On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

0.15 MB·English

by  Marco Ghimenti

ON THE STABILITY OF STANDING WAVES OF
KLEIN-GORDON EQUATIONS IN A SEMICLASSICAL REGIME
2
1 MARCOGHIMENTI,STEFANLECOZ,ANDMARCOSQUASSINA
0
2
Abstract. We investigate the orbital stability and instability of standing

n waves fortwoclassesofKlein-Gordonequations inthesemi-classicalregime.
a
J

9
1 1. Introduction and results

] The nonlinear Klein-Gordon equation
P
A (1.1) ε2utt ε2∆u+mu up−1u=0 (t,x) R+ RN,
− −| | ∈ ×
. where ε,m>0, p>1 for N =1,2 and 1<p<(N +2)/(N 2) for N 3, is one
h − ≥
of the simplest nonlinear partial differential equations invariant for the Poincar´e
t
a
group. We are interested in the study of the nonlinear Klein Gordon equation in
m
presence of a potential depending on the space variable. Two different choices are
[ viable. WecansimplyaddapotentialtermW(x)utoequation(1.1). Thiscasehas

1 beenstudied,forthelinearwaveequation,forexample,byBealsandStraussin[3].
v Otherwise, if we want to fully preserve the invariance for the Poincar´e group of
3 (1.1), we have to change the temporal derivative ε2∂ with a covariant derivative,
tt
7 depending on the potential D2, where D = ε∂ +iV(x). This second approach
0 tt t t
is classical, in quantum electrodynamics, when considering electromagnetic waves.
4
. The first approachleads us to consider the equation
1
0 (1.2) ε2utt ε2∆u+mu Wu up−1u=0, in RN,
− − −| |
2
while the second one to investigate the equation
1
: (1.3) ε2u +2iεVu ε2∆u+mu V2u up−1u=0, in RN.
v tt t− − −| |
i Inthis paper,wetreatsimultaneouslythe twopreviousKlein-Gordonequationsby
X
studying
r
a (1.4) ε2u +2iεVu ε2∆u+mu Wu up−1u=0, in RN,
tt t
− − −| |
where u : RN R C, ε > 0 and V,W are real valued potential functions.
× →
Equation (1.4) formally yields to (1.3) for the choice W = V2 as well as to (1.2)
when V =0. We shall study the stability of standing waves of this equation in the
semiclassicalregimeε 0. Itadmits standingwaves,namelysolutionsofthe form
u(x,t)=eiωt/εϕ (x/ε)→, where ω R and ϕ satisfies
ω ω
∈
(1.5) ∆ϕ + m ω2 2ωV(εy) W(εy) ϕ ϕ p−1ϕ =0, in RN.
ω ω ω ω
− − − − −| |
(cid:0) (cid:1)
Date:January20,2012.
2000 Mathematics Subject Classification. 35D99,35J62,58E05, 35J70.
Key words and phrases. Klein-Gordonequations, stability,semi-classicallimit.
Research supported inpartby2009 MIUR project: “Variational and Topological Methods in
the Study of Nonlinear Phenomena”, ANR project ESONSE and GNAMPA project “Dinamica
e proprieta` di soluzioni concentrate inteorie di campo nonlineari” .
1
2 M.GHIMENTI,S.LECOZ,ANDM.SQUASSINA
To ensure existence of solutions to (1.5) for ε close to 0, we assume the following.
Let V and W be 2. For the function
C
Z(y)=m ω2 2ωV(y) W(y), y RN
− − − ∈
there exists x RN such that
0
∈
(1.6) Z(x )=0, 2Z(x ) is non-degenerate.
0 0
∇ ∇
Furthermore, we assume that
(1.7) Z(x )=m ω2 2ωV(x ) W(x )>0.
0 0 0
− − −
Under these hypotheses, it is well-know (see e.g. [1] or [2, Section 8.2]) that
when ε is close to 0 the equation (1.5) admits a family of positive, exponentially
decaying,solutionsϕ H1(RN)(hiding the dependanceuponε). Moreprecisely,
ω
there exists ξ RN a⊂nd ψ H1(RN) such that ϕ () = ψ ( ξ )+ (ε2) in
ε ω ω ω ε
H1(RN) as ε ∈0, where ξ =∈o(ε) and · ·− O
ε
→
∆ψ +Z(x )ψ = ψ p−1ψ , in RN.
ω 0 ω ω ω
− | |
We are interestedin the (orbital)stability or instability of standing waveswhen
ε goes to 0.
A standing wave of (1.4) is said to be (orbitally) stable if any solution of (1.4)
startingclose to the standing waveremains closefor all time, up to the invariances
of the equation. More precisely, we say that eiωεtϕω xε is stable if for all η > 0
there exists δ > 0 such that for all u0 ∈ H1(RN) ver(cid:0)ify(cid:1)ing ku0−ϕωkH1(RN) < δ
the solution u(x,t) of (1.4) with initial data u satisfies
0
st∈uRpθin∈fRku−eiθϕωkH1(RN) <η.
Since the pioneeringworks[4,6–8,24,25], the study oforbitalstability for standing
wavesof dispersivePDE has attracteda lot ofattention. Among many others,one
canreferto[10,11,13];seealsothebooksandsurveys[5,12,22,23]andthereferences
therein. Relatively few works [9,14,16] are concerned with stability at the semi-
classical limit for Schr¨odinger type equations. For Klein-Gordon equations, after
the ground works [20,21], there has been a recent interest for stability by blow-
up [15,17–19].
To study stability, we first rewrite (1.4) in Hamiltonian form
∂U
(1.8) ε =JE′(U),
∂t
u 0 1
where U = , J = , and
(cid:18)v(cid:19) (cid:18) 1 0(cid:19)
−
1 1 1
E(U)= v iVu 2 + u 2 +m u 2
2k − kL2(RN) 2k∇ kL2(RN) 2k kL2(RN)
1 1
W u2dx u p+1 .
− 2ZRN | | − p+1k kLp+1(RN)
It is easy to see that if u solves (1.4) and v is defined by v := εu +iVu, then
t
u
U = solves (1.8). The charge Q(U) is defined by
(cid:18)v(cid:19)
Q(U)= u¯vdx.
ℑZRN
ON THE STABILITY OF KLEIN-GORDON EQUATIONS 3
In particular, for a standing wave u=eiωt/εϕ (x/ε), the charge is given by
ω
(1.9) Q(ϕ ):=Q(U)=εN ω ϕ 2 + V(εy)ϕ 2 .
ω (cid:18) k ωkL2(RN) ZRN | ω| (cid:19)
Accordingto the theorydevelopedin [7,8], a standing waveeiωεtϕω xε is stable
if two conditions are satisfied. (cid:0) (cid:1)
∂
(i) The Slope Condition: Q(ϕ )<0.
ω
∂ω
(ii) The Spectral Condition: L := ∆+Z(εy) pϕ p−1 has exactly one
ε ω
− − | |
negative eigenvalue and is non degenerate.
On the other hand, denote by n(L ) the number of negative eigenvalues of L and
ε ε
∂ ∂
set p(ω)=0 if Q(ϕ )> 0, p(ω)=1 if Q(ϕ )< 0. Then the standing wave
ω ω
∂ω ∂ω
is unstable if n(L ) p(ω) is odd.
ε
−
Then we have the following
Theorem 1.1. Assumethat conditions (1.6)-(1.7) hold. Then, we have the follow-
ing facts.
(1) If p<1+4/N, then the Slope Condition ∂ Q(ϕ )<0 is fulfilled if
∂ω ω
4
Z(x )<(ω+V(x ))2 N (non-critical case)
0 0
(cid:16)p 1 − (cid:17)
−
or if
Z(x )=(ω+V(x ))2 4 N ,
 0 0 (cid:16)p−1 − (cid:17) (critical case).
 ∆Z(0) ∆V(0) 1+ 2(ω+V(0)) <0,
(cid:16) − (cid:16) Z(0) (cid:17)(cid:17)
(2) Ifp 1+4/N, then we always have ∂ Q(ϕ )>0.
≥ ∂ω ω
(3) We have the equality n(L ) = n( 2Z(x ))+1, where n( 2Z(x )) is the
ε 0 0
∇ ∇
number of negative eigenvalues of 2Z(x ).
0
∇
In particular, the standing waves eiωtϕ are stable if x is non-degenerate local
ω 0
minimum of Z, p<1+4/N and
4
Z(x )<(ω+V(x ))2 N .
0 0
(cid:16)p 1 − (cid:17)
−
Notethat,converselytowhatwashappeninginthecaseofSchr¨odingerequations
studied in [14], the values of the potentials V and W at x comes into play for the
0
SlopeConditioneveninthenoncriticalcase. Notealsothatonlythelocalbehavior
of Z around x influences the stability or instability.
0
Notations : Most of the objects we consider will depend both on ε and ω. We
will emphasize the most important parameter by indicating it as a subscript, the
dependence in the other parameter being understood.
2. Proof of Theorem 1.1
In this section, we prove Theorem 1.1. We start be focusing on the Slope Con-
dition and then we study the Spectral Condition. For the sake of simplicity in
notations and without loss of generality, in the rest of this section we assume that
x =0.
0
2.1. The Slope Condition. We start with the noncritical case.
4 M.GHIMENTI,S.LECOZ,ANDM.SQUASSINA
2.1.1. Noncritical case. We assume that
4
Z(0)=(ω+V(0))2 N .
6 (cid:16)p 1 − (cid:17)
−
We first rewrite Q(ϕ ) by expanding V(εy) and using the exponential decay of
ω
ϕ :
ω
Q(ϕ )=εN(ω+V(0)) ϕ 2 + (εN+1).
ω k ωkL2(RN) O
Therefore, since
∂ ∂
(2.1) Q(ϕ )=εN ϕ 2 +εN(ω+V(0)) ϕ 2 + (εN+1),
∂ω ω k ωkL2(RN) ∂ωk ωkL2(RN) O
to evaluate the sign of the map ω ∂ Q(ϕ ) one should compute the quantity
7→ ∂ω ω
∂
(2.2) ϕ 2 =2 R ϕ ,
∂ωk ωkL2(RN) ZRN ω ω
where R (x):= ∂ϕω(x).
ω ∂ω
We remark that differentiation of (1.5) with respect to ω easily yields
(2.3) L R =2(ω+V(εy))ϕ .
ε ω ω
Ifwenowintroducetherescalingϕω(x)=λp−11ϕλ(√λx),itfollowsthatϕλ satisfies
εy
(2.4) ∆ϕ +λ−1Z ϕ ϕ p−1ϕ =0, in RN.
λ λ λ λ
− (cid:18)√λ(cid:19) −| |
Now, differentiating equation (2.4) with respect to λ and denoting T = ∂ϕλ
λ ∂λ |λ=1
yields
1
(2.5) L T Z(εy)ϕ εy Z(εy)ϕ =0.
ε λ ω ω
− − 2 ·∇
Since 0 is a critical point of Z, a Taylor expansion gives
(2.6) Z(εy)=Z(0)+ (ε2 y 2),
O | |
1
(2.7) εy Z(εy)= (ε2 y 2).
2 ·∇ O | |
Then, from (2.5), as ε 0 we have
→
(2.8) L T =Z(0)ϕ + (ε2 y2 ϕ ), in RN.
ε λ ω ω
O | |
Then, in turn, taking into account identity (2.3) we get
Z(0) R ϕ = R L T + (ε2)
ω ω ω ε λ
ZRN ZRN O
= L R T + (ε2)
ε ω λ
ZRN O
(2.9) = 2(ω+V(εy))ϕ T + (ε2)
ω λ
ZRN O
=2(ω+V(0)) ϕ T + (ε)
ω λ
ZRN O
∂
=(ω+V(0)) ϕ 2 + (ε)
∂λk λkL2(RN)|λ=1 O
N 2
=(ω+V(0)) ϕ 2 + (ε).
(cid:16) 2 − p 1(cid:17)k ωkL2(RN) O
−
ON THE STABILITY OF KLEIN-GORDON EQUATIONS 5
In conclusion, by combining (2.1), (2.2) and (2.8), we have
∂ (ω+V(0))2 4
Q(ϕ )=εN 1+ N ϕ 2 + (εN+1).
∂ω ω (cid:18) Z(0) (cid:16) − p 1(cid:17)(cid:19)k ωkL2(RN) O
−
Then, taking into account the fact that Z(0) > 0 and that ϕ converges to ψ in
ω ω
L2(RN) as ε 0, the sign of ∂ Q(ϕ ) is the sign of
→ ∂ω ω
4
Z(0) (ω+V(0))2 N .
− (cid:16)p 1 − (cid:17)
−
2.1.2. Critical case. We assume now that
4
(2.10) Z(0)=(ω+V(0))2 N .
(cid:16)p 1 − (cid:17)
−
Inthecriticalcase,thetermoforderεN intheexpansionof ∂ Q(ϕ )vanishesand
∂ω ω
we need to calculate the expansion at a higher order. We first refine (2.6)-(2.7).
ε2
Z(εy)=Z(0)+ 2Z(0)(y,y)+ (ε3 y 3)
2 ∇ O | |
1 ε2
εy Z(εy)= 2Z(0)(y,y)+ (ε3 y 3).
2 ·∇ 2 ∇ O | |
Then (2.5) gives
L T =Z(0)ϕ +ε2 2Z(0)(y,y)ϕ + (ε3 y3 )ϕ .
ε λ ω ω ω
∇ O | |
Now, we have
(2.11) Z(0) R ϕ = R L T ε2 2Z(0)(y,y)R ϕ + (ε3).
ω ω ω ε λ ω ω
ZRN ZRN − ZRN ∇ O
From (2.3), we obtain
(2.12) R L T = L R T = 2(ω+V(εy))ϕ T .
ω ε λ ε ω λ ω λ
ZRN ZRN ZRN
Expanding the potential V we get
(2.13) 2V(εy)ϕ T = 2V(0)ϕ T +2ε y V(0)ϕ T
ω λ ω λ ω λ
ZRN ZRN ZRN ·∇
+ε2 2V(0)(y,y)ϕ T + (ε3).
ω λ
ZRN ∇ O
Note that since ϕ =ψ ( ξ )+ (ε2) and ξ =o(ε), we have
ω ω ε ε
·− O
(2.14) 2ε y V(0)ϕ T =2ε y V(0)ψ T +o(ε2)=o(ε2)
ω λ ω λ
ZRN ·∇ ZRN ·∇
where the last cancellation comes from the fact that ψ is radial. Coming back to
ω
(2.12) and as in (2.9), we have
N 2
(2.15) R L T =(ω+V(0)) ϕ 2
ZRN ω ε λ (cid:16) 2 − p 1(cid:17)k ωkL2(RN)
−
+ε2 2V(0)(y,y)ϕ T +o(ε2).
ω λ
ZRN ∇
It remains to compute the integrals involving the Hessians in (2.11) and (2.15).
Since our problem is invariant by an orthonormal transformation, we can assume
6 M.GHIMENTI,S.LECOZ,ANDM.SQUASSINA
without loss of generality that 2V(0) = diag(b ,...,b ). Hence 2V(0)(y,y) =
1 N
∇ ∇
N b y2. Recall also that T can be computed explicitly to have
j=1 j j λ
P
1 1
T = ϕ y ϕ .
λ ω ω
−p 1 − 2 ·∇
−
Therefore,
N
b b ∂ϕ
b y2ϕ T = j y2ϕ2 j y2y ϕ ω.
ZRN j j ω λ −p−1ZRN j ω − 2 kX=1ZRN j k ω ∂yk
We have after integration by parts
N N
∂ϕ
2 y2y ϕ ω = (y2+2δ y2)ϕ2 = (N +2) y2ϕ2.
Xk=1ZRN j k ω ∂yk −Xk=1ZRN j jk j ω − ZRN j ω
Thus
N N
1 N +2
2V(0)(y,y)ϕ T = b y2ϕ T = b y2ϕ2.
ZRN ∇ ω λ Xj=1ZRN j j ω λ −(cid:18)p−1 − 4 (cid:19)Xj=1 jZRN j ω
Recall the following expansion in ε for R and ϕ .
ω ω
∂ψ
ω
ϕ =ψ +o(ε), R = +o(ε).
ω ω ω
∂ω
Therefore, since ψ is radial,
ω
1
y2ϕ2 = y2ψ2 +o(ε)= y ψ 2 +o(ε),
ZRN j ω ZRN j ω Nk| | ωkL2(RN)
and so
(2.16) 2V(0)(y,y)ϕ T =
ω λ
ZRN ∇
1 N +2 1
y ψ 2 ∆V(0)+o(ε).
−(cid:18)p 1 − 4 (cid:19)Nk| | ωkL2(RN)
−
Similarly, we have
(2.17) 2Z(0)(y,y)R ϕ =
ω ω
ZRN ∇
1 N +2 1
y ψ 2 ∆Z(0)+o(ε).
−(cid:18)p 1 − 4 (cid:19)Nk| | ωkL2(RN)
−
Summarizing,using successively (2.11), (2.15), (2.16), (2.17)and(2.10) we have
obtained
1
(2.18) R ϕ = ϕ 2
ZRN ω ω −2(ω+V(0))k ωkL2(RN)
(∆Z(0) ∆V(0)) 1 N +2
+ε2 − y ψ 2 +o(ε2).
NZ(0) (cid:18)p 1 − 4 (cid:19)k| | ωkL2(RN)
−
Now, we compute ∂Q(ϕω). First, recall that, coming back to the definition (1.9)
∂ω
of Q, we have
∂Q(ϕ )
ε−N ω = ϕ 2 +2ω R ϕ +2 V(εy)R ϕ .
∂ω k ωkL2(RN) ZRN ω ω ZRN ω ω
ON THE STABILITY OF KLEIN-GORDON EQUATIONS 7
As in (2.13), (2.14), and (2.16) we can expand in ε and get
2 V(εy)R ϕ =2V(0) R ϕ
ω ω ω ω
ZRN ZRN
1 N +2 1
ε2 y ψ 2 ∆V(0)+o(ε2).
− (cid:18)p 1 − 4 (cid:19)Nk| | ωkL2(RN)
−
Therefore,
∂Q(ϕ )
ε−N ω = ϕ 2 +2(ω+V(0)) R ϕ
∂ω k ωkL2(RN) ZRN ω ω
1 N +2 1
ε2 y ψ 2 ∆V(0)+o(ε2).
− (cid:18)p 1 − 4 (cid:19)Nk| | ωkL2(RN)
−
Using (2.18), we finally get
∂Q(ϕ ) 1 N +2 1
ε−N ω =ε2 y ψ 2
∂ω (cid:18)p 1 − 4 (cid:19)Nk| | ωkL2(RN)×
−
2(ω+V(0))
∆Z(0) ∆V(0) 1+ +o(ε2).
×(cid:18) − (cid:18) Z(0) (cid:19)(cid:19)
2.2. The Spectral Condition. WedefinetheoperatorL := ∆+Z(0) pψp−1.
0 − − ω
It is well known (see e.g. [2]) that the spectrum of L consists of one negative
0
eigenvalue, a N-dimensional kernel (generated by ∂ψω for j = 1,...,N) and the
∂yj
restofthespectrumisboundedawayfrom0. Whenεiscloseto0,thespectrumof
L willbeclosetothespectrumofL . Inparticular,the0eigenvalue,ofmultiplicity
ε 0
N, will transforminto N possibly differenteigenvalues close to 0 but shifted either
to the positive or to the negativeside of the realaxis, depending onthe signof the
eigenvalues of the Hessian of Z at 0. More precisely, the following proposition was
proved in [14] (see [9] for a detailed justification).
Proposition 2.1. The spectrum of L consists of positive spectrum away from 0
ε
and a set of N +1 simple eigenvalues λ ,λ ,...,λ such that
0 1 N
{ }
λ <λ λ .
0 1 N
≤···≤
As ε 0, we have λ < 0 and the following asymptotic expansion holds for the
0
→
other eigenvalues:
λ =c ε2+o(ε2), j =1,...,N,
j j
where c = 1 kψωk2L2(RN) a and a ,...,a are the eigenvalues of the Hessian
j 2k∂ψωk2 j { 1 N}
∂xj L2(RN)
matrix 2Z(0).
∇
Therefore, (3) in Theorem 1.1 is a direct consequence of Proposition 2.1. In
particular, the spectral condition for stability will be satisfied if and only if 0 is a
non-degenerate local minimum of Z.
References
[1] A. Ambrosetti, M. Badiale, and S. Cingolani, Semiclassical states of nonlinear
Schr¨odinger equations,Arch.RationalMech.Anal.,140(1997), pp.285–300.
[2] A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problems
on Rn,vol.240ofProgressinMathematics,Birkh¨auserVerlag,Basel,2006.
8 M.GHIMENTI,S.LECOZ,ANDM.SQUASSINA
[3] M. Beals and W. Strauss, Lp estimates for the wave equation with a potential, Comm.
PartialDifferentialEquations,18(1993), pp.1365–1397.
[4] H. Berestycki and T. Cazenave, Instabilit´e des ´etats stationnaires dans les ´equations de
Schr¨odingeretdeKlein-Gordonnonlin´eaires,C.R.Acad.Sci.ParisS´er.IMath.,293(1981),
pp.489–492.
[5] T. Cazenave,Semilinear Schr¨odinger equations, New YorkUniversity– Courant Institute,
NewYork,2003.
[6] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear
Schr¨odinger equations,Comm.Math.Phys.,85(1982), pp.549–561.
[7] M. Grillakis, J. Shatah, and W. A. Strauss, Stability theory of solitary waves in the
presence of symmetry I,J.Funct.Anal.,74(1987), pp.160–197.
[8] ,Stabilitytheory of solitary waves inthe presence of symmetry II,J.Funct.Anal.,94
(1990), pp.308–348.
[9] I. Ianni and S. Le Coz, Orbital stability of standing waves of a semiclassical nonlinear
Schr¨odinger-Poisson equation,Adv.DifferentialEquations,14(2009), pp.717–748.
[10] L.JeanjeanandS.LeCoz,Anexistenceandstabilityresultforstandingwavesofnonlinear
Schr¨odinger equations,Adv.DifferentialEquations,11(2006), pp.813–840.
[11] ,InstabilityforstandingwavesofnonlinearKlein-Gordonequationsviamountain-pass
arguments,Trans.Amer.Math.Soc.,361(2009), pp.5401–5416.
[12] S.LeCoz,Standing wavesinnonlinear Schr¨odingerequations,inAnalyticalandnumerical
aspectsofpartialdifferentialequations, WalterdeGruyter,Berlin,2009, pp.151–192.
[13] S.LeCoz,R.Fukuizumi,G.Fibich,B.Ksherim,andY.Sivan,Instabilityofboundstates
of a nonlinear Schr¨odinger equation with a Dirac potential, Phys. D,237 (2008), pp. 1103–
1128.
[14] T.-C.LinandJ.Wei,OrbitalstabilityofboundstatesofsemiclassicalnonlinearSchr¨odinger
equations with criticalnonlinearity,SIAMJ.Math.Anal.,40(2008), pp.365–381.
[15] Y.Liu,M.Ohta,andG.Todorova,Instabilit´e forte d’ondes solitairespour des´equations
de Klein-Gordon non lin´eaires et des ´equations g´en´eralis´ees de Boussinesq, Ann. Inst. H.
Poincar´eAnal.NonLin´eaire,24(2007), pp.539–548.
[16] Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr¨odinger equations with
potentials,Comm.Math.Phys.,121(1989), pp.11–33.
[17] M.OhtaandG.Todorova,StronginstabilityofstandingwavesfornonlinearKlein-Gordon
equations,DiscreteContin.Dyn.Syst.,12(2005), pp.315–322.
[18] , Instability of standing waves for nonlinear Klein-Gordon equation and related sys-
tem,inNonlineardispersiveequations, vol.26ofGAKUTOInternat. Ser.Math.Sci.Appl.,
Gakk¯otosho, Tokyo, 2006,pp.189–200.
[19] ,StronginstabilityofstandingwavesforthenonlinearKlein-Gordonequationandthe
Klein-Gordon-Zakharovsystem,SIAMJ.Math.Anal.,38(2007),pp.1912–1931(electronic).
[20] J.Shatah,Unstablegroundstateofnonlinear Klein-Gordonequations,Trans.Amer.Math.
Soc.,290(1985), pp.701–710.
[21] J. Shatah and W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys.,
100(1985), pp.173–190.
[22] C. A. Stuart, Lectures on the orbital stability of standing waves and application to the
nonlinear Schr¨odinger equation,MilanJ.Math.,76(2008), pp.329–399.
[23] T.Tao,Why are solitons stable?,Bull.Amer.Math.Soc.,46(2009), pp.1–33.
[24] M.I.Weinstein,NonlinearSchr¨odingerequationsandsharpinterpolationestimates,Comm.
Math.Phys.,87(1982/83), pp.567–576.
[25] M.I.Weinstein,ModulationalstabilityofgroundstatesofnonlinearSchro¨dingerequations,
SIAMJ.Math.Anal.,16(1985), pp.472–491.
DipartimentodiMatematicaApplicata
Universita` di Pisa
Via F. Buonarroti1/c,56127Pisa
Italia
E-mail address: [email protected]
ON THE STABILITY OF KLEIN-GORDON EQUATIONS 9
Institut de Math´ematiquesde Toulouse,
Universit´e PaulSabatier
118route de Narbonne,31062Toulouse Cedex 9
France
E-mail address: [email protected]
DipartimentodiInformatica
Universita` di Verona
Strada Le Grazie 15,37134Verona
Italia
E-mail address: [email protected]