Strong instability of solitary waves for nonlinear Klein-Gordon equations and generalized Boussinesq equations

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Strong instability of solitary waves for nonlinear Klein–Gordon equations and generalized Boussinesq equations

Instabilité forte d’ondes solitaires pour des équations de Klein–Gordon non linéaires et des équations généralisées de

Boussinesq

Yue Liu

, Masahito Ohta

, Grozdena Todorova

aDepartment of Mathematics, University of Texas, Arlington, TX 76019, USA bDepartment of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan

cDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996, USA Received 28 February 2006; accepted 13 March 2006

Available online 20 September 2006

Abstract

We study here instability problems of standing waves for the nonlinear Klein–Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave’s frequency or the wave’s speed of propagation and on the nonlinearity.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

On étudie des problèmes d’instabilité des ondes stationnaires pour des équations de Klein–Gordon non linéaires et des ondes solitaires pour des équations généralisées de Boussinesq. On établit que ces solutions d’ondes spéciales peuvent être fortement instables par explosion en temps fini, selon le rang de la fréquence des ondes ou la vitesse de propagation des ondes et la non- linéarité.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35L70; 35A15; 35B35; 35R25

Keywords:Nonlinear Klein–Gordon equation; Boussinesq equation; Standing wave; Solitary wave; Strong instability; Blowup

* Corresponding author.

E-mail addresses:yliu@uta.edu (Y. Liu), mohta@rimath.saitama-u.ac.jp (M. Ohta), todorova@math.utk.edu (G. Todorova).

1 Research of the author is supported in part by NSF grant DMS-0245578.

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.03.005

1. Introduction

In this paper, we study strong instability of standing wave solutions e^iωtφ_ω(x)for the nonlinear Klein–Gordon equation

∂_t²u−u+u− |u|^p−1u=0, (t, x)∈R×Rⁿ, (1.1)

and of solitary wave solutionsφ_ω(x−ωt )for the generalized Boussinesq equation

∂_t²u−∂_x²u+∂_x²

∂_x²u+ |u|^p−1u

=0, (t, x)∈R×R, (1.2)

wheren∈N,−1< ω <1,p >1,p <1+4/(n−2)ifn3, andφω is the ground state, i.e., the unique positive radially symmetric solution inH¹(Rⁿ)of the equation

−φ+ 1−ω²

φ− |φ|^p−1φ=0, x∈Rⁿ. (1.3)

See Strauss [28] and Berestycki and Lions [3] for the existence, and Kwong [14] for the uniqueness ofφ_ω.

The stability and instability of the ground state standing waves e^iωtφ_ω(x)for (1.1) have been studied by many authors. Berestycki and Cazenave [1] proved that e^iωtφ_ω(x)is strongly unstable by blowup (see Definition 1.2 below) whenω=0 (see also Payne and Sattinger [24] and Shatah [26]). Shatah [25] proved that e^iωtφω(x)is orbitally stable whenp <1+4/nandω_c<|ω|<1, where

ω_c=

p−1 4−(n−1)(p−1).

On the other hand, Shatah and Strauss [27] proved that e^iωtφ_ω(x)is orbitally unstable whenp <1+4/nand|ω|< ω_c or whenp1+4/nand|ω|<1. Ohta and Todorova [22] proved that e^iωtφ_ω(x)is strongly unstable by blowup when n3 and(p+3)ω²(p−1). Recently, it was proved by Ohta and Todorova [23] that e^iωtφ_ω(x)is strongly unstable by blowup whenn2,p <1+4/nand|ω|ωcor whenn2, 1+4/np <1+4/(n−1)and|ω|<1.

For related results for the nonlinear Schrödinger equations, see [1,9,29,30], and for general theory of orbital stability and instability of solitary waves, see Grillakis, Shatah and Strauss [12,13].

In view of the result of Ginibre and Velo [11], the Cauchy problem for (1.1) is locally well-posed in the energy space X=H¹(Rⁿ)×L²(Rⁿ), that is, for any(u₀, u₁)∈X there exists a unique solutionu=(u, ∂_tu)∈C([0, Tmax), X) of (1.1) withu(0) =(u₀, u₁)such that eitherT_max= ∞orT_max<∞and limt→Tmaxu(t )X= ∞. Moreover, the solutionu(t ) satisfies the conservation laws of energy and charge

E u(t )

=E(u0, u1), Q u(t )

=Q(u0, u1), t∈ [0, Tmax), where

E(u, v)=

Rⁿ

1

2|∇u|²+1

2|u|²+1

2|v|²− 1

p+1|u|^p+1

dx and

Q(u, v)=Im

Rⁿ

¯ uvdx.

In what follows, we putφ_ω=(φ_ω,iωφω). Then, note that Eφω

−ωQφω

=0.

Orbital stability of standing waves for (1.1) refers to stability up to translations and phase shifts. More precisely, Definition 1.1(Orbital stability and instability).We say that a standing wave e^iωtφ_ω(x)is orbitally stable for (1.1) if for anyε >0, there existsδ >0 such that for any(u₀, u₁)∈X=H¹(Rⁿ)×L²(Rⁿ)with(u₀, u₁)− φ_ωX< δ, the solutionu(t ) of (1.1) with initial valueu(0) =(u₀, u₁)exists for allt∈ [0,∞)and satisfies

sup

0t <∞ inf

θ∈R, y∈Rⁿu(t ) −e^iθφ_ω(· +y)

X< ε.

Otherwise, e^iωtφ_ω(x)is said to be orbitally unstable.

Definition 1.2(Strong instability by blowup).We say that a standing wave e^iωtφω(x)is strongly unstable by blowup if for anyε >0, there exists(u₀, u₁)∈Xsuch that(u₀, u₁)− φ_ωX< εand the solutionu(t ) of (1.1) with initial valueu(0) =(u₀, u₁)blows up in finite time.

In view of the above definitions of instability, if the standing wave e^iωtφ_ω(x)is strongly unstable by blowup, then it is orbitally unstable.

The principal result of the present paper for (1.1) is the following.

Theorem 1.3.Assume thatp >1,p <1+4/(n−2)ifn3,−1< ω <1, and 0<2(p+1)ω²< p−1 ifn=1,

0< (p+3)ω²p−1 ifn2. (1.4)

Letφ_ωbe the ground state of (1.3). Then the standing wave solutione^iωtφ_ω(x)of(1.1)is strongly unstable by blowup.

As mentioned above, the strong instability by blowup of standing waves e^iωtφ_ω(x)has been proved by [1] for the caseω=0 and n1, and by [22] for the case(p+3)ω²(p−1)andn3. In the proof of Theorem 1.3, we need to assume thatω=0 for technical reasons (see (2.1) and the proof of Proposition 2.1 below). Theorem 1.3 gives a new result for the casen=1, 2, which is a natural extension of the result for the casen3 by [22]. Although the result for the casen3 is not new, the proof is slightly simpler than the one in [22]. In fact, the essential point in the proof of [22] was to introduce two appropriate invariant sets for the flow of (1.1) (see (2.4) in [22]), while in the present paper we use only one invariant setΣ1which is defined by (2.5).

Now we turn attention to the Boussinesq equation (1.2). The original Boussinesq equation was the first model for the propagation of weakly nonlinear dispersive long surface and internal waves [5,7]. Eq. (1.2) has the equivalent form

u_t=v_x,

v_t=(u−u_xx− |u|^p−1u)_x. (1.5)

It is known in [17] that the Cauchy problem for (1.5) is locally well-posed in the space X=H¹(R)×L²(R).

Moreover, the solutionu(t ) =(u(t ), v(t ))with initial value(u₀, v₀)inC([0, Tmax), X)satisfies the conservation laws E

u(t )

=E(u0, v0), Q u(t )

=Q(u0, v0), 0t < Tmax, where

E(u, v)=

R

1

2|∂_xu|²+1

2|u|²+1

2|v|²− 1

p+1|u|^p+1

dx

and

V (u, v)=

R

uvdx.

Putφ_ω=(φ_ω,−ωφ_ω). Then, a simple computation shows that Eφ_ω

+ωV φ_ω

=0.

The stability of solitary waveφ_ω(x−ωt )up to translations can be defined in the following.

Definition 1.4(Orbital stability and instability).We say that a solitary wave solutionφω(x−ωt )of (1.5) is orbitally stable if for anyε >0, there existsδ >0 such that for(u₀, v₀)∈X=H¹(R)×L²(R)with

(u₀, u₁)− φ_ω

X< δ,

the solutionu(t ) =(u(t ), v(t ))of (1.5) with initial valueu(0) =(u₀, v₀)satisfies sup

0t <∞inf

y∈Ru(t ) − φω(· +y)

X< ε.

Otherwise,φ_ω(x−ωt )is considered to be orbitally unstable.

Definition 1.5(Strong instability by blowup).We say that a solitary wave solutionφω(x−ωt )is strongly unstable by blowup if for anyε >0, there exists(u₀, v₀)∈Xsuch that(u₀, v₀)− φ_ωX< εand the solutionu(t ) of (1.5) with initial valueu(0) =(u₀, v₀)blows up in finite time.

The stability of solitary wavesφ_ω(x−ωt )with|ω|<1 has been the subject of a number of studies, and a satis- factory stability theory is now in hand. For example, Bona and Sachs [4] proved that the solitary waveφ_ω(x−ωt )is orbitally stable if 1< p <5,(p−1)/4< ω²<1. Liu [17] proved the orbital instability under the conditions 1< p <5 andω²< (p−1)/4 orp5 and|ω|<1. On the other hand, Liu [18] showed that solitary waveφ_ω(x−ωt )is strongly unstable by blowup for the wave speedω=0 (see also [19]).

The principal result for (1.5) is stated as follows.

Theorem 1.6.Assume1< p <∞and0<2(p+1)ω²< p−1. Letφ_ωbe the ground state of (1.3). Then the solitary wave solutionφ_ω(x−ωt )=(φ_ω(x−ωt ),−ωφ_ω(x−ωt ))of(1.5)is strongly unstable by blowup.

In next section, we give the proofs of Theorems 1.3 and 1.6. The method of the proofs is based on the idea by Berestycki and Cazenave [1] in the study of strong instability of standing waves for the nonlinear Schrödinger equation as well as the nonlinear Klein–Gordon equation. The crucial point in their proof is to construct some suitable invariant sets under the flow of the evolution equations. Then the strong instability results are obtained by use of the virial identities with the variational characterization of ground states. This method is recently developed by many authors [18–23]. To optimally use these virial identities, we need to construct some particular invariant sets of the flows of (1.1) or (1.5).

Notation.As above and henceforth, we denote the norm of the Lebesgue spaceL^p(Rⁿ)by · pfor 1p∞. The function space in which we shall work is the Sobolev spaceX=H¹(Rⁿ)×L²(Rⁿ).

2. Proof of strong instability

In this section, we prove the main results, Theorems 1.3 and 1.6.

Define functionalsJ_ω andK_ωonH¹(Rⁿ)by J_ω(v)=1

2∇v²₂+1−ω²

2 v²₂− 1

p+1v^p_p⁺₊¹₁, Kω(v)=2α+n−2

2 ∇v²2+(1−ω²)(2α+n)

2 v²2−(p+1)α+n

p+1 v^p_p⁺₊¹₁, where we put

α=(p−1)−(p+3)ω²

2(p−1)ω² . (2.1)

Note that

K_ω(v)=∂_λJ_ω(v_λ)|λ=1, v_λ(x)=λ^αv(x/λ),

and by assumption (1.4) we have 2α >1 ifn=1;α0 ifn2; and 2α+n−2>0 except the casen=2 and (p+3)ω²=p−1.

Moreover, we put d_ω=inf

J_ω(v): v∈H¹(Rⁿ)\ {0}, K_ω(v)=0 , Sω=

v∈H¹(Rⁿ)\ {0}: J_ω(v)=0 , Gω=

w∈Sω: Jω(w)Jω(v)for allv∈Sω

, Mω=

v∈H¹(Rⁿ)\ {0}: J_ω(v)=d_ω, K_ω(v)=0 and

J˜_ω(v)=J_ω(v)− 1

(p+1)α+nK_ω(v)

=(p−1)α+2 (p+1)α+n

1

2∇²2+ (p−1)α

(p−1)α+2·1−ω² 2 v²2

.

Then, we have

J˜_ω(v)=(p−1)α+2 (p+1)α+n

1

2∇v²2+αω²v²2

, (2.2)

dω=infJ˜ω(v)|v∈H¹(Rⁿ)\ {0}, Kω(v)=0

0. (2.3)

Our first goal is to show that the ground stateφω belongs to the setMω under the assumption (1.4). To this end, we need to consider the casen=2 and(p+3)ω²=p−1 separately. For this case, see [2] and [8, Theorem 8.1.5].

In what follows, we assume that(p+3)ω²< p−1 ifn=2.

Proposition 2.1.Ifn=1orn3assume(1.4);ifn=2assume(p+3)ω²< p−1. Then the setMωis not empty.

To prove Proposition 2.1, we need the following lemmas.

Lemma 2.2.Ifv∈H¹(Rⁿ)satisfiesK_ω(v) <0, thenJ˜_ω(v) > d_ω.

Proof. Letv∈H¹(Rⁿ)satisfyK_ω(v) <0. Then, there existsλ1∈(0,1)such thatK_ω(λ1v)=0. Since v=0, we havedωJ˜ω(λ1v)=λ²₁J˜ω(v) <J˜ω(v). 2

Remark.In view of relation (2.3) and Lemma 2.2, it is found that dω=infJ˜ω(v): v∈H¹(Rⁿ)\ {0}, Kω(v)0

.

The following compactness lemmas are obtained by Fröhlich, Lieb and Loss [10], Lieb [16] and Brezis and Lieb [6].

Lemma 2.3. [10,16] Let {f_j}be a bounded sequence in H¹(Rⁿ). Assume that there existsq ∈(2,2^∗) such that lim sup_j→∞f_jq>0, where2^∗= ∞ifn=1,2, and2^∗=2n/(n−2)ifn3. Then, there exist{y_j} ⊂Rⁿ and f ∈H¹(Rⁿ)\ {0}such that{fj(· −yj)}has a subsequence that converges tof weakly inH¹(Rⁿ).

Lemma 2.4.[6]Let1q <∞and{fj}be a bounded sequence inL^q(Rⁿ). Assume thatfj→f a.e. inRⁿ. Then we have

f_j^qq− f_j−f^qq− f^qq→0.

Proof of Proposition 2.1. Let {v_j}be a minimizing sequence of (2.3). By (2.2), we see that {v_j}is bounded in H¹(Rⁿ). Indeed, whenn=1,2, by the assumptions of Proposition 2.1 and (2.1), we haveαω²>0, so{v_j}is bounded inH¹(Rⁿ). Whenn3, by (1.4) and (2.1), we haveα0, so{∇v_j2}is bounded. Since 2< p+1<2^∗, for any ε >0 there existsC_ε>0 such thats^p+1εs²+C_εs^2∗for alls0. Thus, byK_ω(v_j)=0 and the Sobolev inequality, we have

2α+n−2

2 ∇v_j²₂+(1−ω²)(2α+n)

4 v_j²₂Cv_j²₂^∗∗C∇v_j²₂^∗. Since{∇vj2}is bounded, we see that{vj}is bounded inH¹(Rⁿ).

Moreover, by K_ω(v_j)=0 and the Sobolev inequality, there exist positive constants C₁, C₂ andC₃ such that C₁v_j²_H1 C₂v_j^p_p⁺₊¹₁C₃v_j^p_H⁺1¹. Sincev_j =0, we haveC₁/C₃v_j^p_H⁻1¹ and lim sup_j→∞v_jp+1>0.

Therefore, by Lemma 2.3, there exist {yj} ⊂Rⁿ, a subsequence of {vj(· −yj)}(we denote it by {wj}) and w∈ H¹(R)\ {0}such thatwj wweakly inH¹(Rⁿ). By the weakly lower semicontinuity ofJ˜ω, we have

J˜_ω(w)lim inf

j→∞ J˜_ω(w_j)=d_ω.

Moreover, by Lemma 2.4, we have

Kω(wj)−Kω(wj−w)−Kω(w)→0,

which impliesK_ω(w)0. Indeed, suppose thatK_ω(w) >0. SinceK_ω(w_j)=0, we haveK_ω(w_j−w) <0 for largej. By Lemma 2.2, we haveJ˜_ω(w_j−w) > d_ω, and

J˜_ω(w)=lim sup

j→∞

J˜_ω(w_j)− ˜J_ω(w_j−w)

0.

On the other hand, byw=0 and (2.2), we haveJ˜_ω(w) >0. This is a contradiction. Therefore, we see thatK_ω(w)0.

Finally, by Lemma 2.2 andJ˜_ω(w)d_ω, we haveK_ω(w)=0 andd_ω= ˜J_ω(w). Hence,w∈Mω. 2 Proposition 2.5.Ifn=1orn3assume(1.4);ifn=2assume(p+3)ω²< p−1. Then we have

Mω=Gω=

e^iθφω(· +y): θ∈R, y∈Rⁿ if translations and phase shifts are considered or

Mω=Gω=

±φ_ω(· +y): y∈Rⁿ

if only translations are considered, whereφ_ω is the ground state of(1.3).

Proof. First, we show thatMω⊂Gω. Letw∈Mω. Then, there exists a Lagrange multiplierη∈Rsuch thatJ_ω(w)= ηK_ω(w). That is,wsatisfies

−

1−(2α+n−2)η

w+

1−(2α+n)η 1−ω²

w= 1−

(p+1)α+n η

|w|^p−1w (2.4) inH⁻¹(Rⁿ). ByKω(w)=0 andJ_ω(w), w =ηK_ω(w), w, we have

2(p+1)

p−1 ∇w²₂=

n−η(2α+n)

(p+1)α+n

w^p_p⁺₊¹₁.

Sincew=0, we have

η < n

(2α+n){(p+1)α+n},

which implies 1−(2α+n−2)η >0 and 1−(2α+n)η >0 in (2.4). Thus, by [8, Theorem 8.1.1], we havex· ∇w∈ H¹(Rⁿ), and we have

0=Kω(w)=∂λJω(wλ)|λ=1=

J_ω(w), αw−x· ∇w

=η

K_ω(w), αw−x· ∇w

=η∂λKω(wλ)|λ=1, wherewλ(x)=λ^αw(x/λ). Moreover, byKω(w)=0, we have

∂λKω(wλ)|λ=1=(2α+n−2)²

2 ∇w²2+(1−ω²)(2α+n)²

2 w²2−{(p+1)α+n}²

p+1 w^p_p⁺₊¹₁

= −(2α+n−2){(p−1)α+2}

2 ∇w²₂−(1−ω²)(2α+n)(p−1)α

2 w²₂<0.

Thus, we haveη=0, andw∈Sω. Moreover, for anyv∈Sω, we haveK_ω(v)=0. By the definitions ofd_ωandMω, we haveJ_ω(w)=d_ωJ_ω(v). Therefore, we havew∈Gω, and we concludeMω⊂Gω.

On the other hand, by [8, Theorems 8.1.4 to 8.1.6], we haveGω= {e^iθφ_ω(· +y): θ∈R, y∈Rⁿ}. SinceMωis not empty by Proposition 2.1, we haveMω=Gω. 2

Define the setΣ1by Σ₁=

(u, v)∈X|E(u, v)−ωQ(u, v) < d_ω, K_ω(u) <0

. (2.5)

Note thatλ(φ_ω,iωφω)∈Σ₁for anyλ >1.

Lemma 2.6. The set Σ1 is invariant under the flow of (1.1). That is, if the data (u0, u1)∈Σ1, then u(t ) = (u(t ), ∂_tu(t ))∈Σ₁for anyt∈ [0, Tmax), whereu(t ) is the solution of (1.1)with initial value(u₀, u₁)andT_max is the life span ofu(t ).

Proof. For the sake of convenience, we put L_ω(u, v)=E(u, v)−ωQ(u, v).

It is immediately observed that Jω(u)=Lω(u, v)−1

2v−iωu²2Lω(u, v). (2.6)

From the conservation of energy and charge, we have L_ω(u(t )) =L_ω(u₀, u₁) < d_ω for anyt ∈ [0, Tmax). So, to conclude the proof of the lemma, it suffices to show that K_ω(u(t )) <0 for any t∈ [0, Tmax). Suppose that there existst₀∈(0, Tmax)such thatK_ω(u(t₀))=0 andK_ω(u(t )) <0 fort∈ [0, t0). It then follows from Lemma 2.2 that J˜_ω(u(t ))d_ω>0 fort ∈ [0, t0). Thus, we see thatu(t0)=0. Since K_ω(u(t0))=0 andu(t0)=0, it follows from relation (2.6) and the definition ofdω thatdωJω(u(t0))Lω(u(t 0)) < dω, which is a contradiction. 2

Forλ >1, letu_λ(t )=(u_λ(t ), ∂_tu_λ(t ))be the solution of (1.1) with datau_λ(0)=λφ_ω, whereφ_ω=(φ_ω,iωφω)and φ_ωis the ground state of (1.3). LetT_λbe the life span ofu_λ(t ). Define

I_λ(t )=1

2u_λ(t )²

2, 0t < T_λ.

The key lemma to prove the strong instability is the following lower estimate for the virial identity.

Lemma 2.7.For anyλ >1, there exists a constantaλ>0such that d²

dt²I_λ(t )p+3

2 ∂_tu_λ(t )−iωuλ(t )²

2+a_λ, 0t < T_λ. Proof. By simple computations, we have

I_λ(t )=Re

Rⁿ

∂_tu_λ(t )u_λ(t )dx=Re

Rⁿ

∂_tu_λ(t )−iωuλ(t ) u_λ(t )dx

and

I_λ (t )=∂_tu_λ(t )²

2+Re

Rⁿ

∂_t²u_λ(t, x)u_λ(t, x)dx

=p+3

2 ∂_tu_λ(t )²

2+p−1

2 ∇u_λ(t )²

2+

1−ω²u_λ(t )²

2

−(p+1)E u_λ(t )

=p+3

2 iωuλ(t )−∂_tu_λ(t )²

2+(p−1) 1

2∇u_λ(t )²

2+αω²u_λ(t )²

2

−(p+1)Lω

u_λ(t )

+2ωQ u_λ(t )

=p+3

2 ∂_tu_λ(t )−iωuλ(t )²

2+(p−1)((p+1)α+n) (p−1)α+2 J˜_ω

u_λ(t )

−(p+1)Lω

λφ_ω

+2ωQ λφ_ω

. (2.7)

Here, in the last equality, we have used the fact thatL_ω andQare conserved quantities. For anyλ >1,it is easy to see that

L_ω λφ_ω

=J_ω(λφ_ω) < J_ω(φ_ω)=d_ω (2.8)

and ωQ

λφ_ω

=ω²λ²φ_ω²₂> ω²φ_ω²₂. (2.9)

On the other hand, it is found that φ_ω²₂=(n+2)−(n−2)p

(p−1)(1−ω²) d_ω. (2.10)

Here, we put

a_λ=(p−1)((p+1)α+n)

(p−1)α+2 d_ω−(p+1)Lω

λφ_ω

+2ωQ λφ_ω

.

Then, by (2.8)–(2.10), we havea_λ>0. Moreover, by (2.7), we have I_λ (t )p+3

2 ∂_tu_λ(t )−iωuλ(t )²

2+a_λ+(p−1)((p+1)α+n) (p−1)α+2

J˜_ω u_λ(t )

−d_ω

(2.11) for 0t < T_λ. Sinceu_λ(t )is the solution of (1.1) with dataλφ_ω∈Σ₁, it follows from Lemma 2.6 thatu_λ(t )∈Σ₁ for any 0t < T_λ. Hence, it then follows from (2.11) and Lemma 2.2 that

I_λ (t )p+3

2 ∂_tu_λ(t )−iωuλ(t )²

2+a_λ

for 0t < Tλ. This completes the proof of Lemma 2.7. 2

Proof of Theorem 1.3 then follows from Lemma 2.7 and concavity arguments due to Levine [15] as in Payne and Sattinger [24]. For the sake of completeness, we give the proof.

Proof of Theorem 1.3. We use the notation of Lemma 2.7. Sinceλφ_ω→ φ_ωinXasλ→1, it suffices to prove that T_λ<∞for anyλ >1. We prove this by contradiction. Assume thatT_λ= ∞. By Lemma 2.7, we haveI_λ (t )a_λ>0 for anyt∈ [0,∞). This implies that there existst₁∈(0,∞)such thatI_λ(t ) >0 andI_λ(t ) >0 for anyt∈ [t₁,∞). Let β=(p−1)/4.Then by using Lemma 2.7 we obtain the following estimate

I_λ (t )Iλ(t )−(β+1)I_λ(t )²

p+3

4

∂tuλ(t )−iωuλ(t )²

2uλ(t )²

2−

Re

Rⁿ

∂tuλ(t )−iωuλ(t ) uλ(t )

2

dx

0.

Thus, fort∈ [t1,∞), we have I_λ(t )^−β

= −βI_λ(t )^−β−1I_λ(t ) <0, I_λ(t )^−β

= −βI_λ(t )^−β−2

I_λ (t )I_λ(t )−(β+1)I_λ(t )²

0.

Therefore,

Iλ(t )^−βIλ(t1)^−β−βIλ(t1)^−β−1I_λ(t1)(t−t1), t∈ [t1,∞),

so there existst2∈(t1,∞)such thatIλ(t2)^−β0. However, this is a contradiction. This completes the proof. 2 Having established the strong instability by blowup of standing waves for (1.1), attention is now given to the proof of Theorem 1.6, that is, strong instability of solitary waves for (1.5). The proof of Theorem 1.6 is similar to that of Theorem 1.3 and is approached via the following two main lemmas.

Lemma 2.8.Let Σ2=

(u, v)∈X|E(u, v)+ωV (u, v) < dω, Kω(u) <0 .

Then the setΣ2is invariant under the flow of (1.5). That is, if(u0, v0)∈Σ2, thenu(t ) =(u(t ), v(t ))∈Σ2for any t∈ [0, Tmax), whereu(t ) is the solution of (1.5)with initial value(u₀, v₀)andT_maxis the life span ofu(t ).

Proof. We omit the proof because it is similar to that of Lemma 2.6. 2 Note thatλ(φ_ω,−ωφ_ω)∈Σ₂for anyλ >1.

Lemma 2.9.The setA= {w∈H¹(R)|ξ⁻¹wˆ∈L²(R)}is dense inH¹(R), wherewˆ is the Fourier transform ofw.

Proof. See Lemma 4.4 in [18]. 2

Letφω=(φω,−ωφω), whereφωis the ground state of (1.3). By Lemma 2.9, there iswε_i=(wε_i,−ωwε_i)∈Asuch thatw_εi→ φ_ωinH¹asε_i→0. Forλ >1, letu^λ₀=λw_εi. We claim thatu^λ₀∈Σ₂. In fact, we have

Lω

u^λ₀

=E u^λ₀

+ωV u^λ₀

=E λφω

+ωV λφω

+α(εi),

whereα(ε_i)→0 asε_i →0. Since λφ_ω ∈Σ₂, chooseε_i small enough such thatα(ε_i) < d_ω−L_ω(λφ_ω). It is then found that

L_ω u^λ₀

< L_ωφ_ω

=d_ω. (2.12)

Similarly, we have Kω

u^λ₀

=Kω(λφω)+α(εi) < Kω

φω

=0 and

−ωV u^λ₀

= −ωV λφω

+α(εi)=ω²λ²φω²2+α(εi) > ω²φω²2. (2.13)

This in turn implies thatu^λ₀∈Σ2. On the other hand, it is easy to see that φ_ω²₂= 3+p

(p−1)(1−ω²)d_ω. (2.14)

Forλ >1, letu_λ(t )=(u_λ(t ), v_λ(t ))be the solution of (1.5) with initial valueu^λ₀. LetT_λbe the life span ofu_λ. Put Iλ(t )=1

2ξ⁻¹uˆλ(t )²

2, 0t < Tλ.

Since u^λ₀∈A, the function I_λ(t ) is well-defined. The following estimate of the virial identity can be obtained in a similar way as in Lemma 2.7.

Lemma 2.10.For anyλ >1, there is a constanta_λ>0such that I_λ (t )p+3

2 v_λ(t )+ωu_λ(t )²

2+a_λ, t∈ [0, Tλ).

Proof. Since the proof of Lemma 2.10 is similar to that of Lemma 2.7, we only give an outline of the proof. A simple computation shows that

I_λ(t )=Re

R

ξ⁻¹uˆ_λ(t )vˆ_λ(t )dξ=Re

R

ξ⁻¹uˆ_λ(t ) ˆ

v_λ(t )+ωuˆ_λ(t ) dξ

and

I_λ (t )=p+3

2 v_λ(t )²

2+p−1

2 ∂_xu_λ(t )²

2+

1−ω²u_λ(t )²

2

−(p+1)E u_λ(t )

=p+3

2 v_λ(t )+ωu_λ(t )²

2+(p−1) 1

2∂_xu_λ(t )²

2+αω²u_λ(t )²

2

−(p+1)Lω

u_λ(t )

−2ωV u_λ(t )

=p+3

2 v_λ(t )+ωu_λ(t )²

2+(p−1)((p+1)α+1) (p−1)α+2 J˜_ω

u_λ(t )

−(p+1)Lω

u^λ₀

−2ωV u^λ₀

.

Therefore, in view of (2.12)–(2.14), Lemma 2.10 can be obtained by Lemmas 2.2 and 2.8. 2 Proof of Theorem 1.6. The proof follows from Lemma 2.6 and the proof of Theorem 1.3. 2

Acknowledgements

We thank the referees for suggestions and comments.

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