• Aucun résultat trouvé

On Asymptotic Stability in Energy Space of Ground States of NLS in 2D

N/A
N/A
Protected

Academic year: 2022

Partager "On Asymptotic Stability in Energy Space of Ground States of NLS in 2D"

Copied!
26
0
0

Texte intégral

(1)

www.elsevier.com/locate/anihpc

On asymptotic stability in energy space of ground states of NLS in 2D

Scipio Cuccagna

a

, Mirko Tarulli

b,

aDISMI, University of Modena and Reggio Emilia, Padiglione Morselli, Reggio Emilia, 42100, Italy bDepartment of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy Received 25 January 2008; received in revised form 26 November 2008; accepted 5 December 2008

Available online 24 December 2008

Abstract

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.

©2008 Elsevier Masson SAS. All rights reserved.

Résumé

On utilise les travaux de K. Yajima et T. Mizumachi pour prouver des estimations dispersives et régularisantes des solutions de l’équation linéarisée aux états fondamentaux de NLS in 2D. On applique ces résultats pour obtenir des extensions en dimension 2D de la stabilité asymptotique prouvée en littérature pour toutes les dimensions différentes de 2.

©2008 Elsevier Masson SAS. All rights reserved.

Keywords:Nonlinear Schrödinger equation; Asymptotic stability; Ground states of NLS; Wave operators; Fourier integral operators; Fermi Golden Rule (FGR); Strichartz estimates; Smoothing estimates

1. Introduction

We consider even solutions of a NLS iut+u+β

|u|2

u=0, (t, x)∈R×R2, u(0, x)=u0(x). (1.1)

We assume:

(H1) β(0)=0,βC(R,R);

(H2) there exists ap0(1,)such that for everyk=0,1, dk

dvkβ

v2|v|p0k1 if|v|1;

* Corresponding author.

E-mail addresses:cuccagna.scipio@unimore.it (S. Cuccagna), tarulli@mail.dm.unipi.it (M. Tarulli).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.12.001

(2)

(H3) there exists an open intervalOsuch thatuωu+β(u2)u=0 admits aC1-family of ground statesφω(x)for ωO;

(H4) d φω2L2(R)>0 forωO;

(H5) LetL+= −+ωβ(φω2)−2β2ωω2 be the operator whose domain isHrad2 (R2). We assume thatL+has exactly one negative eigenvalue and that it has no (radial) kernel.

By [27] theωφωH1(R2)isC2and by [38,13,14] (H4)–(H5) yields orbital stability of the ground state eiωtφω(x). Here we investigate asymptotic stability. We need some additional hypotheses.

(H6) For anyx∈R,u0(x)=u0(x). That is, the initial datau0of (1.1) are even.

Consider the Pauli matricesσjand the linearizationHωgiven by:

σ1= 0 1

1 0

, σ2=

0 i

i 0

, σ3= 1 0

0 −1

; Hω=σ3

+ωβ φω2

β φω2

φω2 +

φω2

φω2σ2. (1.2)

Then we assume:

(H7) Let Hω be the linearized operator around eit ωφω, see (1.2). Hω has a positive simple eigenvalue λ(ω) for ωOwhose corresponding eigenfunctions are even functions. There exists anN∈Nsuch thatN λ(ω) < ω <

(N+1)λ(ω).

(H8) The Fermi Golden Rule (FGR) holds (see Hypothesis 4.2 in Section 4).

(H9) The point spectrum ofHωconsists of 0 and±λ(ω). The points±ωare not resonances.

Then we prove:

Theorem 1.1. Letω0Oand φω0(x)be a ground state in a family of ground statesφω. Letu(t, x)be a solution to(1.1). Assume(H1)–(H9). In particular assume the (FGR)in Hypothesis4.2. Then, there exist an0>0 and a C >0such that for any(0, 0)and for anyu0withu0e0φω0H1 < ,there existω+O,θC1(R;R), hH1Cand|ω+ω0|C2such that

t→+∞lim u(t,·)eiθ (t )φω+eit h

H1=0.

Theorem 1.1 is the two dimensional version of Theorem 1.1 [10]. The one dimensional version is in [7]. We recall that results of the sort discussed here were pioneered by Soffer and Weinstein [29], see also [24], followed by Buslaev and Perelman [3,4], about 15 years ago. In this decade these early works were followed by a number of results [5,8, 9,15,21–23,25,29–31,33–36]. It was heuristically understood that the rate of the leaking of energy from the so called

“internal modes” into radiation, is small and decreasing when N increases, producing technical difficulties in the closure of the nonlinear estimates. For this reason prior to Gang Zhou and Sigal [12], the literature treated only the case whenN=1 in (H6). [12] sheds light forN >1. The results in [12] deal with all spatial dimensions different from 2 under the so called Fermi Golden Rule (FGR) hypothesis. [10,7] strengthen [12] by considering initial data in H1, by showing that the (FGR) hypothesis is a consequence of what looks generic condition, Hypothesis 4.2 below, if (H8) is assumed. [10] treats also the case when there are many eigenvalues and Hypothesis 4.2 is replaced by a more stringent hypothesis which is a natural generalization of the (FGR) hypothesis in [12]. The same result with many eigenvalues case can be proved also here and in [7], but we skip for simplicity the proof. We recall that Mizumachi [21], resp. [22], extends to dimension 1, resp. 2, the results in [15] valid for small solitons obtained by bifurcation from ground states of a linear equation, while [20] extends in 2D the result in [30]. [7] transposes [21]

to the case of large solitons, with the generalizations contained in [10]. Here we consider the case of dimension 2.

Thanks to the work by [22], it is quite clear how to transpose to dimension 2 the higher dimensional arguments in [10].

The nonlinear arguments in [10] are not sensitive to the dimension except for the lack in 2D of the endpoint Strichartz estimate. Mizumachi [22] shows how to replace it with an appropriate smoothing estimate of Kato type. The estimate and its proof are suggested by [22]. In order to complete the proof of Theorem 1.1 we need some dispersive estimates on the linearizationHω which in spatial dimension 2 are not yet proved in the literature. The main technical task of this paper is the transposition toHω of the proof ofLp boundedness of wave operators of Schrödinger operators in dimension 2 due to Yajima [40]. We use the following notation. We setH0(ω)=σ3(+ω); given normed spaces

(3)

XandY we denote byB(X, Y )the space of operators fromXtoY and givenLB(X, Y )we denote byLX,Y or byLB(X,Y )its norm. We prove:

Proposition 1.2.Assume the hypotheses of Theorem1.1. The following limits are well defined isomorphism, inverse of each other:

W u= lim

t→+∞eit Hωeit H0(ω)u for anyuL2, Zu= lim

t→+∞eit H0(ω)eit Hω for anyuL2c(Hω)(defined in Section2).

For anyp(1,)and anykthe restrictions ofWandZtoL2Wk,pextend into operators such that forC(ω) <semicontinuous inω

WWk,p(R2),Wck,p(Hω)+ ZWk,p

c (Hω),Wk,p(R2)< C(ω) withWck,p(Hω)the closure inWk,p(R2)ofWk,p(R2)L2c(Hω).

We will setL2,sandHm,s

uL2,s = xsuL2(R2) and uHm,s= xsuHm(R2),

wherem∈N,s∈R and x =(1+ |x|2)1/2. For f (x) andg(x)column vectors, their inner product is f, g =

R2tf (x)·g(x) dx. The adjointHis defined by Hf, g = f, Hg.Given an operatorH, its resolvent isRH(z)= (Hz)1.We will writeR0(z)=(z)1.We writeg(t, x)LptLqx = g(t, x)LqxLpt andg(t, x)Lp

tL2,sx = g(t, x)L2,s

x Lpt.

2. Linearization, modulation and set up

We will use the following classical result, [38,13,14], see also [7]:

Theorem 2.1. Suppose thateiωtφω(x)satisfies (H4). Then∃ >0and a A0(ω) >0 such that for anyu(0, x)φωH1< we have for the corresponding solutioninf{u(t, x)eφω(xx0)H1(x∈R2): γ∈Randx0∈R2}<

A0(ω).

We can write the ansatzu(t, x)=eiΘ(t )ω(t )(x)+r(t, x)), Θ(t )=t

0ω(s) ds+γ (t ). Inserting the ansatz into the equation we get

irt= −r+ω(t )rβ φω(t )2

rβ φω(t )2

φω(t )2 rβ φω(t )2

φω(t )2 r¯+ ˙γ (t )φω(t )

iω(t )∂˙ ωφω(t )+ ˙γ (t )r+O r2

.

We settR=(r,r),¯ tΦ=ω, φω)and we rewrite the above equation as iRt=HωR+σ3γ R˙ +σ3γ Φ˙ −iω∂˙ ωΦ+O

R2

. (2.1)

SetH0(ω)=σ3(+ω)andV (ω)=HωH0(ω).The essential spectrum is σe=σe(Hω)=σe

H0(ω)

=(−∞,ω] ∪ [ω,+∞), 0 is an isolated eigenvalue. Given an operatorLwe setNg(L)=

j1ker(Lj). [37] implies that, if{·}means span, Ng(Hω)= {Φ, σ3ωΦ}.λ(ω)has corresponding real eigenvectorξ(ω), which can be normalized so that ξ, σ3ξ =1.

σ1ξ(ω)generates ker(Hω+λ(ω)). The function(ω, x)O×R→ξ(ω, x)isC2;|ξ(ω, x)|< cea|x|for fixedc >0 anda >0 ifωKO,Kcompact. ξ(ω, x)is even inx since by assumption we are restricting ourselves in the category of such functions. We have theHωinvariant Jordan block decomposition

L2=Ng(Hω)

j,±

ker

Hωλ(ω)

L2c(Hω)=Ng(Hω)Ng Hω

(4)

where we setL2c(Hω)= {Ng(Hω)

±ker(Hωλ(ω))}.We can impose R(t )=(zξ+ ¯1ξ )+f (t )

±

ker

Hω(t )λ ω(t )

L2c(Hω(t )). (2.2)

The following claim admits an elementary proof which we skip:

Lemma 2.2.There is a Taylor expansion atR=0of the nonlinearityO(R2)in(2.1)withRm,n(ω, x)andAm,n(ω, x) real vectors and matrices rapidly decreasing inx:

O R2

=

2m+n2N+1

Rm,n(ω)zmz¯n+

1m+nN

zmz¯nAm,n(ω)f+O

f2+ |z|2N+2 .

In terms of the frame in (2.2) and the expansion in Lemma 2.2, (2.1) becomes ift=(Hω(t )+σ3γ )f˙ +σ3γ Φ(ω)˙ −iω∂˙ ωΦ(t )+

zλ(ω)iz˙ ξ(ω)

¯

zλ(ω)+i˙¯z

σ1ξ(ω)+σ3γ (zξ˙ + ¯1ξ )iω(z∂˙ ωξ+ ¯1ωξ )

+

2m+n2N+1

zmz¯nRm,n(ω)+

1m+nN

zmz¯nAm,n(ω)f +O f2

+Olocz2N+2 (2.3) where by Olocwe mean that the there is a factorχ (x)rapidly decaying to 0 as|x| → ∞. By taking inner product of the equation with generators ofNg(Hω)and ker(Hωλ)we obtain modulation and discrete modes equations:

˙ω22

=

σ3γ (zξ˙ + ¯1ξ )iω(z∂˙ ωξ+ ¯1ωξ )+

2N+1 m+n=2

zmz¯nRm,n(ω)

+

σ3γ˙+iω∂˙ ωPc+ N m+n=1

zmz¯nAm,n(ω)

f +O f2

+Olocz2N+2, Φ

,

˙

γdφω22

= same as above, σ3ωΦ,

iz˙−λ(ω)z= same as above, σ3ξ. (2.4)

3. Spacetime estimates forHω

We need analogues of Lemmas 2.1–2.3 and Corollary 2.1 in [22]. We call admissible all pairs(p, q)with 1/p= 1/2−1/qand 2q <∞. We set(p, q)=(p/(p−1), q/(q−1)). In the lemmas below we assume that theHωof the form (1.2) for which hypotheses (H3)–(H5), (H7) and (H9) hold.

Lemma 3.1(Strichartz estimate). There exists a positive numberC=C(ω)upper semicontinuous inωsuch that for anyk∈ [0,2]:

(a) for anyfL2c(ω)and any admissible all pairs(p, q), eit Hωf

LptWxk,q CfHk;

(b) for anyg(t, x)S(R2)and any couple of admissible pairs(p1, q1) (p2, q2)we have t

0

ei(ts)HωPc(ω)g(s,·) ds

Lpt1Wxk,q1

Cg

Lp

2 t Wk,q

2 x

.

Lemma 3.1 follows immediately from Proposition 1.2 sinceW andZintertwineeit HωPc(Hω)andeit H0.

(5)

Lemma 3.2.Lets >1.∃C=C(ω)upper semicontinuous inωsuch that:

(a) for anyfS(R2), eit HωPc(ω)f

L2tL2,xs CfL2; (b) for anyg(t, x)S(R2)

R

eit HωPc(ω)g(t,·) dt L2x

CgL2 tL2,sx . Notice that (b) follows from (a) by duality.

Lemma 3.3.Lets >1.∃C=C(ω)as above such thatg(t, x)S(R2)andt∈R:

t

0

ei(ts)HωPc(ω)g(s,·) ds

L2tL2,xs

CgL2

tL2,sx .

As a corollary from Christ and Kiselev [6], Lemmas 3.2 and 3.3 imply:

Lemma 3.4.Let(p, q) be an admissible pair and lets >1.∃C=C(ω) as above such thatg(t, x)S(R2)and t∈R:

t

0

ei(ts)HωP (ω)g(s,·) ds

LptLqx

CgL2 tL2,sx .

Lemma 3.5.Consider the diagonal matricesE+=diag(1,0),E=diag(0,1).Set P±(ω)=Z(ω)E±W (ω) with Z(ω)andW (ω)the wave operators associated toHω. Then we have foruL2c(Hω)

P+(ω)u= lim

0+

1 2π i lim

M→+∞

M ω

RHω+i)RHωi) u dλ,

P(ω)u= lim

0+

1 2π i lim

M→+∞

ω

M

RHω+i)RHωi)

u dλ (1)

and for anys1ands2and forC=C(s1, s2, ω)upper semicontinuous inω, we have P+(ω)P(ω)Pc(ω)σ3

f

L2,s1 CfL2,s2. (2)

Proof. Formulas (1) hold with P±(ω)replaced byE± andHω replaced by H0 and for anyuL2(R2). Applying W (ω)we get (1) forHω. Estimate (2) follows by the proof of inequality (3) in Lemma 5.12 [7] which is valid for all dimensions. 2

4. Proof of Theorem 1.1

We restate Theorem 1.1 in a more precise form:

Theorem 4.1.Under the assumptions of Theorem1.1we can express u(t, x)=eiΘ(t )

φω(t )(x)+ 2N j=1

pj(z,z)A¯ j x, ω(t )

+h(t, x)

(6)

withpj(z,z)¯ =O(z)near 0, withlimt→+∞ω(t )convergent, with|Aj(x, ω(t ))|Cea|x|for fixedC >0anda >0, limt→+∞z(t )=0,and for fixedC >0

z(t ) N+1

L2N+2t + h(t, x)

Lt Hx1L3tWx1,6< C. (1)

Furthermore, there existshH1(R,C)such that

tlim→∞ ei

t

0ω(s) ds+iγ (t )h(t )eit h

H1=0. (2)

The proof of Theorem 4.1 consists in a normal forms expansion and in the closure of some nonlinear estimates.

The normal forms expansion is exactly the same of [10,7], in turn adaptations of [12].

4.1. Normal form expansion

We repeat [10]. We pickk=1,2, . . . , N and setf =fk fork=1. The otherfk are defined below. In the ODE’s there will be error terms of the form

EODE(k)=O

|z|2N+2 +O

zN+1fk +O

fk2 +O

β

|fk|2 fk

. In the PDE’s there will be error terms of the form

EPDE(k)=Oloc

|z|N+2

+Oloc(zfk)+Oloc

fk2 +O

β

|fk|2 fk

.

In the right-hand sides of Eqs. (2.3)–(2.4) we substitute γ˙ and ω˙ using the modulation equations. We repeat the procedure a sufficient number of times until we can write fork=1 andf1=f

˙ω22

=

2N+1

m+n=2

zmz¯nΛ(k)m,n(ω)+ N m+n=1

zmz¯nA(k)m,n(ω)fk+EODE(k), Φ(ω)

, iz˙−λz=

same as above, σ3ξ(ω) ,

i∂tfk=(Hω+σ3γ )f˙ k+EPDE(k)+

k+1m+nN+1

zmz¯nRm,n(k)(ω),

withA(k)m,n,Rm,n(k) andΛ(k)m,n(ω, x)real exponentially decreasing to 0 for|x| → ∞and continuous in(ω, x). Exploiting

|(mn)λ(ω)|< ωform+nN,m0,n0, we define inductivelyfk withkNby fk1= −

m+n=k

zmz¯nRHω

(mn)λ(ω)

Rm,n(k1)(ω)+fk.

Notice that if R(km,n1)(ω, x) is real exponentially decreasing to 0 for |x| → ∞, the same is true for RHω((mn)λ(ω))R(km,n1)(ω)by|(mn)λ(ω)|< ω. By inductionfk solves the above equation with the above notifications.

Now we manipulate the equation forfN. We fixω1=ω(0). We write i∂tPc1)fN

Hω1+˙+ωω1)

P+1)P1)

Pc1)fN

= +Pc1)EPDE(N )+

m+n=N+1

zmz¯nPc1)Rm,n(N )1) (4.1)

where we split Pc1)=P+1)+P1)with P±1), see Lemma 3.5, whereP+1) are the projections in σc(Hω1)∩ {λ: ±λω1}and with

EPDE(N )=EPDE(N )+

m+n=N+1

zmz¯n

Rm,n(N )(ω)R(N )m,n1)

+ϕ(t, x)fN, ϕ(t, x):=˙+ωω1)

Pc13

P+1)P1) fN+

V (ω)V (ω1) fN +˙+ωω1)

Pc(ω)Pc1)

σ3fN. (4.2)

(7)

By Lemma 3.5 forCN1)upper semicontinuous inω0,∀N we have xN

P+1)P1)Pc13 f

L2x CN1) xNf

L2x. (4.3)

The termϕ(t, x)in (4.2) can be treated as a small cutoff function. We write fN= −

m+n=N+1

zmz¯nRHω

1

(mn)λ(ω1)+i0

Pc1)R(N )m,n1)+fN+1. (4.4) Then

i∂tPc1)fN+1=

Hω1+˙+ωω1)

P+1)P1)

Pc1)fN+1 +

±

O

|z|N+1 RHω

1

±(N+1)λ(ω1)+i0

R±1)+Pc1)EPDE(N ) (4.5) withR+=R(N )N+1,0andR=R(N )0,N+1andEPDE(N )=EPDE(N )+Oloc(zN+1), where we have used that(ωω1)= O()by Theorem 2.1. Notice thatRHω

0(±(N+1)λ(ω0)+i0)R±0)Ldo not decay spatially. In the ODE’s withk=N, by the standard theory of normal forms and following the idea in Proposition 4.1 [5], see [10] for details, it is possible to introduce new unknowns

˜

ω=ω+q(ω, z,z)¯ +

1m+nN

zmz¯n

fN, αmn(ω) ,

˜

z=z+p(ω, z,z)¯ +

1m+nN

zmz¯n

fN, βmn(ω)

, (4.6)

with p(ω, z,z)¯ =

pm,n(ω)zmz¯n and q(z,z)¯ =

qm,n(ω)zmz¯n polynomials in(z,z)¯ with real coefficients and O(|z|2)near 0, such that we get

˙˜ =

EPDE(N ), Φ , iz˙˜−λ(ω)z˜=

1mN

am(ω)zm|2z˜+

EODE(N ), σ3ξ + ¯˜zN

A(N )0,N(ω)fN, σ3ξ

(4.7) witham(ω)real. Next step is to substitutefN using (4.4). After eliminating by a new change of variablesz˜= ˆz+ p(ω,z,ˆ z)¯ˆ the resonant terms, withp(ω,z,ˆ z)¯ˆ =

ˆ

pm,n(ω)zmz¯na polynomial in(z,z)¯ with real coefficients O(|z|2) near 0, we get

˙ˆ =

EPDE(N ), Φ , iz˙ˆ−λ(ω)zˆ=

1mN

ˆ

am(ω)zm|2zˆ+

EODE(N ), σ3ξ

− |ˆzN|2zˆAˆ(N )0,N(ω)RHω

0

(N+1)λ(ω1)+i0

Pc0)RN(N )+1,01), σ3ξ + ¯ˆzNAˆ(N )0,N(ω)fN+1, σ3ξ

(4.8) withaˆm,Aˆ(N )0,N andRN(N )+1,0real. By x1i0=P V1x+iπ δ0(x)and by an elementary use of the wave operators, we can denote byΓ (ω, ω1)the quantity

Γ (ω, ω1)= Aˆ(N )0,N(ω)RHω

1

(N+1)λ(ω1)+i0

Pc1)RN(N )+1,013ξ(ω)

=πAˆ(N )0,N(ω)δ

Hω1(N+1)λ(ω1)

Pc1)R(N )N+1,013ξ(ω) . Now we assume the following:

Hypothesis 4.2.There is a fixed constantΓ >0 such that|Γ (ω, ω)|> Γ.

By continuity and by Hypothesis 4.2 we can assume|Γ (ω, ω1)|> Γ /2.Then we write d

dt

z|2

2 = −Γ (ω, ω1)|z|2N+2+ Aˆ(N )0,N(ω)fN+1, σ3ξ(ω)

¯ˆ zN+1

+

EODE(N ), σ3ξ(ω)

¯ˆ z

. (4.9)

(8)

4.2. Nonlinear estimates

By an elementary continuation argument, the following a priori estimates imply inequality (1) in Theorem 4.1, so to prove (1) we focus on:

Lemma 4.3.There are fixed constantsC0andC1and0>0such that for any0< 0if we have ˆzN+1

L2N+t 22C0 and fN

Lt Hx1L3tWx1,6L

2p0 p0−1

t Wx1,2p0L2tH1,−s

2C1 (4.10)

then we obtain the improved inequalities fN

Lt Hx1L3tWx1,6L

2p0 p0−1

t Wx1,2p0L2tH1,s

C1, (4.11)

ˆzN+1

L2N+2t C0. (4.12)

Proof. Set(t ):=γ+ωω1. First of all, we have:

Lemma 4.4.Letg(0, x)Hx1L2c1)and letω(t )be a continuous function. Considerigt= {Hω1+(t )(P+0)P0))}g+Pc1)F.Then for a fixedC=C(ω1, s)upper semicontinuous inω1ands >1we have

g

Lt Hx1L3tWx1,6L

2p0 p0−1 t Wx1,2p0

C g(0, x)

H1+ FL1

tHx1+L2tHx1,s

.

Lemma 4.4 follows easily from Lemmas 3.1–3.4 and P±1)g(t )=eit Hω1ei

t

0(τ ) dτP±1)g(0)i t

0

ei(ts)Hω1e±i

t

s(τ ) dτP±1)F (s) ds.

Lemma 4.5.Consider Eq.(4.1)forfNand assume(4.10). Then we can splitEPDE(N )=X+O(fN3)+O(fNp0)such thatXL2

tHx1,M 2for any fixedMandO(fN3)+O(fNp0)L1tH1

x 3.

Proof of Lemma 4.5. In the error terms fork=N at the beginning of Section 4.1 we can write EPDE(N )=O()ψ (x)fN+Oloc

|z|N+2

+Oloc(zfN)+Oloc

fN2 +O

fN3 +O

fNp0

withψ (x)a rapidly decreasing function,p0the exponent in (H2) and with O(fNp0)relevant only forp0>3. Denoting Xthe sum of all terms except the last one, settingf =fN, by (4.10) we have:

(1) O()ψ (x)fL2

tHx1,MfL2

tHx1,−M2; (2) Oloc(zf )L2

tHx1,MzfL2

tHx1,M 2; (3) Oloc(f2)L2

tHx1,M f2

L2tHx1,M2. This yields xMXH1

xL2t 2. To bound the remaining term observe:

(4) |f|2fL1tHx1fW1,6

x f2L6

xL1t f3

L3tWx1,63; (5) O(fp0)L1

tHx1fW1,2p0

x fp01

L2px 0 L1

t f

L

2p0 p0−1 t Wx1,2p0

fp01

L

2p0p0−1 p0+1 t Wx1,2p0

p0, where in the last step we usef

L

2p0 p0−1 p0+1 t Wx1,2p0

fα

L

2p0 p01 t L2px 0

f1Lα

t Hx1 for some 0< α <1 byp0>3, interpolation and Sobolev em- bedding. 2

Références

Documents relatifs

The vector fields are not necessarily forward complete but V being a weak common Lyapunov function, the switched system is uniformly stable, and all solutions of (S) starting in

Math. Crist´ ofani, “Orbital stability of periodic traveling-wave solutions for the log-KdV equation”, J. Kiyashko, and A.V. Pelinovsky, “Sharp bounds on enstrophy growth in

In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile.. We next derive a monotonicity formula

Merle, Global well-posedness, scattering and blow-up for the energy- critical, focusing, non-linear Schr¨ odinger equation in the radial case, Invent. Keraani, On the defect

In order to study the asymptotic profile of the ground state solutions we need to develop finer estimates on the ground state energy than those given above and to derive a

Under a Nehari type condition, we show that the standard Ambrosetti–Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.. © 2006

Under some hypothesis on the structure of the spectrum of the linearized operator, we prove that, asymptotically in time, the solution decomposes into a solitary wave with

Lee, Ground state energy of dilute Bose gases in small negative potential case, Journal of Statistical Physics, 134 (2009), pp.. Seiringer, Derivation of the Landau-Pekar equations in