INSTABILITY OF THE STANDING WAVES FOR THE
NONLINEAR KLEIN-GORDON EQUATIONS IN ONE
DIMENSION
arXiv:1705.04216v2 [math.AP] 4 Jun 2017
YIFEI WU
Abstract. In this paper, we consider the following nonlinear Klein-Gordon equation
∂tt u − ∆u + u = |u|p−1 u,
t ∈ R, x ∈ Rd ,
with 1 < p < 1 + 4d . The equation has the standing wave solutions uω = eiωt φω
with the frequency ω ∈ (−1, 1), where φω obeys
−∆φ + (1 − ω 2 )φ − φp = 0.
It was proved by Shatah (1983), and Shatah, Strauss (1985) that there exists a
critical frequency ωc ∈ (0, 1) such that the standing waves solution uω is orbitally
stable when ωc < |ω| < 1, and orbitally unstable when |ω| < ωc . Further, the
critical case |ω| = ωc in the high dimension d ≥ 2 was considered by Ohta,
Todorova (2007), who proved that it is strongly unstable, by using the virial
identities and the radial Sobolev inequality. The one dimension problem was left
after then. In this paper, we consider the one-dimension problem and prove that
it is orbitally unstable when |ω| = ωc .
1. Introduction
In this paper, we consider the stability theory of the following nonlinear KleinGordon equation
∂tt u − ∆u + u = |u|p−1u,
with the initial data
u(0, x) = u0 (x),
t ∈ R, x ∈ Rd ,
ut (0, x) = u1 (x).
(1.1)
(1.2)
4
. The H 1 × L2 -solution (u, ut) of (1.1)–(1.2) obeys
Here d ≥ 1 and 1 < p < 1 + d−2
the following charge, momentum and energy conservation laws,
Z
Q(u, ut) = Im uūt dx = Q(u0 , u1);
(1.3)
Z
P (u, ut) = Re ∇uūt dx = P (u0 , u1);
(1.4)
2
kukp+1
(1.5)
Lp+1 = E(u0 , u1 ).
p+1
The well-posedness for the Cauchy problem for (1.1)–(1.2) was well understand in the
energy space H 1 (Rd ) × L2 (Rd ). More precisely, for any (u0 , u1) ∈ H 1 (Rd ) × L2 (Rd ),
there exists a unique solution (u, ut ) ∈ C([0, T ); H 1(Rd ) × L2 (Rd )) of (1.1)–(1.2),
E(u, ut) = kut k2L2 + k∇uk2L2 + kuk2L2 −
2010 Mathematics Subject Classification. Primary 35L05; Secondary 37K45.
Key words and phrases. Nonlinear Klein-Gordon equation, instability, standing waves, critical
frequency, one dimension.
1
2
YIFEI WU
with the maximal lifetime T = T (k(u0 , u1)kH 1 ×L2 ). If T = ∞, we call that the
Cauchy problem (1.1)–(1.2) is global well-posedness. If T < ∞, we call that the
solution blows up in finite time. See for examples Ginibre and Velo [8, 9] for the
local and global well-posedness, and Payne and Sattinger [26] for the blow-up results.
Further results on the scattering in the case of p ≥ 1 + d4 , see [13, 14, 22] and the
references therein.
The equation (1.1) has the standing waves solution eiωt φω , where φω is the
ground state solution of the following elliptic equation
−∆φ + (1 − ω 2 )φ − φp = 0.
(1.6)
The equation (1.6) exists solutions when the parameter |ω| < 1, see [30] for example. In particular, in one dimension case, the solution to (1.6) is unique up
to the symmetries of the rotation and the spatial transformation. Moreover, the
ground state solution φω is exponential decaying at infinity when |ω| < 1. See also
[1, 4, 5, 6, 7, 15] for some instances on the existence of the multi-solitary waves of
the nonlinear Klein-Gordon and the nonlinear Schrödinger equations.
The stability theory of the the standing waves solution uω (t) = eiωt φω has
been widely studied. In particular, Berestycki and Cazenave [2] proved that it is
4
strong instability when ω = 0 and 1 < p < 1 + d−2
, which is in the sense that
an arbitrarily small perturbation of the initial data can find the perturbed solution
blowing up in finite time. See also Shatah [28] for the related works. One may find
the big difference between the nonlinear Klein-Gordon equation and the nonlinear
Schrödinger equation, because of the lack of the mass conservation law. Further,
when ω 6= 0, Shatah [27] proved that it is orbitally stability when 1 < p < 1 + 4d and
ωc < |ω| < 1, where the frequency ωc equal to
s
p−1
.
ωc =
4 − (d − 1)(p − 1)
The number ωc is critical. Indeed, Shatah and Strauss [29] showed further that
4
when 1 < p < 1 + 4d , |ω| < ωc or 1 + d4 < p < 1 + d−2
, |ω| < 1, the standing waves
iωt
solution e φω is orbitally instability. See also Stuart [31] for the stability of the
solitary waves. The critical cases, |ω| = ωc when 1 < p < 1 + d4 , are degenerate
based on the theories of Grillakis, Shatah and Strauss [10, 11]. The degenerate cases
were further investigated by several authors, such as Comech, Pelinovsky [3], Maeda
[17, 18], and Ohta [23]. In particular, as an application of the theorems established
in [3, 18], the standing waves solution eiωt φω is orbitally unstable in the critical cases
|ω| = ωc when 2 ≤ p < 1 + d4 . The region 1 < p < 2 was not covered because of
the lack of the regularity for the relevant functionals. Further, Ohta and Todorova
[24, 25] (see also [12] for a companion result) proved the strong instability when
4
d ≥ 2, 1 < p < 1 + d4 , |ω| ≤ ωc or d ≥ 2, 1 + d4 ≤ p < 1 + d−2
, |ω| < 1, which
cover the entire instability region in the case of d ≥ 2. The argument the authors
used was the variation argument combining with the virial identities. Hence, the
stability and instability regions were complete division except the one dimension
cases, and the only left problem is the stability theory of the soliton in the case of
1 < p < 2, |ω| = ωc when d = 1. Unfortunately, the argument in [25] is not available
in one dimension problem, because the argument relies on the radial choice of the
instable data, which gives radial compactness and thus the small control of the
remainder terms from the localized virial identities. In one dimension, Liu, Ohta
INSTABILITY OF STANDING WAVES
3
and Todorova [16] also considered the strong instability in some regions which also
has gap from the critical frequency. In present paper, we study the instability of the
standing waves solution in the critical case in one dimension.
Before stating our theorem, we recall some definitions. Let v = ut , ~u = (u, v)T ,
−
→
−
→
~u0 = (u0 , u1)T , and Φω = (φω , iωφω )T . For ε > 0, we denote the set Uε Φω as
−
→
−
→
Uε Φω = {~u ∈ H 1 (R) × L2 (R) : inf 2 k~u − eiθ Φω (· − y)kH 1 ×L2 < ε}.
(θ,y)∈R
Definition 1.1. We say that the solitary wave solution uω of (1.1) is stable if for
−
→
any ε > 0 there exists δ > 0 such that if k~u0 − Φω kH 1 ×L2 < δ, then the solution
−
→
~u(t) of (1.1) with ~u(0) = ~u0 exists for all t > 0, and ~u(t) ∈ Uε Φω for all t > 0.
Otherwise, uω is said to be unstable.
Then the main result in the present paper is
Theorem
q 1.2. Let d = 1, 1 < p < 5, ω ∈ (−1, 1) and φω be the solution of (1.6).
, then the standing waves solution eiωt φω is orbitally unstable.
If |ω| = p−1
4
The method used to prove the theorem is the modulation argument combining
with the virial identity, which is complete different from [3, 18, 25]. The modulation
argument was introduced by Weinstein [32], and strengthened by the mathematicians such as Martel, Merle, Raphaël [19, 20, 21]. In particular, in the Klein-Gordon
setting, we use the modulation method applied by Bellazzini1, Ghimenti, and Le
Coz [1], who considered the total linearized action. Moreover, in the technical level,
we utilize the equality
d −
→ Q Φω = 0.
dω
ω=±ωc
and the term kv − iωuk2L2 in the virial identity to control the scaling parameter.
Further, we believe the strong instability is true in our case and our method
here could be used to prove the strong result. It leaves us an interesting problem to
pursue in the future.
Now the following is the organization of the paper. In Section 2, we give
some preliminaries. It includes some basic definitions and properties, the coercivity
property of the Hessian, the modulation statement and the virial identities. In
Section 3, we control the remainder function and the scaling parameter, and lastly
prove the main theorem.
2. Preliminary
2.1. Notations. For f, g ∈ L2 (R) = L2 (R, C), we define
Z
hf, gi = Re f (x)g(x) dx
R
and regard L (R) as a real Hilbert space. Similarly, for f~, ~g ∈ L2 (R)
we define
Z
~
hf , ~g i = Re f~(x)T · ~g (x) dx.
2
R
2
2
= L2 (R, C) ,
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q
For a function f (x), its L -norm kf kLq =
1
Z
R
q1
and its H 1 -norm
|f (x)|q dx
kf kH 1 = (kf k2L2 + k∂x f k2L2 ) 2 .
Further, we write X . Y or Y & X to indicate X ≤ CY for some constant
C > 0. We use the notation X ∼ Y whenever X . Y . X. Also, we use O(Y ) to
denote any quantity X such that |X| . Y ; and use o(Y ) to denote any quantity X
such that X → 0, if Y → 0.
2.2. Some basic definitions and properties. In the following, we only
q consider
p−1
one dimension problem and the case of 1 < p < 5, in which ωc =
. Let
4
−
→
~u = (u, v)T , Φω = (φω , iωφω )T . Recall that the conserved qualities,
Z
Q(~u) = Im uūt dx,
2
kukp+1
Lp+1 .
p+1
First, we give some basic properties on the charge and energy.
E(~u) = kut k2L2 + kux k2L2 + kuk2L2 −
Lemma 2.1. The following equalities hold,
−
→ d
Q Φω = 0;
(1) dω
ω=±ωc
−
−
→
→
(2) If |ω| = ωc , then (p + 3)E Φω + 8ωQ Φω = 0.
Proof. Note that
−
→
Q Φω = −ωkφω k2L2 .
Moreover, by rescaling, we find,
1
This implies that
φω (x) = (1 − ω 2) p−1 φ0
√
1 − ω2x .
−
2
1
→
Q Φω = −ω(1 − ω 2 ) p−1 − 2 kφ0 k2L2 .
Hence by a direct computation, we have
h
i
2
→
4
d −
− 32
2 p−1
2
1−
Q Φω = −(1 − ω )
ω kφ0 k2L2 .
dω
p−1
This gives (1). For (2), we have
i
−
→ 1 h
2
p+1
2
2
2
E Φω = k∂x φω kL2 + (1 + ω )kφω kL2 −
kφω kLp+1 .
2
p+1
From the equation (1.6), we obtain that
k∂x φω k2L2 + (1 − ω 2)kφω k2L2 − kφω kp+1
Lp+2 = 0;
2
k∂x φω k2L2 − (1 − ω 2 )kφω k2L2 +
kφω kp+1
Lp+2 = 0.
p+1
These give that
−
→
1
E Φω =
p − 1 + 4ω 2 kφω k2L2 .
p+3
−
→
Combining the value of Q Φω above, we obtain (2).
INSTABILITY OF STANDING WAVES
5
Now we define the functional Sω as
Sω (~u) = E(~u) + ωQ(~u).
Then we have
Sω′ (~u)
=
−uxx + u − |u|p−1u
v
+ iω
v
−u
.
Note that Sω′ (Φω ) = 0. Moreover, for the vector f~ = (f, g)T , a direct computation
shows that
p−1
→ ~
−fxx + f − pφp−1
g
′′ −
ω Ref − iφω Imf
Sω Φω f =
+ iω
.
g
−f
−
→
From the invariance of Sω′ Φω in the rotation and spatial transformations, we have
−
→ −
→
−
→ −
→
Sω′′ Φω iΦω = 0,
Sω′′ Φω ∂x Φω = 0.
−
→
Moreover, taking the derivative of Sω′ Φω = 0 gives that
−
→ −
→
−
→
Sω′′ Φω ∂ω Φω = −Q′ Φω .
(2.1)
Then a consequence of Lemma 2.1 (1) is
Corollary 2.2. Let λ ∈ R+ , ω = ±ωc , then
−−→
−
→
Sλω Φλω − Sλω Φω = o (λ − 1)2 .
Proof. From the definition and the Taylor’s type expansion,
−−→
−
→
Sλω Φλω − Sλω Φω
−−→
−
→
−
→
−−→
=Sω Φλω − Sω Φω + (λ − 1)ω Q Φλω − Q Φω
→−−→ −
→ −−→ −
→E
1 D ′′ −
Sω Φω Φλω − Φω , Φλω − Φω
=
2
−−→
−
→
+ (λ − 1)ω Q Φλω − Q Φω + o (λ − 1)2 .
Note that
−−→ −
→
−
→
Φλω − Φω = (λ − 1)ω∂ω Φω + o(λ − 1),
we find that
−
→−−→ −
→ −−→ −
→E
Sω′′ Φω Φλω − Φω , Φλω − Φω
D
→E
→ −
→ −
′′ −
2 2
=(λ − 1) ω Sω Φω ∂ω Φω , ∂ω Φω + o (λ − 1)2
D −
→E
→ −
= − (λ − 1)2 ω 2 Q′ Φω , ∂ω Φω + o (λ − 1)2
−−→
d
= − (λ − 1)2 ω 2 Q Φλω + o (λ − 1)2 ,
dλ
λ=1
here we have used (2.1) in the second step. Using Lemma 2.1 (1), we have
−−→
d
Q Φλω = 0.
dλ
λ=1
Hence,
D
→−−→ −
→ −−→ −
→E
′′ −
Sω Φω Φλω − Φω , Φλω − Φω
= o (λ − 1)2 ,
D
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YIFEI WU
and
−−→
−
→
Q Φλω − Q Φω = o λ − 1 .
Thus we obtain the desirable estimate.
2.3. Coercivity. First, we need the following lemma.
→
~ω = (∂ω φω , iω∂ω φω )T , −
Ψω = (2ωφω , 0)T , then
Lemma 2.3. Let ψ
−
→
−
→ ~
Sω′′ Φω ψ
ω = Ψω .
Moreover, if |ω| = ωc , then
D
Sω′′
−
→ ~ ~ E
Φω ψω , ψω < 0.
(2.2)
(2.3)
Proof. Note that from the equation (1.6), we have
p−1
− ∂xx + (1 − ω 2 ) − pφp−1
ω Re − iφω Im ∂ω φω = 2ωφω .
Then (2.2) follows from a straightforward computation.
For (2.3), we have
D
−
→ ~ ~ E D−
→ ~ E
Sω′′ Φω ψ
,
ψ
=
Ψ
ω
ω
ω , ψω
Z
d
= 2ω φω ∂ω φω dx = ω kφω k2L2
dω
−
→
d
= − Q Φω − kφω k2L2 .
dω
Using Lemma 2.1 (1), when |ω| = ωc ,
D
−
→ ~ ~ E
2
Sω′′ Φω ψ
ω , ψω = −kφω kL2 < 0.
This proves the lemma.
Now we have the following coercivity property.
Lemma 2.4. Let ω = ±ωc . Suppose that ξ~ ∈ H 1 (R) × L2 (R) satisfies
D −
→E D ~ −
→E D ~ −
→E
~
ξ, iΦω = ξ, ∂x Φω = ξ, Ψω = 0.
Then
D
2
−
→~ ~E ξ H 1 ×L2 .
ξ & ~
Sω′′ Φω ξ,
Proof. It was proved in Lemma 7 in [1] that
−
−
→
→ −
→
Ker Sω′′ Φω = Span iΦω , ∂x Φω .
Now using the same manner in the proof of Lemma 8 in [1], and using Lemma 2.3
instead, we can prove the lemma. The details are omitted here.
INSTABILITY OF STANDING WAVES
7
2.4. Modulation. The following modulation lemma says that if the standing wave
solution is stable, then after suitably choosing the parameters, the orthogonality
conditions in Lemma 2.4 can be verified.
Lemma 2.5. Let ω = ±ωc . There exists ε0 > 0, such that for any ε ∈ (0, ε0),
~ ω ), the following properties is verified. There exist C 1 -functions
~u ∈ Uε (Φ
(θ, y) : R2 → R,
λ : R → R+ ,
such that if we define ξ~ by
−−→
~ = e−iθ(t) ~u t, · − y(t) − −
ξ(t)
Φλ(t)ω ,
then ξ~ satisfies the following orthogonality conditions for any t ∈ R,
D −−−→E D
−−−→E D ~ −−−→E
~
~
ξ, iΦλ(t)ω = ξ, ∂x Φλ(t)ω = ξ, Ψλ(t)ω = 0.
(2.4)
(2.5)
Moreover, the following estimates verify
~ H 1 ×L2 + |λ − 1| . ε,
kξk
and for any t ∈ R,
|θ̇(t) − λ(t)ω| + |ẏ| + |λ̇| = O ~
ξ H 1 ×L2 .
Proof. The existence of the parameters follows from classical arguments involving the
implicit function theorem (note that the nonzero of the Jacobian has been obtained
in Lemma 2.4). The estimates of the parameters follow from their dynamics which
can be obtained by the orthogonality conditions (2.5). We cite to Proposition 9 in
[1] for the details of the proof.
2.5. Localized virial identities. To prove main Theorem 1.2, one of the key ingredient is the localized virial identities.
Lemma 2.6. Let ϕ ∈ C 1 (R), then
Z
Z
2
d
Re uūt dx =
|ut| − |ux |2 − |u|2 + |u|p+1 dx;
dt
Z
Z
1
2
d
ux ūt dx = −
ϕ′ |ut|2 + |ux |2 − |u|2 +
|u|p+1 dx.
Re ϕ
dt
2
p+1
Proof. It follows from a direct calculation. See [25] for the details.
3. Proof of the main theorem
We define the smooth cutoff function ϕR ∈ C ∞ (R) as
ϕR (x) = x,
when |x| ≤ R;
ϕR (x) = 0,
when |x| ≥ 2R,
and 0 ≤ ϕ′R ≤ 1 for any x ∈ R. Moreover, we denote
Z
Z
4
Re uūt dx + 2Re ϕR x − y(t) ux ūt dx.
I(t) =
p−1
Then from Lemma 2.6 we have the following lemma.
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YIFEI WU
Lemma 3.1. Let R > 0, if |ẏ| . 1, then
I ′ (t) = −
16ω
8
p+3
· 2E(u0 , u1 ) −
Q(u0 , u1 ) − 2ẏP (u0 , u1) +
kut − iωuk2L2
p−1
p−1
p−1
Z
+O
|ut |2 + |ux |2 + |u|2 + |u|p+1 dx .
|x−y(t)|≥R
Proof. First, we have
Z
Z
d
Re ϕR x − y(t) ux ūt dx = −ẏRe ϕ′R x − y(t) ux ūt dx
dt
Z
d
ux ūt dx.
+ Re ϕR x − y(t)
dt
Then from Lemma 2.6 and the momentum conservation law, we obtain
Z
Z
d
Re ϕR x − y(t) ux ūt dx = −ẏRe ϕ′R x − y(t) ux ūt dx
dt
Z
1
2
−
ϕ′R x − y(t) |ut |2 + |ux |2 − |u|2 +
|u|p+1 dx
2
p+1
Z
′
= −ẏP (u0 , u1) − ẏRe
ϕR x − y(t) − 1 ux ūt dx
Z
2
1
|ut |2 + |ux |2 − |u|2 +
|u|p+1 dx
−
2
p+1
Z
2
1
ϕ′R x − y(t) − 1 |ut|2 + |ux |2 − |u|2 +
|u|p+1 dx.
−
2
p+1
′
Since supp ϕR x − y(t) − 1 ⊂ {x : |x − y(t)| ≥ R}, 0 ≤ ϕ′R ≤ 1 and |ẏ| . 1, we
get
Z
d
Re ϕR x − y(t) ux ūt dx
dt
Z
2
1
|ut |2 + |ux |2 − |u|2 +
|u|p+1 dx
= −ẏP (u0, u1 ) −
2
p+1
Z
2
2
2
p+1
+O
|ut | + |ux | + |u| + |u| dx .
|x−y(t)|≥R
Moreover, from Lemma 2.6,
Z
Z
2
d
Re uūt dx =
|ut | − |ux |2 − |u|2 + |u|p+1 dx.
dt
Combining the two estimates above, we obtain that
4
p+3
I ′ (t) = − 2ẏP (u0, u1 ) +
− 1 kut k2L2 −
kux k2L2
p−1
p−1
p−5
p+3
+
kuk2L2 + 2 2
kukp+1
Lp+1
p−1
p −1
Z
2
2
2
p+1
+O
|ut | + |ux | + |u| + |u| dx .
|x−y(t)|≥R
(3.1)
INSTABILITY OF STANDING WAVES
Note that when |ω| = ωc ,
4
p+3
− 1 kut k2L2 −
kux k2L2 +
p−1
p−1
8
kut − iωuk2L2 −
=
p−1
9
p−5
p+3
kuk2L2 + 2 2
kukp+1
Lp+1
p−1
p −1
p+3
16ω
· 2E(u0 , u1 ) −
Q(u0 , u1 ).
p−1
p−1
Inserting this equality into (3.1), we prove the lemma.
3.1. The choice of the initial data. In this subsection, we choose the initial data
such that it is close to the standing waves solution but leads the instability. We set
−
→
~u0 = (1 + a)Φω ,
(3.2)
here a ∈ (0, a0 ) is an arbitrary small constant, and a0 will be decided later. Then
we have
Lemma 3.2. Let ~u0 be defined in (3.2), then
P (~u0) = 0,
and
−
→
Q(~u0 ) − Q Φω = −2aωkφω k2L2 + O(a2).
Proof. It follows from the definition that P (~u0) = 0. Now consider Q(~u0 ). We write
−
→ D ′ −
→
−
→E
−
→
Q(~u0 ) − Q Φω = Q Φω , ~u0 − Φω + O k~u0 − Φω k2H 1 ×L2
= − ω hφω , u0 − φω i − hiφω , u1 − iωφω i + O a2
= − 2aωkφω k2L2 + O(a2 ).
This finishes the proof of the lemma.
Using the lemma above, we can scale the main part in I ′ (t).
Lemma 3.3. Let ~u0 be defined in (3.2), then
−
p+3
16ω
5−p
· 2E(~u0 ) −
Q(~u0 ) =
· 4aω 2 kφω k2L2 + O(a2 ).
p−1
p−1
p−1
Proof. Making use of Lemma 2.1 (2), we have
−
Since
p+3
16ω
· 2E(~u0) −
Q(~u0 )
p−1
p−1
−
−
p+3 h
→ i
→ i
16ω h
=−
Q(~u0 ) − Q Φω
· 2 E(~u0 ) − E Φω
−
p−1
p−1
−
→
→
p+3
16ω −
−
· 2E Φω −
Q Φω
p−1
p−1
−
−
→ i
→ i
16ω h
p+3 h
Q(~u0 ) − Q Φω .
· 2 E(~u0 ) − E Φω
−
=−
p−1
p−1
−
−
h
−
→ h
→ i
→ i
E(~u0 ) − E Φω = S(~u0 ) − S Φω
− ω Q(~u0 ) − Q Φω ,
10
YIFEI WU
we further write
16ω
p+3
· 2E(~u0 ) −
Q(~u0 )
−
p−1
p−1
−
h
−
→ i 5 − p
→ i
p+3 h
=−
· 2 S(~u0 ) − S Φω
−
· 2ω Q(~u0 ) − Q Φω .
p−1
p−1
By Taylor’s type extension, we have
−
−
→
→
S(~u0) − S Φω = O k~u0 − Φω k2H 1 ×L2 = O(a2 ).
Now using Lemma 3.2, we prove the lemma.
Similar computation also gives
Lemma 3.4. Let λ ∈ R+ with λ . 1, ~u0 be defined in (3.2), then
−
→
Sλω (~u0 ) − Sλω Φω = −2(λ − 1)aω 2 kφω k2L2 + O(a2 ).
Proof. By the definition of Sω , we have
−
→
Sλω (~u0 ) − Sλω Φω
−
h
−
→
→ i
=Sω (~u0) − Sω Φω + (λ − 1)ω Qω (~u0) − Qω Φω .
Since
−
→
S(~u0 ) − S Φω = O(a2 ),
then by Lemma 3.2, we prove the lemma.
Now we control the rest terms in the virial identity in Lemma 3.1. We argue
for contradiction and suppose that the standing wave solution uω is stable. That
is, for any ε > 0, there exists a constant a0 > 0, such that for any a ∈ (0, a0 ), if
−
→
−
→
~ ω)
~u0 ∈ Ua (Φω ), then ~u(t) ∈ Uε (Φω ) for any t ∈ R. We may assume that ~u ∈ Uε (Φ
for ε ≤ ε0 , where ε0 is determined in Lemma 2.5. Hence by Lemma 2.5, we can
write
u = eiθ (φλω + ξ)(· − y);
ut = eiθ (iλωφλω + η)(· − y)
(3.3)
with ξ~ = (ξ, η) satisfying the orthogonal conditions (2.5).
3.2. Lower control of kut − iωukL2 . In this subsection, we prove the following
lemma.
Lemma 3.5. Suppose that ξ~ = (ξ, η) defined in (3.3) satisfying the orthogonal
conditions (2.5), then
kut − iωuk2L2 =(λ − 1)2 ω 2 kφω k2L2 + kη − iωξk2L2
3
3
~
+ O |λ − 1| + a|λ − 1| + kξkH 1 ×L2 .
Proof. By (3.3), we expand it as
kut − iωuk2L2 =kiλωφλω + η − iω(φλω + ξ)k2L2
=ki(λ − 1)ωφλω + η − iωξk2L2
=(λ − 1)2 ω 2 kφλω k2L2 + 2(λ − 1)ω η − iωξ, iφλω + kη − iωξk2L2 .
INSTABILITY OF STANDING WAVES
11
Noting that
kφλω k2L2 = kφω k2L2 + O(|λ − 1|),
then combining with the third orthogonal condition in (2.5), we further get
kut − iωuk2L2 =(λ − 1)2 ω 2 kφω k2L2 + 2(λ − 1)ω η, iφλω
+ kη − iωξk2L2 + O(|λ − 1|3 ).
(3.4)
Now we consider the term hη, iφλω i. First, we use the charge conservation law to
obtain
−
−
−−→
→
→
Q ~u0 − Q Φω +Q Φω − Q Φλω
−−→
= Q ~u − Q Φλω
2
~
= − ξ, λωφλω − η, iφλω + O kξkH 1 ×L2 .
Then by the third orthogonal conditions in (2.5), we have
−−→
−
−
→ h
→ i
~ 21 2 .
η, iφλω = Q Φλω − Q Φω − Q ~u0 − Q Φω
+ O kξk
H ×L
From Lemma 2.1, we have
−−→
−
→
Q Φλω − Q Φω = O |λ − 1|2 ,
and from Lemma 3.2, we have
Q ~u0
−
→
− Q Φω = O(a).
Therefore, we obtain that
~ 21 2 .
η, iφλω = O a + |λ − 1|2 + kξk
H ×L
(3.5)
Now together (3.4) with (3.5), we obtain the desirable result.
~ H 1 ×L2 . In this subsection, we give the following estimate
3.3. Upper control of kξk
~ H 1 ×L2 .
on kξk
Lemma 3.6. Let ξ~ = (ξ, η) be defined in (3.3), then
~ 2 1 2 = O(a|λ − 1| + a2 ) + o (λ − 1)2 .
kξk
H ×L
Proof. From the charge and energy conservation laws,
Sλω ~u0 = Sλω ~u
−−→
−−→
= Sλω ~u − Sλω Φλω + Sλω Φλω
−−→
1 D ′′ −−→ ~ ~E
~ 21 2 .
Sλω Φλω ξ, ξ + Sλω Φλω + o kξk
=
H ×L
2
Hence by Lemma 2.4,
D
−→
E
~
~
~ 2 1 2 . 1 S ′′ −
Φ
ξ,
ξ
kξk
λω
λω
H ×L
h2
−
−−→
−
→ i h
→ i
~ 21 2 .
= Sλω ~u0 − Sλω Φω
− Sλω Φλω − Sλω Φω
+ o kξk
H ×L
12
YIFEI WU
By Lemma 3.4,
−
→
Sλω (~u0 ) − Sλω Φω = −2(λ − 1)aω 2 kφω k2L2 + O(a2 ),
and by Corollary 2.2,
Therefore,
−−→
−
→
Sλω Φλω − Sλω Φω = o (λ − 1)2 .
~ 21 2 .
~ 2 1 2 = O(a|λ − 1| + a2 ) + o (λ − 1)2 + o kξk
kξk
H ×L
H ×L
Absorbing the last term by the left-hand side one, we prove the lemma.
~ ω ),
3.4. Proof of Theorem 1.2. As discussion above, we assume that ~u ∈ Uε (Φ
and thus |λ − 1| . ε. First, we note that from the definition of I(t), we have the
time uniform boundedness of I(t),
−
→
(3.6)
sup I(t) . R kΦω k2H 1 ×L2 + 1 .
t∈R
Now we consider the estimate on I ′ (t). First, by (3.3), the exponential decaying of
φω and 12 ≤ λ ≤ 32 ,
Z
i
h
2
2
2
p+1
dx
|ut | + |ux | + |u| + |u|
|x−y(t)|≥R
Z
h
i
.
|φλω |2 + |∂x φλω |2 + |ξ|2 + |∂x ξ|2 + |ξ|p+1 + |η|2 dx
|x|≥R
2
1
ξ H 1 ×L2 +
= O ~
.
R
Hence by Lemma 3.1,
p+3
16ω
I ′ (t) = −
· 2E(u0 , u1 ) −
Q(u0 , u1 )
p−1
p−1
1
8
2
.
kut − iωuk2L2 + O ~
ξ H 1 ×L2 +
− 2ẏP (u0, u1 ) +
p−1
R
Now by Lemma 3.3, Lemma 3.2, and Lemma 3.5, we have
5−p
I ′ (t) =
· 4aω 2 kφω k2L2 + (λ − 1)2 ω 2 kφω k2L2 + kη − iωξk2L2
p−1
2
1
.
ξ H 1 ×L2 +
+ O a2 + a|λ − 1| + |λ − 1|3 + ~
R
Using Lemma 3.6, we further get
5−p
I ′ (t) =
· 4aω 2 kφω k2L2 + (λ − 1)2 ω 2 kφω k2L2 + kη − iωξk2L2
p−1
+ O a2 + a|λ − 1| + o |λ − 1|2 .
Choosing ε and a0 small enough, we obtain that for any a ∈ (0, a0 ),
5−p
· 2aω 2kφω k2L2 .
I ′ (t) ≥
p−1
This implies that I(t) → +∞ when t → +∞, which is contradicted with (3.6).
Hence we prove the instability of the standing wave uω and thus give the proof of
Theorem 1.2.
INSTABILITY OF STANDING WAVES
13
Acknowledgements
The author was partially supported by the NSFC (No.11571118). The author
would like to thank Professor Masahito Ohta for his kind explanation on the instability theory. The author would also like to express his deep gratitude to Professor
Masaya Maeda for his helpful private discussion, in particular, he introduced me
the references [3, 18, 23], and informed me a flaw in [18] such that the correlative
theorem only applies to the nonlinear Klein-Gordon equation in cases of p ≥ 2.
References
[1] Bellazzini1, J., Ghimenti, M., and Le Coz S., Multi-solitary waves for the nonlinear KleinGordon equation, Comm. Partial Differ. Equ., 39 (2014), 1479–1522.
[2] Berestycki, H., and Cazenave, T., Instabilité desétats stationnaires dans leséquations de
Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris, 293, (1981), 489–492.
[3] Comech A., and Pelinovsky, D., Purely nonlinear instability of standing waves with minimal
energy, Comm. Pure Appl. Math. 56 (2003), 1565– 1607.
[4] Côte, R., and Le Coz, S., High-speed excited multi-solitons in nonlinear Schrödinger equations.
J. Math. Pures Appl. (9) 96 (2011), 135–166.
[5] Côte, R., and Martel, Y., Multi-travelling waves for the nonlinear Klein-Gordon equation. To
appear in Comm. Partial Differ. Equ..
[6] Côte, R., Martel, Y., and Merle, F., Construction of multi-soliton solutions for the L2 supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27 (2011), 273– 302.
[7] Côte, R., and Muñoz, C., Multi-solitons for nonlinear Klein-Gordon equations, Forum Math.
Sigma 2 (2014), 38pp.
[8] Ginibre, J., and Velo, G., The global Cauchy problem for the nonlinear Klein-Gordon equation.
Math. Z. 189 (1985), no. 4, 487–505.
[9] Ginibre, J., and Velo, G., The global Cauchy problem for the nonlinear Klein-Gordon equation.
II. Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), no. 1, 15–35.
[10] Grillakis, M., Shatah, J., and Strauss, W., Stability theorey of solitary waves in the presence
of symmetry, I, J. Funct. Anal., 74 (1987) 160–197.
[11] Grillakis, M., Shatah, J., and Strauss, W., Stability theorey of solitary waves in the presence
of symmetry, II, J. Funct. Anal. 94 (1990) 308–348.
[12] Jeanjean, L., and Le Coz, S., Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, Trans. Amer. Math. Soc., V.361 (2009), 5401–5416.
[13] Ibrahim, S., Masmoudi, N., and Nakanishi, K., Scattering threshold for the focusing nonlinear
Klein-Gordon equation, Analysis and PDE., 4 (2011), 405–460.
[14] Killip, R., Stovall, B., and Visan, M., Scattering for the cubic Klein-Gordon equation in two
space dimensions, Trans. Amer. Math. Soc. 364 (2012), 1571–1631.
[15] Le Coz, S., Li, D., and Tsai, T.-P., Fast-moving finite and infinite trains of solitons for
nonlinear Schrödinger equations. Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1251–1282.
[16] Liu, Y., Ohta, M., and Todorova, G., Strong instability of solitary waves for nonlinear KleinGordon equations and generalized Boussinesq equations. Ann. Inst. H. Poincaré Anal. Non
Linéaire, 24 (2007), 539–548.
[17] Maeda, M., Instability of bound states of of nonlinear Schrödinger equations with Morse index
equal to two, Nonlinear. Anal., 72 (2010), 2100–2113.
[18] Maeda, M., Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct.
Anal., 263 (2012), 511–528.
[19] Martel, Y., and Merle, F., Instability of solitons for the critical generalized Korteweg-de Vries
equation, Geom. Funct. Anal., 11, 74–123 (2001).
[20] Merle, F., Existence of blow-up solutions in the energy space for the critical generalized KdV
equation. J. Amer. Math. Soc., 14(2001), 555–578.
[21] Merle, F., and Raphaël, P., On universality of blow-up profile for L2 critical nonlinear
Schrödinger equation. Invent. Math., 156 (2004), 565–672.
[22] Miao, Ch., and Zheng J., Scattering theory for energy-supercritical Klein-Gordon equation,
Discrete Contin. Dyn. Syst.-S, 9(2016), 2073–2094.
14
YIFEI WU
[23] Ohta, M., Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct.
Anal., 261 (2011), 90–110.
[24] Ohta, M., and Todorova, G., Strong instability of standing waves for nonlinear Klein-Gordon
equations. Discrete Contin. Dyn. Syst. 12 (2005), 315–322.
[25] Ohta, M., and Todorova, G., Strong instability of standing waves for the nonlinear KleinGordon equation and the Klein-Gordon-Zakharov system. SIAM J. Math. Anal. 38 (2007),
1912–1931.
[26] Payne, L.E., and Sattinger, D.H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273–303.
[27] Shatah, J., Stable standing waves of nonlinear Klein-Gordon equations. Comm. Math. Phys.
91 (1983), 313–327.
[28] Shatah, J., Unstable ground state of nonlinear Klein-Gordon equations. Trans. Amer. Math.
Soc. 290 (1985), 701–710.
[29] Shatah, J., and Strauss, W., Instability of nonlinear bound states. Comm. Math. Phys. 100
(1985), 173–190.
[30] Strauss, W., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 2,
(1977), 149C162.
[31] Stuart, D., Modulational approach to stability of non-topological solitons in semilinear wave
equations. J. Math. Pures Appl. (9) 80 (2001), 51–83.
[32] Weinstein, M., Modulational stability of ground states of nonlinear Schrödinger equations,
SIAM J. Math. Anal. 16, 472–491 (1985).
Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R.China
E-mail address: [email protected]