Thomas Duyckaerts

Frank Merle

SCATTERING NORM ESTIMATE NEAR THE THRESHOLD FOR ENERGY-CRITICAL FOCUSING SEMILINEAR WAVE EQUATION

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Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Introduction

We consider the focusing energy-critical wave equation on an interval I (0 ∈ I)

(1.1) ∂ 2 t u -∆u -|u| 4 N-2 u = 0, (t, x) ∈ I × R N u ↾t=0 = u 0 ∈ Ḣ1 , ∂ t u ↾t=0 = u 1 ∈ L 2 ,
where u is real-valued, N ∈ {3, 4, 5}, L 2 := L 2 (R N ) and Ḣ1 := Ḣ1 (R N ). The equation (1.1) is locally well-posed in Ḣ1 × L 2 (see [START_REF] Pecher | Nonlinear small data scattering for the wave and Klein-Gordon equation[END_REF], [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF] and [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF]): if (u 0 , u 1 ) ∈ Ḣ1 × L 2 , there exists an unique solution u, defined on a maximal time of existence I max and such that for all interval J J ⋐ I max =⇒ u S(J) < ∞, where S(J)

:= L 2(N+1) N-2 J × R N .
Furthermore, the solution u of (1.1) scatters forward in time in Ḣ1 × L 2 if and only [0, +∞) ⊂ I max and u S(0,+∞) < ∞.

Thus the norm S(R) measures the nonlinear effect for a given solution. The energy

E(u(t), ∂ t u(t)) = 1 2 |∂ t u(t, x)| 2 dx + 1 2 |∇u(t, x)| 2 dx - N -2 2N |u(t, x)| 2N N-2 dx
is conserved for solutions of (1.1). The defocusing case (equation (1.1) with sign + instead of -in front of the nonlinearity) has been the object of intensive studies in the last decades (see for example [START_REF] Shatah | Geometric wave equations[END_REF] and references therein). In this case the solutions are known to scatter, which implies, for any solution u, a bound of the norm S(R) by an unspecified function of the defocusing energy

E d = 1 2 |∂ t u| 2 + 1 2 |∇u| 2 + N -2 2N |u| 2N N-2 .
In three spatial dimension, an explicit upper bound was proven by T. Tao [START_REF] Tao | Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions[END_REF]: for any solution u of the defocusing equation,

u L 4 t L 12 x ≤ C(1 + E d ) CE 105/2 d ,
which gives, by Strichartz and interpolation estimate, a similar bound for u S(R) .

Going back to the focusing case, consider the explicit Ḣ1 stationnary solution of (1.1)

(1.2)

W := 1 1 + |x| 2 N (N -2) N-2 2
.

In [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF], C. Kenig and F. Merle have described the dynamics of (1.1) below the energy threshold E(W, 0). Namely, if E(u 0 , u 1 ) < E(W, 0), then |∇u 0 | 2 = |∇W | 2 and the solution u scatters (both forward and backward in time) if and only if |∇u 0 | 2 < |∇W | 2 . This implies that for ε > 0 the following supremum is finite:

I ε = sup u∈Fε R×R N |u(t, x)| 2(N+1) N-2 dtdx = sup u∈Fε u 2(N+1) N-2 S(R) ,
where

F ε := u solution of (1.1) such that E(u 0 , u 1 ) ≤ E(W, 0) -ε 2 and |∇u 0 | 2 < |∇W | 2 .
Furthermore, the existence of the non-scattering solution W at the energy threshold shows that lim

ε→0 + I ε = +∞.
The purpose of this note is to give an equivalent of I ε for small ε. Consider the negative eigenvalue -ω 2 (ω > 0) of the linearized operator associated to (1.1) around W :

-ω 2 = inf u∈H 1 u 2 =1 R N |∇u| 2 - N + 2 N -2 R N W 4 N-2 |u| 2 .
(See §3.1 for details). Then Theorem 1. lim

ε→0 + I ε | log ε| = 2 ω R N W 2(N+1) N-2 .
Remark 1.1. It would be interesting to get an explicit value of the limit 2 ω R N W 2(N+1)

N-2 . A straightforward computation gives:

R N W 2(N+1) N-2 = (N (N -2)) N 2 2 2N +1 × N ! ( N 2 )! 2 × π if N is even, R N W 2(N+1) N-2 = (N (N -2))
N 2

However we do not know any explicit expression of ω.

Let us give an outline of the proof of Theorem 1. In Section 2, we show that a sequence of solutions (u n ) such that

E(u n (0), ∂ t u n (0)) < E(W, 0), |∇u n (0)| 2 < |∇W | 2 and lim n→+∞ u n S(R) = +∞
must converge to W up to modulation for a well-chosen time sequence. This relies on the compactness argument of [KM06b, Section 4], using the profile decomposition of Bahouri-Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF], and on the classification of the solutions of (1.1) at the threshold of energy in our previous work [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical wave equation[END_REF]. The second step of the proof is an analysis of the behaviour of solutions whose initial conditions are close to (W, 0), which is carried out in Section 3. We show, as a consequence of the existence of the negative eigenvalue -ω 2 , that such solutions go away from the solution W in a time which is of logarithmic order with respect to the distance of the initial condition to (W, 0). In Section 4 we put together the preceding arguments to prove Theorem 1. Our arguments do not depend strongly on the nature of equation (1.1), and we except that a logarithmic estimate of the scattering norm S(R) near the threshold holds in similar situations, as long as the linearized operator around the ground state admits real nonzero eigenvalues. In Section 5 we give a result and a sketch of proof in the case of the radial, energy-critical focusing nonlinear Schrödinger equation.

Convergence to W and W -near the threshold

In all the article, we will denote by • p the L p norm on R N . Equation (1.1) enjoys the following invariances: if u is a solution and

t 0 ∈ R, x 0 ∈ R N , λ 0 > 0, δ 0 , δ 1 ∈ {-1, +1}, then v(t, x) = δ 0 λ (N -2)/2 0 u t 0 + δ 1 t λ 0 , x + x 0 λ 0
is also a solution. Note that the energy of u and, if u is globally defined, the norm u S(R) are not changed by these transformations. We recall the following classification Theorem, proven in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation[END_REF], for the case E(u 0 , u 1 ) < E(W, 0), and in [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical wave equation[END_REF] for the existence of W -and the case E(u 0 , u 1 ) = E(W, 0): Theorem A (Kenig-Merle,Duyckaerts-Merle). There exists a global solution W -of (1.1) such that

E(W -(0), ∂ t W -(0)) = E(W, 0), ∇W -(0) 2 < ∇W 2 W - S(-∞,0) < ∞, lim t→+∞ ∇(W -(t) -W ) 2 + ∂ t (W -(t) -W ) 2 = 0.
Moreover, if u is a solution of (1.1) such that E(u 0 , u 1 ) ≤ E(W, 0) and ∇u 0 2 ≤ ∇W 2 , then u is globally defined. If furthermore u S(R) = ∞, then u = W -or u = W up to the invariances of the equation.

We will also need the following simple version of long-time perturbation theory results (see e.g. [KM06b, Theorem 2.20]).

Lemma 2.1. Let M > 0. Then there exist positive constants ε(M ) and C(M ) such that for all solutions v and u of (1.1), with initial conditions (v 0 , v 1 ) and (u 0 , u 1 ), if the forward time of existence of v is infinite and

v S(0,+∞) ≤ M and ∇(u 0 -v 0 ) 2 + u 1 -v 1 2 ≤ ε(M ),
then u is globally defined for positive times and u S(0,+∞) ≤ C(M ). A similar statement holds for negative times.

In this section we show the following: Proposition 2.2. Let u n be a family of solutions of (1.1), such that

(2.1) E u n (0), ∂ t u n (0) < E(W, 0), ∇u n (0) 2 < ∇W 2 .
and lim n→+∞ u n S(R) = +∞. Let (t n ) n be a time sequence. Then, up to the extraction of a subsequence there exist δ 0 ∈ {-1, +1} and sequences of parameters

x n ∈ R n , λ n > 0 such that lim n→+∞ δ 0 λ N/2 n ∇u n t n , • -x n λ n -∇W 2 + ∂u n ∂t (t n ) 2 = 0.
(b) Assume that there exists C 0 ∈ (0, +∞) such that

lim n→+∞ u n S(-∞,tn) = +∞ and lim n→+∞ u n S(tn,+∞) = C 0 .
Then, up to the extraction of a subsequence there exist t 0 ∈ R, δ 0 , δ 1 ∈ {-1, +1}, and sequences of parameters

x n ∈ R n , λ n > 0 such that lim n→+∞ δ 0 λ N/2 n ∇u n t n , • -x n λ n -∇W -(t 0 ) 2 + δ 0 λ N/2 n ∂u n ∂t t n , • -x n λ n - ∂W - ∂t (t 0 ) 2 = 0.
Remark 2.3. Case (b) will not be used in the proof of Theorem 1, and is stated only for its own interest.

Sketch of Proof. We will sketch the proof (a), the proof of (b) is similar and left to the reader.

Translating in time all the u n , we may assume that t n = 0 for all n, and thus

(2.2) lim n→+∞ u n S(-∞,0) = lim n→+∞ u n S(0,+∞) = +∞.
In view of (2.1) and (2.2), one can show, using the profile decomposition of [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] as in [KM06b, Proposition 4.2], that there exist (up to the extraction of a subsequence) parameters

λ n > 0, x n ∈ R N , and functions (v 0 , v 1 ) ∈ Ḣ1 × L 2 such that lim n→+∞ 1 λ N/2 n ∇u n 0, • -x n λ n -∇v 0 2 + 1 λ N/2 n ∂u n ∂t 0, • -x n λ n -v 1 2 = 0.
We refer to [KM06a, Section 4] and also [DM07a, Lemma 2.5] for proofs in the case of nonlinear Schrödinger equations that readily apply to our case. Note that

(2.3) ∇v 0 2 ≤ ∇W 2 , E(v 0 , v 1 ) ≤ E(W, 0).
Let v be the solution of (1.1) with initial conditions (v 0 , v 1 ). Theorem A and (2.3) imply that v is globally defined. By Lemma 2.1 and by (2.2),

v S(-∞,0) = +∞, v S(0,+∞) = +∞.
This shows, again by Theorem A, that v = W , up to the invariances of equation (1.1) concluding the proof.

3. Estimates near the threshold 3.1. Preliminaries on the linearized equation. In this subsection, we recall results on the linearized equation near W . We refer to [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical wave equation[END_REF] for the details. Let u be a solution of (1.1) which is close to W . Write u = W + h. Then h is solution to the equation

(∂ 2 t + L)h = R(h). (3.1) L := -∆ - N + 2 N -2 W 4 N-2 , R(h) := |W + h| 4 N-2 (W + h) -W N+2 N-2 - N + 2 N -2 W 4 N-2 h. Let (3.2) W 0 = a N -2 2 W + x • ∇W , W j = b∂ x j W, j = 1 . . . N.
where the constants a and b are chosen so that ∇W j 2 = 1 for j = 0, 1, . . . , N . By the invariances of equation (1.1), L(W j ) = 0 for j = 0, . . . , N . As a consequence the functions W j , j = 0, . . . , N are in the kernel of the quadratic form

Q(h) = 1 2 Lh h = 1 2 |∇h| 2 - N + 2 2(N -2) W 4 N-2 h 2 . Observe that if (h, ∂ t h) is small in Ḣ1 × L 2 , (3.3) E(W + h, ∂ t h) = E(W, 0) + Q(h) + 1 2 |∂ t h| 2 + O h 3 2N N-2
.

Furthermore, one can check that the infimum of Q(h) for h ∈ H 1 , h L 2 = 1, is negative, and thus that L admits a positive, radial eigenfunction Y with eigenvalue -ω 2 < 0. We normalize

Y such that Y 2 = 1. The self-adjointness of L implies (3.4) YW j = 0, j = 0 . . . N.
Consider

G ⊥ := f ∈ Ḣ1 , Yf = ∇W 0 • ∇f = . . . = ∇W N • ∇f = 0 .
The following result (see Proposition 5.5 of [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical wave equation[END_REF]) shows in particular that -ω 2 is the only negative eigenvalue of L:

Claim 3.1. There exists a constant c Q > 0 such that ∀h ∈ G ⊥ , Q(h) ≥ c Q ∇h 2 L 2 .
As a consequence of the Strichartz estimates for the linear wave equation (see [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF] and [START_REF] Lindblad | On existence and scattering with minimal regularity for semilinear wave equations[END_REF]), we easily get the following Strichartz-type estimate for equation (3.1).

Claim 3.2. There exist constants

c, C > 0 such that if h is a solution of (3.1) on an interval [t 0 , t 1 ] such that ∇h 2 + ∂ t h 2 + |t 0 -t 1 | ≤ c. Then h S(t 0 ,t 1 ) ≤ C ( ∇h(t 0 ) 2 + ∂ t h(t 0 ) 2 ) .
3.2. Estimate on the exit time. In this subsection, we consider a sequence u n of solutions of (1.1) such that

lim n→+∞ ∇(u n (0) -W ) 2 + ∂ t u n (0) 2 = 0 (3.5) E(W, 0) -E(u n , ∂ t u n ) = ε 2 n -→ n→+∞ 0 and ∀n > 0, ∇u n (0) 2 < ∇W 2 . (3.6) Let h n = u n -W and decompose h n as (3.7) h n (t) = β n (t)Y + N j=0 γ j,n (t)W j + g n (t), g n (t) ∈ G ⊥ .
For this, observe that the condition g n (t) ∈ G ⊥ is equivalent to

β n (t) = h n (t)Y, (3.8) γ 0,n (t) = ∇ (h n (t) -β n (t)Y) • ∇W 0 , γ j,n (t) = ∇h n (t) • ∇W j , j = 1, . . . , N. (3.9) (we used that Y and W being radial, ∇Y • ∇W j = b ∇Y • ∂ x j ∇W = 0 if j ∈ {1, .
. . , N } and by a similar argument, ∇W j • ∇W k = 0 if j = k). We have: Claim 3.3. Assume (3.5). Then there exists sequences λ n ∈ (0, +∞),

x n ∈ R N such that lim n→+∞ λ n = 1, lim n→+∞ x n = 0,
and for all n, noting c 0 = ∇Y • ∇W 0 ,

1 λ N 2 n ∇u n x -x n λ n • ∇W 0 (x)dx - c 0 λ N-2 2 n u n x -x n λ n -W (x) Y(x)dx = 0 (3.10) ∀j ∈ {1, . . . , N }, 1 λ N 2 n ∇u n x -x n λ n • ∇W j (x) = 0. (3.11)
Sketch of proof. Consider the mapping J : (λ, X, u) → (J 0 , J 1 , . . . , J N ) where

J 0 = 1 λ N 2 ∇u x -X λ • ∇W 0 (x)dx - c 0 λ N-2 2 u x -X λ -W (x) Y(x)dx J k = 1 λ N 2 ∇u x -X λ • ∇W k (x), k ∈ {1, . . . , N }.
A straightforward computation shows that J = 0 and ∂J ∂λ , ∂J ∂X 1 , . . . , ∂J ∂X N is diagonal and invertible at the point (1, 0, . . . , 0, W ). The Claim then follows from (3.5) and the implicit function theorem.

Observe that the conditions (3.10) and (3.11) are equivalent, by (3.8) and (3.9), to the condition that the parameters γ jn corresponding to the modulated solution

1 λ N-2 2 n u n t λn , x
λn vanish at t = 0. By Claim 3.3 we can assume, up to translation and scaling, that (3.12) ∀n, ∀j ∈ {0, 1, . . . , N }, γ j,n (0) = 0.

The main result of this subsection is the following:

Proposition 3.4. There exist a constant η 0 , such that for all η ∈ (0, η 0 ), for all sequence (u n ) satisfying (3.5), (3.6), (3.12) and such that

β n (0)β ′ n (0) ≥ 0, if T n (η) = inf t ≥ 0 : |β n (t)| ≥ η , then for large n, β n (0) = 0, T n (η) ∈ (0, +∞) and (3.13) lim n→+∞ T n (η) log |β n (0)| = 1 ω .
Furthermore,

(3.14) lim inf n→+∞ |β ′ n (T n (η))| ≥ ωη. Remark 3.5. If β n (0)β ′ n (0) < 0, we may achieve the condition β n (0)β ′ n (0) ≥ 0 by considering the solution u n (-t, x) instead of u n (t, x).
The remainder of this subsection is devoted to the proof of Proposition 3.4. We first give, as a consequence of the orthogonality conditions (3.12), a purely variational lower bound on |β n (0)| (Claim 3.6). We then give (Lemma 3.7) precise estimates on β n (t) and ∂ t h n (t) 2 + ∇h n (t) 2 , on an interval (0, t n ) where a priori bounds are assumed. These estimates will give the desired bounds on the exit time T n (η). We will write:

∂ t,x h n (t) 2 = ∇h n (t) 2 + ∂ t h n (t) 2 .
Claim 3.6. There exists M 0 > 0 such that for all sequence (u n ) of solutions of (1.1) satisfying (3.5), (3.6) and (3.12) we have

β n (0) = 0 and lim sup n→+∞ ∂ t,x h n (0) 2 + ε n |β n (0)| ≤ M 0 .
Proof. Developping the energy as in (3.3), we get

E(W, 0) -ε 2 n = E(W + h n , ∂ t h n ) = E(W, 0) + Q(h n ) + 1 2 ∂ t h n 2 2 + O ∇h n 3 2 .
The expression (3.7) of h at t = 0 yields, in view of (3.12)

∇h n (0) 2 ≤ C(|β n (0)| + ∇g n (0) 2 ).
Furthermore, taking into account that Q(Y) < 0 and that the functions W j are in the kernel of Q for j = 0 . . . N , we get

Q(h n (0)) = -β 2 n (0)|Q(Y)| + Q(g n (0)).
Combining the preceding estimates, we obtain

β 2 n (0)|Q(Y)| = ε 2 n + Q(g n (0)) + 1 2 ∂ t h n (0) 2 2 + O β 3 n (0) + ∇g n (0) 3 2 . By Claim 3.1, Q(g n (0)) ≥ c Q ∇g n (0) 2 2 .
This yields for large n,

2β 2 n (0) |Q(Y)| ≥ ε 2 n + c Q ∇g n (0) 2 2 + 1 2 ∂ t h n (0) 2 2 ≥ ε 2 n + c ∇h n (0) 2 2 + 1 2 ∂ t h n (0) 2 2 ,
which concludes the proof of the claim.

Our next result is the following Lemma:

Lemma 3.7 (Growth on [0, T n ]). Let us fix ω + and ω -, close to ω, such that ω -< ω < ω + . There exist positive constants τ 0 , K 0 (depending only on the choice of ω ± ) with the following property. Let (u n ) n be a sequence of solutions of (1.1) satisfying (3.5), (3.6) and such that

β n (0)β n ′ (0) ≥ 0.
Let M > M 0 (where M 0 is given by Claim 3.6). Let η such that

(3.15) 0 < η < 1 K 0 M 3 . Define (3.16) t n = t n (M, η) = inf t ≥ 0 : ∂ t,x h n (t) 2 ≥ M |β n (t)| or |β n (t)| ≥ η .
Then there exists ñ > 0 such that for n ≥ ñ,

∀t ∈ [τ 0 , t n ) , ω -|β n (t)| ≤ |β ′ n (t)| ≤ ω + |β n (t)| (3.17) ∀t ∈ [τ 0 , t n ) , 1 K 0 |β n (0)| e ω -t ≤ |β n (t)| ≤ K 0 |β n (0)| e ω + t (3.18) ∀t ∈ [0, t n ), ∂ t,x h n (t) 2 ≤ K 0 |β n (t)|. (3.19)
Before proving the lemma, we will show that it implies Proposition 3.4. For this we take M = 1 + max{M 0 , K 0 } and apply Lemma 3.7. Then by

(3.19), ∂ t,x h n (t) 2 < M |β n (t)| on [0, t n ] and thus t n (η, M ) = inf t ≥ 0 : |β n (t)| ≥ η = T n (η)
. This shows by (3.18) that T n (η) ∈ (0, +∞) for large n, and by continuity of β n , that β n (T n (η)) = η. In particular, T n (η) must tend to infinity; otherwise, as β n (0) tends to 0, the continuity of the flow would imply that u n (T n (η)) tends to W and β n (T n (η)) to 0, a contradiction. By (3.18), we get for large n, (η) .

1 K 0 |β n (0)| e ω -Tn(η) ≤ η ≤ K 0 |β n (0)| e ω + Tn
By the upper bound inequality we get (noticing that log |β n (0)| is negative for large time)

log |β n (0)| + log η ≤ log(K 0 ) + ω + T n (η).
Hence, using that β n (0) tends to 0, as n goes to infinity,

1 ω + ≤ lim inf n→+∞ T n (η) log |β n (0)| .
Letting ω + tends to ω we get

1 ω ≤ lim inf n→+∞ T n (η) log |β n (0)| .
By the same argument, we get

lim sup n→+∞ T n (η) log |β n (0)| ≤ 1 ω ,
which concludes the proof of (3.13).

To conclude the proof of Proposition 3.4 observe that (3.17) implies, for large n,

ω -η = ω -|β n (T n (η))| ≤ |β ′ n (T n (η))|, which yields (3.14).
In the remainder of this subsection we prove Lemma 3.7.

Proof of Lemma 3.7. In view of Claim 3.6, the fact that β n (0) tends to 0 and the continuity of β n and ∂ t,x h n 2 , the time t n is strictly positive. Furthermore,

∀n, ∀t ∈ (0, t n ), |β n (t)| ≤ η (3.20) ∀n, ∀t ∈ (0, t n ), ∂ t,x h n (t) 2 ≤ M |β n (t)| . (3.21)
Proof of (3.17).

Let

m = 1 2 min ω 2 -(ω -) 2 , (ω + ) 2 -ω 2 .
We first show that if η satisfies (3.15), then

(3.22) β n ′′ -ω 2 β n ≤ m |β n (t)| .
Differentiating twice the equality β n = h n Y, we get, by equation (3.1),

β n ′′ -ω 2 β n = ∂ 2 t h n Y -ω 2 h n Y = R(h n )Y - Lh n + ω 2 h n Y = R(h n )Y.
Thus there exists a constant C 1 , independent of all parameters, such that (3.23)

β n ′′ -ω 2 β n ≤ C 1 ∂ t,x h n 2 2
. By (3.20) and (3.21)

β n ′′ -ω 2 β n ≤ C 1 M 2 β 2 n ≤ C 1 M 2 η|β n |. which yields, if C 1 M 2 η ≤ m (which follows from (3.15) if K 0 is large enough), the desired estimate (3.22).
In what follows, we will assume that β n (0) ≥ 0 and β n ′ (0) ≥ 0 (otherwise, replace β n by -β n in the forthcoming argument). We next show that for t ∈ (0, t n ), (3.24)

β n ′′ (t) > 0, β n ′ (t) > 0, β n (t) > 0.
Indeed by (3.22),

(3.25) (ω 2 -m)β n (t) ≤ β n ′′ (t) ≤ (ω 2 + m)β n (t).
As β n (0) > 0 by Claim 3.6, we get that β, β ′ and β ′′ are (strictly) positive for small positive t. This shows that (3.24) holds near 0, and by an elementary monotonicity argument, that it holds for all t ∈ (0, t n ].

We are now ready to show (3.17). For this we write, as a consequence of (3.25)

β ′ n -ω -β n ′ = β ′′ n -ω -β ′ n ≥ -ω -β ′ n -ω -β n +(ω 2 -(ω -) 2 -m)β n ≥ -ω -β ′ n -ω -β n +mβ n . Hence (using that β n increases with time) d dt e ω -t β ′ n -ω -β n ≥ me ω -t β n (0).
Integrating between 0 and t we get

e ω -t β ′ n -ω -β n ≥ mβ n (0) t 0 e ω -s ds + β ′ n (0) -ω -β n (0) ≥ β n (0) m e ω -t -1 ω - -ω -.
Chosing τ 0 large enough we get a positive right hand side for t ≥ τ 0 , hence the left inequality in (3.17). The right inequality follows similarly by differentiating β ′ nω + β n and we omit the details of the proof.

Proof of (3.18). Assume as in the proof of (3.17) that β n (0) ≥ 0 and β ′ n (0) ≥ 0. By (3.25), and using that β n is positive on (0, T ),

∀t ∈ [0, t n ], β ′′ n (t) -ω + 2 β n (t) ≤ ω 2 + m -(ω + ) 2 β n (t) < 0.
This shows by a standard ODE argument that β n (t) ≤ βn (t), where βn (t) is the solution of the differential equation β′′ nω + 2 βn = 0 with initial conditions βn (0) = β n (0), β′

n (0) = β ′ n (0). Hence (3.26) ∀t ∈ [0, t n ], β n (t) ≤ β n (0) cosh(ω + t) + β ′ n (0) ω + sinh(ω + t). By (3.7), ∂ t h n (0) = β ′ n (0)Y + N j=0 γ ′ j,n (0)W j + g n (0), g n (0) ∈ G ⊥ .
Taking the L 2 -scalar product with Y and recalling that W j , j = 0 . . . N , and g n (0) are orthogonal to Y, we get

|β ′ n (0)| ≤ ∂ t h n (0) 2 .
Thus, in view of Claim 3.6, for large n:

β ′ n (0) ≤ (M 0 + 1)β n (0). By (3.26) (3.27) β n (τ 0 ) ≤ β n (0) cosh(ω + τ 0 ) + M 0 + 1 ω + β n (0) sinh(ω + τ 0 ) ≤ K 1 β n (0)
, for some constant K 1 depending only on the choice of ω + . By (3.17), ∀t ≥ τ 0 , e ω -(t-τ 0 ) β n (τ 0 ) ≤ β n (t) ≤ e ω + (t-τ 0 ) β n (τ 0 ). Using (3.27) for the upper bound and the fact that β n increases for the lower bound , we get ∀t ≥ τ 0 , e ω -(t-τ 0 ) β n (0) ≤ β n (t) ≤ K 1 e ω + (t-τ 0 ) β n (0), which yields (3.18).

Proof of (3.19).

We divide the proof into two steps.

Step 1. Estimates on the coefficients We first show that there exist a constant C 1 > 0, independent of the parameters M and η, such that for all t ∈ [0,

t n ] 1 C 1 |β n | -C 1 ∂ t,x h n 3/2 2 ≤ ∇g n 2 + ∂ t h n 2 + ε n ≤ C 1 β n + C 1 ∂ t,x h n 3/2 2 (3.28) 1 C 1 ∂ t,x h n 2 ≤ |β n | + N j=0 |γ j,n | ≤ C 1 ∂ t,x h n 2 . (3.29)
We have

E (W + h n , ∂ t h n ) = E(W, 0) -ε 2 n
Thus there exists a constant C 2 > 0 (independent of the parameters) such that

Q(h n ) + |∂ t h n | 2 + ε 2 n ≤ C 2 ∂ t,x h n (t) 3 2 .
Furthermore, by (3.7) (and the fact that the functions W j , j = 0 . . . N are in the kernel of Q)

Q(h n ) = -β n 2 |Q(Y)| + Q (g n ) .
Which yields

(3.30) -β n 2 |Q(Y)| + Q (g n ) + |∂ t h n | 2 + ε 2 n ≤ C 2 ∂ t,x h n (t) 3 2 . As g n ∈ G ⊥ , we have Q (g n ) ≈ ∇g n 2
2 , which yields (3.28). Let us show (3.29). Note that the upper bound follows immediately from the definitions of β n and γ j,n (see (3.8) and (3.9)). It remains to show the lower bound. We have

h n (t) = β n (t)Y + N j=0 γ j,n (t)W j + g n (t),
and hence, by (3.28)

∇h n 2 ≤ C   |β n | + N j=0 |γ j,n | + ∇g n 2   ≤ C   |β n | + N j=0 |γ j,n | + ∂ t,x h n 3 2 2   ∂ t,x h n 2 = ∇h n 2 + ∂ t h n 2 ≤ C   |β n | + N j=0 |γ j,n | + ∂ t,x h n 3 2 2   .
As a consequence of (3.20) and (3.21), we obtain

∂ t,x h n 2 =≤ C   |β n | + N j=0 |γ j,n |   + C ∂ t,x h n 2 M 1/2 η 1/2 .
by (3.15), we get the lower bound in (3.29)

Step 2. Bound on γ j,n .

We are now ready to show (3.19). According to (3.29), it is sufficient to show that there exists a constant C 3 independent of M and η ≤ 1

K 0 M 3 such that (3.31) ∀j ∈ {0, . . . , N }, ∀t ∈ [0, t n ] , |γ j,n (t)| ≤ C 3 |β n (t)| .
We have, for j = 0 . . . N .

γ ′ j,n (t) = ∇ ∂ t h n (t) -β n ′ (t)Y ∇W j .
Note that ∇W j ∇Y = 0 if j ≥ 1, but we won't need this fact in the sequel. The preceding inequality yields

(3.32) γ ′ j,n (t) ≤ C ∂ t h n (t) 2 + β n ′ (t) .
By (3.28) and assumptions (3.20) and (3.21),

∂ t h n 2 ≤ C 1 |β n | + ∂ t,x h n 3/2 2 ≤ C 1 |β n | 1 + η 1/2 M 3/2 .
Taking η small enough so that η 1/2 M 3/2 ≤ 1, we get

(3.33) ∂ t h n 2 ≤ 2C 1 |β n |.
By (3.32), taking a larger constant C,

(3.34) γ ′ j,n ≤ C |β n | + β ′ n .
Integrating between 0 and t ≤ τ 0 , and using that γ j,n (0) = 0, that |β n | increases and that the sign of

β ′ n (t) is independant of t ∈ [0, t n ] (see (3.24)), we obtain (3.35) ∀t ∈ [0, τ 0 ] , |γ j,n (t)| ≤ C(t + 1)|β n (t)|.
This yields (3.31) for t ≤ τ 0 . Now by (3.17) and (3.34), and using that the signs of β n and β ′ n do not depend on time,

∀t ≥ τ 0 , γ ′ j,n (t) ≤ C β n ′ (t) .
Integrating between τ 0 and t ∈ [τ 0 , t n ], we get

|γ j,n (t)| ≤ C (|β n (t)| + |γ j,n (τ 0 )|) .
Using (3.35) at t = τ 0 and the fact that |β n | increases, we get (3.31) for t ≥ τ 0 . The proof is complete.

Proof of main result

This section is devoted to the proof of Theorem 1. The proof is divided into 3 steps. In Step 1, we show the lower bound, in the next two steps the upper bound.

Step 1. Lower bound.

We must show (4.1) lim inf

ε→0 + I ε |log ε| ≥ 2 ω R N W 2(N+1) N-2 .
For this we first note that

∇W • ∇Y = -∆W Y = W N+2 N-2 Y > 0,
as Y and W are positive. Consider the family of solutions (u a ) a>0 of (1.1) with initial conditions

u a 0 = W -aY, u a 1 = 0. For small a > 0, |∇u a 0 | 2 = |∇W | 2 -2a ∇W • ∇Y + a 2 |∇Y| 2 < |∇W | 2 .
We have

(4.2) E (u a 0 , u a 1 ) = E(W, 0) + Q (-aY) + O a 3 = E(W, 0) -a 2 |Q (Y)| + O a 3 .
We argue by contradiction. If (4.1) does not hold, there exists a sequence ε n which tends to 0 such that for some ρ > ω

(4.3) ∀n, 2 ρ R N W 2(N+1) N-2 ≥ I εn |log ε n | .
By (4.2), and using that E (u a 0 , u a 1 ) is a continuous function of a, there exists a sequence a n such that

ε 2 n = E(W, 0) -E (u an 0 , u an 1 ) , Furthermore, (4.4) ε n ∼ a n |Q(Y)| as n → +∞.
Let u n = u an . Observe that

∂ t u n (0) = 0, ∇(u n (0) -W ) 2 = a n ∇Y 2 -→ n→+∞ 0.
Furthermore, β n (0) = -a n , β ′ n (0) = 0, which shows that the assumptions of Proposition 3.4 are satisfied. Consider a small η > 0. By Proposition 3.4

lim n→+∞ T n (η) | log a n | = lim n→+∞ T n (η) log |β n (0)| = 1 ω By (4.4), (4.5) lim n→+∞
T n (η) log |ε n | = 1 ω .
Let us give a lower bound for u n S(0,+∞) . From now on we will write T n instead of T n (η) for the sake of simplicity. As u n = W + h n , we have u n S(0,Tn) ≥ W S(0,Tn)h n S(0,Tn) .

Furthermore,

W S(0,Tn) = T N-2 2(N+1) n W 2(N+1) N-2 . Write h 2(N+1) N-2 S(0,Tn) = I∈E Tn h 2(N+1) N-2 S(I) ,
where E Tn is a set of at most Tn c + 1 subinterval of (0, T n ), of length at most c (given by Lemma 3.2) such that (0, T n ) = I∈E Tn I. By Lemma 3.2 and the fact that ∇h n 2 + ∂ t h n 2 ≤ M η on (0, T n ), we get for small η > 0,

h n 2(N+1) N-2 S(0,Tn) ≤ C T n c + 1 η 2(N+1) N-2 . Hence a constant C > 0 such that h n S(0,Tn) ≤ Cη T N-2 2(N+1) n .
Combining the preceding estimates, we obtain

Tn 0 R N |u n | 2(N+1) N-2 ≥ T n W 2(N+1) N-2 -Cη 2(N+1) N-2
.

Hence with (4.5),

lim inf n→+∞ 1 | log ε n | +∞ 0 R N |u n | 2(N+1) N-2 ≥ 1 ω W 2(N+1) N-2 -Cη 2(N+1) N-2
.

Letting η tends to 0 we obtain

lim inf n→+∞ 1 | log ε n | +∞ 0 |u n | 2(N+1) N-2 ≥ 1 ω W 2(N+1) N-2 2(N+1) N-2 .
Next, notice that as ∂ t u(0) = 0, the uniqueness in the Cauchy problem (1.1) implies u(t, x) = u(-t, x) and thus lim inf

n→+∞ 1 | log ε n | 0 -∞ |u n | 2(N+1) N-2 ≥ 1 ω W 2(N+1) N-2 2(N+1) N-2 . Finally, lim inf n→+∞ I εn | log ε n | ≥ lim inf n→+∞ 1 | log ε n | +∞ -∞ |u n | 2(N+1) N-2 ≥ 2 ω W 2(N+1) N-2 2(N+1) N-2 , contradicting (4.3).
Step 1 is complete.

Step 2. Estimate before the exit time.

We next show the upper bound on I ε , i.e that (4.6) lim sup

ε→0 + I ε |log ε| ≤ 2 ω R N W 2(N+1) N-2 .
For this we will show that if ε n > 0 is a sequence that goes to 0 and u n a sequence of solutions of (1.1) such that (4.7)

∇u n (0) 2 < ∇W 2 , E(W, 0) -E(u n , ∂ t u n ) = ε 2 n , then (4.8) lim sup n→+∞ 1 |log ε n | R×R N |u n | 2(N+1) N-2 ≤ 2 ω R N W 2(N+1) N-2 .
Possibly time-translating u n , we may assume (4.9)

u n S(-∞,0) = u n S(0,+∞) -→ n→+∞ +∞
By Proposition 2.2, rescaling and space-translating u n if necessary, we can assume

lim n→+∞ u n = W.
Consider the functions h n and g n , and the parameters β n and γ j,n defined in the beginning of §3.2. Replacing u n (x, t) by u n (x, -t) if it is not the case, we may assume (4.10)

β n (0)β n ′ (0) ≥ 0.
Furthermore, by Claim 3.3, we may also assume (3.12). Fix a small η > 0, and consider T n = T n (η) defined by Proposition 3.4. In this step, we show that there exists a constant C > 0 such that (4.11) lim sup

n→+∞ 1 | log ε n | Tn(η) 0 R N |u n | 2(N+1) N-2 ≤ 1 ω W 2(N+1) N-2 + Cη 2(N+1) N-2
. Indeed, by Claim 3.6, for large n,

ε n ≤ M 0 |β n (0)|.
Hence by Proposition 3.4, (4.12) lim sup

n→+∞ T n | log ε n | ≤ 1 ω .
By the same argument as in Step 1, we get

Tn 0 R N |u n | 2(N+1) N-2 ≤ T n W 2(N+1) N-2 + Cη 2(N+1) N-2 . Hence lim sup n→+∞ 1 T n Tn 0 R N |u n | 2(N+1) N-2 ≤ W 2(N+1) N-2 + Cη 2(N+1) N-2
.

Combining with (4.12), we obtain (4.11).

Step 3. Estimate for large time.

To conclude the proof, we will show that if η is small enough, there exists a constant C(η) > 0 such that for large n (4.13) u n S(Tn(η),+∞) ≤ C(η).

Assuming (4.13), we obtain by (4.11), lim sup

n→+∞ 1 | log ε n | u n 2(N+1) N-2 S(0,+∞) = lim sup n→+∞ 1 | log ε n | u n 2(N+1) N-2 S(0,Tn(η)) ≤ 1 ω W 2(N+1) N-2 + Cη 2(N+1) N-2
.

Letting η tend to 0 we get lim sup

n→+∞ 1 | log ε n | u n 2(N+1) N-2 S(0,+∞) ≤ 1 ω W 2(N+1) N-2 2(N+1) N-2
, which shows, in view of (4.9), the desired estimate (4.8).

It remains to show (4.13). We will argue by contradiction. If (4.13) does not hold, there exist a subsequence of (u n ), still denoted by (u n ) such that In view of (4.14) and (4.15), Proposition 2.2 (a) implies that there exists sequences λ n > 0,

x n ∈ R N , and δ 0 ∈ {-1, +1} such that (4.16) lim

n→+∞ δ 0 λ N/2 n ∇u n T n , • -x n λ n -∇W 2 + ∂u n ∂t (T n ) 2 = 0. By Proposition 3.4, lim inf n→+∞ |β ′ n (T n )| ≥ ωη.
By the decomposition (3.7) of h n ,

∂ t u n (T n )Y = ∂ t h n (T n )Y = β ′ n (T n ).
This shows by (4.16) that β ′ n (T n ) must tend to 0, yielding a contradiction. This concludes the proof of (4.13) and thus of Theorem 1.

Estimate of the scattering norm for energy-critical focusing NLS

In this section we briefly adress the case of the radial energy critical focusing semilinear Schrödinger equation (5.1)

i∂ t u + ∆u + |u| 4 N-2 u = 0, u ↾t=0 = u 0 ∈ Ḣ1 r , where N ∈ {3, 4, 5} and Ḣ1
r is the subset of R N of spherically symmetric functions. The equation (5.1) is locally well-posed (see [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF]) in the energy space Ḣ1 r . Furthermore, if I max ∋ 0 is the maximal interval of definition then

J ⋐ I max =⇒ u S(J) < ∞, where S(J) = L 2(N+2) N-2 ,
and globally defined solutions of (5.1) such that u S(R) is finite scatter (see [START_REF] Bourgain | Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case[END_REF][START_REF] Bourgain | Global solutions of nonlinear Schrödinger equations[END_REF]).

The energy

E(u(t)) = 1 2 |∇u(t)| 2 - N -2 2N |u(t)| 2N N-2
is conserved.

In the defocusing case, all solutions are known to be globally defined and scatter [START_REF] Bourgain | Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case[END_REF][START_REF] Tao | Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data[END_REF]. Furthermore, in [START_REF] Tao | Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data[END_REF], T. Tao gave a bound of u S(R) in term of an exponential of a power of the conserved defocusing energy 1

2 |∇u 0 | 2 + N -2 2N |u| 2N N-2 .
In the focusing case, the function W , defined in (1.2) is still a stationnary solution of W . The following theorem shown in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case[END_REF] for the case E(u 0 ) < E(W ) and in [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF] for the case E(u 0 ) = E(W ), is the analoguous of Theorem A for equation (5.1).

Theorem B (Kenig-Merle,Duyckaerts-Merle).

There exists a global solution W -of (5.1) such that

E W -= E(W ), ∇ W -(0) 2 < ∇W 2 W - S(-∞,0) < ∞, lim t→+∞ ∇ W -(t) -W 2 = 0.
Moreover, if u is a radial solution of (5.1) such that E(u 0 ) ≤ E(W ) and ∇u 0 2 ≤ ∇W 2 , then u is globally defined. If furthermore u S(R) = ∞, then u = W -or u = W up to the invariances of the equation.

Defining

Fε := u radial solution of (5.1) such that

E(u 0 ) ≤ E(W ) -ε 2 and |∇u 0 | 2 < |∇W | 2 .
we get in particular that for ε > 0 the supremum

I ε = sup u∈ Fε R×R N |u(t, x)| 2(N+2) N-2 dtdx = sup u∈ Fε u 2(N+2) N-2 S(R) ,
is finite, and that lim

ε→0 + I ε = +∞.
We wish again to estimate of I ε when ε goes to 0. As in the case of the wave equation, the behavior of I ε is determined by the linearized operator near W . If u = W + h is a solution of (5.1), then, identifying h with the column vector (Re h, Im h) T = (h 1 , h 2 ) T .

∂ t h + L(h) + R(h) = 0, L := 0 ∆ + W 4 N-2 -∆ -N +2 N -2 W 4 N-2 0 ,
where an appropriate norm of R(h) is bounded by ∇h 2 2 when h is small. It is known (see [DM07a, Section 7.1]) that the essential spectrum of L is iR and that L admits only two nonzero real eigenvalues, ω > 0 andω, with eigenfunctions Y ± which are in the space of Schwartz functions. Then:

Theorem 2. lim ε→0 + I ε | log ε| = 2 ω R N W 2(N+2) N-2 .
Our result is restricted to the radial case in spatial dimensions N ∈ {3, 4, 5}. In view of the recent work [START_REF] Killip | The focusing energy-critical nonlinear schrödinger equation in dimensions five and higher[END_REF] on non-radial energy-critical focusing NLS in dimension N ≥ 5, it is natural to expect that the same estimate holds in a more general situation.

The proof of Theorem 2 is very similar to the one of Theorem 1, and we will only sketch it, highlighting the minor differences. In §5.1 we recall a few facts about the operator L and state without proof the analoguous of Propositions 2.2, Propositions 3.4 and Claim 3.6. In §5.2 we briefly explain how to use these results to show Theorem 2. We next recall some spectral properties of the operator L. We refer to [DM07a, §5.1] for the details. We will often identify a complex-valued function f with an R 2 -valued function (f 1 , f 2 ) T , with f 1 = Re f , f 2 = Im f . Developping the energy around W , we get, for small functions h ∈ Ḣ1 ,

E(W + h) = E(W ) + Q(h) + O h 3 2N N-2
, where Q is the quadratic form Q(h) = B(h, h) and B is defined by

B(g, h) = 1 2 ∇g 1 • ∇h 1 - N + 2 2(N -2) g 1 h 1 W 4 N-2 + 1 2 ∇g 2 • ∇h 2 - 1 2 g 2 h 2 W 4 N-2 .
5.2. Sketch of the proof of Theorem 2.

Step 1. Lower bound. We first show (5.9) lim inf

ε→0 + I ε |log ε| ≥ 2 ω R N W 2(N+2) N-2 .
Multiplying Y + and Y -by -1 if necessary, we may assume Re ∇W • ∇Y ± > 0. Consider the family of solutions (u a ) a>0 of (5.1) with initial conditions u a 0 = W -aY + -aY -. Then for small a > 0, |∇u a 0 | 2 < |∇W | 2 . Furthermore (5.10) E (u a 0 ) = E(W ) -2a 2 + O a 3 . We argue by contradiction. If (5.9) does not hold, there exists a sequence ε n which tends to 0 such that for some ρ > ω

(5.11) ∀n, 2 ρ R N W 2(N+2) N-2 ≥ I εn |log ε n | .
We then chose a sequence a n such that (5.12) Writing u n = W + h n , and arguing as in Step 1 of Section 4, we obtain

ε 2 n = E(W ) -E (u an 0 ) , ε 2 n ∼ 2a
T + n (η) 0 R N |u n | 2(N+2) N-2 ≥ T + n (η) W 2(N+2) N-2 -Cη 2(N+2) N-2
.

Hence with (5.13), and letting η tends to 0, lim inf

n→+∞ 1 | log ε n | +∞ 0 |u n | 2(N+2) N-2 ≥ 1 ω W 2(N+2) N-2 2(N+2) N-2 .
Arguing similarly for negative time, we obtain lim inf n→+∞

I εn | log ε n | ≥ lim inf n→+∞ 1 | log ε n | +∞ -∞ |u n | 2(N+2) N-2 ≥ 2 ω W 2(N+2) N-2 2(N+2) N-2
, contradicting (5.11). Step 1 is complete.

Step 2. Upper bound.

To show the upper bound on I ε , we must show that for any sequence ε n > 0 that goes to 0 any sequence u n of solutions of (5.1) such that (5.14)

∇u n (0) 2 < ∇W 2 , E(W ) -E(u n ) = ε 2 n ,
The condition g n (t) ∈ G ⊥ implies that β - n (t) = -B u n (t) -W, Y + . Thus (5.8) contradicts (5.19). This shows u n S(T + n (η),+∞) ≤ C(η). By a similar argument for negative time, we get (5.17). Combining (5.16) and (5.17), we obtain (5.15), which concludes the sketch of the proof of Theorem 2.

  Assume lim n→+∞ u n S(-∞,tn) = lim n→+∞ u n S(tn,+∞) = +∞.

  u n S(-∞,Tn) ≥ u n S(-∞,0) = u n S(0,+∞) -→ n→∞ +∞.

5. 1 .

 1 Convergence to W and estimate on the exit time. In view of Theorem B, and the use of the profile decomposition method in [KM06a, Section 4] (see also [DM07a, Lemma 2.5]), the proof of Proposition 5.1 adapts easily to show: Proposition 5.1. Let u n be a family of radial solutions of (5.1), such that(5.2) E u n (0) < E(W ), ∇u n (0) 2 < ∇W 2 .and lim n→+∞ u n S(R) = +∞. Let (t n ) n be a time sequence. Assumelim n→+∞ u n S(-∞,tn) = lim n→+∞ u n S(tn,+∞) = +∞.Then, up to the extraction of a subsequence, there exist θ 0 ∈ R and a sequence of parameters λ n > 0 such that lim n→+∞

  2 n as n → +∞. Let u n = u an . Then the assumptions of Proposition 5.2 are satisfied. Consider a small η > 0. By Proposition 5.2, noting that β + n (0) = β - n (0) = -a n , we get lim n→+∞

This work was partially supported by the French ANR Grant ONDNONLIN.

Denote by Y + the eigenfunction of L for the eigenvalue ω and Y -= mY + the eigenfunction of L for the eigenvalueω (m = 0 is a real normalization constant), and recall the definition of W 0 in (3.2). One may show that W 0 and iW are in the kernel of Q. Furthermore, Q(Y + ) = Q(Y -) = 0 and we may chose m such that B(Y + , Y -) = -1.

By [DM07a, Lemma 5.2], there exists a constant c > 0 such that ∀h ∈ G⊥ , Q(h) ≥ c ∇h 2 2 . We consider as in §3.2 a sequence u n of radial solutions of (5.1) such that

Arguing as in Claim 3.3, we may assume

Then we have the following analog of Propositions 3.4 and Claim 3.6. We skip the proofs, that are very similar to the previous ones.

Proposition 5.2. There exist a constant η 0 , such that for all η ∈ (0, η 0 ), for all sequence (u n ) satisfying (5.3), (5.4) and (5.6) if

(5.7)

Furthermore,

Observe that in contrast with the wave equation case, there are two eigenfunctions, and that we have distinguished between the coefficient β - n of Y -, which tends to grow for positive times, and the one of Y + , which plays a similar role for negative times.

Claim 5.3. There exists M 0 > 0 such that for all sequence (u n ) of solutions of (5.1) satisfying (5.3), (5.4) and (5.6) we have

we have

In view of Proposition 5.1 and the analoguous of Claim 3.3, we may assume that u n satisfy the assumptions of Proposition 5.2. Fix a small η > 0, and consider T ± n (η) defined by Proposition 5.2. Then by the same proof than in Step 2 of Section 4, one can show that there exists a constant C > 0 such that lim sup

.

By Claim 5.3, for large n,

It remains to show, as in Step 3 of Section 4, that if η is small enough, there exists a constant C(η) > 0 such that for large n (5.17) u n S(-∞,-T - n (η)) + u n S(T + n (η),+∞) ≤ C(η). Combining (5.16) and (5.17) and letting η tends to 0 we would get (5.15).

To show (5.17), we argue by contradiction. Assume that there exists a subsequence of (u n ), such that (from now on, we will write

Then by Proposition 5.1, there exists θ 0 ∈ R and a sequence λ n > 0, such that

As in Step 3 of Section 4, we will get a contradiction by showing that dβ - n dt (T + n (η)) tends to 0. Unlike the case of the wave equation, the convergence to 0 of the time derivative of u is not given directly by the compactness argument of Proposition 5.1. However, (5.18) and the fact that u n is a solution of (5.1) which is in C 0 (R, Ḣ1 ) shows that