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Global dynamics above the ground state for the energy-critical Schrodinger equation with radial data

Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.

In this paper, we consider the semilinear Schrödinger equation on R 3 with the focusing energy-critical power for u = u(t, x) : R 1+3 → C: i∂ t u − ∆u = |u| 4 u, u(0, x) = u 0 (x) (1.1) with radial initial data u 0 ∈Ḣ 1 (or H 1 ). HereḢ 1 (resp. H 1 ) is the standard homogeneous (resp. inhomogeneous) Sobolev space in three dimensions, i.e., the completion of the Schwartz space with respect to the norm f Ḣ1 := ∇f L 2 (resp. f H 1 := f L 2 + ∇f L 2 ). Our consideration is restricted throughout this paper to the radial subspace: A strong solution of (1.1) is a solution that satisfies the Duhamel formula: It enjoys the following energy conservation law Note that this set is invariant for the complex rotation andḢ 1 scaling. Recall (see [1,23]) that W is an extremizer for the Sobolev inequality, i.e., The local well-posedness of (1.1) has been studied in [5,4]. See [11] for a summary of these results. In particular, it is known that on an interval J such that e it△ u 0 L 10 t,x (J×R 3 ) is small enough, there exists a unique solution u of (1.3) in a subspace of C(J,Ḣ 1 ). This allows us to define the maximal time interval of existence I(u) := (−T − (u), T + (u)) with T + (u), T − (u) denoting respectively the forward, backward maximal time of existence (in this class): see again [11] for more detail. The next step is to understand the global behavior of (1.3). Classification of radial solutions of (1.1) was studied for E(u 0 ) < E(W ) in [11], and that for E(u 0 ) = E(W ) in [6]. These results are summarized as follows: For u 0 ∈Ḣ 1 radial with E(u 0 ) ≤ E(W ), • If ∇u 0 L 2 < ∇W L 2 , then the solution is either W − up to symmetry, or scattering as t → ±∞, i.e., T ± (u) = ∞ and there exist u ± ∈Ḣ 1 such that lim t→±∞ u(t) − e −it∆ u ± Ḣ1 = 0.
The goal of this paper is to classify the global behavior of solutions with slightly more energy than the ground states. Our main result is the following. Let (1.13) Theorem 1.1. There is an absolute constant ǫ ⋆ ∈ (0, 1) such that for each ǫ ∈ (0, ǫ ⋆ ], there exist a relatively closed set X ǫ ⊂ H ǫ , and a continuous function Θ : H ǫ \ X ǫ → {±1}, with the following properties. W ⊂ X ǫ ⊂ B Cǫ (W) for some absolute constant C ∈ (0, ∞). The values of Θ are independent of ǫ. For each u 0 ∈ H ǫ and the solution u of (1.1), I 0 (u) := {t ∈ I(u) | u(t) ∈ X ǫ } (1.14) is either empty or an interval. Hence I(u) \ I 0 (u) consists of at most two open intervals. Let σ ∈ {±}. If Θ(u(t)) = +1 for t close to T σ (u), then u 0 ∈ S σ . If Θ(u(t)) = −1 for t close to T σ (u) and u 0 ∈ L 2 (R 3 ), then u 0 ∈ B σ .
In other words, every solution with energy less than E(W ) + ǫ 2 can stay in X ǫ only for an interval of time, though it can be the entire existence time. Once the solution gets out of X ǫ , it has to either scatter or blow-up, according to the sign function Θ(u), though we need an additional condition u 0 ∈ L 2 (R 3 ) to ensure the blow-up.
The above properties hold in both the time directions. Concerning the relation between forward and backward dynamics, we have has non-empty interior in H ǫ ∩ L 2 (R 3 ).
In particular, there are infinitely many solutions which scatter on one side of time and blow up on the other. According to the previous theorem, such transition can occur only by changing Θ(u) from +1 to −1 or vice versa, going through O(ǫ) neighborhood of W, but the change is allowed at most once for each solution. Note however that there may well exist blow-up inside the neighborhood of W, as the equation is energy-critical. It is indeed the case for the energy-critical wave equation. More precise dynamics around W should be studied elsewhere.
In the proof of the above results, we will explicitly construct, in terms of the eigenfunctions of the linearized operator, the functionalsd W and Θ, as well as open initial data sets in the 4 intersections. Now we explain the main ideas of this paper and how it is organized. The proof of Theorem 1.1 relies upon a strategy that was pioneered by the first author and Schlag in [17] in the study of the nonlinear Klein-Gordon equation with the focusing cubic nonlinearity. It relies upon two components: an ejection lemma and a one-pass lemma.
The ejection lemma aims at describing the dynamics of the solution when it is in the exit mode, i.e., when it is close to W and moving away from it. By analogy with the dynamics of solutions of linear differential equations, we would like its dynamics to be ruled by that of its unstable eigenmode of the linearized operator around W. In order to verify this statement, one has to control the orthogonal component of the spectral decomposition of the remainder resulting from the linearization around W. We would like to control this component by using the quadratic terms resulting from the Taylor expansion of the energy around W. This can be done if and only if the remainder satisfies two orthogonality conditions: see Proposition 3.4. In order to satisfy these conditions, one has to give two degrees of freedom to the decomposition of the solution around W: a rotation parameter (this was done in [19]) and a scaling parameter: see Propositions 3.1 and 3.3. Then, one also has to control the evolution of these two parameters. We prove in Proposition 3.7 that we can close the argument. More precisely the dynamic of the solution close to W and in the exit mode is dominated by the exponential growth of the unstable eigenmode; moreover, a relevant functional (denoted by K) grows exponentially and its sign eventually becomes opposite to that of the eigenmode.
The one-pass lemma (see Proposition 3.11 and Section 11 for more details) aims at classifying the fate of the solution. A direct consequence of this lemma is the dichotomy described in the statement of Theorem 1.1. It shows that the orbit cannot cross a neighborhood of W more than once. The proof is by contradiction. Assuming that the solution crosses this neighborhood more than once, then it means that the solution is at two different times t a , t b close to W and in the ejection mode (forward and backward respectively in time). So we can apply the ejection lemma as long as we are not so far from W and then variational estimates (see Proposition 3.9) far from W. The contradiction appears when we integrate by part a localized virial identity (11.10). The left-hand side is much smaller than the right-hand side thanks to the exponential growth of a relevant functional (denoted by K) in the ejection mode and variational estimates far from W. The process involves a parameter m, which is the cut-off radius for the localization. Notice that unlike the subcritical case, one has to take into account the scaling parameter defined by the ground states to which the solution is close to. This requires a much more complicated analysis since we have no control of this parameter. It is also harder than the energy-critical wave equation, for which it is easy to localize virial and energy estimates in space-time, thanks to the finite speed of propagation (see [12]). Indeed, this part of analysis is the main novelty of this paper. In the case where Θ(u(t)) = +1 after ejection (see Section 11), one introduces a radius of the concentration of the kinetic part of the energy (see definition of m + V ) and the hyperbolic parameter (see definition of m H ), estimates K along with some error terms (generated by the cut-off) in the hyperbolic region and the variational region, and compares these estimates. In the worst scenario, one proves a decay estimate (see (11.65)) in the variational region and uses this estimate to implement Bourgain's energy induction method [3]: this allows to construct a solution whose energy is smaller than the original one by a nontrivial amount (in particular it is smaller than that of the ground states), then the theory below the ground state energy (see [11]) implies that it is not close to the ground states, neither is the original solution by a perturbation argument, contradicting the assumption of returning orbit. In the case where Θ(u(t)) = −1 after ejection, we introduce a threshold (see definition of m − V ) that allows to compare K with the main part of the virial identity; then, by integrating the virial identity, we can prove that this threshold must be very large; then, by proving a decay estimate, one can show that this threshold is not so large, which leads to a contradiction.
The fate of the solution depends on Θ(u(t)) when u is ejected. If Θ(u(t)) = −1 and u 0 ∈ L 2 then we prove that it blows up in finite time; if Θ(u(t)) = +1 then we prove that it is scattering: see Section 13. The scattering is proved by a modification of Kenig-Merle approach [11] and arguments from [19]. Unlike the subcritical case, one has to deal with possible blow-up in finite time, although theḢ 1 norm is bounded. The proof is by contradiction. Assuming that scattering fails, then one can find a critical level of energy above which scattering does not hold for solutions that are far from the ground states and Θ(u) = +1. But this means that there exists a sequence (u n ) n≥1 that satisfies the properties that we have just mentioned (in fact, the distance can be upgraded from far to very far, by appealing to the ejection lemma), and, thanks to a concentration compactness procedure, the fact that the energy of u n is just above that of the ground states, one can construct a critical element U c that does not scatter, has energy equal to the critical level of energy, is far from the ground states, and satisfies Θ(U c ) = +1. Moreover its orbit is precompact up to scaling. By using Kenig-Merle's arguments, one sees that U c does not exist.
Acknowledgments : The second author would like to thank W. Schlag and T. Duyckaerts for interesting discussions related to this problem while he visited University of Chicago in April 2012 and IHP in June 2012. The second author was supported by a JSPS fellowship.

Notation
In this section, we set up some notation that appear in this paper. If x is a complex number then x = ℜ(x) + iℑ(x) = x 1 + ix 2 . Here ℜ(x) and x 1 (resp. ℑ(x) and x 2 ) denote the real part (resp. the imaginary part) of x. Given x, y two real numbers, x y (resp. ) means that there exists a universal constant C > 0 such that x ≤ Cy (resp. x ≥ Cy). For any function f on R or [0, ∞), and for any m ∈ (0, ∞), we denote by f m the following rescaled function L p x = L p denotes the standard L p space on R 3 . Some estimates that we establish in this paper require the Littlewood-Paley technology, which we set up now. The Fourier transform of ϕ ∈ S ′ (R 3 ) is denoted by ϕ. Let φ : R → [0, ∞) be a smooth even function satisfying tφ ′ (t) ≤ 0 and The complement of this smooth cut-off is denoted by For any m > 0, Littlewood-Paley operators P <m , P ≥m and P m are defined by The following functionals onḢ 1 (R 3 ) play crucial roles in variational arguments.
It follows from (1.12) and similar arguments to [17] that Let S σ a be the one-parameter group of dilation operators defined as follows S σ a f (x) := e (3/2+a)σ f (e σ x), (2.11) and let S ′ a := ∂ σ S σ a | σ=0 be its generator. It is easy to see that the adjoint is given by (S σ a ) * = S −σ −a , hence by differentiating in σ, We denote by S, W ,S and N the following mixed L p spaces on R 1+3 S := L 10 t,x (R 1+3 ), W := L 10 t (R; L 30/13 x (R 3 )), (2.13) We also use the homogeneous Sobolev spaces defined by completion of the Schwarz space with respect to the norm v X 1 := ∇v X (X = W,S, N ). (2.14) For any interval J ⊂ R and any function space X on R 1+3 , the restriction of X onto J is denoted by X(J). The Sobolev embedding implies We recall the L p decay and the Strichartz estimates (see e.g., [9]). For any p ∈ [2, ∞] 16) where p ′ = p/(p − 1), and for any interval J ∋ 0, In this paper, we constantly use the linearized operator L defined by We recall some spectral properties of L (see [6,8,22]): • It has two resonance functions iW and W ′ .
To see existence of such χ, suppose for contradiction that contradicting the slow decay of W ′ , W by the rapid decay of g 1 , g 2 . Therefore {g 1 , g 2 } ⊥ is not included in {W ′ } ⊥ nor {W } ⊥ . The same conclusion holds under the radial restriction, since orthogonality against radial functions is determined by the spherical average. Hence there exists χ ∈ S radial (R 3 ) satisfying (2.19).

Proof of the main theorem
The proof of Theorem 1.1 relies upon some propositions stated below. The first proposition, proved in Section 4, gives a decomposition of a vector ϕ ∈Ḣ 1 close to the ground states W, taking account of two parameters (the rotation parameter and the scaling parameter) and a constraint (the so-called orthogonality condition). Proposition 3.1 (Orthogonal decomposition of ϕ). There exist an absolute constant 0 < δ E ≪ 1 and a C 1 function (θ,σ) : B δE (W) → (R/2πZ) × R with the following properties. For any ϕ ∈ B δE (W), putting Moreover (θ(ϕ),σ(ϕ)) ∈ (R/2πZ)×R is unique for the above property. Furthermore, if ϕ − W θ,σ Ḣ1 ≪ 1 for some (θ, σ) ∈ R 2 , then The second proposition, proved in Section 5, describes more precisely the decomposition in Proposition 3.1, taking into account the spectral properties of L. This decomposition does not use the radial symmetry.
The third proposition, also proved in Section 5, aims at describing the dynamics of the solution near the ground states, using the decomposition in Proposition 3.1. Again, this does not use the radial symmetry.
Then, letting τ : I → R such that τ ′ (t) := e 2σ(t) , we have where θ τ = ∂θ ∂τ etc., and ) and, decomposing v by Proposition 3.2, (3.11) or equivalently, The next proposition, proved in Section 4, shows that the orthogonal direction γ of v in (3.4) can be controlled by the linearized energy: Proposition 3.4 (Control of orthogonal direction). For any function w ∈Ḣ 1 radial satisfying w 1 , g 2 = 0, we have Hence in the subspace {v ∈Ḣ 1 | (v|χ) = 0}, we can define an equivalent norm E using the decomposition of Proposition 3.2 In particular, in the decomposition of Proposition 3.1, we have Henceforth, we assume that whenever a solution u of (1.1) is in B δE (W), the coordinates σ, θ, v, λ ± , λ 1 , λ 2 and γ are defined by (3.8), (3.4) and (3.7), while τ (t) is a solution ofτ (t) = e 2σ . In short, (3.16) The next proposition, proved in Section 7, ensures the existence of a solution u of (1.1) in a neighborhood of W as long as the scaling parameter σ is bounded from above. Proposition 3.5 (Uniform local existence in τ ). There exists an absolute constant δ L ∈ (0, δ E /2) such that for any solution u of (1. Now we are ready to define the nonlinear distanced W . Let ϕ ∈ B δE (W). Consider the decomposition (3.1) of ϕ. Then we define a local distance d 0 : As observed in [19], this is close to be convex in τ when the solution is ejected out of a small neighborhood of W, but it may have small oscillation around minima in τ . This is a difference for the Schrödinger equation from the Klein-Gordon equation, for which d 2 0 is strictly convex (see [17]). We could treat the possible oscillation as in [19] by waiting for a short time before the exponential instability dominates, which would however bring a certain amount of complication to the statements as well as the proof.
Here instead, we introduce a dynamical mollification of d 2 0 , which yields a strictly convex function in τ . The same argument works in the subcritical setting as in [19]. Let u be the solution of (1.1) with initial data u(0) := ϕ ∈ B 2δL (W). Then Proposition 3.5 ensures that u exists at least for |τ − τ (0)| ≤ 2 in B δE (W). Using the decomposition (3.16) with τ (0) := 0, let where φ is the cut-off function in (2.2). This defines the function The following proposition, proved in Section 8, gives the main static properties of the distance function.
Proposition 3.6 (Nonlinear distance function). The functionald W onḢ 1 radial is invariant for the rotation and scaling, and equivalent to d W . Precisely, there exists an absolute constant C ∈ (1, ∞) such that for all ϕ ∈Ḣ 1 radial and (α, β) ∈ R 2 , Moreover, there exists an absolute constant c D ∈ (0, 1) such that puttinǧ Hence we can used W (ϕ) to measure the distance to W, instead of the standard d W (ϕ). The δ neighborhood with respect to this distance function is denoted bỹ The next proposition, proved in Section 9, describes the dynamics close to the ground states in the ejection mode: Proposition 3.7 (Dynamics in the ejection mode). There is an absolute constant δ X ∈ (0, 1) such thatB δX (W) ⊂ B δL (W), and that for any solution u of (1.1) with sign(λ 1 (t)) is constant, and there exists an absolute constant C K > 0 such that Remark 3.8. By time-reversal symmetry 1 , a similar result holds in the negative time direction, where the last condition of (3.25) is replaced with ∂ tdW (u(t)) ≤ 0.
The next proposition gives a variational estimate away from the ground states, and it is a consequence of (2.8), cf. [13,20]. Proposition 3.9 (Variational estimates). There exist two increasing functions ǫ V and κ from (0, ∞) to (0, 1), and an absolute constant c V > 0, such that for any The next proposition, proved in Section 10, defines a functional Θ that decides the fate of the solution around t = T ± (u), as well as at the exit time t = t X in the above proposition. Proposition 3.10 (Sign functional). There exist an absolute constant ǫ S ∈ (0, 1) and a continuous function Θ : H ǫS ∩Ȟ → {±1}, such that for some 0 < δ 1 < δ 2 < δ X and for any ϕ ∈ H ǫS ∩Ȟ, with the convention sign 0 = +1, Note that the region {ϕ ∈ H ǫS ∩Ȟ | Θ(ϕ) = +1} is bounded inḢ 1 , because but it does not imply global existence for solutions staying in this region, because of the critical nature of (1.1).
The continuity of Θ implies that for any solution u in H ǫ ⊂ H ǫS , Θ(u) ∈ {±1} can change along t ∈ I(u) only if u goes through the small neighborhood H ǫ \ H ⊂B ǫ/cD (W). The next proposition, proved in Section 11, implies that such a transition can happen at most once for each solution.
Proposition 3.11 (One-pass). There exist an absolute constant δ B ∈ (0, δ X ) and an increasing function where Θ is defined by Proposition 3.10.
The above proposition tells that if a solution gets out ofB δ (W), then it can never return there. Moreover, it applies to all δ ∈ (0, δ B ] satisfying E(u) < E(W )+ǫ B (δ) 2 . The solution u stays around W iff t + (δ) = T + (u). The following proposition, proved in Section 12, gives more precise description of such solutions. Proposition 3.14. Under the assumption of Proposition 3.11, suppose that t + (δ) = T + (u). Then there exists We have similar statements in the case t − (δ) = T − (u) by the time-reversal symmetry.
The next proposition, proved in Section 13, describes the asymptotic behavior of solutions which are away from the ground states.

Proposition 3.15 (Asymptotic behavior).
There exists an increasing function ǫ * : x at some t ∈ [t 0 , T + (u)), then T + (u) < ∞. By time-reversal symmetry, the same statements hold for the negative Note that by Remark 3.13 and ǫ * ≤ ǫ B , Θ(u(t)) is well defined for all t ∈ [t 0 , T + (u)) in the above statement.
Armed with the above propositions, we are now ready to prove Theorem 1.1.
Take any δ ∈ (0, δ B ] satisfying ǫ ≤ ǫ * (δ). First suppose that there exists t 0 ∈ I C (u) such thatd If the last condition of (3.36) is replaced with ∂ tdW (u(t 0 )) ≤ 0, then the time reversed version of the above argument implies that (−T − (u), t 0 ) ⊂ I C (u) and the behavior of u towards −T − (u) is determined by Θ(u(t)) there.
Next consider the case where there exist t 1 ∈ I 0 (u) and t 2 ∈ I C (u). Suppose that t 1 < t 2 . Sinced W (u) ≤ ǫ/c D < δ on I 0 (u), we may assumed W (u(t 2 )) < δ by decreasing t 2 if necessary. Then the above argument works with t 0 := t 2 , either forward or backward in time, but the latter case leads to a contradiction with the existence of t 1 ∈ I 0 (u) smaller than t 2 . Hence we have ∂ tdW (u(t 2 )) ≥ 0 and [t 2 , T + (u)) ⊂ I C (u). If t 1 > t 2 , then in the same way, we deduce that (−T − (u), t 2 ] ⊂ I C (u). Therefore, I 0 (u) is either empty or an interval.
Concerning the behavior of u towards T + (u), it only remains to consider the following case: I(u) = I C (u) but (3.36) is never satisfied by any t 0 ∈ I(u). In this case, there are only two possibilities: eitherd W (u(t)) ≥ δ all over I(u), or d W (u(t)) goes below δ and then stays there. In the former case, we can apply Proposition 3.15 to decide the behavior around T ± (u). In the latter case, we can apply Proposition 3.14 on some interval [t 0 , T + (u)) whered W (u) < δ. Then lim sup contradicting I(u) = I C (u). This completes the investigation around T + (u), and the behavior towards −T − (u) is treated in the same way. Theorem 1.1 is proved.
Remark 3.16. The same argument as above works if we replace X ǫ with which is smaller and essentially independent of ǫ. In that case, however, we need to modify our conclusion for the special solutions W ± constructed by Duyckaerts and Merle [6] on the threshold E(u) = E(W ), namely those two solutions (unique modulo the invariance) which are exponentially convergent to W as t → ∞, and scattering or blowing up in t < 0. These solutions are in H ǫ ∩Ȟ for all t ∈ I(u) and ǫ > 0, where Θ = ±1 according to its behavior in t < 0. Thus Θ fails to give the correct prediction for t > 0 in this case. This is exactly the case t 0 = t 1 = T + (u) in Proposition 3.14, namelyd W (u) ց 0 as t ր T + (u). The classification in [6] also implies that it happens only for those special solutions. In other words, X ǫ has been enlarged fromX ǫ in order to eliminate those solutions.

Orthogonal decomposition
In this section, we prove Proposition 3.1. Define a C 1 function F : the implicit function theorem yields δ > 0 and a C 1 function (θ, σ) : To see the uniqueness of (θ,σ) for each ϕ ∈ B δ (W), suppose that we have two ways of decomposition Then by (4.6), we obtain hence the uniqueness of (θ,σ).

Evolution around the ground states
In this section, we prove Proposition 3.2 and Proposition 3.3. First we show that we can normalize g + and g − such that (3.5) holds. Since Hence it is enough to show W, g 2 = 0 (see [19] for a similar argument). Suppose for contradiction that W, g 2 = 0. Then 4 W 5 , g 1 = − L + W, g 1 = W, µg 2 = 0. Let 0 < δ ≪ 1 and v = αW + δg 1 with α = O(δ 2 ) to be chosen shortly. By expansion of the energy and by expansion of K Hence W, g 2 = 0 and we can normalize g ± such that (3.5) holds. Then (3.6) and (3.7) are immediate consequences. Thus we obtain Proposition 3.2.
Next, injecting the decomposition (3.8) into the equation (1.1), we obtain, after straightforward computations using that W θ,σ is a real-valued stationary solution, Applying the change of variable t → τ withτ = e 2σ to the above yields (3.9).

Control by the linearized energy
In this section, we prove Proposition 3.4. First we prove that, for any f ∈Ḣ 1 radial , If the above fails, then for all (a, b) ∈ R 2 , So L + is negative on a two dimensional subspace, which contradicts the fact that L + has only one negative eigenvalue. Hence (6.1) holds.

Nonlinear distance function
In this section, we prove Proposition 3.6. First we proved W ∼ d W . Sinced W = d W for d W (ϕ) ≥ 2δ L , it suffices to consider the case d W (ϕ) ≤ 2δ L . Decompose ϕ by Propositions 3.1 and 3.2. Then we have where we used (3.5). Here C(·) denotes the following functional onḢ 1 : whose Fréchet derivative is N (v). Hence, using Proposition 3.4, Then by Proposition 3.5, we have d 1 (ϕ) ∼ d W (ϕ), and sod W (ϕ) ∼ d W (ϕ).
Hence v is invariant in the rescaled time, namely (8.8) which is inherited by λ * and γ. Therefore d 0 and d 1 are invariant, so isd W .
Next we show the more precise exponential behavior.

One-pass lemma
In this section we prove Proposition 3.11. 11.1. Setting. Let δ, ǫ > 0 and let u be a solution satisfying u(t 0 ) ∈ H ǫ ∩B δ (W) at some t 0 ∈ I(u). The solution u is fixed for the rest of proof, so we denote for brevity,d (t) :=d W (u(t)). (11.1) We will define shortly the hyperbolic and the variational regions in H ǫ . In order to distinguish them, we use small parameters δ V , δ M ∈ (0, δ X ], which will be fixed as absolute constants in the end. First we impose the following upper bounds on δ and ǫ Since ǫ ≤ ǫ S and ǫ < c D δ, we have Put t a := sup{t 1 ∈ (t 0 , T + (u)) | t 0 < t ≤ t 1 =⇒d(t) < δ}. Sinced(t 0 ) < δ, we have t a ∈ (t 0 , T + (u)]. If t a = T + (u) then (3.33) holds with t + = T + (u). Hence we may assume without loss of generality that t a < T + (u) and sod(t a ) = δ. If {t ∈ (t a , T + (u)) |d(t) ≤ δ} is empty, then (3.33) holds with t + = t a . If not, let t b := inf{t ∈ (t a , T + (u)) |d(t) ≤ δ}. Applying Proposition 3.7 at t = t a implies t a < t b . Thus in the remaining case, we have (11.4) from which we will derive a contradiction for small δ > 0 and for small ǫ > 0 with δ−dependent smallness.

Hyperbolic and variational regions.
In particular, the coordinates σ, θ, v, λ + , λ − , λ 1 , λ 2 , γ and τ are defined on I H . Since u is fixed, we regard those as functions of t ∈ I H in the rest of proof.
The soliton size on I H is measured by where E m (t) := m≤|x|≤2m |∇u| 2 + |u/r| 2 + |u| 6 dx. (11.11) Using the decomposition (3.8), (3.14), and Proposition 3.5, placing W σ , S σ −1 v in L 6 x , and placing We need to estimate K(φ m u) and E m on I H for m to be chosen properly. Using the scale invariance of K, (9.11), and Sobolev's inequality as well, we obtain (11.13) wherem(t) := me σ(t) . The decay of W implies W 5 Plugging these into the above yields, for s ∈ L and t ∈ I[s], . (11.14) Similarly, the decomposition (3.5) and Hardy inequality yield for t ∈ I H |x|>m (|∇u| 2 + |u/r| 2 + |u| 6 )dx Putting these estimates into (11.10) yields, on each I[s], Then using the hyperbolic dynamics of λ 1 andτ = e 2σ ∼ e 2σ(s) , we obtain, with some absolute constant C H ≥ 1, In order to control the cut-off error in I V , we introduce (11.19) where the equivalence follows from Hardy's inequality and (11.20) there exists m ∈ (m 0 , m 1 ) for any m 1 > m 0 > 0 such that . (11.21) 11.4. Blow-up region. We start with the simpler case Θ(u) = −1, where the solution will blow up. First considerV m on I V . Since ǫ < ǫ V (δ V ) andd > δ V on I V , Proposition 3.9 implies In order to estimateV m with K(u) on I V , the optimal cut-off radius is given by For any m ≥ m − V , we have from (11.10) and (11.22), (11.25) The last term can be absorbed by the other, using (11.21). Hence for any m 0 ≥ m − V , there exists m ∈ (m 0 , C 1 (δ V )m 0 ) for some constant C 1 (δ V ) > 1 such that The minimal m > 0 satisfying this and (11.18) satisfies (11.27) for which (11.12) (11.28) Now we impose an upper bound on δ by the condition (11.29) which is equivalent to Then the last term in (11.28) is absorbed by the first one, hence In order to bound m − V , we use the equation for |u| 2 : ∂ t |u| 2 = 2ℑ (∇ · (∇uū)) . (11.32) Multiplying it with φ C m/2 /r 2 and integrating on any interval J ⊂ I V , we obtain (11.37) Plugging (11.35), we obtain for t ∈ I V , Hence, imposing another upper bound on δ by the condition: (11.40) we see that (11.39) contradicts (11.31).
In conclusion, the smallness conditions on δ, ǫ in the case Θ = −1 are (11.30), (11.40), and (11.2), which determine δ B and ǫ B . 11.5. Scattering region. Now we consider the case Θ(u) = +1, where the solution will scatter. The argument is similar to that in the previous case, but more involved. In particular, we need several smallness conditions on δ M .
First observe that there exists an absolute constant C E ∼ 1 such that for all t ∈ I(u). Indeed, since E(u) ∼ E(W ) ∼ 1, the upper bound follows from (2.7), while the lower bound follows from E(ϕ) ∼ ϕ 2Ḣ 1 for small ϕ ∈Ḣ 1 . Next we estimate K(φ m u) on I V . Using (2.6) and Proposition 3.9, we see that Then using (1.12) and (1.9), we obtain

(11.44)
In order to decide the cut-off for I V , we put Note that δ V depends on δ M through the condition δ V ≪ δ M in (11.2), which allows us to determine C 2 in terms of δ V only. The minimal m > 0 satisfying both this and (11.18) must satisfy (11.47) for which (11.12) implies (11.48) Now we impose an upper bound on δ by the condition (11.49) which is equivalent to Then the last term in (11.48) is absorbed by the first one, and (11.51) Imposing another upper bound on δ bỹ To compare m H with |I V | 1/2 , use (11.21) with m 0 := m H + |I V | 1/2 and m 1 := m + V (δ M )/2. Then there exists m ∈ (m 0 , m 1 ) such that IV |x|>m m r (|∇u| 2 + |u| 6 + |u/r| 2 )dxdt δ 3 M |I V |, (11.54) because of (11.53). Using Hardy, we have Integrating its square over I V , and using the definition of m + V > 2m for the ∇u 2 L 2 (|x|<2m) term, and (11.54) for the the u/r 2 L 2 (m<|x|<2m) term, we obtain IV |x|<2m Using the radial Sobolev inequality, we have for any ϕ ∈Ḣ 1 radial and m > 0, ϕ 6 Inserting it to the above estimate yields IV |x|<2m [|∇u| 2 + |u/r| 2 + |u| 6 ]dxdt δ 3 M |I V |. (11.58) Noting that m X ≪ m H < m by (11.52) and m H < m 0 , we see from the above estimate that if m 2 H ≫ δ 2 M |I V |, then by (11.10) and (11.18) Next we estimate u/r L 2 (IV ×R 3 ) . By the same argument as for (11.33), we have for any interval J ⊂ I V , where we used m > m 0 > |I V | 1/2 and (11.54). On the other hand, we have from (11.15) and then (11.60), Then using (11.58) for the integral over |x| < 2m, we obtain Decomposing I V into its connected components, we obtain an interval I ⊂ I V such that ∂I ⊂ ∂I V and Now we resort to an argument by Bourgain [3], in order to reduce the problem to energy below the ground state E(W ). Although Bourgain in [3] treated the defocusing case, the perturbative argument works as well for the focusing equation (1.1) under the uniform bound (11.41) inḢ 1 , while the non-perturbative argument with the Morawetz estimate can be replaced with (11.65), as is shown below.
In order to apply the argument to the interval I, the first observation is Proof. Let t 0 := inf I ∈ ∂I ⊂ ∂I V . Then by the definition of I V , we haved(t 0 ) = δ M and Proposition 3.7 from t = t 0 yields some t 1 ∈ I V such thatd(t 1 ) = δ X and ∂ td (t) > 0 on (t 0 , t 1 ) ⊂ I. It suffices to show u S(t0,t1) 1. The scaling invariance reduces it to the case σ(t 0 ) = 0. Then δ M ≪ δ X andτ = e 2σ with (3.28) and (3.26) imply that t 1 > t 0 + 1. Put v := u − W . Then from the equation iv − ∆v = 5W 4 v 1 + iW 4 v 2 + N (v), (11.67) the embedding W 1 ⊂ S, and the Strichartz estimate (2.17), we obtain for any (11.68) Repeating this estimate from t = t 0 on consecutive small intervals, we obtain Hence as in [3], we can decompose the interval I such that for a small fixed constant η > 0. In the following, c denotes a small positive constant, and C(η) denotes a large positive constant which may depend on η, both allowed to change from line to line.
By the perturbation argument from Section 3 to (4.11) in [3], where the sign of nonlinearity is irrelevant, we have for each j, u S1 (Ij ) 1, (11.72) and there exist a subinterval I ′ j ⊂ I j and R j Combining it with (11.57) and (11.65 Hence there exists j ∈ {1, . . . , N } such that By the time reversal symmetry, we may assume without loss of generality Hence using (11.75) and (11.76), we obtain  Letw be the solution of (1.1) with initial dataw(t j+1 ) := w(t j+1 ). Then by the above estimates together with (11.77), (11.80), and (11.81), we get (11.83) where in the first and last steps, we imposed upper bounds on δ M and ǫ respectively: Similarly, plugging (11.77) into (11.82) yields Hence ∇w(t j+1 ) 2 L 2 < ∇W 2 L 2 and therefore by (2.9), K(w(t j+1 )) > 0. (11.87) Hence by the result of Kenig and Merle [11] below the ground state energy, (11.41) and (11.85),w scatters in both time directions with a uniform Strichartz bound: In order to control u by this, we use the long-time perturbation [11, Theorem 2.14]: Lemma 11.1 ( [11]). Let u be a solution of (1.1). Let I be an interval with some t 0 ∈ I ∩ I(u). Let e ∈ N 1 (I) and letũ ∈ C(I;Ḣ 1 ) be a solution of i∂ tũ − △ũ = |ũ| 4ũ + e. (11.89) Assume that for some B 1 , B 2 , B 3 > 0 91) then I ⊂ I(u) and where the implicit constants depend on B 1 , B 2 , B 3 .
13.2. Scattering after ejection. In the case of Θ(u) = +1, the proof of scattering uses arguments from [11] with arguments from [19]. Unlike the subcritical case, we have to take account of the scaling parameter and the fact that the maximal time interval of existence might be finite, even though theḢ 1 norm is bounded by (3.32).
By (13.14) and (13.21), we have as n → ∞, hence we see that for all j ≥ 0 If E(U j ) < E(W ), then we conclude from K(U j ) ≥ 0 in a neighborhood of s j,∞ and [11] that U j exists globally in time and scatters with U j S∩W 1 ∩L ∞ tḢ Assuming that U j S < ∞ for all j = 0, 1, . . . , k, we apply Lemma 11.1 tõ u := k j=0 U j,n + γ k,n , u := u n (13.25) from t 0 := 0 on I := R. (2.7) and (13.23) imply thatũ is bounded in L ∞ tḢ 1 as n → ∞ uniformly in k. From (13.13), the orthogonality conditions (13.10) and a similar argument to that in the proof of Proposition 4.2 in [11],ũ is bounded in S as n → ∞ uniformly in k. (13.22) implies that ũ(0) − u n (0) Ḣ1 → 0 as n → ∞. Hence, in order to apply the lemma for large n, it suffices to make |U j,n | 4 U j,n − |ũ| 4ũ (13.26) small in N 1 (R). Indeed, using U j S∩W 1 < ∞ and the orthogonality conditions (13.10), as well as (13.12), we obtain For a proof, we refer again to Proposition 4.2 in [11] and the references therein (in particular [10]). Thus for large k and large n, Lemma 11.1 yields a bound on u n S(R) uniform in n, contradicting u n S(0,T+(un)) → ∞.
Since E(u n ) → E c = E(U 0 ) and K(U j ) ≥ 0, by (2.7) and (13.23) we have U j = 0 for all j ≥ 1 and γ k,n → 0 inḢ 1 . Hence u n (0) = S −σ0,n −1 e −is0,n∆ V 0 + o(1) inḢ 1 . Hence e isn△ S σn −1 u n (0) → V 0 with σ n := σ 0,n and s n := s 0,n . The following is a corollary of the last part. Proof. If there is no such σ c , then there exists {t n } n≥1 and η > 0 such that inf σ∈R S σ −1 U c (t n ) − U c (t n ′ ) Ḣ1 ≥ η (13.34) for all n = n ′ . Notice that we must have, after possibly passing to a subsequence, t n → T + (U c ): otherwise, we get a contradiction from (13.34) with σ 0 = 0 by continuity of U c (t). Applying Lemma 13.2 to u n (t) := U c (t + t n ) yields a sequence (σ n , s n ) ∈ R 2 such that e isn∆ S σn −1 U c (t n ) is strongly convergent inḢ 1 . After possibly passing to a subsequence, we may assume that s n converges to some s ∞ ∈ [−∞, ∞]. If s ∞ ∈ R, then S σn −1 U c (t n ) is also convergent, contradicting (13.34) for n, n ′ → ∞. If s ∞ = −∞, then U c (t + t n ) S(0,∞) → 0, and if s ∞ = ∞, then U c (t + t n ) S(−∞,0) → 0, since the free solutions with the same data as U c at t = t n are vanishing in that way: see [11] for more detail. In either case, it contradicts U c S(0,T+(Uc)) = ∞.
We are now ready for the final step of the proof of Proposition 3.15.
Claim 13.4. U c does not exist.
Proof. First we consider the case T + (U c ) < ∞. The local wellposedness theory, together with the precompactness of K , implies that blow-up is possible only by concentration σ c (t) → ∞ as t ր T + (U c ) < ∞, see [11] for a proof. For any m > 0 and t ∈ I(U c ), put |T + (U c ) − t| and so U c (0) ∈ L 2 x . Hence by the L 2 conservation, we get U c (0) L 2 = U c (t) L 2 → 0 as t ր T + (U c ). So U c = 0 and it contradicts T + (U c ) < ∞.
Thus we have obtained another critical elementŪ ω (−t), that is the time inversion of U ω , with the scale bound σ ω ≥ 0. Hence the above argument for A > −∞ applied to this new critical element yields a contradiction.