the functionsvn:=χunandv= (v1, . . . , vm) :=χu, we compute v_jⁿ(0) =χϕⁿ_j and

iv˙ⁿ_j + ∆²v_jⁿ= ∆²χuⁿ_j + 2∇∆χ∇uⁿ_j + ∆χ∆uⁿ_j + 2∇χ∇∆uⁿ_j + 2 ∇∆χ∇uⁿ_j +∇χ∇∆uⁿ_j + 2

N

X

i=1

∇∂iχ∇∂iuⁿ_j

+χ

m

X

k=1

|uⁿ_k|^p|uⁿ_j|^p−2uⁿ_j .

Similarly,vj(0) =χφj and

iv˙j+ ∆²vj= ∆²χuj+ 2∇∆χ∇uj+ ∆χ∆uj+ 2∇χ∇∆uj

+ 2 ∇∆χ∇u_j+∇χ∇∆u_j+ 2

N

X

i=1

∇∂_iχ∇∂_iu_j

+χ

m

X

k=1

|uk|^p|uj|^p−2uj .

Denotingw_n= (w₁ⁿ, . . . , wⁿ_m) :=v_n−v andz_n = (z₁ⁿ, . . . , z_mⁿ) :=u_n−u, we have iw˙ⁿ_j + ∆²wⁿ_j = ∆²χz_jⁿ+ 4∇∆χ∇zⁿ_j + ∆χ∆z_jⁿ+ 4∇χ∇∆z_jⁿ

+ 4

N

X

i=1

∇∂_iχ∇∂_iz_jⁿ+χ

m

X

k=1

|uⁿ_k|^p|uⁿ_j|^p−2uⁿ_j −

m

X

k=1

|u_k|^p|u_j|^p−2u_j .

Thanks to Strichartz estimate, we obtain kwnk

L^∞_T(L²)∩L

8p N(p−1)

T (L^2p)^(m) .kχ(ϕ_n−ϕ)k_(L2)^(m)+k∆²χz_nk_(L1

T(L²))^(m)+k∇∆χ∇z_nk_(L1 T(L²))^(m)

+k∇χ∇∆znk_(L1(L²))^(m)+k∇∂iχ∇∂iz_nk_(L1(L²))^(m)

+

m

X

j,k=1

χ |uⁿ_k|^p|uⁿ_j|^p−2uⁿ_j − |u_k|^p|u_j|^p−2u_j

L

8p p(8−N)+N

T (L

2p 2p−1)

.

Thanks to the Rellich Theorem, up to subsequence extraction, we have :=kχ(ϕn−ϕ)k_L2

x→0 asn→ ∞.

Moreover, by the conservation laws via properties ofχ, I₁:=k∆²χz_nk_(L1

T(L²))^(m)+k∇∆χ∇z_nk_(L1

T(L²))^(m)+k∇χ∇∆z_nk_(L1 T(L²))^(m)

+k∇∂iχ∇∂iz_nk_(L¹

T(L²))^(m)