Thomas Duyckaerts

Frank Merle

DYNAMIC OF THRESHOLD SOLUTIONS FOR ENERGY-CRITICAL WAVE EQUATION

We consider the energy-critical non-linear focusing wave equation in dimension N = 3, 4, 5. An explicit stationnary solution, W , of this equation is known. In [KM06b], the energy E(W, 0) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u0, u1) = E(W, 0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to [DM07], which treats the energy-critical non-linear focusing radial Schrödinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.

). An explicit solution of (1.1) is the stationnary solution in Ḣ1 (but in L 2 only if N ≥ 5) (1.2)

.

Introduction and main results

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}, and Ḣ1 := Ḣ1 (R N ). The theory of the Cauchy problem for (1.1) was developped in many papers (see [Pec84, GSV92, LS95, SS94, SS98, Sog95, Kap94]). Namely, if (u 0 , u 1 ) ∈ Ḣ1 × L 2 , there exists an unique solution defined on a maximal interval I = (-T -(u), T + (u)) and the energy

The works of Aubin and Talenti [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF], give the following elliptic characterization of W (throughout the paper we denote by

• p the L p norm on R N ) ∀u ∈ Ḣ1 , u 2 * ≤ C N ∇u 2 (1.3) u 2 * = C N ∇u 2 =⇒ ∃ λ 0 > 0, x 0 ∈ R N , δ 0 ∈ {-1, +1} u(x) = δ 0 λ (N -2)/2 0 W x + x 0 λ 0 , (1.4)
where C N is the best Sobolev constant in dimension N .

The dynamical behavior of some solutions of (1.1) was recently described in [START_REF] Schlag | On the focusing critical semi-linear wave equation[END_REF], [START_REF] Schlag | Slow blow-up solutions for the h 1 (R 3 ) critical focusing semi-linear wave equation[END_REF] (in the radial three-dimensional case) and [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]. In [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF], Kenig and Merle has shown the important role of W , whose energy E(W, 0) = 1 N C N N is an energy threshold for the dynamics in the following sense. Let u be a solution of (1.1), not necessarily radial, such that (1.5) E(u 0 , u 1 ) < E(W, 0).

Then

• if ∇u 0 2 < ∇W 2 , we have I = R and u

L 2(N+1) N-2 t,x
< ∞, which implies from the linear theory of (1.1) that the solution scatters;

• if ∇u 0 2 2 > ∇W 2
2 then T + < ∞ and T -< ∞. Our goal (as is [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF] for the nonlinear Schrödinger equation in the radial case) is to give a classification of solutions of (1.1), not necessarily radial, with critical energy, that is with initial condition (u 0 , u 1 ) ∈ Ḣ1 × L 2 such that E(u 0 , u 1 ) = E(W, 0).

The stationnary solution W belongs to this energy level, is globally defined and does not scatter. Another example of special solutions is given by the following.

Theorem 1 (Connecting orbits). Let N ∈ {3, 4, 5}. There exist radial solutions W -and W + of (1.1), with initial conditions W ± 0 , W ± 1 ∈ Ḣ1 × L 2 such that E(W, 0) = E(W + 0 , W - 1 ) = E(W - 0 , W - 1 ), (1.6)

T + (W -) = T + (W + ) = +∞ and lim t→+∞ W ± (t) = W in Ḣ1 , (1.7) ∇W - 2 < ∇W 2 , T -(W -) = +∞, W - L 2(N+1) N-2 ((-∞,0)×R N ) < ∞, (1.8) ∇W + 2 > ∇W 2 , T -(W + ) < +∞. (1.9)
Remark 1.1. Our construction gives a precise asymptotic development of W ± near t = +∞. Indeed there exists an eigenvalue e 0 > 0 of the linearized operator near W , such that, if Y ∈ S(R N ) is the corresponding eigenfunction with the appropriate normalization, (1.10) ∇ W ± (t) -W ± e -e 0 t Y L 2 + ∂ t W ± (t) -W ± e -e 0 t Y L 2 ≤ Ce -2e 0 t .

We refer to (6.1) and (6.9) for the development at all orders in e -e 0 t .

Remark 1.2. Similar solutions were constructed for NLS in [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF]. However, in the NLS case, we were not able to prove that T -(W + ) < ∞ except in the case N = 5. We see this fact, in particular in the case N = 3, as a nontrivial result. Note that W + is not in L 2 except for N = 5, so that case (c) of Theorem 2 below does not apply.

Our next result is that W , W -and W + are, up to the symmetry of the equation, the only examples of new behavior at the critical level.

Theorem 2 (Dynamical classification at the critical level). Let N ∈ {3, 4, 5}. Let (u 0 , u 1 ) ∈ Ḣ1 × L 2 such that

(1.11) E(u 0 , u 1 ) = E(W, 0) = 1 N C N N .
Let u be the solution of (1.1) with initial conditions (u 0 , u 1 ) and I its maximal interval of definition. Then the following holds: The constant C N is defined in (1.3). In the theorem, by u equals v up to the symmetry of the equation, we mean that there exists t 0 ∈ R, x 0 ∈ R N , λ 0 > 0, δ 0 , δ 1 ∈ {-1, +1} such that

u(t, x) = δ 0 λ (N -2)/2 0 v t 0 + δ 1 t λ 0 , x + x 0 λ 0 .
Remark 1.3. Case (b) is a direct consequence of the variational characterization of W given by (1.4). Furthermore, using assumption (1.11), it shows (by continuity of u in Ḣ1 and the conservation of energy) that the assumptions |∇u(t

0 )| 2 < |∇W | 2 , |∇u(t 0 )| 2 > |∇W | 2
do not depend on the choice of the initial time t 0 . Of course, this dichotomy does not persist when E(u 0 , u 1 ) > E(W, 0).

Remark 1.4. Theorem 2 is also the analoguous, for the wave equation, of Theorem 2 of [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF] for NLS, but without any radial assumption. The nonradial situation carries various problems partially solved in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF], the major difficulty being a sharp control in time of the space localization of the energy. We conjecture that the NLS result also holds in the nonradial situation. Note that case (a) implies (W being radial) that any solution of (1.1) satisfying (1.11) and whose initial condition is not radial up to a space-translation must scatter if

|∇u 0 | 2 < |∇W | 2 .
Remark 1.5. In dimension N = 3 or N = 4, W + is not in L 2 , and case (c) means that any critical-energy solution such that |∇u 0 | 2 > |∇W | 2 and u 0 ∈ L 2 blows-up for t < 0 and t > 0.

It seems a delicate problem to get rid of the assumption u 0 ∈ L 2 .

Remark 1.6. As a corollary, in dimension N = 5, a dynamical characterization of W is obtained. It is, up to the symmetry of the equation, the only L 2 -solution such that E(u 0 , u 1 ) ≤ E(W, 0) that does not explode and does not scatter neither for positive nor negative time.

The paper is organized as follows. In Section 2 we recall previous results on the Cauchy Problem for (1.1) and give preliminary properties of solutions of (1.1) at the energy threshold such that |∇u 0 | 2 < |∇W | 2 and which do not scatter for positive times. These properties mainly follow from [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]. In Section 3, we show that these solutions converge exponentially to W as t → +∞, which is the first step of the proof of Theorem 2 in case (a). In Section 4, we show the same result for energy-threshold solutions such that |∇u 0 | 2 > |∇W | 2 , u 0 ∈ L 2 and that are globally defined for positive time. In Section 5, we study the linearized equation around the solution W . Both theorems are proven in Section 6. The main tool of the proofs is a fixed point giving the existence of the special solutions and, by the uniqueness property, the rigidity result in Theorem 2.

Preliminaries of subcritical threshold solutions

2.1. Quick review on the Cauchy problem. We recall some results on the Cauchy Problem for (1.1). We refer to [KM06b, Section 2], for a complete overview. If I is an interval, write

S(I) := L 2(N+1) N-2 (I × R N ), N (I) := L 2(N+1) N+3 (I × R N ) (2.1) u ℓ(I) := u S(I) + D 1/2 x u L 2(N+1) N-1 (I×R N ) + ∂ t D -1/2 x u L 2(N+1) N-1 (I×R N ) . (2.2)
We first consider the free wave equation:

∂ 2 t u -∆u = f, t ∈ (0, T ) (2.3) u ↾t=0 = u 0 , ∂ t u ↾t=0 = u 1 , (2.4) where D 1/2 x f ∈ N (0, T ), u 0 ∈ Ḣ1 , u 1 ∈ L 2 . The solution of (2.3), (2.4) is given by u(t, x) = cos t √ -∆ u 0 + sin t √ -∆ √ -∆ u 1 + t 0 sin (t -s) √ -∆ √ -∆ f (s)ds.
Then we have the following Strichartz estimates (see [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF][START_REF] Lindblad | On existence and scattering with minimal regularity for semilinear wave equations[END_REF]).

Proposition 2.1. Let u and f be as above. Then u ∈ C 0 (0, T ; Ḣ1 ) and ∂ t u ∈ C 0 (0, T ; L 2 ). Furthermore, for a constant C > 0 independent of T ∈ [0, ∞]

(2.5)

u ℓ(0,T ) + sup t∈[0,T ] ∇u(t) 2 + ∂ t u(t) 2 ≤ C ∇u 0 2 + u 1 2 + D 1/2 x f N (0,T ) . Furthermore, if D 1/2 x f ∈ N (T, +∞), (2.6) ∀t ≥ 0, ∇ +∞ T sin (t -s) √ -∆ √ -∆ f (s)ds 2 + ∂ t +∞ T sin (t -s) √ -∆ √ -∆ f (s)ds 2 ≤ C D 1/2 x f N (T,+∞) .
A solution of (1.1) on an interval I ∋ 0 is a function u ∈ C 0 (I, Ḣ1 ) such that ∂ t u ∈ C 0 (I, Ḣ1 ) and u ∈ S(J) for all interval J ⋐ I and

u(t, x) = cos t √ -∆ u 0 + sin t √ -∆ √ -∆ u 1 + t 0 sin (t -s) √ -∆ √ -∆ |u(s)| 4 N-2 u(s)ds. Proposition 2.2. (see [Pec84, GSV92, SS94]) (a) Existence. If u 0 ∈ Ḣ1 , u 1 ∈ L 2
, there exists an interval I ∋ 0 and a solution u of (1.1) on I with initial conditions (u 0 , u 1 ).

(b) Uniqueness. If u and ũ are solutions of (1.1) on an interval I ∋ 0 such that u(0) = ũ(0) and ∂ t u(0) = ∂ t ũ(0), then u = ũ on I.

According to Proposition 2.2, if (u 0 , u 1 ) ∈ Ḣ1 × L 2 , there exists a maximal open interval of definition for the solution u of (1.1), that we will denote by (-T -(u), T + (u)). The following holds Proposition 2.3.

(a) Finite blow-up criterion.

If T + := T + (u) < ∞ then u S(0,T + ) = ∞.
A similar result holds for negative times. (b) Continuity. Let u be a solution of (1.1) on an interval I with initial condition

(u 0 , u 1 ) ∈ Ḣ1 × L 2 . If (u k
) is a sequence of solution of (1.1) with initial conditions

(u k 0 , u k 1 ) -→ k→+∞ (u 0 , u 1 ) in Ḣ1 × L 2
and J ⋐ (-T -, T + ), then for large k, J ⊂ -T -u k , T + u k , and

(u k , ∂ t u k ) -→ k→+∞ (u, ∂ t u) in C 0 (J, Ḣ1 ) × C 0 (J, L 2 ), u k -→ k→+∞ u in S(J). (c) Scattering. If u is a solution of (1.1) such that T + (u) = ∞ and u S(0,∞) < ∞, then u scatters. (d) Finite speed of propagation. (see [KM06b, Lemma 2.17]) There exist ε 0 , C 0 > 0, de- pending only on ∇u 0 2 and u 1 2 , such that if there exist M, ε > 0 satisfying ε < ε 0 and |x|≥M |∇ x u 0 | + 1 |x| 2 |u 0 | 2 + |u 0 | 2 * + |u 1 | 2 ≤ ε, then, ∀t ∈ [0, T + (u)), |x|≥ 3 2 M +t |∇u(t, x)| 2 + 1 |x| 2 |u(t, x)| 2 + |u| 2 * + |∂ t u(t, x)| 2 dx ≤ C 0 ε.

2.2.

Properties of subcritical threshold solutions. We are now interested in solutions of (1.1) with maximal interval of definition (T -, T + ) and such that

E(u 0 , u 1 ) = E(W, 0), ∇u 0 2 < ∇W 2 (2.7) u S(0,T + ) = ∞. (2.8)
We start with the following claim (see [KM06b, Theorem 3.5]).

Claim 2.4 (Energy Trapping). Let u be a solution of (1.1) satisfying (2.7). Then for all t in the interval of existence (-T -, T + ) of u.

(2.9)

∇u(t) 2 2 + N 2 ∂ t u 2 2 ≤ ∇W 2 2 . Proof.
Recall the following property which follows from a convexity argument

(2.10) ∀v ∈ Ḣ1 , ∇v 2 2 ≤ ∇W 2 2 and E(v, 0) ≤ E(W, 0) =⇒ ∇v 2 2 ∇W 2 2 ≤ E(v, 0) E(W, 0) .
(See [DM07, Claim 2.6]). Let u be as in the claim. Note that by remark 1.3 ∇u(t) 2 < ∇W 2 for all t in the domain of existence of u. Now, according to (2.10) and the fact that

E(u(t), ∂ t u(t)) = E(W, 0), ∇u(t) 2 2 ∇W 2 2 ≤ E(u, ∂ t u) -1 2 ∂ t u(t) 2 2 E(W, 0) = E(W, 0) -1 2 ∂ t u(t) 2 2 E(W, 0) ,
which yields (2.9), recalling that N E(W, 0) = ∇W 2 2 . We recall now some key results from [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]. In their work, these results are shown for a critical element u, where assumptions (2.7) are replaced by E(u 0 , u 1 ) < E(W, 0), and ∇u 0 2 < ∇W 2 . Rather than recalling the proofs which are long and far from being trivial, we will briefly explain how they adapt in our case.

If (f, g) is in Ḣ1 × L 2 , we write (f, g) λ 0 ,x 0 (y) =   1 λ N 2 -1 0 f y λ 0 + x 0 , 1 λ N 2 0 g y λ 0 + x 0   , f λ 0 ,x 0 (y) = 1 λ N 2 -1 0 f y λ 0 + x 0 .
Lemma 2.5. Let u be a solution of (1.1) satistisfying (2.7) and (2.8). Then there exist continuous functions of t, (λ(t), x(t)) such that

K := u(t), ∂ t u(t) λ(t),x(t) , t ∈ [0, T + ) has compact closure in Ḣ1 × L 2 .
The proof of Lemma 2.5, which corresponds to Proposition 4.2 in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF], is very close to the proof of Proposition 4.1 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] and of Proposition 2.1 in [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF]. The two main ingredients are the fact, proven in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF] that a solution of (1.1) such that E(u 0 , u 1 ) < E(W, 0) and ∇u 0 2 < ∇W 2 is globally defined and scatters, a Lemma of concentration-compactness for solution to the linear wave equation due to Bahouri and Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF], and variational estimates as in Claim 2.4.

Proposition 2.6 ([KM06b]

). Let u be a solution of (1.1) satisfying (2.7) and (2.8). Assume that there exist functions (λ(t), x(t)) such that

K := u(t), ∂ t u(t) λ(t),x(t) , t ∈ [0, T + )
has compact closure in . . . × L 2 and that one of the following holds (a) T + < ∞, or (b) T + = +∞ and there exists λ 0 > 0 such that ∀t ∈ [0, +∞), λ(t) ≥ λ 0 .

Then u 1 ∇u 0 = 0.

Proof. If inf t∈(-T -,T + ) ∂ t u(t) 2 2 = 0, then, using that ∇u(t) 2 is bounded, and that ∂ t u∇u(t) is conserved, we get immediately that u 1 ∇u 0 = 0. Thus we may assume (2.11)

∃δ 0 > 0, ∀t ∈ (-T -, T + ), ∇u(t) 2 2 ≤ ∇W 2 2 -δ 0 .
In this case, the proof is the same as in [KM06b, Propositions 4.10 and 4.11] which is shown under assumption (2.8) and

(2.12)

∇u 0 2 < ∇W 2 , E(u 0 , u 1 ) < E(W, 0).
This implies by variational estimates (2.11), which is what is really needed in the proof of the proposition.

Proposition 2.7 ( [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]). Let u be a solution of (1.1) satisfying (2.7) and (2.8). Assume

(2.13) u 1 ∇u 0 = 0.

Then T + = ∞.
This result is proven in [KM06b, Section 6] under the assumptions (2.8) and (2.12), but assumption (2.12) is only used to show that ∇u(t) 2 is bounded, which is, in our case, a consequence of (2.7) (see [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]Remark 6.14]).

As a consequence of Proposition 2.6 and 2.7 we have, following again [KM06b]:

Proposition 2.8. Let u be a solution of (1.1) satisfying (2.7) and (2.8). Let λ(t), x(t) given by Lemma 2.5. Then

(a) T + = ∞. (b) lim t→+∞ tλ(t) = +∞. (c) R N u 1 ∇u 0 = 0. (d) lim t→+∞ x(t) t = 0.
Corollary 2.9. Let u be a solution of (1.1) with maximal interval of definition (-T -, T + ) and such that E(u 0 , u 1 ) ≤ E(W, 0) and ∇u 0 2 ≤ ∇W 2 . Then

T + = T -= +∞.
Proof of Corollary 2.9. It is a consequence of (a). Indeed, if ∇u 0 2 = ∇W 2 , then by Claim 2.4, u 1 = 0. Furthemore by (2.10), E(W, 0) = E(u 0 , 0) = E(u 0 , u 1 ). Thus u 0 2 * = W 2 * , and, by the characterization (1.4) of W , u 0 = ±W λ 0 ,x 0 for some parameters λ 0 , x 0 . By uniqueness in (1.1), u is one of the stationnary solutions ±W λ 0 ,x 0 , which are globally defined. Let us assume now ∇u 0 2 < ∇W 2 . Then if E(u 0 , u 1 ) < E(W, 0), we are in the setting of [KM06b, Theorem 1.1], which asserts than T + = T -= +∞. On the other hand, if E(u 0 , u 1 ) = E(W, 0), then if u S(0,T + ) < ∞, we know by the finite blow-up criterion of Proposition 2.3, that T + = ∞, and if u S(0,T + ) = ∞, then by (a), T + = ∞. The same argument for negative times shows that T -= ∞.

Proof of Proposition 2.8. Proof of (a). Let u be as in Lemma 2.5. By Proposition 2.6, if T + < ∞, then u 1 ∇u 0 = 0, but then by Proposition 2.7, T + = ∞ which is a contradiction. Thus T + = ∞, which shows (a).

Proof of (b). Assume that (b) does not hold. Then there exists a sequence t n → +∞ such that

(2.14) lim n→+∞ t n λ(t n ) = τ 0 ∈ [0, +∞).
Consider

w n (s, y) = 1 λ(t n ) N-2 2 u t n + s λ(t n ) , x(t n ) + y λ(t n ) . (2.15) w n0 (y) = w n (0, y), w n1 (y) = ∂ s w n (0, y).
(2.16) By the compactnees of K, (w n0 , w n1 ) n converges (up to the extraction of a subsequence) in Ḣ1 × L 2 . Let (w 0 , w 1 ) be its limit, and w be the solution of (1.1) with initial condition (w 0 , w 1 ). Note that E(w 0 , w 1 ) = E(W, 0) and ∇w 0 2 ≤ ∇W 2 . Thus by Corollary 2.9

T -(w 0 , w 1 ) = ∞.
Furthermore, as -t n λ(t n ) → -τ 0 , and by the continuity of the Cauchy Problem for (1.1) ((b) in Proposition 2.3), 1

λ(t n ) N-2 2 u 0 x(t n ) + y λ(t n ) = w n (-t n λ(t n ), y) -→ n→∞ w(-τ 0 , y) in Ḣ1 1 λ(t n ) N 2 u 1 x(t n ) + y λ(t n ) = ∂ s w n (-t n λ(t n ), y) -→ n→∞ ∂ s w(-τ 0 ) in L 2 .
Since λ(t n ) tends to 0, we obtain that w(-τ 0 ) = 0 and ∂ s w(-τ 0 ) = 0, which contradicts the equality E(w 0 , w 1 ) = E(W, 0). The proof of (b) is complete.

Proof of (c). According to (a), T + = ∞.

λ n (s) N-2 2 w n s, x n (s) + y λ n (s) = 1 λ(t) N-2 2 u t, x(t) + y λ(t) , (2.22) v n1 (s, y) := 1 λ n (s) N 2 (∂ s w n ) s, x n (s) + y λ n (s) = 1 λ(t) N 2 (∂ t u) t, x(t) + y λ(t) . (2.23)
which shows that (v n0 (s), v n1 (s)) ∈ K. In view of (2.19) and the continuity property (b) in Proposition 2.3, (w n (s), ∂ s w n (s)) converges in Ḣ1 × L 2 . By the compactness of K and (2.21) this shows that there exists λ(s) ∈ [1, +∞), x(s) ∈ R N such that for some subsequences,

lim n→+∞ λ n (s)) = λ(s), lim n→+∞ x n (s)) = x(s)
and 1

λ(s) N-2 2 w s, x(s) + • λ(s) , 1 λ(s) N 2 (∂ s w) s, x(s) + • λ(s) ∈ K.
Thus w fullfills all the assumptions of Proposition 2.6, case (b), which shows that w 1 ∇w 0 = 0. By (2.19) and the conservation of ∂ t u(t)∇u(t)

u 1 ∇u 0 = ∂ t u(t n )∇u(t n ) = w n1 ∇w n0 -→ n→∞ w 1 ∇w 0 = 0.
The proof of (c) is complete.

Proof of (d).

We follow the lines of the proof of Lemma 5.5 in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF]. Assume that (d) does not hold, and consider

t n → +∞ such that |x(t n )| t n ≥ ε 0 > 0.
In particular, x(t) is not bounded. We may assume that x(0) = 0, λ(0) = 1 and that x and λ are continuous. For R > 0, let

t 0 (R) := inf t ≥ 0, |x(t)| ≥ R ∈ [0, +∞). Thus t 0 (R) is well defined, t 0 (R) > 0, |x(t)| < R for 0 ≤ t < t 0 (R) and |x(t 0 (R))| = R. As a consequence, if R n := |x(t n )|, then t n ≥ t 0 (R n ), which implies (2.24) R n t 0 (R n ) ≥ ε 0 . Let (2.25) e(u) := 1 2 |∂ t u| 2 + 1 2 |∇u| 2 - 1 2 * |u| 2 * , r(u) := |∂ t u| 2 + |∇u| 2 + 1 |x| 2 |u| 2 + |u| 2 * .
By compactness of K, we know that for each ε > 0, there exists R 0 (ε) such that (2.26) ∀t ≥ 0,

λ(t)|x-x(t)|≥R 0 (ε) r(u)dx ≤ ε.
Let ϕ ∈ C ∞ 0 (R N ) be radial, nonincreasing and such that ϕ(x) = 1 for |x| ≤ 1 and ϕ(x

) = 0 if |x| ≥ 2. Let ψ R (x) := xϕ( x R
). Let ε > 0, to be chosen later independently of n and

R n := R 0 (ε) inf t∈[0,t 0 (Rn)] λ(t) + |x(t n )| = R 0 (ε) inf t∈[0,t 0 (Rn)] λ(t) + R n z Rn (t) := R N ψ Rn (x)e(u)(t, x)dx. Note that (2.27) 0 ≤ t ≤ t 0 (R n ), |x| ≥ R n =⇒ λ(t)|x -x(t)| ≥ λ(t)( R n -R n ) ≥ R 0 (ε).
Step 1. Bound on z ′ Rn (t). Let us show

(2.28)

∃C 1 > 0, ∀n, 0 ≤ t ≤ t 0 (R n ) =⇒ |z ′ Rn (t)| ≤ C 1 ε.
Indeed by explicit calculation and equation (1.1) (see [KM06b, Lemma 5.3]), we get recalling that by (c) and the conservation of the moment ∂ t u(t)∇u(t) = u 1 ∇u 0 = 0

(2.29) z ′ Rn (t) = R N ∂ t u∇u + O |x|≥ Rn r(u)dx = O |x|≥ Rn r(u)dx
Estimate (2.28) then follows from (2.26) and (2.27)

Step 2. Bounds on z Rn (0) and z Rn (t 0 (R n )). We next show

|z Rn (0)| ≤ 2ε R n + M R 0 (ε) (2.30) R n E(W ) -ε -2 R n ε -M R 0 (ε) λ(t 0 (R n )) ≤ |z Rn (t 0 (R n ))|, (2.31)
where M := sup t≥0 r(u)(t, x)dx ≤ C ∇W 2 2 by Claim 2.4. We have.

|z Rn (0)| = |x|≥R 0 (ε)
ψ Rn e(u)dx

+ |x|≤R 0 (ε)
ψ Rn e(u)dx.

Recall that |ψ

λ(t)|x-x(t)|≤R 0 (ε) xe(u)dx = λ(t)|x-x(t)|≤R 0 (ε) x(t)e(u)dx + λ(t)|x-x(t)|≤R 0 (ε) (x -x(t))e(u)dx.
The second term in the right-hand side of (2.33) is bounded by R 0 (ε)

λ(t)
r(u)(t, x)dx. On the other hand

λ(t)|x-x(t)|≤R 0 (ε) x(t)e(u)dx = x(t)E(W ) - λ(t)|x-x(t)|≥R 0 (ε)
x(t)e(u)dx and thus by (2.26) (2.34)

λ(t)|x-x(t)|≤R 0 (ε) x(t)e(u)dx ≥ |x(t)|(E(W ) -ε).
Combining (2.33) and (2.34) with t = t 0 (R n ) we get (2.31).

Step 3. Conclusion of the proof of (d). According to the two precedent steps

C 1 εt 0 (R n ) ≥ R n (E(W ) -ε) -4 R n ε - 2M R 0 (ε) λ(t 0 (R n )) C 1 ε ≥ R n t 0 (R n ) (E(W ) -ε) -4 R n t 0 (R n ) ε -M R 0 (ε) t 0 (R n )λ(t 0 (R n )) . (2.35) As a consequence of (b) t 0 (R n )λ(t 0 (R n )) → 0 as n → ∞. Furthermore Rn t 0 (Rn) = Rn t 0 (Rn) + R 0 (ε) t 0 (Rn) inf t∈[0,t 0 (Rn)] λ(t) . Again by (b), t 0 (R n ) inf t∈[0,t 0 (Rn)] λ(t) tends to +∞. Thus R n t 0 (R n ) = R n t 0 (R n ) + o(1), n → ∞.
Together with (2.24) and (2.35), we get

C 1 ε ≥ R n t 0 (R n ) (E(W ) -5ε) + o(1) ≥ ε 0 (E(W ) -5ε) + o(1), n → ∞.
Chosing ε small enough, so that

C 1 ε ≤ ε 0 4 E(W ) and E(W )-5ε > 1 2 E(W )
we get a contradiction.

Convergence to W in the subcritical case

The aim of this section is to prove the following result in the subcritical situation ( ∇u 0 2 < ∇W 2 ), which is the nonradial version of the result of [DM07, Proposition 3.1] in the NLS radial setting. The main difficulty here compared to [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF] and [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation[END_REF] is to control the space localization of the energy (see §3.3).

Proposition 3.1. Let u be a solution of (1.1) such that (2.7) and (2.8) hold. Then there exist λ 0 , x 0 such that

(3.1) ∇(u(t) -W λ 0 ,x 0 ) 2 + ∂ t u 2 ≤ Ce -ct .
Furthermore,

(3.2) u S(-∞,0) < ∞.
Remark 3.2. From Corollary 2.9, (2.7) implies that u is defined on R.

3.1.

Convergence for a subsequence. Let

(3.3) d(t) := |∇u(t, x)| 2 dx -|∇W (x)| 2 dx + |∂ t u(t, x)| 2 dx.
Then the equality

E(u(t), ∂ t u(t)) = E(W, 0) implies u 2 * 2 * -W 2 * 2 * ≤ Cd(t).
It is known (see [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case[END_REF]) that the variational characterization (1.4) of W by Aubin and Talenti implies that there exists a function ε 0 (δ) such that ε 0 (δ) → 0 as δ → 0 and, for any fixed t

(3.4) inf µ,X,± ∇ u µ,X (t) ± W 2 ≤ ε 0 d(t) .
The key point of the proof of Proposition 3.1 is to show that d(t) tends to 0, which by (3.4) implies that there exists (λ(t), x(t)) t≥0 such that u λ(t),x(t) (t) -W tends to 0 in Ḣ1 as t tends to +∞. We first show:

Lemma 3.3. Let u be a solution of (1.1) such that (2.7) and (2.8) hold. Then

(3.5) lim T →+∞ 1 T T 0 d(t)dt → t→+∞ 0.
Corollary 3.4. There exists an increasing sequence τ n → +∞ such that

lim n→+∞ d(τ n ) = 0. Proof. Let ϕ be a C ∞ function such that ϕ(x) = 1 if |x| ≤ 1 and ϕ(x) = 0 if |x| ≥ 2. For R > 0, write (3.6) ϕ R (x) = ϕ(x/R) and ψ R (x) = xϕ(x/R). Let ε > 0. Consider as in [KM06b, §5] (3.7) g R (t) := ψ R .∇ x u∂ t udx + ( N -1 2 ) ϕ R u∂ t udx.
Step 1. Bound on g R (t). We first show

(3.8) ∃C 1 > 0, ∀t ≥ 0, |g R (t)| ≤ C 1 R. Indeed, note that supp ϕ R ∪ supp ψ R ⊂ {|x| ≤ 2R}, so that |ψ R (x)| ≤ 2R and |ϕ R (x)| ≤ 2R |x| . Hence |g R (t)| ≤ 2R |∇ x u∂ t u|dx + N 2 2R |x| |u∂ t u|dx,
which yields (3.8) by Hardy's inequality and Claim 2.4.

Step 2. Bound on g ′ R (t). There exists C 2 > 0, c > 0, such that for all ε > 0, there exists

t 1 = t 1 (ε) > 0 such that (3.9) ∀t ∈ [t 1 , T ], g ′ εT (t) ≤ -cd(t) + C 2 ε.
By explicit computation and the equality E(u 0 , u 1 ) = E(W, 0) (see Claim C.1 in the appendix)

g ′ εT (t) = 1 N -2 |∂ t u| 2 dx - 1 N -2 |∇W | 2 dx -|∇u| 2 dx + O |x|≥εT r(u)dx ,
where r(u) is defined in (2.25). Note that by Claim 2.4 an elementary calculation

(3.10) - 1 N -2 |∂ t u| 2 dx + 1 N -2 |∇W | 2 dx -|∇u| 2 dx ≥ 1 N + 2 |∂ t u| 2 dx + |∇W | 2 dx -|∇u| 2 dx .
Thus there exists C 2 > 0 such that

(3.11) g ′ R (t) ≤ - 1 N + 2 d(t) + C 2 |x|≥εT r(u).
By Proposition 2.8, tλ(t) → +∞ and |x(t)|/t → 0. Thus we may chose

t 1 = t 1 (ε) such that t ≥ t 1 =⇒ |x(t)| ≤ ε 2 t and |λ(t)| ≥ 2R 0 (ε) εt , where R 0 (ε) is defined in (2.26) Then for t 1 ≤ t ≤ T , |x| ≥ εT =⇒ λ(t)(|x| -|x(t)|) ≥ 2R 0 (ε) εT εT - εT 2 ≥ R 0 (ε),
which yields, together with (2.26) and (3.11), our expected estimate (3.9).

Step 3. End of the proof. Integrating (3.9) between t 1 and T , one gets

g εT (T ) -g εT (t 1 ) = T t 1 g ′ εT (t)dt ≤ -c T t 1 d(t)dt + C 2 (T -t 1 )ε.
and thus, by

(3.8), c T T t 1 d(t) ≤ 2C 1 ε + C 2 ε, hence lim sup T →+∞ 1 T T 0 d(t)dt ≤ 2C 1 + C 2 c ε
which yields the result.

3.2. Modulation of solutions. We will now precise, using modulation theory, the dynamics of solutions of (1.1) near W . We will only suppose

(3.12) E(u 0 , u 1 ) = E(W, 0),
without any further assumption on the size of ∇u 0 2 . We have the following development of the energy near W :

(3.13)

E(W + f, g) = E(W, 0) + Q(f ) + 1 2 g 2 2 + O ∇f 3 2 , f ∈ Ḣ1 , g ∈ L 2
where Q is the quadratic form on Ḣ1 defined by

(3.14) Q(f ) := 1 2 |∇f | 2 - N + 2 2(N -2) W 4 N-2 f 2 .
Let us specify an important coercivity property of Q. Consider the orthogonal directions W , W , W j , j = 1 . . . N in the real Hilbert space Ḣ1 = Ḣ1 (R N , C) where W and W j are defined by

W = c ∂ ∂µ (W µ,X ) ↾(µ,X)=(1,0) = -c N -2 2 W + x • ∇W W j = c j ∂ ∂X j (W µ,X ) ↾(µ,X)=(1,0) = c j ∂ x j W.
(3.15)

and the constants c, c 1 , . . . , c N are chosen so that

(3.16) ∇ W 2 = ∇W 1 2 = . . . ∇W N 2 = 1.
We have

(3.17) Q(W ) = - 2 (N -2)C N N , Q ↾span{ W ,W 1 ,...,Wn} = 0,
where C N is the best Sobolev constant in dimension N . The first assertion follows from direct computation and the fact that

W 2 * 2 * = ∇W 2 2 = 1 C N N
. From (3.13), (3.15), and the invariance of E by all transformations f → f µ,X , we get that

Q( W ) = Q(W 1 ) = . . . = Q(W N ) = 0.
Furthermore, it is easy to check that W , W 1 , . . . , W N are Q-orthogonal, which gives the second assertion.

Let H := span{W, W , W 1 , . . . , W N } and H ⊥ its orthogonal subspace in the real Hilbert space Ḣ1 . The quadratic form Q is nonpositive on H. By the following claim, Q is positive definite on H ⊥ (see [START_REF] Rey | The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent[END_REF]Appendix D] for the proof).

Claim 3.5. There is a constant c > 0 such that for all radial function f in

H ⊥ Q( f ) ≥ c ∇ f 2
2 . Now, let u be a solution of (1.1) satisfying (3.12) and define d(t) as in (3.3). We would like to specify (3.4). We start by chosing µ and X.

Claim 3.6. There exists δ 0 > 0 such that for all solution u of (1.1) satisfying (3.12), and for all t in the interval of existence of u such that

(3.18) d(t) ≤ δ 0 , there exists (µ(t), X(t)) ∈ (0, +∞) × R N such that u µ,X ∈ { W , W 1 , . . . , W N } ⊥ .
We omit the proof of Claim 3.6, which follows from (3.4) and the implicit function Theorem. We refer for example to [DM07, Claim 3.5] for a similar proof.

If a and b are positive, we write a ≈ b when C -1 a ≤ b ≤ Ca with a positive constant C independent of all parameters of the problem. Now, consider u satisfying (3.12) and assume that on an open subset J of its interval of definition, u also satisfies (3.18).

According to the preceding claim, there exist (µ(t), X(t)) such that ∀t ∈ J, u µ,X (t) ∈ { W , W 1 , . . . , W N } ⊥ .

We will prove the following lemma, which is a consequence of Claim 3.5, in Appendix A.

Lemma 3.7 (Estimates on the modulation parameters). Let u, µ, X be as above. Changing u into -u if necessary, write

u µ,X (t) = (1 + α(t))W + f (t), 1 + α := 1 ∇W 2 2 ∇W • ∇u µ,X dx, f ∈ H ⊥ . Then |α| ≈ ∇(αW + f ) ≈ ∇ f 2 + ∂ t u 2 ≈ d(t) (3.19) α ′ µ + µ ′ µ 2 + X ′ (t) ≤ Cd(t). (3.20)
3.3. Exponential convergence to W . Using Subsections 3.1 and 3.2, we are now ready to prove Proposition 3.1

Let u be as in the proposition. By Lemma 2.5, we may assume that There exist functions λ(t), x(t) continuous on [0, +∞) such that

K := u(t), ∂ t u(t) λ(t),x(t) , t ∈ [0, +∞) is relatively compact in Ḣ1 × L 2 . (3.21)
Let µ(t) and X(t) be the modulation parameters of Subsection 3.2, defined for d(t) ≤ δ 0 . It is easy to see that the compactness of K implies that the set

K 1 := u(t), ∂ t u(t) [µ(t),X(t)] , t ∈ [0, +∞), d(t) < δ 0 has compact closure in Ḣ1 × L 2
. By an elementary construction, one can find continuous functions λ(t) and x(t) of t ∈ [0, +∞) such that λ(t), x(t) = (X(t), µ(t)) if d(t) ≤ δ 0 . The set K defined as in (3.21) has compact closure in Ḣ1 × L 2 . We will still denote by x(t) and λ(t) the new parameters that satisfy, in addition to (3.21),

(3.22) d(t) < δ 0 =⇒ λ(t) = µ(t), x(t) = X(t).
The proof of Proposition 3.1 relies on the two following Lemmas.

Lemma 3.8 (Virial type estimates on d(t)). Let u be a solution of (1.1) satisfying (2.7), (3.21), and (3.22). Then there is a constant C > 0 such that

0 ≤ σ < τ =⇒ τ σ d(t)dt ≤ C sup σ≤t≤τ |x(t)| + 1 λ(t) d(σ) + d(τ ) .
Remark 3.9. We will also need the following variant of Lemma 3.8, whose proof is exactly the same: if u satisfies the assumptions of Lemma 3.8 for all t ∈ R, then there is a constant C > 0

-∞ < σ < τ < +∞ =⇒ τ σ d(t)dt ≤ C sup σ≤t≤τ |x(t)| + 1 λ(t) d(σ) + d(τ ) .
Lemma 3.10 (Parameters control). Let u be a solution of (1.1) fullfilling the assumptions of Lemma 3.8. Then there exists a constant C > 0 such that

0 ≤ σ and σ + 1 λ(σ) ≤ τ =⇒ |x(τ ) -x(σ)| + 1 λ(τ ) - 1 λ(σ) ≤ C τ σ d(t)dt.
Remark 3.11. The technical assumption σ + 1 λ(σ) < τ is needed because of the infinite choice of parameters x(t) and λ(t) when d(t) > δ 0 .

Let us first assume Lemma 3.8 and 3.10 to show Proposition 3.1.

Proof of Proposition 3.1. Step 1. Let us show that, (replacing u by u(• -x ∞ ) for some x ∞ ∈ R N if necessary), there exist c, C > 0 and λ ∞ ∈ (0, ∞) (3.23) ∞ t d(s)ds + |λ(t) -λ ∞ | + |x(t)| ≤ Ce -ct .
We first show that x(t) and 1 λ(t) are bounded. According to Lemmas 3.8 and 3.10, there exists a constant C 0 > 0 such that for all 0 ≤ σ ≤ s < t ≤ τ with s + 1 λ(s) < t, we have

(3.24) |x(s) -x(t)| + 1 λ(s) - 1 λ(t) ≤ C 0 sup σ≤r≤τ |x(r)| + 1 λ(r) (d(σ) + d(τ )).
Now consider the increasing sequence τ n → +∞, given by Corollary 3.4, and chose n 0 such that

(3.25) n ≥ n 0 ⇒ d(τ n ) ≤ 1 4C 0
Using (3.24) with σ = s = τ n 0 , and τ = τ n for some large n we get, in view of (3.25)

τ n 0 + 1 λ(τ n 0 ) < t =⇒ |x(τ n 0 ) -x(t)| + 1 λ(τ n 0 ) - 1 λ(t) ≤ 1 2 sup τn 0 ≤r |x(r)| + 1 λ(r)
.

Thus sup

τn 0 + 1 λ(τn 0 ) ≤t |x(t)| + 1 λ(t) ≤ 1 2 sup τn 0 ≤t |x(t)| + 1 λ(t) + |x(τ n 0 )| + 1 λ(τ n 0 )
,

which
, if σ + 1 λ(σ) < τ , |x(σ) -x(τ )| + 1 λ(σ) -1 λ(τ ) ≤ Ce -cσ . Thus there exist x ∞ ∈ R N , ℓ ∞ ∈ [0, +∞) such that |x(t) -x ∞ | + 1 λ(t) -ℓ ≤ Ce -ct .
Translating u, we will assume x ∞ = 0. It remains to show that ℓ ∞ > 0. Assume that ℓ ∞ = 0. Let 0 ≤ σ ≤ s. Let τ n be the sequence such that d(τ n ) → 0. Then, by (3.24), if n is large enough (so that s

+ 1 λ(s) < τ n ), |x(s) -x(τ n )| + 1 λ(s) - 1 λ(τ n ) ≤ C 0 sup σ≤r≤τn |x(r)| + 1 λ(r) (d(s) + d(τ n )).
Letting n tends to infinity, we get, by the assumptions ℓ ∞ = 0, and

x ∞ = 0 0 ≤ σ ≤ s =⇒ |x(s)| + 1 λ(s) ≤ C 0 sup σ≤r |x(r)| + 1 λ(r) d(σ).
Taking the supremum in s in the preceding inequality, we get, if

σ = τ n sup τn≤s |x(s)| + 1 λ(s) ≤ C 0 sup τn≤s |x(s)| + 1 λ(s) d(τ n ).
Recalling that d(τ n ) tends to 0, we get a contradiction, showing that ℓ ∞ > 0. The proof of (3.23) is now complete.

Step 2. Proof of (3.1). Let us first show by contradiction

(3.27) lim t→+∞ d(t) = 0.
Indeed, if it does not hold, there exists a subsequence of (τ n ) n (that we still denote by (τ n ) n ), and a sequence (τ n ) n such that

τ n < τn , ∀t ∈ [τ n , τn ), d(t) < δ 0 /2, d(τ n ) = δ 0 /2.
On [τ n , τn ] the parameters α(t), µ(t) and X(t) of Lemma 3.7 are well-defined. By (3.23) and Lemma 3.7.

(3.28)

|α(τ n ) -α(τ n )| ≤ τn τn α ′ (t) µ(t) dt ≤ C τn τn d(t)dt ≤ Ce -τn .
By Lemma 3.7, α(t) ≈ d(t). As d(τ n ) → 0 and d(τ n ) = δ 0 /2, this contradicts (3.28), showing (3.27).

In view of (3.27), there exists T > 0 such that for t ≥ T , d(t) < δ 0 , so that u is close to ±W for t ≥ T . By continuity of u, the sign before W does not change for large t. Changing u into -u if necessary, we can make it a +. Write as in Lemma 3.7

(3.29) u µ,X (t) = (1 + α(t))W + f (t).
Integrating

+ ∂ t u 2 d(t) ≈ |α(t)|. Thus (3.30) ∀t ≥ T, |α(t)| + |µ(t) -λ ∞ | + |X(t)| + ∇ f 2 + ∂ t u(t) 2 ≤ Ce -ct .
This implies (3.1) in view of (3.29).

Step 3. Proof of (3.2). Assume, in addition to the assumption of Proposition 3.1, that we have

(3.31) u S(-∞,0) = +∞.
By Lemma 2.5 there exist λ(t) and x(t), defined for t ∈ R such that

K := u(t), ∂ t u(t) λ(t),x(t) , t ∈ R has compact closure in Ḣ1 × L 2 .
As a consequence of the preceding steps, applied to u(t, x) and u(-t, x), we get that d(t) tends to 0 as t goes to +∞ and -∞, and that 1 λ(t) and x(t) are bounded independently of t ∈ R. By Remark 3.9,

(3.32) σ < τ ⇒ τ σ d(t)dt ≤ C (d(σ) + d(τ )) .
Letting σ go to -∞ and τ to +∞, we get that d(t) = 0 for all t. Thus u = W up to the invariance of the equation, which contradicts the assumption ∇u 0 2 < ∇W 2 .

Proof of Lemma 3.8. Let R > 0 to be chosen later and g R the function defined by (3.7)

Step 1. Bound on g R . Let us show that there is a constant C 0 independent of t ≥ 0 such that

(3.33) |g R (t)| ≤ C 0 Rd(t).
Indeed by the explicit expression of g R , the fact that ψ R ≤ 2R and ϕ R ≤ 2R/|x| and Hardy's inequality we get

|g R (t)| ≤ CR ∂ t u(t) 2 ∇u(t) 2 ≤ CR ∂ t u(t) 2 . By Lemma 3.7 ∂ t u 2 ≤ Cd(t) for t such that d(t) ≤ δ 0 . As ∂ t u 2 is bounded by 2E(W ) (Claim 2.4
), this bounds is valid for any t, which concludes the proof of (3.33).

Step 2. Bound on g ′ R . In this step we show that there exist ρ 0 > 0, c > 0, independent of σ and τ such that if for some t ∈ [σ, τ ],

(3.34) R ≥ ρ 0 1 λ(t) + |x(t)| , then (3.35) g ′ R (t) ≤ -cd(t).
Indeed by Claim C.1 in the appendix,

g ′ R (t) = 1 N -2 |∂ t u| 2 dx - 1 N -2 |∇W | 2 dx -|∇u| 2 dx + A R (u, ∂ t u),
where A R is defined in (C.1). We first claim the following bounds on A R (u, ∂ t u):

∀ε > 0, ∃ρ ε > 0, ∀t ≥ 0, R ≥ 2|x(t)| + ρ ε λ(t) =⇒ |A R (u, ∂ t u)| ≤ ε. (3.36) ∃C 1 > 0, ∀ρ ≥ 1, ∀t ≥ 0, R ≥ 2|x(t)| + 2ρ λ(t) and d(t) < δ 0 (3.37) =⇒ |A R (u(t), ∂ t u(t))| ≤ C 1 1 ρ N-2 2 d(t) + d(t) 2 .
By (C.1), there exists C 2 > 0 such that

(3.38) A R (u, ∂ t u) ≤ C 2 |x|≥R r(u)dx, where r(u) is defined in (2.25). Let ρ ε := 2R 0 (ε/C 2 ), where R 0 is defined in (2.26). Assume that R ≥ 2|x(t)| + ρ ε λ(t) . Then |x| ≥ R =⇒ |x -x(t)| ≥ R -|x(t)| ≥ R 2 ≥ R 0 (ε/C 2 ) λ(t) .
By (3.38) and the definition of R 0 , we get (3.36).

Let us show (3.37). Let t such that d(t) < δ 0 , where δ 0 is the parameter given by §3.2. Recall that by (3.22), λ(t) = µ(t) and X(t) = x(t).

For any λ 0 , x 0 , we know that W λ 0 ,x 0 is a solution of (1.1) independent of t, so that g R (t) = 0, and g ′ R (t) = 0, which shows by Claim C.

1 that A R (W λ 0 ,x 0 , 0) = 0. Thus A R (u, ∂ t u) = A R (u, ∂ t u) -A R W 1 µ ,-X , 0 .
By the change of variable x = X + y µ we get

(3.39) a jk R (x)∂ j u(x)∂ k u(x)dx -a jk R ∂ ∂ x j W 1 µ ,-X (x) ∂ ∂ x k W 1 µ ,-X (x)dx = a jk R X + y µ ∂ j (W + f )∂ k (W + f )dy -a jk R X + y µ ∂ j W ∂ k W dy = a jk R X + y µ (∂ j W ∂ k f + ∂ j f ∂ k W ) dy + a jk R X + y µ ∂ j f ∂ k f dy.
where f = u µ,X -W , is such that ∇f (t) 2 ≤ C 0 d(t) by Lemma 3.7. Now, a similar calculation on the other terms of A R (u,

∂ t u) -A R (W 1 µ ,-X ) yields the bound (3.40) |A R (u, ∂ t u)| = A R (u, ∂ t u) -A R W 1 µ ,-X , 0 ≤ C X+ y µ ≥R |∇f | 2 + |∇W • ∇f | + W 2 * -1 |f | + |f | 2 * + 1 µ 2 X + y µ 2 W |f | + |f | 2 dy.
Let us bound the terms of the right-hand side of (3.40) that are linear in f Recall that

µ(t) = λ(t) and X(t) = x(t). Using that R ≥ 2|x(t)| + 2ρ µ(t) , we get, if X + y µ ≥ R |y| µ ≥ 2 ρ µ =⇒ |y| ≥ 2ρ and X + y µ ≥ y µ -X ≥ y µ - R 2 ≥ y µ - ρ µ ≥ 1 2 y µ .
Thus, recalling that W (y) ≈ |y| 2-N for large |y|.

X+ y µ ≥R |∇W • ∇f |dy ≤ |y|≥2ρ |∇W | 2 1/2 ∇f 2 ≤ C ρ N-2 2 d(t) X+ y µ ≥R W 2 * -1 |f |dy ≤ |y|≥2ρ W 2 * N+2 2N f 2 * ≤ C ρ N+2 2 d(t) X+ y µ ≥R W |f | µ 2 X + y µ 2 dy ≤ X+ y µ ≥R 4 W |f | y 2 ≤ C |y|≥2ρ 1 |y| 2 |W | 2 1/2 ∇f 2 ≤ C ρ N-2 2 d(t).
By (3.40), we get (3.37).

We are now ready to show (3.35). Note that by Claim C.1, we have, for a small constant c > 0,

(3.41) g ′ R (t) ≤ -cd(t) + |A R (u, ∂ t u)|. Chose δ 1 := min δ 0 , c 4C 1 and ρ 1 > 1 such that C 1 ρ N-2 2 1 ≤ c
4 where C 1 is the constant in (3.37). By (3.37)

d(t) < δ 1 and R ≥ 2 |x(t)| + ρ 1 λ(t) =⇒ |A R (u, ∂ t u)| ≤ c 2 d(t).
According to (3.36) with ε := c 2 δ 1 ,

d(t) ≥ δ 1 and R ≥ 2|x(t)| + ρ ε λ(t) =⇒ |A R (u, ∂ t u)| ≤ c 2 δ 1 ≤ c 2 d(t).
In view of (3.41), we get (3.35) under the assumption (3.34) for ρ 0 := max(2ρ 1 , ρ ε , 2).

Step 2 is complete.

Step 3. End of the proof. Take

R := 2ρ 0 sup σ≤t≤τ 1 λ(t) + |x(t)| ,
where ρ 0 is given by Step 2. Then by (3.35)

∀t ∈ [σ, τ ], cd(t) ≤ -g ′ R (t)
Integrating between σ and τ , we get, in view of (3.33)

c τ σ d(t)dt ≤ |g R (σ)| + |g R (τ )| ≤ C 0 R(d(σ) + d(τ )),
which yields the conclusion of Lemma 3.8.

Proof of Lemma 3.10.

Step 1. Bounds by compactness on a short time interval. We show that there exists C 1 > 0 such that

(3.42) ∀τ, σ ≥ 0, |τ -σ| ≤ 1 λ(τ ) =⇒ λ(τ )|x(τ ) -x(σ)| + λ(τ ) λ(σ) + λ(σ) λ(τ ) ≤ C 1 .
If not, we may find sequences τ n , σ n ≥ 0 such that

(3.43) |τ n -σ n | ≤ 1 λ(τ n ) , λ(τ n )|x(τ n ) -x(σ n )| + λ(τ n ) λ(σ n ) + λ(σ n ) λ(τ n ) -→ n→+∞ +∞.
Extracting subsequences, we may assume

(3.44) lim n→+∞ λ(τ n )(σ n -τ n ) = s 0 ∈ [-1, 1].
Consider the solution of (1.1)

v n (s, y) := 1 (λ(τ n )) N-2 2 u s λ(τ n ) + τ n , y λ(τ n ) + x(τ n ) .
By compactness of K, extracting subsequences if necessary, v n , ∂vn ∂s ↾s=0 has a limit (v 0 , v 1 ) in Ḣ1 × L 2 . Let v be the solution of (1.1) with initial condition (v 0 , v 1 ), which is globally defined according to Corollary 2.9. By Proposition 2.3 (b),

w n (y) := v n (λ(τ n )(σ n -τ n ), y) = 1 (λ(τ n )) N-2 2 u σ n , y λ(τ n ) + x(τ n ) -→ n→+∞ v(s 0 , y) in Ḣ1 .
Furthermore the compactness of K implies that the following sequence stays inside a compact set of Ḣ1 .

1

(λ(σ n )) N-2 2 u σ n , y λ(σ n ) + x(σ n ) = λ(τ n ) λ(σ n ) N-2 2 w n λ(τ n ) λ(σ n ) y + λ(τ n )(x(σ n ) -x(τ n )) .
Thus λ(τn) λ(σn) , λ(τn) λ(σn) and λ(τ n )(x(τ n ) -x(σ n )) must be bounded, contradicting (3.43).

Step 2. Control of the variations of d. Let δ 0 > 0 be given by Subsection 3.2. Let us show (3.45) ∃δ 1 > 0, ∀τ ≥ 0, sup

τ ≤t≤τ + 1 λ(τ ) d(t) > δ 0 =⇒ inf τ ≤t≤τ + 1 λ(τ ) d(t) > δ 1 .
Indeed, assume that it does not hold, so that (extracting if necessary), we may find sequences

(τ n ) n , (t n ) n , (t ′ n ) n , such that (3.46) t n , t ′ n ∈ τ n , τ n + 1 λ(τ n ) , d(t n ) → 0 and d(t ′ n ) > δ 0 . Let v n (s, y) := 1 (λ(t n )) N-2 2 u s λ(t n ) + t n , y λ(t n ) + x(t n ) .
By the compactness of K, and the fact that d(t n ) tends to 0, we may assume that (v n (0), ∂ s v n (0)) tends to some W λ 0 ,x 0 .

By Step 1, 1 λ(tn) ≤ C 1 λ(τn) , thus λ(t n )(t n -t ′ n ) is bounded. Extracting if necessary, we may assume lim n λ(t n )(t n -t ′ n ) = s 0 ∈ [-1, 1]. By Proposition 2.3 (b), (3.47) lim n→∞ v n λ(t n )(t n -t ′ n ) = ±W λ 0 ,x 0 in Ḣ1 . Furthermore, v n (λ(t n )(t n -t ′ n )) = 1 λ(t n ) N-2 2 u t ′ n , y λ(t n ) + x(t n ) . Thus by (3.46), ∇W 2 2 - ∇v n 2 2 > δ 0 , which contradicts (3.47).
Step 2 is complete.

Step 3. End of the proof We first show that , there exists C > 0 such that

(3.48) 0 ≤ σ ≤ σ ≤ τ ≤ τ = σ + 1 C 1 λ(σ) =⇒ |x(τ ) -x(σ)| + 1 λ(τ ) - 1 λ(σ) ≤ C τ σ d(
|x(σ) -x(τ )| + 1 λ(σ) - 1 λ(τ ) = τ σ x ′ (t)dt + τ σ λ ′ (t) λ 2 (t) dt ≤ C τ σ d(t)
(σ) -x(τ )| ≤ C 1 λ(σ) and 1 λ(σ) -1 λ(τ ) = 1 λ(σ) 1 -λ(σ) λ(τ ) ≤ 2C 1 λ(σ) . |x(σ) -x(τ )| + 1 λ(σ) - 1 λ(τ ) ≤ 3C 1 λ(σ) ≤ 3C 2 1 λ(σ) = 3C 2 1 |σ -τ | ≤ 3C 2 1 δ 1 τ σ d(t)dt.
The proof of (3.48) is complete.

It is straightforward to deduce the conclusion of Lemma 3.10 from (3.48) , dividing the interval [σ, τ ] into small subintervals, and we omit the details.

Supercritical case for L 2 solutions

In this section we study a solution u of (1.1) such that

u 0 ∈ L 2 (4.1) E(u 0 , u 1 ) = E(W, 0), ∇u 0 2 > ∇W 2 (4.2) T + (u) = +∞. (4.3)
Our main result is the following. Proposition 4.1. Let u be a solution of (1.1) with N ∈ {3, 4, 5} satisfying (4.1), (4.2) and (4.3). Then N = 5 and changing u into -u if necessary, there exist c, C > 0 and λ 0 , x 0 such that

(4.4) ∀t ≥ 0, ∇u(t) -∇W λ 0 ,x 0 2 + ∂ t u(t) 2 ≤ Ce -ct .
Remark 4.2. In dimension N = 3 or N = 4, Proposition 4.1 asserts than any solution of (1.1) satisfying (4.1) and (4.2) must blow-up in finite time for positive and negative time. We are not able to prove (4.4). Nevertheless, one can show the weaker property lim

T →+∞ 1 T T 0 d(t)dt = 0. Let y(t) := R N (u(t)) 2 dx,
and define d(t) by (3.3). We first prove the following. Proof of Corollary 4.4. Indeed by (4.5), y ′ (t) < 0. But if T -(u) = +∞, (4.5) applied to the solution u(-t, x) of (1.1) (which also satisfies the assumptions of Proposition 4.1) shows that y ′ (t) > 0, which is a contradiction.

Proof of Lemma 4.3. By direct calculation (and using equation (1.1) and assumption (4.2) to compute y ′′ )

y ′ (t) = 2 R N u(t)∂ t u(t)dx (4.8) y ′′ (t) = 2 (∂ t u(t)) 2 -|∇u(t)| 2 + |u(t)| 2 * (4.9) = 4 N -1 N -2 (∂ t u(t)) 2 + 4 N -2 |∇u(t)| 2 -|∇W | 2 ≥ d(t).
Furthermore, by Cauchy-Schwarz inequality, (4.10)

y ′ (t) 2 ≤ 4 (u(t)) 2 (∂ t u(t)) 2 ≤ N -2 N -1 y(t)y ′′ (t).
Proof of (4.5).We argue by contradiction. Note that by Remark 1.3, assumption (4.2) implies that ∇u(t) 2 > ∇W 2 for all t. By (4.9), y ′′ (t) > 0 for any t ≥ 0. Assume that for some t 0 , y ′ (t 0 ) ≥ 0.

(4.11) ∀t > t 0 , y ′ (t) > 0.

Hence by (4.10), N -1

N -2 y ′
y ≤ y ′′ y ′ , which yields by integration

∀t ≥ t 0 + 1, y ′ (t) y ′ (t 0 + 1) ≥ y(t) y(t 0 + 1) N-1 N-2
, which leads to blow-up in finite time from the fact that N -1 N -2 > 1, contradicting (4.3).

Proof of (4.6). The function y is positive and, by (4.5), decreasing. Thus (4.12) lim t→+∞ y(t) = y ∞ ∈ [0, +∞).

We

u(t) -u ∞ 2 = 0. Indeed we have, if 0 ≤ t 1 < t 2 , (4.17) |u(t 1 , x) -u(t 2 , x)| 2 = t 2 t 1 ∂ t u(t, x)dt 2 ≤ (t 2 -t 1 ) |∂ t u(t, x)| 2 dt.
Integrating (4.17) in space, we get by (4.7)

u(t 1 ) -u(t 2 ) 2 2 ≤ |t 1 -t 2 | t 2 t 1 ∂ t u(t) 2 2 dt ≤ C|t 1 -t 2 | t 2 t 1 d(t)dt ≤ C|t 1 -t 2 |e -ct 1 .
By an elementary summation argument, we obtain, taking a larger constant C, the bound u(t 1 ) -u(t 2 ) 2 2 ≤ Ce -ct1 for t 1 < t 2 . Thus u satisfies the Cauchy criterion in L 2 as t → +∞, which yields (4.16).

Step 2. End of the proof By (4.7), there exists a sequence t n → ∞ such that d(t n ) tends to 0. Thus, extracting a subsequence and changing u into -u if necessary, there exists λ 0 , x 0 such that u(t n ) tends to W λ 0 ,x 0 in Ḣ1 , thus in D ′ (R N ). In view (4.16), u(t n ) tends also to

u ∞ in D ′ (R n ). Thus W λ 0 ,x 0 = u ∞ ∈ L 2 .
(4.20) |α ′ (t)| + |µ ′ (t)| + |X ′ (t)| ≤ Cd(t).
In view of (4.7), this shows as in the end of the proof of Proposition 3.1 that d(t), α(t), µ(t) and X(t) converges exponentially when t → +∞, which implies (4.4). The proof of Proposition 4.1 is complete.

Preliminaries on the linearized equation near W

This section is similar to the corresponding one in the NLS case [DM07, Section 5].

Let u be a solution of (1.1), defined on [0, +∞), and close to W . Let h := u -W . Then

∂ 2 t h -∆h -|W + h| 4 N-2 (W + h) + W N+2 N-2 = 0,
which we rewrite as

∂ 2 t h + Lh = R(h), (5.1) 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. Note that 1 2 (Lu, u) L 2 = Q(u)
where Q is the quadratic form defined in (3.14).

Preliminary estimates.

Recall the definition of the spaces ℓ(I) and N (I) defined in (2.1), (2.2).

Lemma 5.1. There exists C > 0 such that if f ∈ L 2 * , I is a time interval and u, v ∈ ℓ(I).

D 1/2 x (W 4 N-2 u) N (I) ≤ C |I| 2 N+1 + |I| 5 2(N+1) u ℓ(I) (5.2) R(f ) 2N N+2 ≤ C f 2 2 * + f N+2 N-2 2 *
(5.3)

D 1/2 x (R(u) -R(v)) N (I) (5.4) ≤ C 1 + |I| 6-N 2(N+1) u -v ℓ(I) u ℓ(I) + v ℓ(I) + u 4 N-2 ℓ(I) + v 4 N-2 ℓ(I)
.

We postpone the proof of Lemma 5.1 to Appendix B. We will also need the following version of Lemma 5.1 with exponentially decreasing norms.

Corollary 5.2. Let u, v ∈ ℓ(t 0 , +∞), t 0 ∈ R, such that for some γ > 0, and some constant M > 0, ∀t ≥ t 0 , u ℓ(t,+∞) + v ℓ(t,+∞) ≤ M e -γt . Then there exists C = C(γ, M ) > 0 such that

D 1/2 x (W 4 N-2 u) N (t,+∞) ≤ Ce -γt , R(u(t)) 2N N+2 ≤ Ce -2γt ∀t ≥ t 0 , D 1/2 x (R(u) -R(v)) N (t,+∞) ≤ Ce -γt u -v ℓ(t,+∞)
. Proof. This is an immediate consequence of Lemma 5.1 and the following elementary Claim, which is Claim 5.8 in [DM07]: Claim 5.3 (Sums of exponential). Let t 0 > 0, p ∈ [1, +∞[, a 0 = 0, E a normed vector space, and f ∈ L p loc (t 0 , +∞; E) such that (5.5) ∃τ 0 > 0, ∃C 0 > 0, ∀t ≥ t 0 , f L p (t,t+τ 0 ,E) ≤ C 0 e a 0 t .

Then for t ≥ t 0 ,

(5.6) f L p (t,+∞,E) ≤ C 0 e a 0 t 1 -e a 0 τ 0 if a 0 < 0; f L p (t 0 ,t,E) ≤ C 0 e a 0 t 1 -e -a 0 τ 0 if a 0 > 0.

Corollary 5.4 (Strichartz estimates for the perturbative equation). Let h be a solution of (5.1)

on [0, ∞) such that ∇h(t) 2 + ∂ t h(t) 2 ≤ Ce -γt . Then h ℓ(t,+∞) + D 1/2 x W 4 N-2 h N (t,+∞) ≤ Ce -γt , R(h(t)) 2N N+2 + D 1/2 x R(h) N (t,+∞) ≤ Ce -2γt .
Proof. The proof is the same than the one of [DM07, Lemma 5.6]. We sketch it for the sake of completness. Note that all desired estimates are, by Corollary 5.2, a consequence of h ℓ(t,+∞) ≤ Ce -γt , so that we only need to show this last estimate We have

∂ 2 t h -∆h = N + 2 N -2 W 4 N-2 h + R(h).
Let t > 0 and τ ∈ (0, 1). First note that W + h is solution of (1.1), and thus, by the standard Cauchy problem theory for (1.1), h ℓ(t,t+τ ) is finite. By Strichartz inequality (Proposition 2.1) and Lemma 5.1,

h ℓ(t,t+τ ) ≤ C D 1/2 x W 4 N-2 h N (t,t+τ ) + D 1/2 x R(h) N (t,t+τ ) + e -γt ≤ C τ 2 N+1 h ℓ(t,t+τ ) + h 2 ℓ(t,t+τ ) + h N+2 N-2 ℓ(t,t+τ ) + e -γt .
By a standard argument (see the proof of [DM07, Lemma 5.7]), we deduce from the preceding inequality, τ is small, h ℓ(t,t+τ ) ≤ Ce -γt .

The conclusion follows from Claim 5.3.

5.2. Spectral theory for the linearized operator. The following Proposition sums up spectral properties of L (see [START_REF] Schlag | On the focusing critical semi-linear wave equation[END_REF], [START_REF] Schlag | Slow blow-up solutions for the h 1 (R 3 ) critical focusing semi-linear wave equation[END_REF] for the radial case in R 3 ).

Proposition 5.5. The operator L on L 2 with domain H 2 is a self-adjoint operator with essential spectrum [0, +∞), no positive eigenvalue and only one negative eigenvalue -e 2 0 , with a radial, exponentially decreasing, smooth eigenfunction Y. Furthermore, if

G ⊥ := f ∈ Ḣ1 , Yf = ∇f • ∇ W = ∇f • ∇W 1 = . . . = ∇f • ∇W N = 0 .
Then there exists c Q > 0 such that

(5.7) ∀f ∈ G ⊥ , Q(f ) ≥ c Q ∇f 2 2 . Remark 5.6. The proposition shows {u ∈ Ḣ1 , Lu = 0} = span{ W , W 1 , . . . , W N }.
Indeed, the inclusion ⊃ is already known. For the other inclusion, note that if the dimension of the space {u ∈ Ḣ1 , Lu = 0} was strictly higher than N + 1, we could find Z ∈ Ḣ1 , Z = 0, such that LZ = 0, and orthogonal to W , W 1 , . . . , W N . By self-adjointness of L, ZY = -1 e 0 ZLY = 0. Thus on one hand Z ∈ G ⊥ \ {0}, and on the other Q(Z) = (LZ, Z) = 0 contradicting (5.7).

Proof of the proposition. Step 1. Existence of a negative eigenvalue. The fact that L is a selfadjoint operator on L 2 with domain H 2 is well-known. Note that L = -∆ -N +2 N -2 W 4 N-2 with W 4 N-2 ≈ 1/|x| 4 for large x. In particular, W 4 N-2 is bounded and tends to 0 at infinity, which shows that the essential spectrum of L is [0, ∞) (see e.g. [RS78, Theorem XIII.14]). Furthermore, |x|W 4 N-2 tends to 0 at infinity, so that by Kato's Theorem [RS78, Theorem XII.58], L does not have any positive eigenvalue.

By the equation

-∆W = W N+2 N-2 , we have LW = -4 N -2 W N+2 N-2 and thus LW W = - 4 N -2 W 2 * < 0.
By approximating W by compactly supported functions inf

u 2 =1,u∈H 2 Luu < 0,
Thus L has at least one negative eigenvalue -e 2 0 , and the corresponding eigenfunction Y is exponentially decreasing by Agmon estimate. We chose -e 2 0 to be the first eigenvalue of L, which implies that Y is radial and -e 2 0 is a simple eigenvalue Step 2. Proof of (5.7). We first show

(5.8) ∀f ∈ G ⊥ , Q(f ) > 0. If not ∃f ∈ G ⊥ , Q(f ) ≤ 0. Let g ∈ span f, Y, W , W 1 , . . . , W N .
Taking into account that L W = LW 1 = . . . = LW N = 0, and that LYf = -e 2 0 Yf = 0 we get

Q(g) = Lg g = L(αY + βf )(αY + βf ) = β 2 Q(f ) -α 2 e 2 0 Y 2 ≤ 0. Note that span f, Y, W , W 1 , . . . , W N is a subspace of Ḣ1 of dimension N + 3, whereas H ⊥ = span{W, W , W 1 , W 2 , . . . , W N } ⊥ (the orthogonal is taken in Ḣ1 ) is of codimension N + 2 in Ḣ1 .
Thus there exists a nonzero g ∈ span f, Y, W , W 1 , . . . , W N ∩ H ⊥ . By Claim 3.5, Q(g) > 0, whereas we have just shown that Q(g) ≤ 0 yielding a contradiction. The proof of (5.8) is complete.

We now turn to the proof of (5.7). We argue again by contradiction. If (5.7) does not hold, there exists a sequence (f n ) such that (5.9)

f n ∈ G ⊥ , ∇f n 2 = 1, Q(f n ) -→ n→∞ 0.
Extracting a subsequence from (f n ), we may assume

f n ⇀ f in Ḣ1 .
The weak convergence of f n ∈ G ⊥ to f implies that f ∈ G ⊥ . By Cauchy-Schwarz inequality for the positive quadratic form Q on G ⊥ , we get

Q(f n )Q(f ) ≥ Lf f n .
Thus by (5.9),

Q(f ) = lim n→+∞ Lf f n = 0.
As f ∈ G ⊥ , (5.8) shows that f = 0 and thus

f n ⇀ 0 in Ḣ1 .

Now, by compactness W

4 N-2 |f n | 2 tends to 0. Using that by (5.9), Q(f n ) tends to 0, we get that ∇f n 2 tends to 0, contradicting (5.9). The proof of (5.7) is complete.

Step 3. Uniqueness of the negative eigenvalue. Assume that L has a second eigenfunction Y 1 , with eigenvalue -e 2 1 ≤ 0. As -e 2 0 is the first eigenvalue of L, we have that -e 2 0 < -e 2 1 and YY 1 = 0. The same argument than above shows that

∀f ∈ span{Y, Y 1 , W , W 1 , . . . W N }, Q(f ) ≤ 0,
which yields a subspace of Ḣ1 of dimension N + 3 where Q is nonpositive, contradicting the fact that Q is positive on the subspace H ⊥ , which is of codimension N + 2 in Ḣ1 .

In the sequel, we will chose Y such that (5.10) Y 2 = 1.

5.3. Properties of the nonhomogeneous linearized equation. Let t 0 ≥ 0. We are now interested by the following problem (5.11)

∂ 2 t h + Lh = ε, t ≥ t 0 , where h ∈ C 0 [t 0 , +∞), Ḣ1 , ∂ t h ∈ C 0 [t 0 , +∞), L 2 , ε ∈ C 0 [t 0 , +∞), L 2N N+2
and D

1/2

x ε ∈ N (t 0 , +∞) (see (2.1) for the definition of N (t 0 , +∞)).

Proposition 5.7. Let h and ε be as above. Assume that for some constant c 0 , c 1 such that 0 < c 0 < c 1 ,

∂ t h(t) 2 + ∇h(t) 2 ≤ Ce -c 0 t (5.12) ε(t) 2N N+2 + D 1/2 x ε N (t,+∞) ≤ Ce -c 1 t . (5.13) Then, if c - 1 is any arbitrary number < c 1 . • if c 1 > e 0 , there exists A ∈ R such that (5.14) ∂ t h(t) -Ae -e 0 t Y 2 + ∇(h(t) -Ae -e 0 t Y 2 ≤ Ce -c - 1 t ; • if c 1 ≤ e 0 , (5.15)
∂ t h(t) 2 + ∇h(t) 2 ≤ Ce -c - 1 t . Proof. Write h(t) = β(t)Y + γ(t) W + N j=1 γ j (t)W j + g(t), g(t) ∈ G ⊥ .
By the definition of G ⊥ , the condition g(t) ∈ G ⊥ is equivalent to

(5.16) β(t) := h(t)Y, γ(t) := ∇ (h(t) -β(t)Y) ∇ W , γ j (t) := ∇ (h(t) -β(t)Y) ∇W j .
Step 1. Reduced case.

In this case, we assume, in addition to the hypothesis of Proposition 5.7

(5.17)

∀t ≥ t 0 , β(t) := h(t)Y = 0.
And show that (5.14) (with A = 0) or (5.15) hold. It is sufficient to show (5.18)

∂ t h(t) 2 + ∇h(t) 2 ≤ Ce -(c 0 +c 1 ) 2 t .
An iteration argument will give the desired result. We first prove

(5.19) 1 2 d dt Q(h(t)) + ∂ t h(t) 2 2 = R N ε(t)∂ t h(t). Indeed, recalling that Q(h) = Lh h, we get 1 2 d dt Q(h(t)) + ∂ t h(t) 2 2 = R N Lh(t)∂ t h(t) + R N ∂ 2 t h(t)∂ t h(t),
which gives directly (5.19) from equation (5.11), .

We now turn to the proof of (5.18). Note that h is exponentially decreasing in the Strichartz norms:

(5.20)

h ℓ(t,+∞) ≤ Ce -c 0 t Indeed ∂ 2 t h -∆h = N +2 N -1 W 4 N+2
h + ε and by Corollary 5.2, assumptions (5.12) and (5.13),

D 1/2 x N + 2 N -1 W 4 N+2 h + ε N (t,+∞) ≤ Ce -c 0 t .
By Strichartz estimates (see Proposition 2.1), we get (5.20). Now, by (5.19),

d dt Q(h(t)) + ∂ t h(t) 2 2 ≤ C D 1/2 x ε(t) 2(N+1) N+3 D -1/2 x ∂ t h(t) 2(N+1) N-1
.

Integrating between t and +∞, we get, combining assumption (5.13) on ε, estimate (5.20), and Hölder inequality in time,

Q(h(t)) + ∂ t h(t) 2 2 ≤ C D 1/2 x ε N (t,+∞) h ℓ(t,+∞) ≤ Ce -(c 0 +c 1 )t
. By Claim 3.5, and (5.17), ∇g(t

) 2 2 ≤ CQ(g(t)) = CQ(h(t)) ≤ Ce -(c 0 +c 1 )t . Hence (5.21) ∂ t h(t) 2 + ∇g(t) 2 ≤ Ce -(c 0 +c 1 ) 2 t .
Furthermore, note that by the definition of γ in (5.16) and (5.21).

γ(t) = R N h(t)∆ W -→ 0 t→+∞ , |γ ′ (t)| = R N ∂ t h(t)∆ W ≤ Ce -c 0 +c 1 2 . Hence (5.22) |γ(t)| ≤ Ce -(c 0 +c 1 )t 2
.

By an analogous argument

(5.23)

N j=1 |γ j (t)| ≤ Ce -(c 0 +c 1 )t 2 .
This gives (5.18) and concludes Step 1.

Step 2. General case. We no longer assume β(t) = 0. We have:

β ′′ (t) = e 2 0 β(t) + η(t), where η(t) := R N ε(t)Y. (5.24) Indeed, β ′′ (t) = R N ∂ 2 t h(t)Y = - R N Lh(t)Y + R N ε(t)Y = e 2 0 R N h(t)Y + η(t)Y.
We will show that h(t) := h(t) -β(t)Y and ε(t) := ε(t) -η(t)Y satisfy the hypothesis of Step 1.

By assumption (5.13), |η(t)| ≤ Ce -c 1 t . We distinguish two cases. First case: e 0 < c 1 . Then e e 0 t |η(t)| ≤ Ce -(c 1 -e 0 )t with c 1 -e 0 > 0. Solving (5.24), we see that there exist real parameters β + , β -such that

β(t) = β -e -e 0 t + β + e e 0 t - 1 2e 0 +∞ t e e 0 (t-s) η(s)ds + 1 2e 0 +∞ t
e -e 0 (t-s) η(s)ds.

Note that

+∞ t e e 0 (t-s) η(s)ds

+ +∞ t
e -e 0 (t-s) η(s)ds ≤ Ce -c 1 t . Furthermore β(t) tends to 0 by (5.12) which shows that β + = 0. Hence (5.25) β(t) = β -e -e 0 t + O(e -c 1 t ).

Second case: c 1 ≤ e 0 . Solving again (5.24), we get real parameters β + , β -such that

β(t) = β -e -e 0 t + β + e e 0 t - 1 2e 0 +∞ t e e 0 (t-s) η(s)ds - 1 2e 0 t 0
e -e 0 (t-s) η(s)ds.

Note that

1 2e 0 +∞ t e e 0 (t-s) η(s)ds + 1 2e 0 t 0 e -e 0 (t-s) η(s)ds ≤ C e -c 1 t if c 1 < e 0 te -c 1 t if c 1 = e 0 ,
so that we must have again β + = 0. As a conclusion

(5.26)

c 1 < e 0 =⇒ β(t) = O(e -c 1 t ), c 1 = e 0 =⇒ β(t) = O(te -c 1 t ).
In view of (5.24), it is easy to check that in both cases, h and ε satisfy the assumptions of Step 1, which implies, together with (5.25) or (5.26), the conclusions (5.14) or (5.15) of Proposition 5.7.

Proof of main results

In this section we conclude the proofs of Theorem 1 an 2. We start, in Subsection 6.1, by constructing approximate solutions U a k of (1.1) which converge to W as t → +∞. Subsection 6.2 is devoted to a fixed point argument near U a k for large k. The proof of Theorems 1 and 2 is the object of Subsection 6.3, except for the blow-up of W + for negative times, which is shown in Subsection 6.4. 6.1. A family of approximate solutions converging to W . Lemma 6.1. Let a ∈ R. There exist functions (Φ a j ) j≥1 in S(R N ), such that Φ a 1 = aY and if

(6.1) U a k (t, x) := W (x) + k j=1
e -je 0 t Φ a j (x), then as t → +∞, (6.2)

ε k := ∂ 2 t U a k -∆U a k -U a k 4 N-2 U a k = O(e -(k+1)e 0 t ) in S(R N ).
Proof. The proof is identical to the proof of [DM07, Lemma 6.1]. We sketch it for the sake of completness. Note that (6.3)

∂ 2 t (W + h) -∆(W + h) + |W + h| 4 N-2 (W + h) = ∂ 2 t h + Lh -R(h)
where L and R are defined in (5.1). We have

R(h) = W N+2 N-2 J(W -1 h), J(t) := |1 + t| 4 N-2 (1 + t) -1 - N + 2 N -2 t.
The function J is real analytic for |t| < 1 and J(0) = J ′ (0) = 0. Thus J(t) is a power series of the form j≥2 c j t j which has radius of convergence 1. In particular, if h ∈ S(R N ) and satisfies

|h(x)W -1 (x)| ≤ 1/2, for all x ∈ R N , then (6.4) R(h) = +∞ j=2 c j W N+2 N-2 hW -1 j ,
where the series converges in S(R N ).

Let us fix a ∈ R. We will omit most superscripts a to simplify notations. Clearly, by (6.3), if

U 1 = W + aYe -e 0 t , ∂ 2 t U 1 -∆U 1 -U 1 4 N-2 U 1 = e 2 0 ae -e 0 t Y + ae -e 0 t L(Y) -R aYe -e 0 t = -R aYe -e 0 t ,
which yields (6.2) for k = 1. Now assume that for some k ≥ 1, there exist Φ 1 ,. . . ,Φ k in S(R N ) such that (6.2) holds with (6.5)

U k = W + h k , where h k := k j=1 e -je 0 t Φ j . Let Φ k+1 ∈ S(R N ). Then (6.6) ∂ 2 t h k + e -(k+1)e 0 t Φ k+1 + L h k + e -(k+1)e 0 t Φ k+1 -R h k + e -(k+1)e 0 t Φ k+1 = (k + 1) 2 e 2 0 + L e -(k+1)e 0 t Φ k+1 + ε k + R(h k ) -R h k + e -(k+1)e 0 t Φ k+1 .
By (6.4) we see that ε k must be, for large t > 0, an infinite sum of the form j≥0 e -je 0 t Ψ j,k (x), with convergence in S(R N ). Furthermore, the induction hypothesis (6.2) shows that Ψ j,k = 0 for j ≤ k. Thus

ε k (t, x) = j≥k+1 e -je 0 t Ψ j,k (x).
Furthermore, R(h k ) and R h k + e -(k+1)e 0 t have similar asymptotic developments, and if t is large enough, using that h k = O(e -e 0 t ), we get R(h k ) -R(h k + e -(k+1)e 0 t ) ≤ Ce -e 0 (k+2)t . This shows (6.7)

ε k + R(h k ) -R h k + e -(k+1)e 0 t Φ k+1 = e -(k+1)e 0 t Ψ k+1,k + O e -(k+2)e 0 t in S(R N ).
By Proposition 5.5, -(k + 1) 2 e 2 0 is not in the spectrum of L. It is classical that the resolvent (k + 1) 2 e 2 0 + L -1 maps S into S (see e.g. [DM07, §7.2.2] for the proof of a similar fact). In view of (6.6) and (6.7), it suffices to take

Φ k+1 := (k + 1) 2 e 2 0 + L -1 Ψ k,k+1
to get (6.2) at rank k + 1.

6.2. Contraction argument near an approximate solution of large order. Proposition 6.2. Let a ∈ R. There exists k 0 > 0 such that for any k ≥ k 0 , there exists t k ≥ 0 and a solution U a of (1.1) such that for t ≥ t k , (6.8) U a -U a k ℓ(t,+∞) ≤ e -(k+ 1 2 )e 0 t .

Furthermore, U a is the unique solution of (1.1) satisfying (6.8) for large t. It is independent of k and satisfies, for large t, (6.9)

∇(U a (t) -U a k ) 2 + ∂ t (U a (t) -U a k ) 2 ≤ e -(k+ 1
2 )e 0 t .

Proof. We follow the lines of the proof of [DM07, Proposition 6.3].

Step 1. Transformation into a fixed-point problem. As in the proof of Lemma 6.1, we will fix a ∈ R and omit most of the superscripts a. Recall the definition of ε k and h k from (6.1) and (6.5). The proof relies on a fixed point argument to construct

w := U a -W -h k .
The function U a is solution of (1.1) if and only if U a -W is solution of (5.1). Substracting equation (5.1) on U a -W and the equation

∂ 2 t h k + Lh k = R(h k ) + ε k , we get that U a satisfies (1.1) if and only if w satisfies ∂ 2 t w + Lw = R(h k + w) -R(h k ) -ε k .
This may be written as

∂ 2 t w -∆w = N + 2 N -2 W 4 N-2 w + R(h k + w) -R(h k ) -ε k .
Thus the existence of a solution U a of (1.1) satisfying (6.8) for t ≥ t k may be written as the following fixed-point problem on w (6.10) ∀t ≥ t k , w(t) = M k (w)(t) and w ℓ(t,+∞) ≤ e -(k+ 1 2 )e0t , where M k (w)(t) :=

- +∞ t sin (t -s) √ -∆ √ -∆ N + 2 N -2 W 4 N-2 w(s) + R(h k (s) + w(s)) -R(h k (s)) -ε k (s) ds.
Let us fix k and t k . Consider

E k ℓ := w ∈ ℓ(t k , +∞); w E k ℓ := sup t≥t k e (k+ 1 2 )e0t w ℓ(t,+∞) < ∞ B k ℓ := w ∈ E k ℓ , w E k ℓ ≤ 1 .
The space E k ℓ is clearly a Banach space. In view of (6.10), it is sufficient to show that if t k and k are large enough, the mapping M k is a contraction on B k ℓ .

Step 2. Contraction property. Note that by Strichartz inequality on the free equation (Lemma 2.5), there is a constant

C * > 0 such that if w, w ∈ E k ℓ , k ≥ 1, M k (w) ℓ(t,+∞) ≤ C * D 1/2 x W 4 N-2 w N (t,+∞) (6.11) + D 1/2 x R(h k + w) -R(h k ) N (t,+∞) + D 1/2 x ε k N (t,+∞) M k (w) -M k ( w) ℓ(t,+∞) ≤ C * D 1/2 x W 4 N-2 (w -w) N (t,+∞)
(6.12)

+ D 1/2 x R(h k + w) -R(h k + w) N (t,+∞)
. Claim 6.3. There exists a constant C k > 0, depending only on k such that for all w, w ∈ B k ℓ and t ≥ t k D 1/2 x ε k N (t,+∞) ≤ C k e -(k+1)e 0 t , (6.13)

D 1/2 x R(h k + w) -R(h k + w) N (t,+∞) ≤ C k e -(k+ 3
2 )e 0 t w -w E k ℓ . (6.14) Furthermore, there exists k 0 > 0 such that for all k ≥ k 0 and all w ∈ E k ℓ (6.15)

D 1/2 x W 4 N-2 w N (t,+∞) ≤ 1 4C * e -(k+ 1 2 )e 0 t w E k ℓ .
Proof of Claim 6.3. The proof is very close to [DM07, Claim 6.4]. Estimate (6.13) follows immediately from (6.2), and (6.14) follows immediately from Corollary 5.2. Let us show (6.15). Let τ 0 ∈ (0, 1). By Lemma 5.1, there exists a constant C 2 > 0 such that

D 1/2 x (W 4 N-2 w) N (t,t+τ 0 ) ≤ C 2 τ 2 N+1 0 w ℓ(t,t+τ 0 ) ≤ C 2 τ 2 N+1 0 e -(k+ 1 2 )e0t w E k ℓ .
By Claim 5.3,

D 1/2 x (W 4 N-2 w) N (t,+∞) ≤ C 2 e -(k+ 1
2 )e 0 t

1 -e -(k+ 1

2 )e 0 τ 0

τ 2 N+1 0 w E k ℓ .
Chosing τ 0 and k 0 such that C 2 τ 2 N+1 0 = 1 8C * , and e -(k 0 + 1 2 )e 0 τ 0 ≤ 1 2 , we get (6.15) for k ≥ k 0 . Chose k ≥ k 0 . By (6.11) and Claim 6.3, we get, if

g ∈ B k ℓ M k (w) ℓ(t,+∞) ≤ 1 4 e -(k+ 1 2 )e 0 t w E k ℓ + C * C k e -(k+ 3 2 )e 0 t w E k ℓ + C * C k e -(k+1)e 0 t k ≤ e -(k+ 1 2 )e 0 t 1 4 + C * C k e -e 0 t k + C * C k e -1 2 e 0 t k . Chosing t k so large that C * C k e -e 0 t k + C * C k e -1 2 e 0 t k ≤ 1 2 , we get that for large k, M k (w) is in B k ℓ . Now, let w, w ∈ B k ℓ .
Similarly, by (6.12) and Claim 6.3,

M k (w) -M k ( w) E k ℓ ≤ w -w E k ℓ 1 4 + C * C k e -e 0 t k , which shows, chosing a larger t k if necessary, that M k is a contraction of B k ℓ .
Thus, for each k ≥ k 0 , (1.1) has an unique solution U a satisfying (6.8) for t ≥ t k . The preceding proof clearly remains valid taking a larger t k , so that the uniqueness still holds in the class of solutions of (1.1) satisfying (6.8) for t ≥ t ′ k , where t ′ k is any real number larger than t k . Using the uniqueness in the Cauchy problem (1.1), it is now straightforward to show that U a does not depend on k ≥ k 0 .

It remains to show (6.9). Let k > 0 be a large integer and w ∈ B k ℓ . By Strichartz inequality (2.6), and the definition of M k , we have, for t ≥ t k ,

∇M k (w)(t) 2 + ∂ t M k (w)(t) 2 ≤ D 1/2 x N + 2 N -2 W 4 N-2 w + R(h k + w) -R(h k ) -ε k N (t,+∞) .
As a consequence of Claim 6.3 and the fact that w k E ℓ ≤ 1, we get

M k (w)(t) Ḣ1 ≤ C e -(k+ 1 2 )e0t w E k ℓ + e -(k+1)e 0 t ≤ Ce -(k+ 1 2 )e0t .
Applying the preceding inequality to the solution w = U a -U a k of the fixed point w = M k (w), we get directly (6.9). The proof of Proposition 6.2 is complete. 6.3. Conclusion of the proofs. At this levels, the proof are similar to the one of [START_REF] Duyckaerts | Dynamic of threshold solutions for energy-critical nls[END_REF], except for the blowing-up of W + which is proven in the next subsection.

Proof of Theorem 1. The function Y is an eigenfunction for the first eigenvalue of L, thus it must have constant sign. Replacing Y by -Y if necessary, we may assume that Y(x) > 0 for all x and thus (6.16) ∇Y • ∇W > 0.

Let W ± := U ±1 , which yields two solutions of (1.1) for large t ≥ t 0 . Then all the conditions of Theorem 1 are satisfied. Indeed, W ± is globally defined gor t ≥ t 0 and by (6.9), (W ± , ∂ t W ± ) tends to (W, 0) in Ḣ1 × L 2 , which yields (1.7). The energy condition (1.6) then follows from the conservation of the energy. Furthermore, again by (6.9), ∇U a 2 2 = ∇W 2 2 + 2ae -e 0 t (∇W • ∇Y + O(e -3 2 e 0 t ), which shows, together with (6.16), that for large t > 0,

∇W + (t) 2 > 0, ∇W -(t) 2 < 0.
From Remark 1.3, this inequalities remain valids for every t in the intervals of existence of W + and W -.

Finally T -(W -) = -∞ by (a) in Proposition 2.8 and u S(-∞,0) < ∞ by (3.2) in Proposition 3.1 .

Except for the proof of the finite time blow-up of W + for negative time, which we postpone to Subsection 6.4, the proof of Theorem 1 is complete.

Proof of Theorem 2. Let us first prove: Lemma 6.4. If u is a solution of (1.1) satisfying (6.17)

∇(u(t) -W ) 2 + ∂ t u(t) 2 ≤ Ce -γ 0 t , E(u 0 , u 1 ) = E(W ) then ∃!a ∈ R, u = U a .
Corollary 6.5. For any a = 0, there exists T a ∈ R such that (6.18)

U a = W + (t + T a ) if a > 0 U a = W -(t + T a ) if a < 0.
Proof of Lemma 6.4. Let u = W + h be a solution of (1.1) for t ≥ t 0 satisfying (6.17). Recall that h satisfies equation (5.1).

Step 1. We show that there exists a ∈ R such that

(6.19) ∀η > 0, ∇(h(t) -ae -e 0 t Y) 2 + ∂ t h(t) + ae 0 e -e 0 t Y 2 + h(s) -ae -e 0 s Y ℓ(t,+∞) ≤ C η e -(2-η)e 0 t .
Indeed we will show (6.20)

∇h(t) 2 + ∂ t h 2 ≤ Ce -e 0 t , R(h(t)) 2N N+2 + D 1/2 x (R(h)) N (t,+∞) ≤ Ce -2e 0 t .
Assuming (6.20), we are in the setting of Proposition 5.7, with ε = R(h), c 0 = e 0 and c 1 = 2e 0 . The conclusion (5.15) of the lemma would then yield (6.19). It remains to prove (6.20). By Corollaries 5.2 and 5.4 the bound on R(h) in (6.20) follows from the bound on ∇h(t) 2 + ∂ t h(t) 2 , so that we only need to show this first bound. By Corollary 5.4, assumption (6.17) implies ∇h(t) 2 + ∂ t h(t) 2 + h ℓ(t,+∞) ≤ Ce -γ 0 t . By Corollary 5.2

R(h(t))

L 2N N+2 + ∇(R(h)) N (t,+∞) ≤ Ce -2γ 0 t .
Thus we can apply Proposition 5.7, showing that h(t) Ḣ1 ≤ C e -e 0 t + e -3 2 γ 0 t .

If 3 2 γ 0 ≥ e 0 the proof of (6.20) is complete. If not, assumption (6.17) on v holds with 3 2 γ 0 instead of γ 0 , and an iteration argument yields (6.20).

Step 2. Let us show (6.21)

∀m > 0, ∃t 0 > 0, ∀t ≥ t 0 , ∂ t u(t) -U a (t) 2 + ∇ u -U a ℓ(t,+∞) ≤ e -mt .

By

Step 1, (6.21) holds for m = 3 2 e 0 . Let us assume (6.21) holds for some m = m 1 > e 0 . We will show that it holds for m = m 1 + e 0 2 , which will yield (6.21) by iteration and conclude the proof.

Write

h(t) := u(t) -W , h a (t) := U a (t) -W (so that in particular u -U a = h -h a ). Then ∂ 2 t (h -h a ) + L(h -h a ) = -R(h) + R(h a ). By induction hypothesis ∂ t (h(t) -h a (t)) 2 + ∇(h(t) -h a (t)) 2 + ∇ h -h a ℓ(t,+∞) ≤ e -m 1 t . According to Corollary 5.2 D 1/2 x R(h) -R(h a ) N (t,+∞) + R(h(t)) -R(h a (t)) L 2N N+2 ≤ Ce -(m 1 +e 0 )t .
Step 1. Estimates for large positive t. Let us show that there exists R 0 , t 0 , c 0 > 0 such that for all R ≥ R 0 ,

∀t ≥ t 0 , y ′′ R (t) ≥ 4 N -1 N -2 ∂ t W + 2 + 2 N -2 ∇W + 2 -|∇W )| 2 ≥ 2 N -2 d(t). (6.23) y ′ R (t 0 ) ≤ -2c 0 , y R (t 0 ) ≤    C R if N = 3 C log R if N = 4 C if N = 5 (6.24)
By explicit computations and E(W + , ∂ t W + ) = E(W, 0), we have

y ′ R (t) = 2 W + ∂ t W + ϕ R , (6.25) y ′′ R (t) = 4 N -1 N -2 R N ∂ t W + 2 + 4 N -2 R N ∇W + 2 - R N |∇W | 2 (6.26) + ∆ϕ R W + 2 + 2 (1 -ϕ R ) ∇W + 2 -W + 2 * -∂ t W + 2 .
Replacing W by W + in the preceding expressions, we see that the corresponding y R must be constant, so that in particular, ∆ϕ

R (W ) 2 + 2 (1 -ϕ R ) |∇W | 2 -|W | 2 * = 0.
By (6.8) in Proposition 6.2, W + = W +e -e 0 t Y+r 1 with ∇r 1 2 + ∂ t r 1 2 ≤ Ce -2e 0 t . Developping W + , we get, recalling that Y is in S and that ϕ R

(x) = 1 for |x| ≤ R, ∆ϕ R W + 2 + 2 (1 -ϕ R ) ∇W + 2 -W + 2 * -∂ t W + 2 ≤ C e -e 0 t R + e -2e 0 t , ∇W + (t) 2 -|∇W | 2 = 2e -e 0 t W Y + O(e -2 0 t ).
Thus by (6.26)

y ′′ R (t) ≥ 4 N -1 N -2 ∂ t W + 2 + 2 N -2 ∇W + (t) 2 -|∇W | 2 + 4 N -2 e -e 0 t W Y -C e -e 0 t R + e -2e 0 t .
As W Y > 0, we get (6.23) for R ≥ R 0 . Now, fixing R ≥ R 0 , we get, using that y ′ R (t) tends to 0 at infinity,

y ′ R (t 0 ) = - +∞ t 0 y ′′ R (t)dt ≤ -4 N -1 N -2 +∞ t 0 |∂ t W + (t)| 2 dt,
which yields the first assertion in (6.24). It remains to shows the second assertion in (6.24). Note that W ≈ C |x| N-2 at infinity, so that (6.27) lim

t→+∞ y R (t) = W 2 ϕ R ≈        R if N = 3, log R if N = 4, 1 R if N = 5. Furthermore, |y ′ R (t)| ≤ C ∂ t W + 2 y R (t) ≤ Ce -e 0 t y R (t), and thus y R (t 0 ) -lim t→∞ y R (t) ≤ C ∞ t 0
e -e 0 t dt ≤ C, which yields together with (6.27), the second assertion in (6.24).

Step 1 is complete

Step 2. Estimates for t 0 -ε 0 R ≤ t ≤ t 0 . As a consequence of the preceding estimates, we show that there exists C 0 > 0 such that for R ≥ R 0 and t

0 -ε 0 R ≤ t ≤ t 0 , y ′′ R (t) ≥ 4 N -1 N -2 ∂ t W + 2 + 4 N -2 ∇W + 2 -|∇W | 2 - C 0 R N -2 , (6.28) y ′ R (t) ≤ -c 0 , (6.29)
where ε 0 := c 0 2C 0 . Estimate (6.29) follows from (6.28) by integration in time and the fact that y ′ R (t 0 ) ≤ -2c 0 for R large. Let us show (6.28).

By (6.26), it is sufficient to show (6.30

) |x|≥R r(W + )(t)dx ≤ C R N -2 .
where r(W + ) is defined in (2.25). Writing W + = W + O(e -e 0 t ), and using that W

≈ 1 |x| N-2 , |∇W | ≈ 1 |x| N-1 , as |x| → +∞, we get for large R |x|≥ R 6 r(W + )(t 0 + R/4) ≤ C R N -2 + Ce -R 4 e 0 ≤ C R N -2 .
Hence by finite speed of propagation (Proposition 2.3 (d)), and taking R large, (6.31)

∀t ≤ t 0 , |x|≥ R 2 +t 0 -t r(W + )(t) ≤ C R N -2 ,
which yields (6.30), and thus (6.28).

Step 3. Differential inequalities. Let us show that there exists a constant C > 0 such that (6.32)

∀t ∈ t 0 - ε 0 2 R, t 0 , 0 ≤ -y ′ R (t) ≤ C R y R (t 0 ).
By (6.24), if N = 4 or N = 5, y R (t 0 ) R → 0 as R → ∞ and 2c 0 ≤ -y ′ R (t 0 ). Thus (6.32) gives an immediate contradiction in this cases. The remaining case N = 3, which is the limit case in (6.32) will be treated in Steps 4 and 5.

By Step 2 and the fact that N ≥ 3 we have, for

t 0 -ε 0 R ≤ t ≤ t 0 , (6.33) y ′ R (t) 2 = 4 ϕ R W + ∂ t W + 2 ≤ 4 ϕ R W + 2 ϕ R (∂ t W + ) 2 ≤ N -2 N -1 y R (t) y ′′ R (t) + C 0 R . Claim 6.6 (Differential inequality argument). Let T > 0 and y ∈ C 2 ([0, T ]). Assume ∀t ∈ [0, T ], y ′ (t) ≥ c 0 > 0, y(t) > 0 and for some C 1 > 0, (6.34) ∀t ∈ [0, T ], y ′ (t) 2 ≤ N -2 N -1 y(t) y ′′ (t) + C 1 T .
Then there is a constant C > 0 (depending only on N , c 0 and C 1 , but not on T , such that ∀t ∈ 0, T 2 , T y ′ (t) ≤ Cy(0).

Proof. By (6.34),

y ′ (t) y(t) ≤ N -2 N -1 y ′′ (t) y ′ (t) + C 1 T y ′ (t) ≤ N -2 N -1 y ′′ (t) y ′ (t) + C 1 T c 0 .
Then, integrating between s and t, 0

≤ s ≤ t ≤ T , log y(t) y(s) ≤ N -2 N -1 log y ′ (t) y ′ (s) + C 1 c 0 , i.e e - C 1 c 0 y ′ (s) y(s) N-1 N-2 ≤ y ′ (t) y(t) N-1 N-2
.

Integrating with respect to t between s and T , we get

e - C 1 c 0 y ′ (s) y(s) N-1 N-2 (T -s) ≤ -(N -2) 1 y(T ) 1 N-2 - 1 y(s) 1 N-2 ≤ N -2 y(s) 1 N-2 , which yields (6.35) ∀s ∈ [0, T ], y ′ (s) y(s) (T -s) ≤ (N -2)e C 1 c 0 ≤ C.
Integrating (6.35) between 0 and t ∈ 0, T 2 we get log(y(t)) ≤ log(y(0)) + C log T T -t ≤ log(y(0)) + C log 2, i.e. y(t) ≤ Cy(0), which, gives together with (6.35), the announced result.

By (6.33) and the preceding claim on the function y = y R (t 0 -t), with R large, T = ε 0 R and C 1 = C 0 ε 0 , we get (6.32).

Step 4. Let us show that there exists a constant C > 0 such that (6.36) ∀R ≥ 1, ∀t ∈ R, |y ′′ R (t)| ≤ Cd(t). Indeed (6.37) y ′′ R (t) = 8

∂ t W + 2 ϕ R + 2 ∇W + 2 -W + 6 -∂ t W + 2 ϕ R + W + 2 ∆ϕ R .
Let us fix t ∈ R. First assume that d(t) ≤ δ 0 . Then by Lemma 3.7, there exists λ 0 , x 0 such that ∇(W + (t) -W λ 0 ,x 0 ) 2 + ∂ t W + (t) 2 ≤ Cd(t). Noting that 8 (∂ t W λ 0 ,x 0 ) 2 ϕ R + 2 |∇W λ 0 ,x 0 | 2 -|W λ 0 ,x 0 | 6 ϕ R + (W λ 0 ,x 0 ) 2 ∆ϕ R = 0, we get (6.36) for d(t) ≤ δ 0 by developping (6.37). By the energy equality E(W + , ∂ t W + ) = E(W, 0), W + 6 6 = 3 ∇W + 2 + 3 ∂ t W + 2 + E(W, 0). Thus there exists a constant C > 0 such that d(t 0 ) ≥ δ 0 =⇒ Cd(t) ≥ ∇W + 2 2 + W + 6 6 + ∂ t W + 2 2 , which shows, together with (6.37), implies estimate (6.36) in the case d(t) ≥ δ 0

Step 5. End of the proof in the case N = 3. Let us first show (6.38)

+∞ -∞ d(t)dt < ∞.
Estimates (6.24) and (6.32) give a constant C, independant or R, such that 0 ≤ -y ′ R (t 0 -ε 0 R) ≤ C. Thus by (6.23) and (6.28),

+∞ t 0 -ε 0 R d(t)dt = t 0 t 0 -ε 0 R d(t) + +∞ t 0 d(t) ≤ t 0 t 0 -ε 0 R y ′′ R (t) + C R + +∞ t 0 y ′′ R (t) ≤ Cε 0 + +∞ t 0 -ε 0 R y ′′ R (t)dt = Cε 0 -y ′ R (t 0 -ε 0 R) ≤ C.
Letting R tends to ∞ we get (6.38). Let M ≫ 1. Let t ∈ [t 0 -M, t 0 ]. By (6.28), for R ≥ R 0 , y ′′ R (t) ≥ 4d(t) -C 0 R . Taking R M ≫ 1 so that min t 0 -M ≤t≤t 0 d(t) ≥ C 0 R M , we get that y ′′ R M (t) ≥ 2d(t) for t 0 -M ≤ t ≤ t 0 and thus, in view of (6.23), (6.39) ∀t ≥ t 0 -M, y ′′ R M (t) ≥ 2d(t). By (6.38), there exists a sequence t n → -∞ such that d(t n ) → 0. As a consequence, y ′ R (t n ) → 0. We have y ′ R (t n ) = - x F (f ) p F ′ (f ) p 1 D 1/2 x f p 2 (c) Let m ∈ N * and r 1 , r 2 , r 3 ∈ (1, ∞) such that 1 r 1 + 1 r 2 + 1 r 3 = 1. Then

D 1/2 x (f 2m -g 2m ) p f 2m-1 p 1 + g 2m-1 p 1 D 1/2 x (f -g) p 2 + f 2m-2 r 1 + g 2m-2 r 1 D 1/2 x f r 2 + D 1/2 x g r 2 f -g r 3 .
Points (a) and (b) follows from [KPV93, Theorems A.7, A.8, A.9 and A.12]. Point (c) is a consequence of (a). , which gives (5.2), using Hölder inequality in time.

We will skip the proof of (5.3) which is a direct consequence of Hölder inequality.

Proof of (5.4).

Fixing t, we will show x u + D 1/2 x v 2(N+1)

N-1

+ D 1/2 x u + D 1/2 x v 4 N-2 2(N+1) N-1 .
Hölder inequality in time will yield the desired result. We have R(u) = W N+2 N-2 J W -1 u , where J(s) = |1 + s|

  R. Furthermore, either u = W -up to the symmetry of the equation, or u If |∇u 0 | 2 = |∇W | 2 then u = W up to the symmetry of the equation. (c) If |∇u 0 | 2 > |∇W | 2 , and u 0 ∈ L 2 then either u = W + up to the symmetry of the equation, or I is finite.

Lemma 4. 3 .

 3 Let u satisfying the assumptions of Proposition 4.1. Then ∀t ≥ 0, y ′ (t) < 0 (4.5) lim t→+∞ y(t) = y ∞ ∈ (0, +∞) (4.6) +∞ t d(s)ds ≤ Ce -ct . (4.7) Corollary 4.4. Under the assumptions of Proposition 4.1, T -(u) < ∞.

+∞-

  ∞ d(t)dt = 0. This shows that d(t) = 0 for all t, which is a contradiction.D 1/2 x (f g) p f p 1 D 1/2 x g p 2 + D 1/2 x f p 3 g p 4 . (a) (b) Let F ∈ C 1 (R) such that F (0) = 0. Then D 1/2

.

  Furthermore, W is a C ∞ function on R N such that W 4 N-2 ≈ 1/|x| 4 , D x W

(B. 2 )

 2 D 1/2 x (R(u) -R(v)) 2(N+1) N+3 D 1/2 x (u -v) 2(N+1) N-1 |u| + |v| 2(N+1)

4N- 2 N..N.F (W -1 ψ k h) 3 .F (W -1 ψ k h) 4 W -1 ψ k h 4 2 3k h 4 ThusFF (W -1 ψ k h) 3 23

 234443 (1 + s) -1 -N +2 N -2 s, J ′ (s) = N +2 N -2 |1 + s| 4In view of the integral expression of I(u, v), (B.2) will follow from the estimates Let us first show (B.4). By the pointwise bound |W + h|4 Choose also ψ ∈ C ∞ 0 (R 5 ) such that supp ψ ⊂ {1/2 ≤ |x| ≤ 8} and ψ(x) = 1 on {2 ≤ |x| ≤ 4}. Let ψ k (x) := ψ(x/2 k-1 ) for k ≥ 1 and ψ 0 (x) := ϕ(x/2). Then supp ψ k ⊂ 1 2 k-2 ≤ |x| ≤ 1 2 k+2 , k ≥ 1, supp ψ 0 ⊂ {|x| ≤ 4}; x ∈ supp ψ k =⇒ ψ k (x) = 1. We have (B.8) W 4/3 F W -1 h = k≥0 ψ k W 4/3 F W -1 h = k≥0 ψ k W 4/3 F W -1 ψ k (x)h .We leave the proof of the following estimates which follow from the explicit expression of W and scaling arguments to the reader.Claim B.2. For all p ∈ [1, ∞] and for all k ≥ 0, 2+5/p)k .By Claim B.1 (a), (B.9) D 1/2 x ψ k W 4/3 F (W -1 ψ k h) Note that |F (s)| |s|, so that, in view of Claim B.2, (W -1 ψ k h) 4 2 (-9/2+5/6+3)k = 2 -2 3 k h 4 .Furthermore, F ′ being bounded, by Claim B.1 (b), then (a)(-4+5/12)k 2 (3+5/12)k h 4 + 2 3k D 1/2 x h 3 2 -1 6 k h 4 + 2 -7 12 k D 1/2 x h 3 . By (B.9), (B.10) and (B.11), (B.12) D 1/2 x W 4/3 F (W -1 h) 4/3 F (W -1 ψ k h) 12 5 h 4 + D 1/2x h 3 .In view of (B.6), we get, by (B.7) and (B.12) + h 4 + D 1/2 x h 3 .

  shows the boundedness of x and λ. By Lemma 3.8 between σ = t and τ = τ n , and taking into account the fact that x(t) and

	1 λ(t) are bounded, we get +∞ t d(s)ds ≤ Cd(t). Thus for some constants c, C > 0, τn t d(s)ds ≤ C(d(t) + d(τ n )). Letting n goes to infinity we obtain
	(3.26)

+∞ t d(s)ds ≤ Ce -ct , which is the first bound in (3.23).

By (3.26), Lemma 3.10 and the fact that x(t) and 1 λ(t) are bounded, we have

  the estimate |α ′ (t)| ≤ Cµ(t)d(t) of Lemma 3.7, we get, by (3.23), |α(t)| ≤ Ce -ct . Furthermore, again by Lemma 3.7, ∇ f 2

  must show y ∞ > 0. Let us first show that for t ≥ 0 Proof of Proposition 4.1. Step 1. Convergence in L 2 . We first show that there exits u ∞ ∈ L 2

	(4.13)			|y ′ (t)| ≤ C ∂ t u(t) 2 ≤ Cd(t).
	By Cauchy-Schwarz, |y ′ (t)| ≤ u(t) 2 ∂ t u(t) 2 . By (4.12), u(t) 2 = y(t) is bounded, which shows the first bound in (4.13). According to Lemma 3.7, if d(s) ≤ δ 0 then ∂ t u(t) 2 ≤ Cd(t). Furthermore, if d(s) ≥ δ 0 , ∂ t u(t) 2 2 ≤ d(t) ≤ 1 δ 0 d(t) 2 , hence the bound ∂ t u(t) 2 ≤ Cd(t), which concludes the proof of (4.13).
	To show that y ∞ > 0, we argue by contradiction. Assume that y ∞ = 0. By(4.13),
	(4.14)	y(t) = -(y ∞ -y(t)) = -	t	+∞	y ′ (s)ds ≤ C	t	+∞	d(s)ds.
	Note that
	(4.15)
									1/2
	and thus, by (4.13),	+∞ t	d(s)ds	1/2
	Proof of (4.7). By (4.13) and (4.15) ,
				∀t ≥ 0,	t	∞	d(s)ds ≤ |y ′ (t)| ≤ Cd(t),
	which implies (4.7).
	such that
	(4.16)				lim t→+∞

+∞ t d(s)ds ≤ |y ′ (t)|. Indeed T t y ′′ (s)ds = y ′ (T ) -y ′ (t) ≤ -y ′ (t), which yields (4.15) in view of (4.9). Combining (4.14) and (4.15), we get

+∞ t d(s)ds ≤ |y ′ (t)| ≤ 2 ∂ t u(t) 2 y(t) 1/2 ≤ C ∂ t u(t) 2 +∞ t d(s)ds

≤ Cd(t) for t ≥ 0. This is a differential inequality of the form √ Y ≤ -CY ′ , which can not be valid on [0, ∞) if ∀t ≥ 0, Y > 0. The proof of (4.6) is complete.

  are well-defined. Furthermore, by (4.18), µ(t) must be bounded on n [t n , t ′ n ]. Thus by (4.7) and the same argument as in the proof of (3.27), α(t ′

		This shows that N = 5 and
	(4.18)	lim t→+∞	u(t) -W λ 0 ,x 0 2 = 0.
	Let us show
	(4.19)		lim t→+∞	d(t) = 0.

If (4.19) does not hold, there exist increasing sequences

(t n ) n , (t ′ n ) n such that t n < t ′ n , d(t n ) → 0, d(t ′ n ) = δ 0 and d(t) < δ 0 for t ∈ [t n , t ′ n ). On [t n , t ′ n ]

, the modulation parameters µ(t) and X(t) n ) tends to 0 which contradicts the estimate

d(t ′ n ) ≈ α(t ′ n ) of Lemma 3.7. Hence (4.19

). Thus there exists T > 0 such that for t ≥ T , d(t) ≤ δ 0 . By (4.18), µ converges to λ -1 0 . In view of estimate (3.20) of Lemma 3.7 and the boundedness of µ,

  Now assume that d(t) ≥ δ 0 . Thus

		2 ∇W 2 2 δ 0	d(t 0 ) ≥ ∇W 2 2 . As a consequence,
	1 +	2 ∇W 2 2 δ 0	d(t) ≥ ∇W + 2 2 + ∂ t W + 2 2 .

  Note that by (6.38), both integrals in the preceding inequality are finite. Letting M tends to +∞, we get

	t 0 -M tn	y ′′ R (t)dt -	+∞ t 0 -M y ′′ R (t)dt, which yields by (6.36) and (6.39)
		+∞ t 0 -M	d(t)dt ≤ C	t 0 -M tn	d(t)dt +	1 2	|y ′ R (t n )|.
	Letting n tends to infinity, we obtain
				+∞			t 0 -M
	∀M ≫ 1,	t 0 -M	d(t)dt ≤ C	-∞	d(t)dt.

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

1

Then by Proposition 5.7 ∂ t (h(t) -h a (t)) 2 + ∇(h(t) -h a (t)) 2 + ∇ h -h a ℓ(t,+∞) ≤ Ce -m 1 + 3 4 e 0 t , which yields (6.21) with m = m 1 + e 0 2 . By iteration, (6.21) holds for any m > 0. Taking m = (k 0 + 1)e 0 (where k 0 is given by Proposition 6.2), we get that for large t > 0 ∇ u -U a k 0 Z(t,+∞) ≤ e -(k 0 + 1 2 )e 0 t .

By uniqueness in Proposition 6.2, we get as announced that u = W a which concludes the proof of the lemma.

Proof of Corollary 6.5. Let a = 0 and chose T a such that |a|e -e 0 Ta = 1. By (6.9), (6.22) ∂ t U a (t + T a ) -W ∓ e -e 0 t Y 2 + ∇ U a (t + T a ) -W ∓ e -e 0 t Y 2 ≤ Ce -3 2 e 0 t .

Furthermore, U a (• + T a ) satisfies the assumptions of Lemma 6.4, which shows that there exists a ′ ∈ R such that U a (• + T a ) = U a ′ . By (6.22), a ′ = 1 if a > 0 and a ′ = -1 if a < 0, hence (6.18).

Let us turn to the proof of Theorem 2. Point (b) is an immediate consequence of the variational characterization of W ([Aub76], [START_REF] Talenti | Best constant in Sobolev inequality[END_REF]).

Let us show (a). Let u be a solution of (1.1) such that E(u 0 , u 1 ) = E(W, 0) and ∇u 0 2 < ∇W 2 . Assume that u S(R) = ∞. Replacing if necessary u(t) by u(-t), we may assume that u S(0,+∞) = ∞. Then (replacing u by -u if necessary), Proposition 3.1 shows that there exist

. This shows that u µ -1 0 ,-µ -1 0 x 0 fullfills the assumptions of Lemma 6.4 with ∇u 0 2 < ∇W 2 . Thus there exists a < 0 such that u µ -1 0 ,-µ -1 0 x 0 = U a . By Corollary 6.5, u(t) = W - µ 0 ,x 0 (t + T a ), which shows (a).

The proof of (c) is similar. Indeed if u is a solution of (1.1) defined on [0, +∞) and such that E(u 0 , u 1 ) = E(W, 0), ∇u 0 2 > ∇W 2 and u 0 ∈ L 2 , then by Proposition 4.1, u(t) -W µ 0 ,x 0 Ḣ1 ≤ Ce -ct , which shows using Lemma 6.4 and the same argument as before that for some T ∈ R, u(t) = W + µ 0 ,x 0 (t + T ). The proof of Theorem 2 is complete. 6.4. Blow-up of W + . In this section we prove that the function W + blows-up in finite negative time.

We will argue by contradiction, assuming that W + is globally defined. As before, we will write

We start with some estimates on y ′ R and y ′′ R .

Appendix A. Estimates on the modulation parameters

In this appendix we prove Lemma 3.7. Proof of (3.19). In the proof of estimate (3.19), t is just a parameter that we will systematically omit.

Developping the equality

By the orthogonality of f with W in Ḣ1 , and the equation ∆W + W N+2 N-2 = 0 we have

Thus W and f are Q-orthogonal and Q(αW

If δ 0 is small, so is ∇(αW + f ) L 2 , so that by (A.2) and Claim 3.5, there exists c > 0 such that

2), we get

We have

which concludes the proof of (3.19).

Proof of (3.20). Let v(t) := u µ,X (t). Then

Differentiating (A.4), and writing y = µ(x -X(t)), we get

Recall that v = W + αW + f . Multiplying the preceding equation by µ -N 2 , we obtain

Hence (using the orthogonality in Ḣ1 of W , W , W 1 ,. . . ,W N )

Let

Then, noting that for all t, ∂ t f (t) ∈ H ⊥ and using estimate (3.19), we have

Summing up estimates (A.5) and using that w(t) 2 = ∂ t u(t) 2 d(t), we get

Appendix B. Some technical estimates

In this appendix we proof Lemma 5.1. We will need the following version of Hölder estimate for fractional Sobolev spaces Claim B.1. Let p, p 2 , p 3 , p 4 ∈ (1, ∞), p 1 ∈ (1, ∞], such that 1 p 1 + 1 p 2 = 1 p , and

< ∞, we get (B.4). It remains to show (B.5). We will distinguish two cases.

First case: N = 3 or N = 4. Then 4 N -2 ∈ {2, 4}. By Claim B.1, (c),

. This yields (B.5), and concludes the proof of (B.2) (thus of (5.4)) when N = 3 or N = 4.

Second case: N = 5.

In this case, we must bound D

by sum of powers of h 4 and D 1/2

x h 3 . Note that Claim B.1 (c) is no longer available. We have

This yields (B.5), concluding the proof of (5.4) in the case N = 5.

Appendix C. Derivative of g R Claim C.1. Let u be a solution of (1.1) such that E(u 0 , u 1 ) = E(W, 0) and g R be defined by (3.7). There exist

where

Proof.

Furthermore, denoting by ψ Rj , j = 1 . . . N , the coordinates of ψ R ,

where the ãRjk are bounded independently of R, C ∞ , and supported in |x| ≥ R. By similar integration by parts on the other terms of (C.2), we get

where A R (u, ∂ t u) is a sum of integrals of the desired form (C.1). Furthermore, N -1 2

Noting that for |x| ≤ R, ϕ R = 1 and ∆ϕ R = 0, and that if |x| ≥ 2R, ∆ϕ R = 0, we get

where A R (u, ∂ t u) is again of the form (C.1).

Summing up (C.3) and (C.4), we obtain

where A R is defined by (C.1) for some functions a jk R , b 1 R , b 2 R , b 3 R . To conclude the proof, note that E(u, ∂ t u) = E(W, 0) implies

and recall that E(W, 0) = 1 N ∇W 2 2 , which gives (C.2).

E-mail address: thomas.duyckaerts@u-cergy.fr Thomas Duyckaerts, Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint Martin, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise cedex, France.