Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger equations with combined power-type nonlinearities

J. Math. Anal. Appl. 392 (2012) 111–122

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Global existence and asymptotic behavior of the Cauchy problem for fourth-order Schrödinger equations with combined power-type nonlinearities Cuihua Guo School of Mathematical Science, Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China

article

abstract

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Article history: Received 17 October 2011 Available online 11 April 2012 Submitted by Jie Xiao

We study the global existence of solutions in Sobolev spaces and the asymptotic behavior for the Cauchy problem of the following fourth-order Schrödinger equation with combined 8

power-type nonlinearities iut + ∆2 u + λ1 |u|p u + λ2 |u| n−4 u = 0, where t ∈ R, x ∈ Rn , 8 ≤ p < n−8 4 , λ1 , λ2 are nonzero real numbers. n © 2012 Elsevier Inc. All rights reserved.

Keywords: Fourth-order Schrödinger equation Combined power-type nonlinearities Cauchy problem Global existence Asymptotic behavior

1. Introduction In this paper we study the Cauchy problem of the following energy-critical fourth-order Schrödinger equation with a subcritical perturbation in R × Rn :



8

iut + ∆2 u + λ1 |u|p u + λ2 |u| n−4 u = 0, u(0, x) = ϕ(x), x ∈ Rn ,

for t ∈ R, x ∈ Rn ,

(1.1)

8 where λ1 , λ2 are nonzero real numbers, and 8n ≤ p < n− is a positive constant. 4 Fourth-order Schrödinger equations are very important equations which arise in many physical application fields. They have been introduced by Karpman [1] and Karpman and Shagalov [2] to take into account the effects of small fourthorder dispersion terms on the solitons. They have also been studied from a mathematical viewpoint by many authors. For instance, the energy-critical focusing fourth-order Schrödinger equation (λ1 = 0, λ2 < 0) has been studied by Miao et al. [3]. They obtained that the solution is global and scatters in the radial case under some conditions. In [4] Miao et al. proved that any finite energy solution is global and scatters for defocusing case (λ1 = 0, λ2 > 0) and n ≥ 9. Pausader [5] has obtained the global well-posedness and scattering in the defocusing case (λ1 = 0, λ2 > 0) for radially symmetrical initial data and n ≥ 5. In [6] Pausader investigated the cubic defocusing fourth-order Schrödinger 8

equation (λ1 = 0, λ2 > 0, |u| n−4 u → |u|2 u) and proved that the equation is globally well-posed for n ≤ 8 and the solution scatters for 5 ≤ n ≤ 8. For the case λ1 ̸= 0, λ2 ̸= 0 and n = 8, Zhang and Zheng [7] proved that the solution is global for 1 < p < 2, scattering will occur either λ2 > 0, 1 < p < 2 or when the mass of solution is 8

small enough, 1 ≤ p < 2. But they only discuss the case n = 8 (obviously |u| n−4 u = |u|2 u). As far as we know, there are few results about global existence and scattering for the energy-critical fourth-order Schrödinger equations

E-mail address: [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.03.028

112

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

with combined power-type nonlinearities in arbitrary dimensions. The aim of this paper is to study the global existence of solutions and scattering of the Cauchy problem of (1.1) for any dimensions. In [8] Tao et al. studied a Schrödinger equation with combined power-type nonlinearities. So we will utilize the ideas and techniques of [8]. The method of [7,8] is the ‘‘perturbative’’ method. The idea of this method is to obtain the global well-posedness by the good local well-posedness combining with the global kinetic energy control. The same method is used in [9]. The aim of this paper is to improve the global existence and scattering for fourth-order Schrödinger equations in arbitrary dimensions. As for the local wellposedness, we can obtain it by similar techniques in [10]. Here we omit it. We also refer the reader to [11–18] for other results about fourth-order Schrödinger equations. The main results of this work are the following theorems: Theorem 1.1 (Global Solvability). Assume that 4 < n ≤ 8. Let ϕ(x) ∈ H 2 (Rn ) and λ2 > 0, then there exists a unique global solution u(t , x) of (1.1) which satisfies

∥u∥Lq

loc

(R;H 2,r (Rn ))

  ≤ C ∥ϕ∥H 2 (Rn )

for all biharmonic admissible pairs (q, r ) (biharmonic admissible pairs will be introduced in Section 2). Theorem 1.2 (Scattering Results). Assume that 4 < n ≤ 8. For any ϕ(x) ∈ H 2 (Rn ), let u be the unique global solution of (1.1), there exist u± such that lim ∥u(t ) − W (t )u± ∥H 2 (Rn ) = 0,

t →±∞

provided

λ1 > 0 ,

λ2 > 0,

λ2 > 0,

8

8 n


8 n−4

,

or n

≤p<

8 n−4

and ∥ϕ∥L2 (Rn ) ≤ C ∥1ϕ∥L2 (Rn ) ,





(where the operator W (t ) will be introduced in Section 2, C ∥1ϕ∥L2 (Rn ) of Theorem 1.2).





is a small enough parameter, see the proof

Obviously, (1.1) has two conservation laws: M (u)(t ) = ∥u(t , ·)∥L2 (Rn ) ,

(1.2)

and E (u)(t ) =

  1 Rn

2

|1u(t , x)|2 +

 2n λ1 (n − 4)λ2 |u(t , x)|p+2 + |u(t , x)| n−4 dx. p+2 2n

(1.3)

We recall the dispersive estimates for the linear equation related to the Eq. (1.1) in Section 2 and we will present the nonlinear estimates in Section 3. In Section 4 we present the proofs of the theorems. 2. Notations and fundamental solution operator estimates Given T > 0 and a function space on Rn , we denote by ∥ · ∥Lq ([−T ,T ];X ) and Lq ([−T , T ]; X ) respectively the following norm and the corresponding function space on [−T , T ] × Rn : For 1 ≤ q < +∞,



T

∥f (·, t )∥

∥f ∥Lq ([−T ,T ];X ) =

q X dt

 1q

.

−T

And for q = ∞,

∥f ∥L∞ ([−T ,T ];X ) = ess. sup−T
T x

T

x

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

113

For any time interval T , we denote γ

γ

γ

γ

X˙ 0 (T ) = LT 1 Lρx 1 ∩ LT 2 Lρx 2 ∩ LT 3 Lρx 3 ∩ LT 4 Lρx 4 ,

  γ γ γ  LT 1 Lρx 1 ∩ LT 2 Lρx 2 ∩ LT 5 Lρx 5 ,

Y˙ 0 (T ) =

  LγT 1 Lρx 1 ∩ LγT 2 Lρx 2 ∩ LγT 6 Lρx 6 ,

8 n


8(n − 1) n2 − 3n + 8

8(n − 1)

n2 − 3n + 8


, 8

n−4

,

γ

γ

Z˙ 0 (T ) = LT 1 Lρx 1 ∩ LT 2 Lρx 2 where

(γ1 , ρ1 ) =



2(n + 4) 2(n + 4)

,







(γ2 , ρ2 ) = , 2 , n−4 n + 16    2n 8 8(p + 2) n(p + 2) (γ3 , ρ3 ) = , , (γ4 , ρ4 ) = , , (n − 4)p − 4 n + 4 − (n − 4)p (n − 4)p n + 2p   2(n + 2) 2n(n + 2) (γ5 , ρ5 ) = , , n − 6 + (n − 4)p n2 − 2n + 24 − 4(n − 4)p   2n(n − 4) 2(n − 4) , 2 . (γ6 , ρ6 ) = n + 4 − np n − 8n − 16 + 4np n

n

,

2(n + 4) 2n(n + 4)



And X˙ 1 (T ) = {u : 1u ∈ X˙ 0 (T )},

X 1 (T ) = X˙ 0 (T ) ∩ X˙ 1 (T ),

Y˙ 1 (T ) = {u : 1u ∈ Y˙ 0 (T )},

Y 1 (T ) = Y˙ 0 (T ) ∩ Y˙ 1 (T ),

Z˙ (T ) = {u : 1u ∈ Z˙ (T )},

Z 1 (T ) = Z˙ 0 (T ) ∩ Z˙ 1 (T ).

1

0

The fundamental solution of the linear equation related to (1.1) is given by the following oscillatory integral: I (x, t ) = (2π )−n



eix· ξ +it |ξ | dξ . 4

Rn

We denote by W (t )(t ∈ R) the fundamental solution operator W (t )ϕ = I (x, t ) ∗ ϕ(x),

ϕ(x) ∈ S ′ (Rn ).

Definition 2.1. For two integers 2 ≤ q ≤ ∞ and 2 ≤ r < ∞, we say that (q, r ) is a Schrödinger admissible pair if the following condition is satisfied: 2 q

 =n

1 2

−

1



r

.

Definition 2.2. For two integers 2 ≤ γ ≤ ∞ and 2 ≤ ρ < ∞, we say that (γ , ρ) is an biharmonic admissible pair if the following condition is satisfied: 4

γ

 =n

1 2

−

1

ρ



.

Lemma 2.1 (See [19]). For any biharmonic admissible pairs (q, r ) and (γ , ρ), there hold the following estimates

∥W (t )ϕ∥Lq Hxs,r ≤ C ∥ϕ∥Hxs , T  t     W (t − τ )f (·, τ )dτ   q 0

s,r

≤ C ∥f ∥ γ ′

s,ρ ′

L T Hx

LT Hx

,

where ϕ ∈ H s (Rn ), f ∈ Lγ ([−T , T ]; H s,ρ (Rn )). ′

′

For any Schrödinger admissible pairs (a, b) and (c , d), there hold the following estimates

∥W (t )ϕ∥La H˙ s,b ≤ C ∥ϕ∥ s− 2a , ˙x T x H  t     W (t − τ )f (·, τ )dτ    a ˙ s,b ≤ C ∥f ∥Lc ′ H˙ s− 2a − 2c , d′ , 0

LT Hx

s− 2a

c′

T

x

s− 2a − 2c , d′

˙ where ϕ ∈ H (R ), f ∈ L ([−T , T ]; H˙ (Rn )). By Sobolev inequality and Lemma 2.1, we have n

114

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

Corollary 2.2. For any biharmonic admissible pair (q, r ), there holds the following estimate

 t     W (t − τ )f (·, τ )dτ  ,   q ˙ 2,r ≤ C ∥f ∥ 2 ˙ 1, n2n +2 0

LT Hx

LT Hx

˙ where f ∈ L2 ([−T , T ]; H

1, n2n +2

(Rn )).

Proof. By Sobolev inequality, we have

 t   t       W (t − τ )f (·, τ )dτ      q ˙ 2,r ≤  W (t − τ )f (·, τ )dτ  q ˙ s,r1 , 0

0

LT Hx

where r1 and s satisfy

2 q

=n



1 2

1 r1

−



LT Hx

1 r1

and

1 r

−

s−2 . n

=

in Lemma 2.1, the desired result is obtained. Taking a = q, b = r1 and c = 2, d = n2n −2



Using Lemma 2.1 and Corollary 2.2, by Duhamel formula, we get the following lemma: Lemma 2.3. If u is the solution of the problem (1.1), then for any biharmonic admissible pares (q, r ), (γ1 , ρ1 ), (γ2 , ρ2 ), we have 8

∥u∥Lq H 2,r ≤ C0 ∥u0 ∥H 2 + C1 ∥ |u|p u∥ γ1′ T

+ C1 ∥ |u| n−4 u∥ γ2′

ρ′ LT Lx 1

x

ρ′ LT Lx 2

+ C1 ∥∇(|u|p u)∥

8

2n n+2 L2T Lx

+ C1 ∥∇(|u| n−4 u)∥

2n n+2

L2T Lx

.

Lemma 2.4 (See [4,20]). If u is a solution of the problem (1.1), then the following holds for λ1 > 0, λ2 > 0 and n > 4

∥ |∇|−

n−5 4

u∥4L4 L4 ≤ C sup ∥u(t )∥2 1 ∥u(t )∥2L2 . T x

[0,T ]

x

˙ x2 H

Using Lemma 2.4 and interpolation theorem [21], we obtain Lemma 2.5. If u is a solution of the problem (1.1), then the following holds for λ1 > 0, λ2 > 0 and n > 4

∥ u∥

2(n−1) n−1 n−3 LT Lx

≤ C ∥ |∇|−

n−5 4

4

n−5

LT Lx

LT Lx

−1 u∥ n4−14 ∥∇ u∥ n∞ 2.

3. Nonlinear estimates Lemma 3.1. Assume that 4 < n ≤ 8, then we have

∥ |u|p u∥ γ4′

ρ′ LT Lx 4

∥∇(|u|p u)∥

≤ CT 1−

2n n+2 L2T Lx

(n−4)p 8

∥u∥pγ4

2,ρ4

˙x LT H

≤ CT 1−

(n−4)p 8

∥u∥Lγ4 Lρx 4 ,

(3.1)

T

∥u∥pL∞ H˙ 2 ∥1u∥Lγ3 Lρx 3 ,

(3.2)

T

x

T

where (γi , ρi ), i = 3, 4 are as in Section 2. Proof. Using Hölder inequality and Sobolev inequality, we have

∥ |u|p u∥ γ4′

ρ′ LT Lx 4

≤ CT 1− ≤ CT 1−

∥∇(|u|p u)∥

2n n+2 L2T Lx

(n−4)p 8

(n−4)p 8

≤ CT 1− ≤ CT 1−



∥u∥pγ4 ∥ u∥

(n−4)p 8

(n−4)p 8

γ ρ ρ ∗ u L 4 Lx 4 T LT Lx p γ ρ γ ˙ 2,ρ4 u L 4 Lx 4 T LT 4 H x

∥ ∥

∥ ∥

∥u∥p

2n n−4

ρ∗ =

∥∇ u∥

n−4

T

2n

γ n+2−(n−4)p LT 3 Lx

∥u∥pL∞ H˙ 2 ∥1u∥Lγ3 Lρx 3 . x



,

L∞ L T x T

n(p + 2)



C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

115

Lemma 3.2. Assume 4 < n ≤ 8, we have 8

8

∥ |u| n−4 u∥

≤ C ∥u∥ nγ−24 2,ρ2 ∥u∥Lγ1 Lρx 1 ,

2(n+4) 2(n+4) n+8 n+8 LT Lx

(3.3)

T

n+4

8

∥∇(|u| n−4 u)∥

˙x LT H

2n n+2 L2T Lx

≤ C ∥u∥ nγ−24 2,ρ2

(3.4)

˙x LT H

where (γi , ρi ), i = 1, 2 are as in Section 2. Proof. Using Hölder inequality and Sobolev inequality, we have 8

∥ |u| n−4 u∥

8

≤ C ∥ |u| n−4 ∥

2(n+4) 2(n+4) n+8 n+8 LT Lx

n+4 n+4 LT 4 Lx 4

∥ u∥

2(n+4) n

2(n+4) n

LT

Lx

8

≤ C ∥u∥ n2−(4n+4) LT

n−4

∥ u∥

2(n+4) n−4

2(n+4) n

LT

Lx

2(n+4) n

Lx

8

≤ C ∥u∥ n2−(4n+4) LT

n−4

∥ u∥

2n(n+4) n2 +16

2,

˙x H

2(n+4) n

LT

2(n+4) n

Lx

8

= C ∥u∥ nγ−24 2,ρ2 ∥u∥Lγ1 Lρx 1 . ˙x LT H

T

Similarly we can obtain 8

8

∥∇(|u| n−4 u)∥

2n n+2 L2T Lx

≤ C ∥u∥ n2−(4n+4) LT

2(n+4) n−4

n−4

∥∇ u∥

2(n+4) n−4

LT

Lx

2n(n+4) n2 −2n+8

Lx

8

≤ C ∥u∥ n2−(4n+4) LT

2,

n−4

˙x H

2n(n+4) n2 +16

∥ u∥

2(n+4) n−4

LT

2,

˙x H

2n(n+4) n2 +16

n+4

= C ∥u∥ nγ−24 2,ρ2 .  ˙x LT H

Lemma 3.3. Assume that 4 < n ≤ 8. (i) For the case p

∥ |u| u∥

≤ C ∥ u∥

2n n+4

L2T Lx

∥∇(|u|p u)∥

2n n+2

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 LT Lx

≤ C ∥u∥

L2T Lx

(ii) For the case p

∥ |u| u∥


8 n

∥∇(|u|p u)∥

x

we have the following results:

∥u∥Lγ5 Lρx 5 ,

(3.5)

T

(n2 −3n+8)p−8(n−1) 2(n+2)

∥1u∥L∞ L2 T

8 , n −4

∥1u∥Lγ5 Lρx 5 .

∥ u∥

−(n2 −3n+8)p+8(n−1) 2(n−4) L∞ L2 T

x

∥u∥Lγ6 Lρx 6 ,

(3.7)

T

x

≤ C ∥u∥

L2T Lx

−(n2 −3n+8)p+8(n−1) 2(n−4)

(np−8)(n−1) 2(n−4) 2(n−1) n−1 n−3 LT Lx

∥u∥L∞ L2 T

∥1u∥Lγ6 Lρx 6 ,

(3.8)

T

x

where (γi , ρi ), i = 5, 6 are as in Section 2. Proof. (i) For the case

∥ |u|p u∥

2n n+4

8(n−1) n2 −3n+8

≤ C ∥ u∥ p

L2T Lx


LT

≤ C ∥ u∥

(3.6)

T

x

we have the following results:

(np−8)(n−1) 2(n−4) 2(n−1) n−1 n−3 L L T

2n n+2


(n2 −3n+8)p−8(n−1) 2(n+2) L∞ L2 T

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 LT Lx

8(n−1) , n2 −3n+8

≤ C ∥ u∥

2n n+4

L2T Lx

∥1u∥

8(n−1) n2 −3n+8

8 , n −4

using Hölder inequality, we have

n(n+2)p 4n−8+2(n−4)p

∥ u∥

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 LT Lx

∥ u∥

2(n+2) n−6+(n−4)p

LT

Lx

(n2 −3n+8)p−8(n−1) 2(n+2) 2n n−4 L∞ L T x

2n(n+2) n2 −2n+24−4(n−4)p

Lx

∥u∥

2(n+2) n−6+(n−4)p

LT

2n(n+2) n2 −2n+24−4(n−4)p

Lx

116

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122 (n2 −3n+8)p−8(n−1) 2(n+2) L∞ L2

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 L L

∥ 1 u∥

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 LT Lx

∥1u∥L∞ L2

≤ C ∥ u∥

∥ u∥

x

T

= C ∥ u∥

(n2 −3n+8)p−8(n−1) 2(n+2)

2n(n+2) n2 −2n+24−4(n−4)p

Lx

∥u∥Lγ5 Lρx 5 . T

x

T

2(n+2) n−6+(n−4)p

LT

x

T

Similarly we can obtain

∥∇(|u|p u)∥

≤ C ∥ u∥ p

2n n+2 L2T Lx

2(n+2)p 8−(n−4)p

LT

≤ C ∥ u∥

= C ∥ u∥ 8 n


∥ |u| u∥

≤ C ∥ u∥

2n n+4

L2T Lx

(n2 −3n+8)p−8(n−1) 2(n+2)

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 L L

∥1u∥L∞ L2

8(n−1) , n2 −3n+8

x

T

∥1u∥

2(n+2) n−6+(n−4)p

LT

(n2 −3n+8)p−8(n−1) 2(n+2)

2n(n+2) n2 −2n+24−4(n−4)p

Lx

∥1u∥Lγ5 Lρx 5 . T

x

T

2n(n+2) n2 −4n+20−4(n−4)p

Lx

x

using Hölder inequality, we have

2p(n−4) np−8

LT

2(n+2) n−6+(n−4)p

LT

∥1u∥L∞ L2

p

p

∥∇ u∥

[8−(n−4)p](n−1) 2(n+2) 2(n−1) n−1 n−3 LT Lx

T

(ii) For the case

n(n+2)p 4n−8+2(n−4)p

Lx

p(n−4) 4−2p

∥ u∥

2(n−4) 4+n−np

LT

Lx

2n(n−4) n2 −8n−16+4np

Lx

−(n2 −3n+8)p+8(n−1) 2(n−4)

(np−8)(n−1) 2(n−4) 2(n−1) n−3 n−1 LT Lx

∥u∥L∞ L2

(np−8)(n−1) 2(n−4) 2(n−1) n−1 n−3 LT Lx

∥u∥L∞ L2

≤ C ∥ u∥

2n(n−4) n2 −8n−16+4np

Lx

∥u∥Lγ6 Lρx 6 . T

x

T

2(n−4) 4+n−np

LT

−(n2 −3n+8)p+8(n−1) 2(n−4)

= C ∥ u∥

∥ u∥

x

T

Similarly we can obtain

∥∇(|u|p u)∥

≤ C ∥u∥p 2p(n−4)

2n n+2

L2T Lx

LT

≤ C ∥ u∥

= C ∥ u∥

Lemma 3.4. If

∥ |u|p+1 ∥

8 n

≤p<

8 , n −4

2(n+4) 2(n+4) n+8 n+8 LT Lx

np−8

2(n−4) 4+n−np

LT

2n(n−4) n2 −10n−8+4np

Lx

−(n2 −3n+8)p+8(n−1) 2(n−4)

∥u∥L∞ L2

(np−8)(n−1) 2(n−4) 2(n−1) n−3 n−1 LT Lx

∥u∥L∞ L2

x

T

2(n−4) 4+n−np

LT

−(n2 −3n+8)p+8(n−1) 2(n−4)

2n(n−4) n2 −8n−16+4np

Lx

∥1u∥Lγ6 Lρx 6 .  T

x

T

∥1u∥

then 12−(n−4)p 4

≤ C ∥ u∥ γ 1

ρ LT Lx 1

≤ C ∥u∥ γ1

2n ˙ 1, n+2 L H

∥∇ u∥

(np−8)(n−1) 2(n−4) 2(n−1) n−1 n−3 LT Lx

8−(n−4)p 4 ρ LT Lx 1

∥ |u|p+1 ∥ 2

p(n−4) 4−2p

Lx

np−8

∥1u∥ γ24

ρ LT Lx 2

np−4

∥1u∥ γ24

ρ LT Lx 2

,

(3.9)

,

(3.10)

where (γi , ρi ), i = 1, 2 are as in Section 2. Proof. Using Hölder inequality, interpolation inequality and Sobolev inequality, we have

∥ |u|p+1 ∥

2(n+4) n+8

LT

2(n+4) n+8

≤ C ∥ |u|p ∥

Lx

≤ C ∥ u∥ ≤ C ∥ u∥

n+4 n+4 LT 4 Lx 4

T

p (n+4)p (n+4)p LT 4 Lx 4 8−(n−4)p 4 γ ρ LT 1 Lx 1 8−(n−4)p 4

≤ C ∥ u∥ γ 1

ρ LT Lx 1

= C ∥ u∥

∥u∥Lγ1 Lρx 1

∥ u∥

∥u∥Lγ1 Lρx 1 T

np−8 4 2(n+4) 2(n+4) n−4 n−4 LT Lx np−8

∥1u∥ γ24

12−(n−4)p 4 γ ρ LT 1 Lx 1

∥u∥Lγ1 Lρx 1

ρ LT Lx 2

∥1u∥

∥u∥Lγ1 Lρx 1

np−8 4 γ ρ LT 2 Lx 2

T

.

T

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

117

Similarly we obtain

∥ |u|p+1 ∥ 2

≤ C ∥ |u|p ∥

1, 2n n+2

˙ L H

n+4 n+4 LT 4 Lx 4

≤ C ∥u∥p (n+4)p LT

4

(n+4)p Lx 4

8−(n−4)p 4

≤ C ∥ u∥ γ 1

ρ LT Lx 1

∥∇ u∥

2(n+4) n−4

LT

2n(n+4) n2 −2n+8

Lx

∥1u∥Lγ2 Lρx 2 T

np−4

∥1u∥ γ24

ρ LT Lx 2

. 

4. Proofs of Theorems Let u be the solution of problem (1.1). We can decompose the solution u into u = v + w, where v stands for the solution of the initial value problem



8

ivt + ∆2 v + λ2 |v| n−4 v = 0, v(0, x) = ϕ(x), x ∈ Rn ,

t ∈ R, x ∈ Rn ,

(4.1)

and w is the solution of the initial value problem



8

8

iwt + ∆2 w = −λ1 |v + w|p (v + w) − λ2 |v + w| n−4 (v + w) + λ2 |v| n−4 v, w(0, x) = 0, x ∈ Rn .

t ∈ R, x ∈ Rn ,

(4.2)

By the remark of [6], we know that (4.1) is globally well-posed if λ2 > 0. And we have

∥v∥Lq (R;H˙ 2,r ) ≤ C (∥ϕ∥H˙ 2 ),

∥v∥Lq (R;Lr ) ≤ C (∥ϕ∥H 2 ),

(4.3)

where (q, r ) is a biharmonic admissible pair. In the following, for any given time interval [0, T ], we prove the existence of a solution w for problem (4.2) and make estimates on some norms of it. For any given ε (which will be specified later), by (4.3), we can divide R into subintervals I0 , I1 , . . . , IJ −1 such that on each Ij

∥v∥X 1 (Ij ) ∼ ε,

0 ≤ j ≤ J − 1.

So for any given time interval [0, T ], there exists J ′ ≤ J such that (renumbering, if necessary) ′

[0, T ] = ∪Jj=−01 (Ij ∩ [0, T ]),

Ij = [tj , tj+1 ],

t0 = 0,

tJ ′ = T .

Lemma 4.1. For any given time interval [0, T ], there is a unique solution w(t , x) for the problem (4.2)

∥w∥X 1 [0,T ] ≤ C , where C will be dependent on p. Proof. First we prove that there exists a solution w(t , x) for problem (4.2) on I0 = [0, t1 ] (renumbering if necessary). We shall prove it by Banach fixed point theorem. For this purpose we rewrite problem (4.2) into integral form, namely

w(t ) = i

t1







8

8

W (t − τ ) λ1 |v + w|p (v + w) + λ2 |v + w| n−4 (v + w) − λ2 |v| n−4 v (τ )dτ .

0

We denote



B0 = w : ∥w∥L∞ (I0 ;H 2 (Rn )) + ∥w∥X 1 (I0 ) ≤ η0 = |I0 |1−

(n−4)p 8



,

and define a mapping S as follows S w(t ) = i

t1





8

8



W (t − τ ) λ1 |v + w|p (v + w) + λ2 |v + w| n−4 (v + w) − λ2 |v| n−4 v (τ )dτ .

0

In the sequel we prove that S is well-defined and it maps B0 into B0 .

118

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

By Lemma 2.3 and Lemmas 3.1–3.2, we have

  8 8   ∥S w∥B0 ≤ C  |v + w| n−4 (v + w) − |v| n−4 v  + C ∥ |v + w|p (v + w)∥ γ ′

2(n+4) L n+8



2(n+4) I0 ;L n+8

2n ρ′ ˙ 1, n+2 L 4 (I0 ;L 4 )∩L2 I0 ;H





  1, 2n ∩L2 I0 ;H˙ n+2



  8 8 (n−4)p n−4 n−4 ≤ C1 ∥v∥X 1 (I ) + ∥w∥X 1 (I ) ∥w∥X 1 (I0 ) + C1 |I0 |1− 8 (∥v∥pX 1 (I ) + ∥w∥pX 1 (I ) )∥w∥X 1 (I0 ) 0 0 0 0   8 (n−4)p 8 ≤ C1 ε n−4 + η0n−4 η0 + C1 |I0 |1− 8 (εp + η0p )η0 , and

  8 8   ∥S w1 − S w2 ∥B0 ≤ C  |v + w1 | n−4 (v + w1 ) − |v + w2 | n−4 (v + w2 )

2(n+4) L n+8



+ C ∥ |v + w1 |p (v + w1 ) − |v + w2 |p (v + w2 )∥ γ ′

2(n+4) I0 ;L n+8



  1, 2n ∩L2 I0 ;H˙ n+2

2n ρ′ ˙ 1, n+2 L 4 (I0 ;L 4 )∩L2 I0 ;H

8

8





8

≤ C1 (2∥v∥Xn−1 (4I ) + ∥w1 ∥Xn−1 (4I ) + ∥w2 ∥Xn−1 (4I ) )∥w1 − w2 ∥X 1 (I0 ) 0

0

0

(n−4)p

+ C1 |I0 |1− 8 (2∥v∥pX 1 (I ) + ∥w1 ∥pX 1 (I ) + ∥w2 ∥pX 1 (I ) )∥w1 − w2 ∥X 1 (I0 ) 0 0  0  8 (n−4)p 8 n − 4 1 − 8 (2ε p + 2η0p )∥w1 − w2 ∥X 1 (I0 ) . ∥w1 − w2 ∥X 1 (I0 ) + C1 |I0 | ≤ C1 2ε n−4 + 2η0 8

1− n−4 p

8

(1− n−4 p)

(p+1)(1− n−4 p)

8 8 + t1 ) < 12 , then S is a Hence if we first take 2C1 ε n−4 < 12 , then take t1 such that 2C1 (ε p t1 8 + t1n−4 contraction mapping of B0 in itself. We see that there is a solution on I0 = [0, t1 ] from Banach’s fixed point theorem. Secondly, we prove that there exists a solution w(t , x) for problem (4.2) on I1 = [t1 , t2 ]. Taking w(t1 ) as an initial value, we have

w(t ) = W (t )w(t1 ) + i



t2



8



8

W (t − τ ) λ1 |v + w|p (v + w) + λ2 |v + w| n−4 (v + w) − λ2 |v| n−4 v (τ )dτ .

t1

We denote



B1 = w : ∥w∥L∞ (I1 ;H 2 (Rn )) + ∥w∥X 1 (I1 ) ≤ η1 = 2C0 |I1 |1−

(n−4)p 8



,

and define a mapping S as follows S w(t ) = W (t )w(t1 ) + i



t2



8

8



W (t − τ ) λ1 |v + w|p (v + w) + λ2 |v + w| n−4 (v + w) − λ2 |v| n−4 v (τ )dτ .

t1

Similarly by Lemma 2.3 and Lemmas 3.1–3.2, we have

  8 8   ∥S w∥B1 ≤ C0 ∥w(t1 )∥H 2 + C  |v + w| n−4 (v + w) − |v| n−4 v  + C ∥ |v + w|p (v + w)∥ γ ′

2n ρ′ ˙ 1, n+2 L 4 (I1 ;L 4 )∩L2 I1 ;H



2(n+4) L n+8



2(n+4) I1 ;L n+8



  1, 2n ∩L2 I1 ;H˙ n+2



  8 8 (n−4)p ≤ C0 ∥w(t1 )∥H 2 + C1 ∥v∥Xn−1 (4I ) + ∥w∥Xn−1 (4I ) ∥w∥X 1 (I1 ) + C1 |I1 |1− 8 (∥v∥pX 1 (I ) + ∥w∥pX 1 (I ) )∥w∥X 1 (I1 ) 1 1 1 1   8 (n−4)p 8 ≤ C0 ∥w(t1 )∥H 2 + C1 ε n−4 + η1n−4 η1 + C1 |I1 |1− 8 (εp + η1p )η1 , and

  8 8   ∥S w1 − S w2 ∥B1 ≤ C  |v + w1 | n−4 (v + w1 ) − |v + w2 | n−4 (v + w2 )

2(n+4) L n+8

+ C ∥ |v + w1 |p (v + w1 ) − |v + w2 |p (v + w2 )∥ γ ′



2(n+4) I1 ;L n+8



  1, 2n ∩L2 I1 ;H˙ n+2

2n ρ′ ˙ 1, n+2 L 4 (I1 ;L 4 )∩L2 I1 ;H





C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122



8

8



8

≤ C1 2∥v∥Xn−1 (4I ) + ∥w1 ∥Xn−1 (4I ) + ∥w2 ∥Xn−1 (4I 1

1)

1

119

∥w1 − w2 ∥X 1 (I1 )

(n−4)p

+ C1 |I1 |1− 8 (2∥v∥pX 1 (I ) + ∥w1 ∥pX 1 (I ) + ∥w2 ∥pX 1 (I ) )∥w1 − w2 ∥X 1 (I1 ) 1 1   1 8 (n−4)p 8 ≤ C1 2ε n−4 + 2η1n−4 ∥w1 − w2 ∥X 1 (I1 ) + C1 |I1 |1− 8 (2εp + 2η1p )∥w1 − w2 ∥X 1 (I1 ) . 8

n−4

8

(1− n−4 p)

n−4

8 Hence if we take t2 such that 2C1 (ε p (t2 − t1 )1− 8 p + (2C0 ) n−4 (t2 − t1 ) n−4 + (2C0 )p (t2 − t1 )(p+1)(1− 8 p) ) < 12 , then S is a contraction mapping of B1 in itself. We see that there is a solution on I1 = [t1 , t2 ] from Banach’s fixed point theorem. Using an induction argument, we take



Bj = w : ∥w∥L∞ (Ij ;H 2 (Rn )) + ∥w∥X 1 (Ij ) ≤ ηj = (2C0 )j |Ij |1−

(n−4)p 8



,

and let S w(t ) = W (t )w(tj ) + i

tj+1





8

8



W (t − τ ) λ1 |v + w|p (v + w) + λ2 |v + w| n−4 (v + w) − λ2 |v| n−4 v (τ )dτ .

tj 8

n−4

8

(1− n−4 p)

n−4

8 + (2C0 )pj (tj+1 − tj )(p+1)(1− 8 p) ) < 12 , then If we take tj+1 such that 2C1 (ε p (tj+1 − tj )1− 8 p + (2C0 ) n−4 (tj+1 − tj ) n−4 S is a contraction mapping of Bj in itself. We see that there is a solution on Ij = [tj , tj+1 ] from Banach’s fixed point theorem. Finally we get a unique solution of (4.2) on [0, T ] such that

∥w(t )∥X 1 ([0,T ]) ≤

J ′ −1 

∥w(t )∥X 1 (Ij ) ≤

j =0

J ′ −1 

j

4 1 − n− p 8

(2C0 )j Ij

′

≤ J ′ (2C0 )J T 1−

n−4 p 8

≤ C.

j =0

Thus we have ∥u∥X 1 ([0,T ]) ≤ ∥v∥X 1 ([0,T ]) + ∥w∥X 1 ([0,T ]) ≤ C (∥ϕ∥H 2 ).



Lemma 4.2. For the problem (1.1), if λ2 > 0, then

∥u(·, t )∥H 2 ≤ C (E , M ), where E , M are as in (1.2) and (1.3). Proof. It is obvious in the case λ1 ≥ 0. So we only discuss the case λ1 < 0. Using Young’s inequality with ε , we have

|u|p+2 ≤ ε ·

p(n − 4) 8

2n

|u| n−4 + ε

p(n−4) − 8−( n−4)p

8 − (n − 4)p 8

|u|2 ,

ε=

4(n − 4)(p + 2)λ2 n(n − 4)p|λ1 |

.

So

|u|p+2 ≥ −

(n − 4)λ2 2n

2n

|u| n−4 + C (λ1 , λ2 , n, p)|u|2 ,

substituting it into (1.2) and (1.3), we obtain the desired result.



Proof of Theorem 1.1. We obtain Theorem 1.1 by Lemmas 4.1 and 4.2.



Proof of Theorem 1.2. Firstly, we prove that ∥u∥Y 1 (R) for the case λ1 , λ2 > 0 and ∥u∥Z 1 (R) for the case λ2 > 0, are bounded. Case 1: λ1 , λ2 > 0. By Theorem 1.1 and Lemma 2.5, we have

∥ u∥

2(n+1) Ln−1 R,L n−1 (Rn )





8 n

≤p<

8 n−4

≤ C ∥u∥L∞ (R,H 2 (Rn )) ≤ C (E , M ).

So we take ε (which will be specified later), divide R into subintervals I0 , I1 , . . . , IJ −1 such that on each Ij

∥ u∥

2(n+1) Ln−1 Ij ,L n−1 (Rn )





∼ ε.

(4.4)

On the other hand, by (4.3), we take η (which will also be specified later), divide R into subintervals I0′ , I1′ , . . . , IK′ −1 such that on each Ik′

∥v∥Y (Ik′ ) ∼ η.

(4.5)

120

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

For convenience, we denote one of Ij by [a, b]. Without loss of generality, we may assume that ′

[a, b] = ∪kK=−01 Ik′ ,

K′ ≤ K.

Similar to Lemma 4.1, by inductive arguments, if we take the above ε and η small enough, using Lemma 3.3, we can 8(n−1) obtain that for the case 8n < p ≤ n2 −3n+8 :

∥w∥Y 1 (I ′ ) ≤ (2C )k ε

(np−8)(n−1) 2(n+2)

;

(4.6)

k

and for the case

8(n−1) n2 −3n+8


∥w∥Y 1 (I ′ ) ≤ (2C )k ε

8 : n −4

[8−(n−4)p](n−1) 2(n+2)

k

.

(4.7) 8(n−1) , n2 −3n+8

In the following, we only prove the case 8n < p ≤ Indeed, by (4.2) and Lemmas 2.3 and 3.3, we have

∥w∥Y 1 (I ′ ) ≤ C ∥w(tk )∥H 2 + C ∥ |u|p u∥ 2  ′ k

2n



Ik ;L n+4

L

8

8(n−1) n2 −3n+8

the case

+ C ∥ |u|p u∥ 2  ′

˙ Ik ;H

L

1, 2n n+2


8 n−4

is similar, so here we omit it.



8

+ C ∥ |v + w| n−4 (v + w) − |v| n−4 v∥

2(n+4) L n+8



2(n+4) Ik′ ;L n+8



8

8

+ C ∥ |v + w| n−4 (v + w) − |v| n−4 v∥ 2  ′ L

≤ C ∥w(tk )∥H 2 + C ε

(np−8)(n−1) 2(n+2)

˙ Ik ;H

1, 2n n+2

 n+4

8

8

(η + ∥w∥Y 1 (I ′ ) ) + C η n−4 ∥w∥Y 1 (I ′ ) + C η∥w∥Yn−1 (4I ′ ) + C ∥w∥Yn−1 (4I ′ ) . k

k

k

k

So (4.6) holds by taking ε and η small enough. Thus K ′ −1

∥w∥Y 1 [a,b] ≤



∥w∥

Y 1 (I ′ )

≤

k

k=0

 ′ K −1  (np−8)(n−1)   (2C )k ε 2(n+2) ,   k =0 K ′ −1

  [8−(n−4)p](n−1)    (2C )k ε 2(n+2) ,  k =0

8 n


8(n − 1) n2

− 3n + 8

8(n − 1) n2 − 3n + 8


, 8

n−4

.

Furthermore we have

∥u∥Y 1 [a,b] ≤ ∥w∥Y 1 [a,b] + ∥v∥Y 1 [a,b] ≤ C (∥ϕ∥H 2 ). Noting that [a, b] is arbitrary, we have that

∥u∥Y 1 (R) ≤

J −1 

∥u∥Y 1 (Ij ) ≤ JC (∥ϕ∥H 2 ).

j =0

Using Lemma 2.3 and Lemmas 3.2–3.3, we obtain

∥u∥Lq (R;H 2,r ) ≤ C (∥ϕ∥H 2 ) for any biharmonic admissible pair (q, r ). Remark. Obviously, we have ∥u∥Z 1 (R) is bounded. We will use this fact later. Case 2: λ2 > 0,

8 n

≤p<

8 . n −4

By (4.3), we take ε (which will also be specified later), divide R into subintervals I0 , I1 , . . . , IK −1 such that on each Ik

∥v∥Z˙ 1 (Ik ) ∼ ε.

(4.8)

We claim that for small M and ε

∥v∥Z 1 (Ik ) ≤ C ε.

(4.9)

Indeed, by Lemma 2.1 and Corollary 2.2, we have

  8   ∥v∥Z˙ 0 (Ik ) ≤ C ∥v∥L2x + C  |v| n−4 v 

8

2(n+4) 2(n+4) n+8 n+8 LT Lx

≤ CM + C ε n−4 ∥v∥Z˙ 0 (Ik ) .

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122 8

Obviously if we take C ε n−4 <

1 , 2

121

then we have ∥v∥Z˙ 0 (Ik ) ≤ 2CM. Furthermore, if we take M small enough, then we have

∥v∥Z 1 (Ik ) = ∥v∥Z˙ 0 (Ik ) + ∥v∥Z˙ 1 (Ik ) ≤ C ε. Similar to Lemma 4.1, by inductive arguments, using Lemma 3.4, if we take the above ε and M small enough, we have 8

∥w∥Z 1 (Ik ) ≤ (2C )k M n−4 −p ≤ C . Thus

∥u∥Z 1 (Ik ) ≤ ∥w∥Z 1 (Ik ) + ∥v∥Z 1 (Ik ) ≤ C (∥ϕ∥H 2 ), ∥u∥Lq (R,H 2,r ) ≤ C (∥ϕ∥H 2 ), for all biharmonic admissible pair (q, r ). Secondly, we prove the asymptotic state. For 0 < t < +∞, we define u+ ( t ) = ϕ + i

t





8





8



W (−s) λ1 |u|p u + λ2 |u| n−4 u (s)ds, 0

u+ = ϕ + i

+∞



W (−s) λ1 |u|p u + λ2 |u| n−4 u (s)ds, 0

and for −∞ < t < 0, we define u− ( t ) = ϕ + i

0







8

W (−s) λ1 |u|p u + λ2 |u| n−4 u (s)ds, t

u− = ϕ + i



0



8



W (−s) λ1 |u|p u + λ2 |u| n−4 u (s)ds. −∞

For 0 < t < τ , by Lemmas 2.3 and 3.4, we have

 t      8 p n−4 u (s)ds ∥u+ (t ) − u+ (τ )∥Hx2 =  W (− s ) λ | u | u + λ | u | 1 2   2 τ H  t  x     8 p  n−4 ≤  W (−s) λ1 |u| u + λ2 |u| u (s)ds τ

2 n L∞ t ([τ ,t ];Hx (R ))

  8   ≤ C λ1 |u|p u + λ2 |u| n−4 u

2(n+4) L n+8



2(n+4) [τ ,t ];L n+8 (Rn )



  8   + λ1 |u|p u + λ2 |u| n−4 u 2  L

[τ ,t ];H˙

1, 2n n+2 (Rn )



  n+4 1 n−4 ≤ C ∥u∥pZ+ + ∥ u ∥ , 1 ([τ ,t ]) Z 1 ([τ ,t ]) which combines the fact ∥u∥Z 1 (R) is bounded, implying u+ (t ) is a Cauchy sequence in H 2 (Rn ), thus u+ (t ) will converge to some function in H 2 (Rn ) as t → +∞. Obviously, this function must be function u+ . Moreover, we have

∥u(t ) − W (t )u+ ∥H 2 (Rn ) = ∥W (−t )u(t ) − u+ ∥H 2 (Rn )  +∞      8 p  ≤ W (−s) λ1 |u| u + λ2 |u| n−4 u (s)ds ∞ t Lt ([t ,+∞);Hx2 (Rn ))   8    ≤ C λ1 |u|p u + λ2 |u| n−4 u 2(n+4)  2(n+4) L n+8

  8   + λ1 |u|p u + λ2 |u| n−4 u 2  L

[t ,+∞);L

[t ,+∞);H˙

1, 2n n+2 (Rn )

  n+4 1 n−4 ≤ C ∥u∥pZ+ + ∥ u ∥ → 0, 1 ([t ,+∞)) Z 1 ([t ,+∞)) so we have lim ∥u(t ) − W (t )u+ ∥H 2 (Rn ) = 0.

t →+∞

(Rn )

n+8



t → +∞,

122

C. Guo / J. Math. Anal. Appl. 392 (2012) 111–122

Using similar arguments, we can obtain lim ∥u(t ) − W (t )u− ∥H 2 (Rn ) = 0. 

t →−∞

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