Center-stable manifold of the ground state in the energy space for the critical wave equation

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.


Introduction
We study global dynamics of the critical wave equation (CW) u − ∆u = f ′ (u) := |u| 2 * −2 u, 2 * := 2d d which also preserves the energy E(u λ ) = E(u), making (CW) special and critical. It also gives rise to the ground state solutions in the explicit form ∀λ > 0, −∆W λ + f ′ (W λ ) = 0, (1.8) which has the minimal energy among all the stationary solutions. The scale and translation invariance of (CW) generates a family of ground states as a smooth manifold in H with dimension 1 + d: and the more subtle type-II blow-up, for which u(t) H is bounded but u(t) fails to be strongly continuous in H beyond some t < ∞.
(1. 15) It was proved that if u ∈ C([0, T + ); H) is a strong solution up to the maximal existence time T + ∈ (0, ∞], which does not stay close to the ground state solitons near t = T + , then u either blows up away from the ground state solitons, or it scatters (to 0) as t → ∞. We have the same for t < 0, and moreover, the 2 × 2 combinations of scattering and blow-up in t > 0 and in t < 0 respectively are realized by initial-data sets in H which have non-empty interior. The key ingredient for proof is the existence of a small neighborhood of the ground states such that any solution exiting from it can never come back again, called the one-pass theorem. A missing piece in the above result of [14] is the global dynamics around the ground states, compared with the corresponding results for the subcritical Klein-Gordon equation [21] and for the Schrödinger equation in the radial symmetry [20], where we have 3 × 3 complete classification of (1.14) including the scattering to the ground states on a center-stable manifold of codimension 1.
On the other hand, there have been many papers [16,17,9,10,4,1,3] for (CW) constructing various types of solutions around the ground states, including centerstable manifolds in some stronger topology than the energy space, on which the solutions scatter to the ground states, type-II blow-up at prescribed power law rate or with eternal oscillations between such rates, type-II blow-up at time infinity. The latter phenomena clearly distinguish (CW) from the dynamics of the subcritical equation.
In this paper, we construct a smooth center-stable manifold of codimension 1 in the energy space, which embraces all the solutions scattering to, or staying close to the ground state solitons. Indeed the last property is the defining characterization of the manifold. Plugging it into the result in [14], we complete the 3 × 3 classification for (CW) in the region (1.14), which is now described.
Denote H-distance to the ground states by dist W (ϕ) := inf{ u(t) − ψ H | ψ or −ψ ∈ S tatic (W )}, (1.16) and the time inversion for any initial data ϕ ∈ H and any initial data set A ⊂ H by (2) u(0) ∈ ±M, T < ∞ and lim t→T dist W ( u(t)) > δ > Cε. as t ր T . The gap between Cε and δ is actually huge in the proof. The above theorem except for the existence of M was essentially proved in [14]. Next we can exploit the Lorentz transform to include all the ground state solitons. Let u be any solution with E( u(0)) < |E(W ) + ε 2 | 2 + |P ( u(0))| 2 and let T ∈ (0, ∞] be its maximal existence time. Then we have only one of the following (1)- (3).
(3) u(0) ∈ ±M L and u(t) ∈ ±O 1 for all t near T . The initial data sets and M L enjoy the same properties as in the previous theorem.
The manifolds M and M L are center-stable manifolds of S tatic (W ) and S oliton (W ) respectively, but they contain solutions blowing up in finite time. The invariance by the flow should be understood that the solutions starting on the manifold stay there as long as they exist, and similarly for the Lorentz transform. Again by [6], in case (3) with T < ∞, we have a smooth (λ, c), p ∈ R d and ϕ ∈ H such that, as t ր T , Our argument to construct the center-stable manifold is somehow similar to the numerical bisection in [24,2], where the center-stable manifold was searched for as the threshold between scattering and blowup. Indeed, our proof does not touch the delicate dynamics of those solutions on the manifold, but relies on the behavior of those off the manifold. In particular, we do not need any dispersive estimate on the linearized operator as in [16,21,20,1], which makes our proof much simpler. In this respect, it is similar to [22] in the subcritical case. On the other hand, the criticality or the concentration phenomenon forces us to work in the space-time rescaled according to the solution itself. For that part we employ the same argument as in the previous paper [14].
The next question is if we can remove the energy restriction (1.14). Concerning it, Duyckaerts, Kenig and Merle [8] recently established an outstanding result of asymptotic soliton resolution for d = 3: Every solution with radial symmetry is either type-I blow-up, or decomposes into a sum of ground states with time-dependent scaling and a free solution (1.20) where T is the maximal existence time of u. Given this expansion, one might expect that the dividing manifold of dynamics could be extended as the collection of all such solutions with N > 0. However, it is very hard to prove such a statement even if we know the above asymptotics, because of the instability of the ground state. Moreover, one can easily observe that the above naive guess is not correct when T < ∞ and the energy is larger, as one can construct such blow-up solutions in the deep interior of blow-up solutions, by using finite speed of propagation (see Appendix A). Instead of pursuing that approach, we extend our center-stable manifold globally by adding large radiation, thereby including at least all solutions (1.20) with N = 1 and T = ∞, as well as some of them with T < ∞. A simple procedure is proposed to reduce the analysis to the previous case E < E(W )+ε 2 by detaching large radiation, which relies on the asymptotic Huygens principle, valid for all d ∈ N and without radial symmetry. The extended manifold splits the energy space into the scattering and blow-up regions locally around itself, although the entire dynamical picture is still far beyond our analysis.  (2) u(0) ∈ ±M D , T < ∞ and u(t) ∈ ±O 4 for t near T .
More detailed statements are given in the main body of paper. This paper is organized as follows. In the rest of this section, we introduce some notation and coordinates, together with a few basic facts and estimates, mostly overlapping with the previous paper [14].
In Part I starting with Section 2, we deal with the solutions with energy slightly above the ground state. The center-stable manifold is constructed as a threshold between scattering and blowup, which completes the 9-set classification of dynamics, in a form similar to the subcritical case [21]. The main new ingredient is the ignition Lemma 2.2, which roughly says that for any solution staying close to the ground states, any arbitrarily small perturbation in the unstable direction eventually leads to the ejection from a small neighborhood as in the ejection lemma of [14]. These are extended by the Lorentz transform in the end of Section 3.
In Part II starting with Section 4, we extend the results in Part I to large energy by adding out-going radiation. The extended manifold contains all the solutions scattering to the ground state solitons, while it is still a dynamical threshold between the scattering and the blowup. The main ingredient is the detaching Lemma 4.4, which allows one to detach out-going radiation energy from a solution to produce another solution with smaller energy but the same behavior. We also extend the one-pass theorem of [14] by allowing out-going large radiation.
1.1. Strichartz norms and strong solutions. For any I ⊂ R, we use the Strichartz norms for the wave equation with the following exponents where q ′ := q/(q − 1) denotes the Hölder conjugate. Slightly abusing the notation, we often apply these norms to the first component of vector functions such as (1.23) The small data theory using the Strichartz estimate implies that there is a small ε S > 0 such that for any T > 0 and ϕ ∈ H satisfying there is a unique solution u of (CW) on [0, T ) satisfying which is scattering to 0 if T = ∞. The uniqueness holds in and for any a ∈ R, (S σ a ϕ)(x) := e σ(d/2+a) ϕ(e σ x). Their generators are T ′ = −∇, S ′ = S ′ −1 ⊗ S ′ 0 , and S ′ a = r∂ r + d/2 + a. For any u ∈ S olution (I) and (σ, c) ∈ R 1+d , T c S σ u(e σ t) ∈ S olution (e −σ I). (1.29) The linearization around the ground state W is written by the operator as well as the nonlinear term The matrix version of the linearization is given by JL with The generator of each invariant transform of (CW) gives rise to generalized null vectors, namely with A = S ′ −1 or T ′ = −∇, It is well known that for the ground state W , there is no other generalized null vector. Note however that AW is not an eigenfunction but a threshold resonance, i.e. AW ∈ L 2 for d ≤ 4. Besides L −1 + (0) = {∇W } and the absolutely continuous spectrum [0, ∞), L + has only one negative eigenvalue and the ground state, for which the orthogonal subspace and projection are denoted by Henceforth, the L 2 inner product is denoted by and for vector functions (ϕ 1 , ϕ 2 )|(ψ 1 , ψ 2 ) := ϕ 1 |ψ 1 + ϕ 2 |ψ 2 ∈ R, which may be applied to column vectors as well as higher dimensional vectors. Throughout the paper, a pair with a comma (·, ·) denotes a vector, but never an inner product.

1.3.
Coordinates around the ground states. We recall from [14] our dynamical coordinates for the solution u around the ground states S tatic (W ): where v(t) = (v 1 (t), v 2 (t)) ∈ H does not generally satisfy v 2 (t) =v 1 (t) because of the modulation (σ(t), c(t)). The unstable and stable modes are denoted by λ ± : for which we introduce linear functionals Λ ± : H → R by If u(t) is close to S tatic (W ), then we can uniquely choose (σ(t), c(t)) such that the orthogonality condition holds 2 by the implicit function theorem. Note that it is not preserved by the linearized equation, since neither S ′ 0 ρ nor T ′ ρ is an eigenfunction of L + . The linearized energy norm E is defined on the entire H by ( (1.44) For any δ > 0, define open neighborhoods of 0 and S tatic (W ) in H, by Here is a precise statement on the orthogonality (1.42) Lemma 1.4. There exist δ Φ ∈ (0, 1) and a smooth map ( σ, c) : where we used the identity T c S σ a = S σ a T e σ c . Hence we have [14, (2.26)] for the identity), and where a W := −∆W |ρ /d = f ′ (W )|ρ /d > 0 and I denotes the identity matrix acting on R d . Then the implicit function theorem implies that there is a unique (s, y) ∈ R 1+d such that (α, µ)(s, y) = 0, |s − σ| + e σ |y − c| provided that δ > 0 is small enough. Since (α, µ) is obviously smooth in ψ, the implicit function is also smooth in Ψ σ,c (B δ ). For the uniqueness on N δ , suppose that Ψ σ,c (ψ) ∈ Ψ s,y (B δ ) for some (s, y) ∈ R 1+d , then Hence the uniqueness on N δ follows from the implicit function theorem.
Remark 1.1. Φ σ,c and Ψ σ,c are not smooth in (σ, c) for the γ component, since the derivative in (σ, c) induces S ′ γ and T ′ γ, which are not generally in H. Indeed Φ σ,c is continuous for (σ, c) at each fixed point on H, but not uniformly on any ball in H.
In [22] this was remedied by introducing a topology ("mobile distance") in which translations are also Lipschitz continuous. Instead of that, we will fix (σ, c) with respect to perturbation of the initial data, even though we modulate it in time.
Next we change the time variable from t to τ by (1.53) Then we get the equation of v as an evolution in τ : and differentiating the orthogonality condition (1.42) yields (1.55) This is linear in v (or γ), because the orthogonality (1.42) is not preserved by the linearized equation, a notable difference from the standard modulation analysis in the subcritical case. In the original time t, it yields For the eigenmode we have where we used α = 0 = µ only in the last step, since we will consider the case (α, µ) = 0 as well. In the unstable/stable modes, the equation reads We also recall the distance function d W : H → [0, ∞) defined in [14], which satisfies d W (ϕ) ∼ dist W (ϕ), and, for some constant for either sign ±.
Part I: Slightly above the ground state energy In the first part of paper, we study the global dynamics in the region E(u) < E(W ) + ε 2 , and its Lorentz extension, completing the picture in [14] with a centerstable manifold and the dynamics around it.

Center-stable manifold around the ground states
First we construct a center-stable manifold around the ground states S tatic (W ). This will be later extended in three ways: (1) By the backward flow, to the region E < E(W ) + ε 2 , (2) By the Lorentz transform, to the region E < E(W ) 2 + ε 4 + |P ( u)| 2 (3) By adding large radiation, which may have arbitrarily large energy. In order to define the manifold as a graph of (λ − , γ) → λ + , we define Then we have the trichotomy: (1) If λ + > m + (ϕ), then u blows up away from the ground states. More precisely

4)
and stays there for the rest of t < T .
Moreover, in the cases (1) and (3), there exists T X ∈ (0, T ) such that with the sign + for (1) and − for (3). In addition, m + (0) = m ′ + (0) = 0 and Obviously, the three asymptotics in (1)-(3) are distinctive. From the preceding results around the ground states, we know that the case (2) contains type-II blowup and global solutions scattering to the ground states. Type-I blowup is contained in the case (1), but it may also contain type-II blowup. (2.6) comes from the one-pass theorem proved in [14].
Thus we obtain a manifold of codimension 1 in H: which contains S tatic (W ) and is invariant by the forward flow within N δm,δm . It is also invariant by T and S by definition. Then (2.7) implies that it is tangent to the center-stable subspace of the linearized evolution at each point on S tatic (W ) = {Ψ σ,c (0)} σ,c ⊂ M 0 , and that M 0 is transverse to its time inversion M † 0 , since ϕ → ϕ † exchanges λ + and λ − . More explicitly is a local center manifold of codimension 2, on which every solution u satisfies . The solutions starting from the first set blow up away from the ground states in t < 0, while those starting from the second set scatters to 0 as t → −∞. A It is clearly non-degenerate in the direction T ϕ g + by (2.7). Moreover, M + > 0, M + = 0 and M + < 0 respectively give the trichotomy (1)-(3). The proof of the above theorem goes as follows. First we observe that if ϕ H ≪ |λ + | ≪ 1 then we can apply the ejection lemma and the one-pass theorem from [14], and obtain (1) for λ + ≫ ϕ H and (3) for −λ + ≫ ϕ H . Moreover, the ejection lemma implies that every solution ejected at t = T X from a small neighborhood of Hence there is at least one λ + in between, for which the solution is never ejected, i.e. the case (2). The uniqueness of such λ + follows also from the instability of W , or the exponential growth of the unstable component λ + for the difference of two solutions. The next section is devoted to its estimate, which is essentially the only ingredient in addition to [14].
As in [22], we abbreviate the differences by the following notation: for any symbol X and any function F .

2.1.
Igniting the unstable mode. In this subsection, we prove the following: For any solution trapped by the ground states, an arbitrarily small perturbation leads to the ejection from the small neighborhood unless the perturbation is almost zero in the unstable direction. More precisely, Then there exist t I ∈ (0, T ), λ + ∈ R, and u 1 ∈ S olution ([0, t I ]) such that (2.14) In particular, we have This lemma is proved by exponential growth in the unstable direction of the difference u 0 − u 1 in the rescaled coordinate for u 0 . It may take very long depending on the initial size of the perturbation, but in the rescaled time τ , where the solution u 0 is (forward) global in both the scattering and the blow-up cases. The difference is estimated mainly by the energy argument, rather than dispersive estimates. The nonlinearity is too strong to be controlled solely by Sobolev, for which we employ Strichartz norms which are uniform on unit intervals of τ .
Hence the main idea is similar to [22], but we do not use the mobile distance, but instead the same modulation parameters (σ(t), c(t)) for both u 0 and u 1 , in order to avoid destroying the energy structure for the difference. This is indeed much simpler, whereas the former idea seems hard to apply in the critical setting because of the change of time variable.
The main difference from the ejection lemma in [14] is that there is no bound on the time for the unstable mode to grow to some amount, and we estimate the difference of two solutions rather than the difference from the ground state. In particular, the equation for the difference naturally contains linear terms coming from the nonlinearity, which prevents us from a crude Duhamel argument as in [14].
Before starting the proof, we see that the Strichartz norms can be uniformly bounded on unit time intervals in the rescaled variables: for 0 < t < T . Then we have τ (T ) > η l and Proof. We obtain from the inequality on (σ ′ , c ′ ) that for 0 < η < 1, and w solves on (0, T ) Hence by Strichartz we have for small η > 0 In particular there is some η l < 1/8 such that the above estimates hold for η ≤ 2η l , and so τ (T ) > τ (2e −σ(0) η l ) > η l . Under the scaling property and similarly for v 2 and in L ∞ H. Hence putting and the last norm is bounded by which concludes the estimate on v.
For the difference, the same change of variable as in (2.24) yields for which leads to the estimate on v 1 − v.
In the following, we regard all the dynamical variables as functions of τ rather than t, unless explicitly specified. We have the equations for the difference Hence the third equation follows from the fourth one in (2.38). By the assumption, Suppose that for some τ 0 > 0 we have Choosing C I > 1 large enough, we have (2.41) at τ 0 = 0. We will prove that the second condition of (2.41) is preserved until the first one is broken. The linearized energy in (1.43) implies that at each time Using (1.55) as well, we derive from the difference equations (2.45) To control ⊳γ ⊲ , we use the linearized energy identity where W := f ′′ (W ). The terms on the right except for the first one are simply bounded by δ ⊳γ ⊲ H ⊳v ⊲ H . The nonlinear term can be bounded only via τ -integral.

2.2.
Construction of the manifold. Now we are ready to prove Theorem 2.1. Let where δ * ≫ ε * > 0 are the small constants in the one-pass theorem [14,Theorem 5.1]. For each fixed ϕ, we divide the set B + δ + for λ + according to the behavior of u. Let A ± be the totality of λ + ∈ B + δ + for which there exists t 0 ∈ [0, T ) such that Then the ejection lemma [14, Lemma 3.2] followed by the one-pass theorem [14, Theorem 5.1] implies the following. If λ + ∈ A ± then the solution u is exponentially ejected out of the small neighborhood d W < d W ( u(t 0 )) and never comes back again. Moreover, if λ + ∈ A + then u blows up in t > t 0 , and if λ + ∈ A − then u scatters to 0. By the local wellposedness of (CW) in H, both A ± are open. To see that both are non-empty, consider the case δ + ∼ |λ + | ≫ ϕ H . Let Then (1.46) implies that and, by definition of d W , we have at t = 0 (2.69) Hence we deduce and thus λ + ∈ A sign λ + for |λ + | ∼ δ + . In particular, both A ± are non-empty, which implies that the remainder is also non-empty. Every solution u for λ + ∈ A 0 must violate (2.66) for each t ∈ [0, T ), to avoid the ejection. Hence at each t ∈ (0, T ), one of the following holds where the last condition in (2.66) is not considered, since it is implied by the others, due to the ejection lemma. Since d W ( u(0)) δ + ≪ δ * , either (1) or (2) holds for small t > 0. Since the ejection lemma can be applied with ∂ t d W ( u) = 0 as well, we deduce that d W ( u(t)) is strictly decreasing in t > 0 until (2) is satisfied, where u spends its remaining life (hence never reaching (3)).
Choosing δ + ≪ ι 6 I small enough, we can ensure that If there are more than one λ + ∈ A 0 for the same ϕ, say λ 0 + = λ 1 + , then we can apply Lemma 2.2 to the corresponding solutions with the initial data u j (0) = W + λ j + g + + ϕ and with ι ∈ (0, ι I ] satisfying Then its conclusion together with (2) leads to a contradiction Thus we can define the functional m + by putting A 0 = {m + (ϕ)}, and then A + = (m + (ϕ), δ + ), A − = (−δ + , m + (ϕ)). The same reasoning as above implies that we can never apply Lemma 2.2 for different ϕ 0 , ϕ 1 ∈ B ′ δ ′ with u j (0) = W + m + (ϕ j )g + + ϕ j and d 2 W ( u j (0)) + ⊳ϕ ⊲ H ≪ ι 6 . Therefore max j=0,1 which implies the Lipschitz continuity (2.7). Since m + (0) = 0 is obvious, it also implies that |m + (ϕ)| = o( ϕ H ). In particular, we may restrict both the domain and the range of m + to have the same radius δ m as in the statement of the theorem, though there is no merit for that besides reducing the number of parameters. The trichotomic dynamics readily follows from the ejection lemma and the onepass theorem in [14]. The estimate on the L 2 * distance in (1) is derived from the bound K(u(t)) ≤ −κ(δ * ) in the variational region [14, Lemma 4.1] as follows. Choose δ m ≪ κ(δ * ). Let u be a solution in the case (1) and let t be after the ejection, namely which implies κ(δ * ) ϕ L 2 * and so dist L 2 * (u(t), S tatic (W ) 1 ) κ(δ * ) ≫ δ m . Thus it only remains to prove that m + is C 1 .

Then equation (2.82) has a unique global solution
at some τ 0 ≥ 0, then there exists τ I ∈ (τ 0 , ∞), such that for all τ > τ I we have On the other hand, if (2.84) fails for all τ 0 ≥ 0, then for all τ > 0 Proof. We only sketch the proof for (2.86), since the rest is essentially the same as Lemma 2.2. In the same way as for (2.52), we obtain, for any 0 < τ 0 < τ < τ 0 + η l , and so, using (1.43), we obtain (2.86).
To show the continuity of m ′ + for ϕ 0 , take any sequence ϕ 0 n → ϕ 0 in B ′ δm , let u 0 n be the solution starting from u 0 n (0) = Ψ σ(0),c(0) (m + (ϕ 0 n )g + + ϕ 0 n ) and let v 0 n := Ψ −1 σ,c ( u 0 n ). The local wellposedness implies that v 0 n → v 0 in L ∞ τ H ∩St τ (0, S) for any S ∈ (0, ∞). Let (2.82) n be the equation obtained by replacing v 0 with v 0 n in (2.82). For any small ζ > 0, Lemma 2.4 allows one to choose S ≫ 1 such that the solution of (2.82) with λ ′ for all τ > S/2. On the other hand, for any ϕ ′ ∈ B ′ 1 , the solution of (2.82) with λ ′ for all τ ≥ 0. Since S ≫ 1 and k ≫ ι I , combining these two estimates yields that the solution of (2.82) with |λ ′ for all τ > S/2. The local uniform convergence of v 0 n implies that the solution of (2.82) n with the same initial data also satisfies (2.94) around τ = S for large n. Moreover, since ( σ( u 0 n ), c( u 0 n )) → (σ, c) uniformly on [0, 2S] as n → ∞, we have the same estimate (2.94) around τ = S also in the coordinate associated with the solution u 0 n , for large n. Then it implies that This concludes the proof of Theorem 2.1.

Extension of the manifold and the 9-set dynamics
where T + is the maximal existence time of u. Conversely, if a solution u satisfies the above condition and E( u) − E(W ) ≪ δ 2 m , then its orbit is included in M 1 . This is because every solution getting close enough to the ground states is classified by the trichotomy of Theorem 2.1, whereas those solutions which never approach the ground states have been classified into the 4 sets of scattering to 0 and blowup away from the ground states in [14]. The one-pass theorem in [14] implies that every solution on M 1 ∩ M † 1 satisfies d W ( u(t)) δ m all over its life, and so it is essentially the same as the center manifold M 0 ∩ M † 0 , and in particular with codimension 2. The rest of M 1 is split into two parts, scattering to 0 or blowup away from the ground states, in the negative time direction. Each set is non-empty and relatively open in M 0 . Therefore, we have all 3 × 3 combinations of dynamics in t > 0 and in t < 0: (1) blowup away from the ground states, (2) trapping by the ground states (or by M 0 for t > 0 and by M † 0 for t < 0), and (3) scattering to 0. It was already shown in [14] that the combinations of (1) and (3) have non-empty interior. Moreover, those 9 sets exhaust all possible dynamics in the region E(u) − E(W ) ≪ min(δ m , ε * ) 2 . Thus we obtain Theorem 1.1.
Before going to the next step using the Lorentz transform, it is convenient to consider the space-time maximal extension of each solution of (CW).

Space-time extension and restriction.
To solve the equation locally in and out of light cones, and in more general space-time sets, we introduce restricted energy semi-norms. Let B d be the totality of Borel sets in R d . For any B ∈ B d and any a ≥ 0, we define two sets B ±a ∈ B d by It is clear that for any a, b, t ≥ 0, we have where ≃ means the isometry, with the quotient norm Henceforth, we denote for brevity We also use the more explicit semi-norms for ϕ ∈ H: (3.10) All of these three semi-norms are increasing for B and invariant for T , S, namely We may have the reverse inequalities when B is smooth. In particular, we have (3.14) Restriction of energy-type functionals is denoted as follows The finite propagation speed implies that if a solution u of (CW) satisfies By the Strichartz estimate, there is C > 0 such that if C ψ H⇂B ≤ ε S then there is a free solution v satisfying v(0) = ψ on B and v St(R) ≤ ε S , and so is u ∈ S olution (R) satisfying u(0) = ψ on B and u St(R) ε S , which is unique on B −|t| (R). Now we introduce the space-time maximal extension of a solution of (CW). For any ϕ ∈ H, c ∈ R d and R > 0, consider the local solution in the light cone K c,R := {|x − c| + t < R, t ≥ 0} with the initial data u(0) = ϕ on |x − c| < R. Let t + (ϕ, c) be the supremum of such R that there is a unique solution u in K c,R satisfying u L qm (K c,R ) < ∞. The uniqueness in cones implies that we have a unique solution u in the space-time region (3.18) as well as the Lipschitz continuity We also write for any strong solution u (either before or after the above space-time maximal extension), The maximal existence time is then given by The small data theory in interior and exterior cones implies that for any ϕ ∈ H, there are a(ϕ), b(ϕ) > 0 such that The definition of t + implies that (3.23) and so the small data theory implies lim inf In particular, the number of first blow-up points is bounded in the case of type-II Similarly we can define t − (ϕ, c) < 0 to be the maximal extension in the negative direction, and thus a unique solution u in the maximal space-time domain satisfying for some a(ϕ), b(ϕ) > 0 and for all x, y ∈ R d , Since the Lorentz transforms preserve light cones as well as the measure and the topology of R 1+d , the property (3.23) of t + is also preserved. Hence, each strong solution u defined on its maximal space-time domain D(u) is transformed by any Lorentz transform into another solution defined on the maximal domain. This process can produce a solution with no Cauchy time slice, namely inf t + < sup t − , but we ignore such solutions, in order to keep the dynamical viewpoint in terms of the Cauchy problem or the flow in H. In other words, the Lorentz transforms should be restricted to the range where inf t + < sup t − is kept. For the blow-up solutions on the center-stable manifold, we have Lemma 3.1. Let 0 < T < ∞ and u ∈ S olution ([0, T )) satisfy Then there exists c * ∈ R d and ε > 0 such that Proof. Let (σ(t), c(t)) := ( σ( u(t)), c( u(t)). First we show σ(t) → ∞ as t → T − 0.
If not, there exist a sequence t n ր T with sup n σ(t n ) < ∞ and R > 0 such that which ensures solvability in the cone |x − c| + |t − t n | < R for all c ∈ R d and all t n , thereby extending the solution u beyond T , a contradiction. Hence σ(t) → ∞. Then the modulation equation (1.56) implies convergence c(t) → ∃c * ∈ R d as well as the first estimate of (3.29). Since u(t) − T c S σ W H ∼ d W ( u(t)) < δ, that behavior of (σ, c) implies that for any R > 0, which ensures solvability in the exterior cone |x − c * | + |t − T ′ | > R for all T ′ < T close to T , and so t + ( u(0)) ≥ T ′ + |x − c * | − R and t − ( u(0)) ≤ T ′ − |x − c * | + R.
Letting R ց 0 and T ′ ր T , we obtain It can not be better for t + as u blows up at (T, c * ), so we obtain the identity in (3.29). The finite propagation implies u(t) H(|x−c * |>R+|T −t|) δ, so that we can solve in a slightly larger exterior cone starting from t = T ′ ∈ (0, T ). Thus we obtain the last estimate in (3.29). To see that M 2 is locally C 1 diffeo to M 1 , it suffices to see that the Lorentz transform gives a local C 1 mapping around any solution. Let u ∈ S olution ([−T, T ]) for some T > 0, then the local wellposedness yields a neighborhood O ∋ u(0) and R > 0 such that for any ϕ ∈ O we have

Extension by the
Then there is a neighborhood U ∋ 1 in the Lorentz group such that every transform in U maps the region X to a set containing [−T /4, T /4] × R d . Since the space rotation plays no role, we may restrict to those transforms defined on (t, x 1 ) ∈ R 2 , which can be parametrized as for each θ ∈ (−Θ, Θ). The continuity of S θ can be seen by the linear energy identity. For any smooth function u defined on X, we have by the divergence theorem where E F (ϕ) := ϕ 2 H /2 denotes the free energy. Applying this to the difference of two solutions u, w starting from O yields where the implicit constants depend on Θ. The standard perturbation argument implies that the last term is also bounded by u(0) − w(0) 2 H ≪ 1. The existence and continuity of the derivative of S θ is shown similarly by applying the energy estimate to the linearized equation Therefore M 2 is also a C 1 manifold with codimension 1 in H. It is clear from the construction that all these manifolds M 0 ⊂ M 1 ⊂ M 2 are connected. For the trichotomy around M 2 , it is obvious from the energy estimate that every scattering (to 0 as t → ∞) solution is transformed into another such solution by any Lorentz transform. The solutions in the other part of the neighborhood blow up away from a (much) bigger neighborhood, which is generated from the δ X neighborhood of S tatic (W ) by the above extensions. Thus we obtain the 9-set dynamics classification in the region (3.40) Reduction of this region to E(u) < E(W ) + ε 2 is done as in [14] by the Lorentz transform and the identities for (CW) Indeed, if E(u) < |P (u)| then we can transform it (in some space-time region) to another solution with negative energy, which has to blow up in both time directions by the classical argument of Levine [18], or more precisely by [11]. Hence the solution before the transform should also blow up in both directions. If E(u) = |P (u)| and it is global for t > 0, then there is a sequence of solutions u n ∈ S olution ([0, ∞)) given by Lorentz transforms such that E(u n ) → 0. Then the classical argument of Payne-Sattinger [23] implies that K(u n ) ≥ 0 as soon as E(u n ) < E(W ), and so E(u n ) ∼ u n 2 L ∞ t H → 0. The small data scattering implies that u n L qm (R 1+d ) E(u n ) 1/2 → 0, but since the Lorentz transform is measure preserving on R 1+d , it implies that u L qm (R 1+d ) = 0. In short, all the solutions with E(u) ≤ |P (u)| blow up in both time directions except for the trivial solution 0.
If E(u) > |P (u)|, then we can transform it to another solution u with P ( u) = 0 and E( u) < E(W ) + ε 2 , and so u should either scatter to 0 as t → ∞, blow up away from the ground state in the positive time direction, or live on M 1 . Each of those properties is transferred back to the original solution u. Note that if D( u) contains no time slice then the original solution u must blow up in both time directions. Thus we complete the 9-set dynamics classification slightly above the ground states, and the proof of Theorem 1.2.

Part II: Large radiation
The goal in the rest of paper is to extend the center-stable manifold to the entire energy space H, together with the dynamics around it, by a simple argument which allows one to reduce the problem to M 0 in the region E(u) < E(W ) + ε 2 , using the asymptotic Huygens principle together with the finite speed of propagation.

Detaching the radiation
For any B ∈ B d and any T ∈ (0, ∞], we define a semi-norm Smallness in R T B will imply that we can detach the exterior component using the wave starting from ψ which is out-going dispersive in the sense of the energy on the interior cone B +t , and also the Strichartz norm on R d , both for 0 ≤ t ≤ T .
The lower semi-continuity of the norms implies that the infimum in defining R T B is achieved by some ψ ∈ H such that ψ = ϕ on B ∁ and For the last equivalence, ≥ is by definition of H ⇂ B ∁ , while follows from The following laws of order are trivial by definition as is the invariance T c S σ ϕ R T B = ϕ R e σ T e σ B+c , but it is not invariant under the time inversion ϕ → ϕ † . The space-time continuity of the norms implies Also note the trivial identity ϕ R T R d = 0. The following "asymptotic Huygens principle" plays a crucial role in using the dispersive property of (CW) in the above function space. Proof. Since B is included in some ball, it suffices to prove lim Since the statement is obviously stable in the energy norm, we may restrict the initial data to a dense set, say C ∞ 0 (R d ). Multiplying the equation with (t 2 + r 2 )v + 2trv r + (d − 1)tv, we obtain conservation of the conformal energy where ∇ ⊥ v := ∇v − xv r denotes the derivative in the angular directions. Since in the region the L 2 norm of the left tends to 0 uniformly as T → ∞, as well as those of v/(t − T ) and ∇ ⊥ v, while v L 2 * (R d ) → 0 by the free dispersive decay.
The asymptotic Huygens principle implies the following decay of R ∞ B : For any ϕ ∈ H and any bounded B ∈ B d , In other words, every free solution in H will eventually gets into any small ball of R ∞ B around 0, as well as every scattering solution of (CW) in H. Also, when the solution is around the ground states with large dispersive remainder, we can take the semi-norm R T B small by the following. for some interval I. Then for any t 0 , t 1 ∈ I we have Proof. Let ψ(t) := T c(t) S σ(t) W , v(t) := U(t)( F (0) + R(0)), and w(t) := u(t) − ψ(0). Then we have and v St(0,S) F St(0,S) + R(0) H ≤ ς, for any S ∈ (0, T ). Choosing ε D ≪ 1, we deduce that as long as 0 < e σ(0 Now define a time sequence inductively by t 0 = 0 and t j+1 = t j + e −σ(t j ) ε D . Then applying the above argument from t j yields for t j ≤ t ≤ t j + e −σ(t j ) ε D , and induction on j yields the desired estimate (4.12).
If I = [0, T ) and T < ∞, then σ(t) → ∞ as t ր T , since otherwise there is a sequence t n ր T with sup n σ(t n ) < ∞ and δ > 0 such that (4.20) Choosing ε D ≪ ε S , this ensures solvability in the cone |x − c| + |t − t n | < δ for all c ∈ R d and all t n , thereby extending the solution beyond T , a contradiction. Hence σ(t) → ∞. Then and ψ(s) H⇂B ∁ → 0. Choosing R small enough yields (4.13).
The following Sobolev-type inequality implies that R ∞ B controls L 2 * , which is a notable difference from R T B with T < ∞.
Remark 4.1. It is obvious from the homogeneous nature that the above lemma fails on any bounded set in R for any q < ∞. If such an inequality would hold, then it must be uniform for the rescaling u(t, x) → u(λt, λx), but the L q t (I) norm decays as I shrinks to a point t 0 ∈ R, while the L ∞ t (I) norm converges to the value at t 0 . The following lemma allows us to detach exterior radiation which is small in R T R from any solution of (CW).
and v(0) H u(0) H⇂B ∁ . More precisely, there is w ∈ C([0, T ); H) such that ς. (4.33) Note that no energy bound is required on u, while the condition in R T B can be satisfied either by localization as in (4.5) or by dispersion as in (4.10), which is useful respectively for concentrating blow-up and for scattering solutions. In addition, if T = ∞ then combining the above with (4.24) yields (4.33) for u x . The propagation speed of (CW) implies that u x = u on (B +t ) ∁ = (B ∁ ) −t . If T < ∞, then let w be the solution of and so U(t − T ) w( T ) St(0, T ) ς by the Strichartz estimate. Also we have To define u θ in (II), let w θ be the solution of (4.38) with w θ ( T ) = θ w( T ) and let u θ := u − w θ . Then obviously w 0 = 0, w 1 = w, w = 0 on B +t (0, T ), and so u θ = f ′ (u θ ) in the same way as (4.41). The same estimate as above yields In the case T = ∞, we define a sequence w n ∈ C([0, n]; H) with T = n as above. Then the uniform bound allows us to take a weak limit along a subsequence to w ∈ C([0, ∞); H) ∩ St(0, ∞) solving (4.38), the estimates and w = 0 on B +t (0, ∞). For (II), let w θ be the solution of the integral equation obtained by the iteration starting from θw. Then u θ := u − w θ satisfies the desired properties, which is seen by the same argument as above.
To define the map A : u(0) → u d (0) in (III), we perturb u(0) around some fixed u 0 (0) ∈ H satisfying the assumption. Let v 0 be the free solution chosen as above for u 0 . For u(0) ∈ H close to u 0 (0), we have Then in the same way as above, if T < ∞, let u x ∈ S olution ([0, T ]) with u x (0) = v(0), let w be the solution of (4.38), and let u d = u − w. By the Strichartz estimate, we see that the maps where u x and w are Lipschitz with respect to u(0) ∈ H ⇂ B ∁ , leading to (4.34).
The following lemma is crucial to show that the ground state component is small in the region where the solution is dispersive. (4.48) The estimate on g ± follows from that ρ/W ∈ L ∞ ∩ W 1,d (R d ). Indeed, let χ := ρ/W and let ϕ = W on B. Then we have g ± = (2k) −1/2 (1, ±k)χϕ 1 on B, and Hence g ± H⇂B W H⇂B . The estimate in H(B) is similar.

Center-stable manifold with large radiation
Now we can extend the center-stable manifold by adding large radiation. Fix ς m > 0 such that ς m ≪ δ m . Let M 3 be the totality of u(0) ∈ H such that for the solution u we can apply the detaching Lemma 4.4 with for t close to T . On the other hand, if u(0) ∈ M 3 with u ∈ S olution ([0, T )), then for t close to T . Hence M + (A(ψ)) > 0 implies that ψ ∈ M 3 . If M + (A(ψ)) < 0, then (4.32) with Strichartz implies that the solution u starting from ψ also scatters to 0, contradicting (5.3), and so ψ ∈ M 3 . In order to conclude that M 3 is a C 1 manifold of codimension 1, it now suffices to show that the C 1 functional M + • A does not degenerate on its zero set M 3 . Indeed, if M + (A(ψ)) = 0 then ∂ h M + (A(ψ + hT A(ψ) g + )) ∼ 1, because by the Lipschitz property of A (4.34) and m + (2.7), and the last condition of (5.1) together with (4.47), we have By Lemma 4.4(II), we can connect each ϕ ∈ M 3 with some ψ ∈ M 0 by a C 1 curve, which is included in an enlarged M 3 for which the last bound in (5.1) is replaced with Cς m for some constant C > 1. Including those curves connecting M 3 to M 0 , we obtain a slightly bigger manifold M 3 , which is C 1 and connected with codimension 1. Let and all those starting from the other scatter to 0 as t → ∞.
On the other hand, if u ∈ S olution ([0, ∞)) scatters to S tatic (W ), namely then u(0) ∈ M 4 . This is because Lemma 4.2 implies that we can detach the free radiation U(t)ϕ at some large t = T , so that u d (t) = T u(t) W + O(ς) in H for all t ≥ T with ς ≪ ς m ≪ δ m . The last condition of (5.1) is ensured by (4.13). Hence p(t) converges to some p * ∈ R d , and then W λ(t),c(t),p(t) − W λ(t),c(t),p * → 0 in H. Take a Lorentz transform which maps W 0,0,p * to W , and apply it to u. Then we obtain another global solution satisfying (1.21) with p(t) ≡ 0, namely scattering to S tatic (W ), and so belonging to M 4 . Hence u is on M 5 . Since each Lorentz transform defines a local C 1 diffeo around each solution, there is a neighborhood of M 5 transformed from a neighborhood of M 4 , such that all solutions starting off the manifold within the neighborhood either scatter to 0 as t → ∞ or blow up in t > 0 away from a bigger neighborhood. Thus we obtain Theorem 1.3.
We are now ready to prove the first one-pass theorem in the radiative distance.
Proof. We will reduce it to [14] by the detaching Lemma 4.4. Choose ς * < ε S and let B, v, u x and u d be as in the above lemma, with ψ ∈ ± S tatic (W ) satisfying (6.8).
Combining it with (6.6) yields Choose ς * ≪ ε * of [14, Theorem 5.1], so that we can apply that one-pass theorem to u d , both from t = 0 forward in time, and from t = T backward in time. Specifically, there are ε, δ > 0 such that 0)), d W ( u d (T ))) < δ (6.23) so that we can deduce from the theorem that max 0≤t≤T d W ( u d (t)) < δ, and then, by the above lemma, d R ( u) δ ∼ ς for 0 < t < T . Next, if d R ( u(t 0 )) ≥ C * ς at some t 0 , with C * > 1 large enough, then the above lemma implies that d W ( u d (t 0 )) > δ and so, by the classification of the dynamics after ejection in [14], we conclude that d W ( u d (t)) δ * after some t 1 ∈ (t 0 , T ), and then u d either blows up in finite time, or scatters to 0, so does u by (6.6). The blow up occurs away from the ground states in the sense of (5.2). Moreover, this dichotomy is determined by sign(K(u d (t 1 ))). Choosing ς * smaller if necessary, and using (6.6), we have ±δ * ∼ K(u d (t 1 )) = K B +t 1 (u(t 1 )) + O(ς), (6.24) which implies sign K B +t 1 (u(t 1 )) = sign K(u d (t 1 )).
The above one-pass theorem does not preclude oscillation between ς < d R < C * ς. In the case of d W in [14], it was possible to exclude such oscillations completely thanks to the convexity in time of d 2 W near S tatic (W ), which is not inherited by d R . However, we can make an exact version of the above one-pass theorem by the flow. Theorem 6.5 (One-pass theorem with large radiation). There exist constants C * > 1 > ς * > 0, and an open set X(ς) ⊂ H for each ς ∈ (0, ς * ] satisfying: (1) X(ς) is increasing, i.e. ς 1 < ς 2 =⇒ X(ς 1 ) ⊂ X(ς 2 ).