Global dynamics of the nonradial energy-critical wave equation above the ground state energy

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in the natural energy space with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as time approaches infinity. In our previous paper arxiv:1010.3799 we treated the radial case, and this paper provides the natural nonradial extension of these results. However, the present paper is self-contained and in fact develops a somewhat different formalism in order to handle the more complex nonradial situation.

For a comprehensive review of these basic issues we refer the reader to [11].
In a previous paper [12] the authors studied the global dynamics of radial solutions to (1.1). To state that result as well as the main result of this paper, we recall some of the basic structures associated with the critical equation. First, one has the conserved energy of (1.1) as well as the conserved momentum P ( u) := u|∇u . (1.7) Remarkably, (1.1) admits the static Aubin solutions of the form (1.8) where S σ −1 denotes theḢ 1 preserving dilation (S σ −1 ϕ)(x) = e (d/2−1)σ ϕ(e σ x). which are unique, up to dilation and translation symmetries, amongst the non-negative, non-zero (not necessarily radial) C 2 solutions, see [2]. They also minimize the static energy among all non-trivial static solutions. The work of Kenig, Merle [10,11] and Duyckaerts, Merle [6,7] allows for a characterization of the global-in-time behavior of solutions with E( u) ≤ J(W ).
In this paper we study the behavior of solutions with for some small ε > 0. Solutions of subcritical focusing NLKG and NLS equations with radial data in R 3 of energy slightly above that of the ground state were studied by the latter two authors in [18,19]. The nonradial subcritical Klein-Gordon equation in three dimensions was treated in [20].
The key feature of (1.1) by contrast to NLKG is the scaling invariance of (1.1) manifested by u(t, x) → e σ(d/2−1) u(e σ t, e σ x) = S σ −1 u(e σ t), (1.13) which leaves the energy unchanged. In particular, the analogue of the "one pass theorem" proved in [21] needs to be modified, specifically by replacing the discrete set of attractors {Q, −Q} there by a (2d + 1)-parameter family of solitons. For any (σ, p, q) ∈ R × R d × R d , denote the scaling-Lorentz transform of W by W σ (p, q) = W σ (x − q + p( p − 1)|p| −2 p · (x − q)) (1.14) where p := 1 + |p| 2 . Then for any fixed (p, q) ∈ R 2d , u(t, x) = W σ (p, q + tp/ p ) (1. 15) gives a ground state soliton of (1.1). Hence the ground state soliton family is S := {(W σ (p, q), −∇W σ (p, q) · p/ p ) | (σ, p, q) ∈ R 1+2d } ⊂ H. (1.16) Note that in the subcritical NLS case [20], the scaling parameter σ is essentially fixed or at least bounded from above and below by the L 2 conservation law, but in the critical case there is no factor which a priori prevents the scale from going to 0 or +∞. On the other hand, by using the Lorentz transform, we can reduce the problem to the case of zero momentum, where the soliton family is S 0 := { W σ (x − q) | (σ, q) ∈ R 1+d } ⊂ H, W σ = (W σ , 0), (1.17) with the energy constraint E( u) < J(W ) + ε 2 (slightly changing ε > 0). Introduce the "virial functional" and note that K(W ) = 0. The following positivity is crucial for the variational structure around W Note that the derivative of J(ϕ) with respect to any scaling except for S σ −1 gives a non-zero constant multiple of K(ϕ). This is a special feature of the scaling critical case, which allows us to work with a single K, whereas in the subcritical case [21] we needed two different functionals and their equivalence.
The main result of this paper is summarized as follows.
such that the following properties hold: For any solution u in H ε on the maximal existence interval I(u), let Then I 0 (u) is an interval, I + (u) consists of at most two infinite intervals, and I − (u) consists of at most two finite intervals. u(t) scatters to 0 as t → ±∞ if and only if ±t ∈ I + (u) for large t > 0. Moreover, there is a uniform bound M < ∞ such that For each σ 1 , σ 2 ∈ {±}, let A σ 1 ,σ 2 be the collection of initial data u(0) ∈ H ε , and for some T − < 0 < T + , Then each of the four sets A ±,± has non-empty interior, exhibiting all possible combinations of scattering to zero/finite time blowup as t → ±∞, respectively.
The radial version of this exact theorem was proved in [12]. The main difference from that paper is of course the presence of the translation and Lorentz symmetries which need to be taken into account. Actually, the Lorentz symmetry does not play much role under the energy constraint E( u) < J(W ) + ε 2 , where the solution can approach to ±S only if |P ( u)| ε. In contrast, the translational freedom is not a priori controlled by conserved quantities, and so we instead eliminate it by suitable orthogonality conditions. In other words, the modulation theory here amounts to a system of d + 1 ODEs corresponding to the dilation and translation symmetries.
By using the Lorentz transform, we can extend the above result to bigger energy, depending on the size of momentum.  Actually, we can reduce the corollary to the theorem only for |E( u)| > |P ( u)|, where we can find a Lorentz transform from u to another solution w with E( w) = E( u) 2 − |P ( u)| 2 and P (w) = 0, see (7.30).
The other case |E( u)| ≤ |P ( u)| is treated separately, which is essentially known. Indeed, for such a solution u, there is a Lorentz transform to another solution w with E(w) < J(W ). If the original solution u is global in one direction, then so is w (see Lemma 7.3). Then we have K(w(0)) ≥ 0, otherwise the classical result of Payne-Sattinger [22] (or more precisely by Kenig-Merle [11] in the current setting) implies that w blows up in both directions. Then This is already a contradiction if |E( u)| < |P ( u)|, since then we can make E(w) < 0. In the remaining case |E( u)| = |P ( u)|, we can make E( w) as small as we wish. Hence the above inequality implies that the energy norm can be made arbitrarily small. Then the small data scattering implies that The rest of the paper is devoted to prove the main theorem. We differ strongly from [12] in terms of the basic formalism which defines our approach. To be more precise, we perform a change of coordinates in the time variable which allows us to work with a fixed reference Hamiltonian in the perturbative analysis rather than a moving one as in [11]. This leads to some simplifications in the ejection lemma, for example, see Lemma 3.2. We remark that the formalism is also different from the one used in the nonradial subcritical equation [20], where a complex formulation was chosen, and more essentially, in the choice of orthogonality conditions, which also brings some simplification.
One application of Theorem 1.1 is the following corollary, which removes the radial assumption from [5,Corollary 6.3]. The solution W + is the one discovered by Duyckaerts, Merle [7]. It is a radialḢ 1 × L 2 solution, exists globally in forward time and approaches W inḢ 1 , and blows up in finite negative time. As above, the dimension satisfies d = 3 or d = 5. In [7, Theorem 2, (c)] this result is proved (nonradially) under the additional condition that u 0 ∈ L 2 . Using [12], this L 2 condition was removed in [5], but only in the radial setting. As noted in [5,Remark 6.5], the removal of the radial assumption in [12] would then complete [7] in the sense that the L 2 -condition can be removed even nonradially. This is what we accomplish in this paper, whence Corollary 1.3. For the proof, we refer the reader to [5].
is locally wellposed in H, with the conservation of energy: The equation is the Hamiltonian flow in H with conserved Hamiltonian E relative to the symplectic form 2.2. The translation and scaling symmetries. Another feature of equation (1.1) is its invariance with respect to the scaling: which is a unitary group acting on H, with the generator With Λ * a denoting the adjoint relative to L 2 (R d ) one has Λ * a = −Λ −a and thus Λ * = −Λ 1 ⊗ Λ 0 . Similarly, the unitary group of translations is denoted by (2.6) Our analysis in this paper is around the static Aubin solution whose vector and scaled versions are denoted by be a solution with σ = σ(t) and c = c(t). In general, v need not have the structure (1.2), which is why we do not write v. Noting that where the linearized and superlinear operators are defined by The structure of the spectrum of L + over L 2 (R d ) is as follows: the discrete spectrum consists of a unique negative eigenvalue of L + which we denote by −k 2 . The associated eigenfunction is the ground state of L + , denoted by ρ: The essential spectrum of L + is [0, ∞), and it is purely absolutely continuous. At the threshold 0, one has an eigenvalue of multiplicity d, with eigenfunctions ∇W , and a resonance function W ′ = Λ −1 W which is unique.

2.3.
A change of time and the static linearized operator. The timedependent coefficient on the linearized operator is removed by the standard change of time variable from t to τ : The "generalized" eigenvectors of JL are JLg ± = ±kg ± , g ± := (1, ±k)ρ/ √ 2k, (2.14) where ρ is the aforementioned ground state of L + . The normalization here is such that ω(g + , g − ) = 1. Define ρ := (ρ, 0). Note that Λ W ∈ L 2 for d < 5. Hence we decompose where By construction, The more natural µ := ω(v, J∇ W ) = v 1 |∇W is problematic since the latter inner product is not well-defined. Note that we did not extract the remaining root-mode J∇ W from γ, which corresponds to Lorentz "boosts", i.e., translations in momentum.
2.4. Energy expansion. Using (2.15) the energy is expanded as where the superquadratic part is given by One has the estimate Then and where the implicit constants in (2.22) only depend on the dimension.
Proof. The first statement follows from the self-adjointness of L + and the description of the spectrum of L + . For the second, we need to invoke the calculus of variations and the concentration-compactness method. It is clear form Sobolev imbedding and Hölder's inequality that the left-hand side of (2.22) dominates the right-hand side. Suppose the reverse inequality of (2.22) fails. Then there exists a sequence {f n } ⊂Ḣ 1 (R d ) with ∇f n 2 = 1 and f n ⊥ ρ which further satisfies After passing to a subsequence we may assume that From the local convergence in L 2 we conclude that By the first condition in (2.23) the left-hand side in (2.25) also tends to 1.
In what follows we denote so that (2.22) implies the following: for any γ with γ 1 ⊥ ρ, ∇ρ we have Orthogonality conditions near the ground state. Now we introduce the crucial orthogonality conditions near the family of static solutions ±S 0 . Indeed, we claim that for any u ∈ H with (2.30) (2.30) are the orthogonality conditions which we use in this paper. To verify this claim, take any v 1 small inḢ 1 (and thus small in L 2 * (R d )), and define (taking +S 0 for simplicity) It follows from the inverse function theorem that there exists (σ, c) small with F (σ, c) = 0. But thenṽ 1 defined by means of satisfies ṽ 1 |Λ 0 ρ = 0 and ṽ 1 |∇ρ = 0. The orthogonality conditions (2.30) are equivalent to α = 0 and µ = 0 in (2.15), which "eliminate" the dilation and translation symmetries, respectively.
2.6. Linearized energy. Change variables from (λ + , λ − ) to (λ 1 , λ 2 ) as follows: (2.33) Note that in these variables we have λ j = v j |ρ (j = 1, 2) and Now we define the nonlinear energy distance near ±S 0 by means of the equations with v H small and some choice of ±, and we use the decomposition (2.15). Lemma 2.1 together with Hence it is natural to define the linearized energy norm by where a W , b W > 0 are as in (2.15) and (2.26), I d is the d-dimensional unit matrix, and ∇ 2 ρ is the Hessian. The modulation equations (2.38) determine the evolution of (σ, c) as long as v remains small in H. For future reference, we remark that in the notation of (3.3) as long as v E is small.

Hyperbolic drivers.
On the other hand, the unstable/stable modes evolve by the following equations derived from (2.13) with In terms of λ 1 and λ 2 these equations become The relation between these systems is given by (2.33 This will be important to guarantee convexity of the distance function in the ejection lemma. However it should not be confused with the definition ∂ t u 1 = u 2 , even though λ j = v j , ρ , as t and τ are different.

Distance function, λ dominance, ejection
The following lemma establishes the existence of a distance function associated with the soliton manifold ±S 0 in such a way that near this manifold the distance function is proportional to the unstable mode in a suitable sense.

1)
and so that for any we have In addition, if Proof. There are 0 < δ A ≪ 1 and C ≥ 1 such that putting d 0 (u) := Cdist H (±S 0 , u), the above arguments starting from (2.29) work in the re- Now we choose a smooth cutoff function χ ∈ C ∞ (R) satisfying χ(r) = 1 for |r| ≤ 1 and χ(r) = 0 for |r| ≥ 2 and set The stated properties now follow easily from the considerations in the previous section.
The following lemma is the analogue of the "ejection lemma" in our previous papers, see [21]. As usual, we shall need the Payne-Sattinger functional in our analysis of the global dynamics, which is why it appears below.

11)
and Apply the decomposition from Lemma 3.1. Then for t > t 0 in I and as long as (3.14) for some absolute constant C * > 0 and s = ±1 is fixed on the time interval.
Proof. By Lemma 3.1 and (3.11), we conclude that |λ 1 (t 0 )| ≃ δ 0 . Furthermore, as long as d W ( u(t)) remains sufficiently small and one has d W ( u(t)) ≥ δ 0 , the relation is increasing this relation is preserved. We shall therefore assume (3.15) in our argument that establishes the monotonicity and (3.13). The logic here is that once we have shown these properties to be correct, then the validity of (3.15) follows a posteriori by the method of continuity. Differentiating (3.6) using (2.42) and (2.38) as well as (3.2) yields, and In conjunction with the previous lemma we conclude from (3.17) that d 2 W (u) is increasing and convex in τ , as long as it remains sufficiently small. Next, we remark that λ 1 and ∂ τ λ 1 = λ 2 have the same sign, since The evolution of λ + in τ is determined by (2.41), which states that Since d W ( u) |λ + |, we see that |λ + | ≃ e kτ δ 0 , and as claimed. As for the scaling parameters, (2.38) yields |∂ τ σ| γ H d W ( u), and hence integrating the exponential bound in τ implies |σ−σ(t 0 )| d W ( u).
The γ-part is estimated as in Lemma 4.3 of [21]. To this end define Then as well as (where v d,1 denotes the first component of v d ) where we used (2.40) to bound σ τ and c τ e σ . In view of the preceding, where we used the exponential growth of d W ( u) to pass to the last line. Furthermore, In conclusion, as claimed. Finally, expanding the K-functional, one checks that Inserting the expansion v 1 = λ 1 ρ + γ 1 into (3.27) and using the bounds on λ 1 (t) and γ(t) that we just established implies the desired properties of K.
We remark that unlike the subcritical nonradial paper [20] the distance function is convex near a minimum and thus increasing in Lemma 3.2. The difference lies with the choice of orthogonality conditions corresponding to the translational symmetry, which in our case insure that ∂ τ λ 1 = λ 2 . This coincides with the behavior in the radial subcritical case, see [21].

The variational structure in the energy critical setting
We recall the following characterization of the ground state: where J(ϕ) is the static energy defined in (1.11), and ±W are the unique minimizer up to the dilation (as in W σ ) and translation symmetries. In other words, W is the unique (up to the same symmetries) extremizer of the Sobolev embeddingḢ 1 (R d ) ֒→ L 2 * (R d ). We need the following variational structure outside of the soliton tube.
Proof. We first eliminate the u 2 component from u: if u 2 2 ≪ δ, then it follows that d W (u 1 , 0) > δ/2. On the other hand, if u 2 ≃ δ, then assuming ε 1 (δ) ≪ δ as we may, it follows that with some absolute constant c. But then we must have u 1 −W σ (·−c) Ḣ1 δ for all σ, c. Hence in all cases. In the rest of proof we regard u = u 1 ∈Ḣ 1 (R d ) with dist(u, S 0 ) δ.
By the critical Sobolev imbedding, the statement holds provided ∇u 2 < c 0 where c 0 > 0 is some absolute constant. Thus, assume the lemma fails and let {u n } ∞ n=1 ⊂Ḣ 1 be a sequence with as well as dist(u n , S 0 ) δ 0 . Since we see that {u n } ∞ n=1 is bounded inḢ 1 ∩ L 2 * and so c < ∞. Then the latter two conditions of (4.5) implies that {u n } is an extremizing sequence for the critical Sobolev embeddingḢ 1 (R d ) ⊂ L 2 * (R d ), and so, by the celebrated theorem of P.-L. Lions [15,Theorem I.1], it is compact inḢ 1 up to scaling and translation, hence converging strongly to the unique minimizer W up to scaling and translation. But this clearly contradicts dist(u n , S 0 ) δ 0 .
As in the previous works [21], we can define a sign functional by combining the ejection lemma with the variational structure exhibited in the previous lemma.
where ε 1 (δ) is defined in Lemma 4.1. Then there exists a unique continuous function S : where we set sign 0 = +1.
Proof. The proof is the same as in the subcritical radial case, see [18].

The one-pass theorem
A key step in the proof of our main theorem is to show that the sign S(u(t)) can change at most once for any solution of (1.1). This goes by the name of one-pass theorem, see [21]. The current section is entirely devoted to this theorem: Theorem 5.1. There exist 0 < ε * ≪ δ * ≪ δ H with the following properties: Let u ∈ C(I; H) be a solution of (1.1) on an open interval I, satisfying for some ε ∈ (0, ε * ], δ ∈ ( √ 2 ε, δ * ] and T 1 < T 2 ∈ I Then d W ( u(t)) > δ for all t > T 2 in I.
Proof. By increasing T 1 and decreasing T 2 if necessary, we may assume in addition that √ 2 ε < d W ( u(T 1 )) and ∂ t d W ( u(t))| t=T 1 ≥ 0. Then Lemma 3.2 applies for all t ∈ [T 1 , T 2 ] and so d W ( u(t)) is increasing for t > T 1 until it reaches δ H (the small absolute scale in the ejection lemma). Arguing by contradiction, we assume that for some t > T 2 we have d W ( u(t)) ≤ δ. Such a t can occur only away from T 2 (this will be made more precise shortly), and after d W ( u(t)) has increased to size δ H ≫ δ. Moreover, by applying Lemma 3.2 backward in time, we can find T 3 > T 2 such that d W ( u(t)) decreases from δ H down to δ as t ր T 3 , and so that for T 2 < t < T 3 . We may further assume σ(u(T 2 )) = 0 ≤ σ(u(T 3 )), (5.2) by rescaling and reversing time, if necessary. Here σ is defined in Lemma 3.1. We now proceed by combining the proof ideas of the analogous theorem for the critical radial wave equation [12] with that for the subcritical nonradial Klein-Gordon equation, see [20] (with slight improvement). Following the latter reference, we first show that the centers of the ground state as given by the path c(t), diverge from each other between times T 2 and T 3 by an amount ≪ T 3 −T 2 . Once this is done, we shall adapt the virial argument from [12] to the nonradial context, which will then allow us to exclude almost homoclinic orbits. It will be understood that all times t belong to the interval I.
By spatial translation, we may assume that c(T 2 ) = 0. By the ejection we have and by the finite speed of propagation where c(t) = c(u(t)) ∈ R d is defined by Lemma 3.1 as long as u(t) is close to S 0 , which is true when t is close to T 2 or T 3 . Consider a localized center of energy defined by (with u = (u 1 , u 2 )) where w(t, x) is the cut-off function onto a light cone defined by w(t, x) = χ(|x|/(t − T 2 + S)) (5.6) for some 1 ≪ S = S(δ) < δ −2 to be determined, and some χ ∈ C ∞ (R) satisfying χ(r) = 1 for |r| ≤ 1.5 and χ(r) = 0 for |r| ≥ 2. Then using the equation of u, we havė where E ext (t) := u(t) 2 H(|x|>t−T 2 +S) denotes the exterior free energy. Hence, The conserved momentum is small because Using the finite propagation as in [20] and [12] we have for all t ≥ T 2 On the other hand, the radial symmetry and the rate of decay of W imply that The contribution of W σ(T 3 ) (x − c(T 3 )) at t = T 3 is estimated as follows. Denote c 3 := c(T 3 ), σ 3 := σ(T 3 ). Then Now, using (5.2), where o(1) is with respect to the limit S → ∞ (uniformly for the other parameters c 3 , σ 3 , T 2 and T 3 ). Exploiting (5.4) and the obvious cancellation yields On the other hand, Combining these estimates yields and therefore To obtain the desired contradiction, we use the localized virial identity with the same choice of w as above. By similar considerations as above, one has the upper bounds We claim that integrating the differential equation in (5.18) and exploiting the ejection dynamics and the variational structure (cf. [21]) leads to the lower bound where 0 < ν(δ, δ H ) → ∞ as δ → +0 and δ H fixed. This clearly contradicts (5.20), provided that we choose δ * ≪ δ 2 H small enough. It remains to prove (5.21). Let T be the set of times at which the distance d W ( u(t))| [T 2 ,T 3 ] reaches a local minima in [δ, δ S ]. In particular, T ∋ T 2 , T 3 by the choice of T 2 and T 3 . For every t * ∈ T , we can apply the ejection Lemma 3.2 from t = t * in both time directions. Then we get an interval is decreasing for t < t * and increasing for t > t * , and d W ( u) = δ H on ∂I(t * ) \ {T 2 , T 3 }. Moreover, imposing 0 < ε * < ε 1 (δ S ), (5.22) we can ensure that u stays in H δ S for t ∈ [T 2 , T 3 ], so that the sign s in the ejection lemma is the same for all I(t * ) by Corollary 4.2. Furthermore, the exponential behavior allows us to estimate Summing this over all t * ∈ T including T 2 and T 3 , we get If s = +1, then the same argument encounters the difficulty that outside of the δ H -ball the lower bound of Lemma 4.1 may become degenerate due to smallness of ∇u 2 2 . Indeed, replacing κ(δ S ) in the above argument by min(κ(δ S ), c ∇u 2 2 ) and using the uniform bound on u H in the region S = +1 yields This leads to (5.21) for s = +1 if Therefore assume that with some absolute constant κ 2 . To lead (5.29) to a contradiction, we use the (localized) energy equipartition where w, E ext and S = δ −1 are as before. Taking δ * , κ 2 ≪ J(W ), we obtain On the other hand, the same argument as for (5.20) yields The above result has some important implications for the sign functional from Corollary 4.2. To be specific, let It is easy to see that H * \ H X is a small neighborhood of ±S 0 . The proof is the same as in the radial case [12], so we omit it. Note that H * \ H X is included in d W < δ * , and that (3)-(4) completely determine S(ϕ), since we have chosen ε * < ε 1 (δ S ) in (5.22). Moreover, S(ϕ) depends only on ϕ 1 .
It remains to determine the fate of the solutions in H * with d W ≥ δ * . We will do this in the following two sections for S = ±1 , respectively.

Blow-up after ejection
In analogy to [12], we now prove 1 the following 1 The proof is essentially the same as in [12], but since we employ somewhat different notation, we provide the details for the reader's convenience.

Scattering after ejection
Here we essentially repeat the argument given in [12] for the reader's convenience, with the small changes necessitated by the presence of space and momentum translations. In the region S = +1, we already know that all solutions are uniformly bounded in H, but it is not sufficient for global existence of strongly continuous solution in the critical case. Now we resort to the recent result by Duyckaerts-Kenig-Merle [4] to preclude concentration (type II) blow-up. This is the only place where we have to restrict the dimensions 2 to 3 or 5 Proof. First, Lemma 3.2 precludes blow-up in the hyperbolic region, since the scaling parameter is a priori bounded during the ejection process, which is valid when reversing the time direction. Hence a blow-up may happen only when d W ( u(t)) > δ H , where K(u(t)) ≥ 0 and so the energy assumption in Theorem 5.1 implies This allows us to employ the main result in [4], after reducing ε * if necessary. Suppose u is a solution on [0, T + ) in H X with S = +1 and d W ( u(t)) > δ H with the blow-up time T + < ∞. According to their result, we can then write for t sufficiently near T + for some σ(t) → ∞, c(t) ∈ R d and some fixed ϕ ∈ H. It is then easily checked that as t → T + − 0 we have from which we infer in particular that K(ϕ 1 ) ≥ 0. Similarly, we obtain which implies via K(ϕ 1 ) ≥ 0, This however contradicts d W ( u(t)) > δ H ≫ ε * near T + .
Next we employ the Kenig-Merle scheme from [10,11] to improve the above result. The one-pass theorem will be incorporated in the same way as in the subcritical case [21]. Extinction of the critical element requires a little extra work due to the possibility of concentration, which will be however reduced to the above proposition. The restriction d W ≥ δ * is essential for the uniform Strichartz bound, since the latter does not hold for all scattering solutions, even for E( u) < J(W ).
Proof. We argue by contradiction. Let u n be solutions on [0, ∞) in H X satisfying E( u n ) → E * ≤ J(W ) + ε 2 * , u n L q t,x (0,∞) → ∞, d W ( u n (t)) ≥ δ * , S( u n (t)) = +1, (t > 0) (7.6) where we choose q = 2(d + 1)/(d − 2) so that L q t,x is an admissible Strichartz norm for the wave equation on R d . Here and after, X(I) denotes the restriction to I × R d of the Banach function space X on R × R d . It is well-known that L q t,x and the energy norm are sufficient to control all the other Strichartz norms, such as L p tḂ 1/2 p,2 with p = 2(d + 1)/(d − 1), as well as the nonlinear term in some dual admissible norm such as in L p ′ tḂ 1/2 p ′ ,2 (see, for example, [8]).
We may assume that E * is the minimum for the above property. Following the Kenig-Merle argument, the proof consists of two parts: construction and exclusion of a critical element.
Part I: Construction of a critical element.
Assuming the existence of (7.6), we are going to show that there is a critical element u * , that is a solution on [0, ∞) in H X satisfying and that its trajectory is precompact modulo dilations and translations in H.
If d W ( u n (0)) < δ H , then by Lemma 3.2, we have d W ( u n (t)) ≥ δ H at some later t > 0. Since the Strichartz norm on the ejection time interval is uniformly bounded, we may time-translate each u n so that without losing (7.6).
Since we chose ε * < ε 1 (δ S ) ≤ ε 1 (δ H ), Lemma 4.1 implies Now apply 3 the Bahouri-Gérard decomposition from [1], see also Lemma 4.3 in [11], to { u n (0)} n≥1 . Let U (t) denotes the free wave propagator. We conclude that there exist λ j n > 0, t j n ∈ R, x j n ∈ R d , ϕ j ∈ H and free waves w J n such that for any J ≥ 1 where T j n := T −x j n S log λ j n , such that | log(λ j n /λ k n )| + λ j n |t j n − t k n | + λ j n |x j n − x k n | → ∞ (7.11) for each j = k, for each J, and The last property applies to any other non-sharp Strichartz norm by interpolation, since those free waves are all uniformly bounded.
large and fixed, and n ≥ n 0 (J) is sufficiently large: t,x (R) = 0, (7.20) which implies u n is bounded in L q t,x , contradicting (7.6). Thus we have obtained where (T − , T + ) is the maximal existence interval of U 1 . We now distinguish three cases (a)-(c) by s 1 ∞ = lim n→∞ λ 1 n t 1 n : (a) s 1 ∞ = ∞. Then by definition (7.16), U 1 is a local solution around t = ∞ with finite Strichartz norms, and Hence we can use the long-time perturbation argument on (0, ∞), which gives a contradiction via (7.20) as above.
(7.24) (c) s 1 ∞ ∈ R. Then by the same perturbative arguments as above, the nonlinear profile decomposition (7.20) holds on any compact interval in (T − , T + )/λ 1 n −t 1 n . Thus, as in the case (b), we deduce from inf t≥0 d W ( u n (t)) ≥ δ * that inf Then the same argument as in (b) implies that T + = ∞ and U 1 L q t,x (s 1 ∞ ,∞) = ∞, since otherwise U 1 scatters and the nonlinear profile decomposition holds on [0, ∞), contradicting (7.6).
Thus we arrive at the conclusion that s 1 ∞ < ∞ and U 1 is a critical element after time translation. This implies E(U 1 ) = E * by the minimality, which extinguishes the other profiles U j (j > 1) as well as the remainder w J n as n → ∞, through the nonlinear energy decomposition.
Having constructed a critical element u * , we apply the above argument to the sequence u n (t) = u * (t − t n ), t n → ∞. (7.26) Then the vanishing of all but one (free) profile implies that for some contin- is precompact, concluding the first part of the proof. Before proceeding to the extinction, we show that Suppose towards a contradiction that |P ( u * )| = |P 1 ( u * )| ≫ ε * . Then, since J(W ) ≤ E( u * ) < J(W ) + ε 2 * , we can use the Lorentz transform to reduce the energy below J(W ). Indeed, let w be any global strong energy solution of (1.1) and Lorentz transform it as follows: with a parameter ν ∈ R, w(t, x) → w ν (t, x) := w(t cosh ν + x 1 sinh ν, x 1 cosh ν + t sinh ν, x 2 , x 3 ). (7.29) Then one checks that w ν is again a strong energy solution of (1.1) which satisfies E(w ν ) = E(w) cosh ν + P 1 (w) sinh ν, P 1 (w ν ) = P 1 (w) cosh ν + E(w) sinh ν, P α (w ν ) = P α (w), (α > 1). (7.30) Now we claim that we can apply the above transform to the forward global solution u * , and then with some ν = O(ε * ) we can construct another forward global solution u ⋆ with E( u ⋆ ) < J(W ) and u ⋆ L q t,x (0,∞) = ∞. This contradicts Kenig-Merle's result [11] for E < J(W ). In order to transform a solution with infinite Strichartz norm, we argue in the same way as in the subcritical case using the finite propagation speed: The proof is also the same as for [20, Lemmas 6.1 and 6.2], so we omit it.
Part II: Exclusion of a critical element.
Let u * be a critical element (7.7), hence for t ≥ 0 is precompact in H. We proceed in three steps.
Step 1: lim sup t→∞ ρ(t)/t < ∞. To see this, note that by finite propagation speed, we have If for some sequence of times {s n } n≥1 we had ρ(s n )/s n → ∞, then by precompactness of { w * (t)} t≥0 , we get w * (s n ) H → 0, whence also u * (s n ) H → 0, which would force E * = 0, a contradiction.
Step 3: Construction of a blow up solution via a re-scaling of u * . Pick a sequence s n → ∞ with lim n→∞ ρ(s n )/s n = c ∈ (0, ∞), as well as w * (s n ) → ∃ϕ in H. Define a sequence of solutions u n (t, x) := s d/2−1 n u * (s n t, s n x) (7.46) whence we have u n (1) → c −d/2 ϕ(x/c) in H. The above two steps imply that u n is precompact in C([τ, 1]; H) for any 0 < τ < 1, and so, after replacement by a subsequence, it converges to some u ∞ in C((0, 1]; H). By the local wellposedness theory, it has finite Strichartz norms locally in time, and so u ∞ is the unique strong solution on (0, 1] with the initial condition u ∞ (1) = ϕ. Clearly we also have d W ( u ∞ (t)) ≥ δ * and S( u ∞ (t)) = +1 for 0 < t ≤ 1.
In order to complete the proof of Theorem 1.1, we now exhibit open data sets at time t = 0 such that we have blow up/scattering at t = ±∞, four possibilities in all. However, this has been done in the radial case [12] by producing four solutions starting from the neighborhood H * \H X and exiting from it in finite time in both time directions, for all four combinations of S at the exiting times. Since such behavior is obviously stable in the energy space H, we get an open set around each solution by the local wellposedness.