Scattering theory for energy-supercritical Klein-Gordon equation

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\R^d)\times H_x^{s_c-1}(\R^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schr\"odinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.

Moreover, it leaves the Sobolev normḢ sc x (R d ) with s c = d 2 − 1 invariant. Since s c > 1, it is called the energy-supercritical.
The scattering theory for the Klein-Gordon equation with f (u) = µ|u| p−1 u has been intensively studied in [3,4,12,14,26,27]. For µ = 1 and (1. 3) 1 Brenner [4] established the scattering results in the energy space H 1 x (R d ) × L 2 x (R d ), which does not contain all subcritical cases for d 10. Thereafter, Ginibre and Velo [12] exploited the Birman-Solomjak space ℓ m (L q , I, B) in [2] and the delicate estimates to improve the results in [4], which covered all subcritical cases. Finally K. Nakanishi [26] obtained the scattering results for the critical case (p = 1 + 4 d−2 ) by the strategy of induction on energy [9] and a new Morawetz-type estimate. And recently, S. Ibrahim, N. Masmoudi and K. Nakanishi [14,15] utilized the concentration compactness ideas to give the scattering threshold for the focusing (µ = −1) nonlinear Klein-Gordon equation. Their method also works for the defocusing case.
In this paper, we consider the cubic Klein-Gordon equation in dimension d 5, which is the energy-supercritical case. Such results have been recently established for many other equations including the nonlinear wave equation (NLW) and the nonlinear Schrödinger equation (NLS), since Kenig-Merle [18] on NLW for radial solutions in R 3 . We also refer to [5-7, 10, 19-22].
To be more precise, let us recall the results for the energy-supercritical nonlinear wave equationü − ∆u + |u| p u = 0, (t, x) ∈ R × R d , p > 4 d − 2 .
In dimension three, Kenig and Merle [18] proved that if the radial solution u is apriorily bounded in the critical Sobolev space, that is, (u, u t ) ∈ L ∞ t (I;Ḣ sc x (R d ) ×Ḣ sc−1 x (R d )) with s c := d 2 − 2 p > 1, then u is global and scatters. In [19], they also considered the radial solutions in odd dimensions. Later, Killip and Visan [21] showed the result in R 3 for the non-radial solutions by making use of Huygens principal and so called "localized double Duhamel trick". Further, they [22] proved the radial solution in all dimensions in some ranges of p. Thereafter, Bulut [5][6][7] proved the results in dimensions d 5 for the cubic nonlinearity (i.e. p = 2). Recently, Duyckaerts, Kenig and Merle [10] obtain such result for the focusing wave equation with radial solution in three dimension. Their proof relies on the compactness/rigidity method, pointwise estimates on compact solutions obtained in [18], and channels of energy arguments used by the authors in previous works [11] on the energy-critical equation.
Before stating the main result, we introduce some background materials.
) and u ∈ [W ](J) (defined in (1.8)) for any compact interval J ⊂ I and for each t ∈ I, it obeys the Duhamel formula: We refer to the interval I as the lifespan of u. We say that u is a maximal-lifespan solution if the solution cannot be extended to any strictly larger interval. We say that u is a global solution if I = R.
The solution lies in the space [W ](I) locally in time is natural since by Strichartz estimate, the linear flow always lies in this space. Also, if a solution u to (1.1) is global, with u W (R) < +∞, then it scatters in both time directions in the sense that there exist solutions v ± of the free Klein-Gordon equation In view of this, we define as the scattering size of u, where Closely associated with the notion of scattering is the notion of blowup: Definition 1.2 (Blowup). Let u : I × R d → C be a maximal-lifespan solution to (1.1). If there exists a time t 0 ∈ I such that S [t 0 ,sup I) (u) = +∞, then we say that the solution u blows up forward in time. Similarly, if there exists a time t 0 ∈ I such that S (inf I,t 0 ] (u) = +∞, then we say that u(t, x) blows up backward in time.

Then the solution u is global and scatters.
The outline of the proof of Theorem 1.1: For any 0 E 0 < +∞, we define where the supremum is taken over all solutions u : is a non-decreasing function. Moreover, from the small data theory, see Theorem 2.1, we know that is the threshold from the small data theory.
From the stability theory (see Theorem 2.5), we see that L is continuous. Therefore, there must exist a unique critical E c ∈ (0, +∞] such that L(E 0 ) < +∞ for E 0 < E c and L(E 0 ) = +∞ for E 0 E c . In particular, if u : H sc ×H sc−1 < E c , then u is global and moreover, The proof of Theorem 1.1 is equivalent to show E c = +∞. We argue by contradiction. We show that if E c < +∞, then there exists a nonlinear global solution of (1.1) with )-norm be exactly E c . Moreover, this solution satisfies some strong compactness properties. This is completed in Section 4 where we utilize the profile decomposition that was established in Ibrahim, Masmoudi and Nakanishi [14], and a strategy introduced by Kenig and Merle [17]. Finally, we utilize the finiteness of the energy to show that the solutions obtained in Section 4 are not possible. More precisely, by Morawetz inequality [4,23,25] (1.10) we know that the left-hand side of (1.10) is bounded by the energy for any T > 0.
On the other hand, notice that by finite speed of propagation and concentration of potential energy, the left-hand side of (1.10) should grow logarithmical in time T . This gives a contradiction by choosing T sufficiently large. The paper is organized as follows. In Section 2, we deal with the local theory for the equation (1.1). In Section 3, we give the linear and nonlinear profile decomposition and show some properties of the profile. Thereafter, we extract a critical solution in Section 4. Finally in Section 5, we preclude the critical solution, which completes the proof of Theorem 1.1.
We conclude the introduction by giving some notations which will be used throughout this paper. We always assume the spatial dimension d 5 and f (u) = |u| 2 u. For any r, 1 r ∞, we denote by · r the norm in L r = L r (R d ) and by r ′ the conjugate exponent defined by 1 r + 1 r ′ = 1. For any s ∈ R, we denote by H s (R d ) the usual Sobolev space. Let ψ ∈ S(R d ) be such that supp ψ ⊆ ξ : 1 2 |ξ| 2 and j∈Z ψ(2 −j ξ) = 1 for ξ = 0. Define ψ 0 by ψ 0 = 1 − j 1 ψ(2 −j ξ). Thus supp ψ 0 ⊆ ξ : |ξ| 2 and ψ 0 = 1 for |ξ| 1. We denote by ∆ j and P 0 the convolution operators whose symbols are respectively given by ψ(ξ/2 j ) and ψ 0 (ξ). For s ∈ R, 1 r ∞, the Besov spaces B s r,2 (R d ) andḂ s r,2 (R d ) are defined by For details of Besov space, we refer to [1]. For any interval I ⊂ R and any Banach space X we denote by C(I; X) the space of strongly continuous functions from I to X and by L q (I; X) the space of strongly measurable functions from I to X with u(·); X ∈ L q (I).
We denote by ·, · the scalar product in L 2 .

Preliminaries
2.1. Strichartz estimate and local theory. In this section, we consider the Cauchy problem for the equation (1.1) The integral equation for the Cauchy problem (2.1) can be written as where Let U (t) = e itω , theṅ Now we recall the following dispersive estimate for the operator U (t) = e itω .
Lemma 2.1 ( [4,12]). Let 2 r ∞ and 0 θ 1. Then According to the above lemma, the abstract duality and interpolation argument(see [13], [16]), we have the following Strichartz estimates. Lemma 2.2 ( [4,12,23,24]). Let 0 θ i 1, ρ i ∈ R, 2 q i , r i +∞, i = 1, 2. Assume that (θ i , d, q i , r i ) = (0, 3, 2, +∞) satisfy the following admissible conditions where the subscript R stands for retarded, and Now it is useful to define several spaces and give estimates of the nonlinearities in terms of these spaces. Define where In addition to the ST -norm, we also need the corresponding dual norm Then we have by Strichartz estimate where the time interval I contains zero. [8,28]). Let s 0, and 1 < r, p j , q j < +∞ be such that . As a direct consequence, we have the following nonlinear estimate.

Lemma 2.4 (Nonlinear estimate). Let I be a time slab, one has
Proof. It follows from the above product rule and Sobolev embedding: Using Hölder's inequality and Sobolev embedding, we obtain Plugging this into (2.9), we get (2.8).
We can now state the local well-posedness for (1.1) with large initial data and small data scattering in the space H sc (R d ) × H sc−1 (R d ), which is the first step to obtain the global time-space estimate and then lead to the scattering.
There exists a small constant δ = δ(E) such that if (u 0 , u 1 ) H sc ×H sc−1 E and I is an time interval containing zero such that where C is the Strichartz constant as in Lemma 2.2.
In particular, if (u 0 , u 1 ) H sc ×H sc−1 δ, then the solution u is global and scatters.
Proof. We apply the Banach fixed point argument to prove this lemma. First we define the solution map It suffices to prove that the operator defined by the RHS of (2.12) is a contraction map on B for I. If u ∈ B, then by Strichartz estimate (2.7), (2.8) and (2.10), we have Thus we can choose δ small depending on E and the Strichartz constant C such that On the other hand, by a same argument as before and Lemma 2.4, we have for which allows us to derive by taking δ small such that A standard fixed point argument gives a unique solution u of (1.1) on I × R d which satisfies the bound (2.11).
Using Theorem 2.1 as well as its proof, one easily derives the following local theory for (1.1). We omit the standard detail here.
This solution also has the following properties: ( there is a unique solution to (1.1) in a neighborhood of infinity so that (1.6) holds.

Perturbation lemma.
In this part, we give the perturbation theory of the solution of (1.1) with the global space-time estimate.
With any real-valued function u(t, x), we associate the complex-valued function u(t, x) by where ℜz denotes the real part of z ∈ C. Then the free and nonlinear Klein-Gordon equations are given by Lemma 2.5. Let I be a time interval, t 0 ∈ I and u, w ∈ C(I; L 2 (R d )) satisfy for some function eq(u), eq(w). Assume that for some constants M, E > 0, we have Assume also that we have smallness conditions where ǫ is as above and . Then we conclude that Proof. Since w ST (I) M , there exists a partition of the right half of I at t 0 : such that N C(L, δ) and for any j = 0, 1, · · · , N − 1, we have The estimate on the left half of I at t 0 is analogue, we omit it. Let then γ satisfies the following difference equation which implies that t j e i ∇ (t−s) ∇ sc−1 γ(γ 2 + 3γω + 3ω 2 ) + eq(u) − eq(w) ds.
By Strichartz estimate (2.7) and nonlinear estimate (2.8), we have . Therefore, assuming that (2.23) γ ST (I j ) δ ≪ 1, ∀ j = 0, 1, · · · , N − 1, then by (2.20) and (2.22), we have for some absolute constant C > 0. By (2.17) and iteration on j, we obtain provided we choose ǫ 1 sufficiently small. Hence the assumption (2.23) is justified by continuity in t and induction on j. Then repeating the estimate (2.22) once again, we can get the ST-norm estimate on γ, which implies the Strichartz estimates on u.

Profile decomposition
In this section, we first recall the linear profile decomposition of the sequence of L 2 x -bounded solutions of (i∂ t + ∇ ) v = 0 which was established in [14]. And then we show the nonlinear profile decomposition which will be used to construct the critical element and obtain its compactness properties in the next section.
3.1. Linear profile decomposition. First, we give some notation. For any triple (t j n , x j n , h j n ) ∈ R × R d × (0, 1] with arbitrary suffix n and j, let τ j n , T j n , and ∇ j n respectively denote the scaled time shift, the unitary and the self-adjoint operators in Now we can state the linear profile decomposition as follows Lemma 3.1 (Linear profile decomposition, [14]). Let v n (t) = e i ∇ t v n (0) be a sequence of free Klein-Gordon solutions with uniformly bounded L 2 x (R d )-norm. Then after replacing it with some subsequence, there exist K ∈ {0, 1, 2 . . . , ∞} and, for each integer j ∈ [0, K), ϕ j ∈ L 2 (R d ) and {(t j n , x j n , h j n )} n∈N ⊂ R × R d × (0, 1] satisfying the following. Define v j n and ω k n for each j < k K by then we have and for any l < j < k K and any t ∈ R, where µ ∈ MC and MC is defined to be Moreover, each sequence {h j n } n∈N is either going to 0 or identically 1 for all n. Remark 3.1. We call { v j n } n∈N a free concentrating wave for each j, and w k n the remainder. From (3.5), we have the following asymptotic orthogonality We remark the following estimates for 1 < p < ∞, hold uniformly for 0 < h n 1, by Mihlin's theorem on Fourier multipliers.

3.2.
Nonlinear profile decomposition. After the linear profile decomposition of a sequence of initial data in the last subsection, we now show the nonlinear profile decomposition of a sequence of the solutions of (1.1) with the same initial data in the space H sc (R d ) × H sc−1 (R d ).
First we construct a nonlinear profile associated with a free concentrating wave. Let v j n be a free concentrating wave for a sequence (t j n , x j n , h j n ) ∈ R × R d × (0, 1], (3.9) Then by Lemma 3.1, for a sequence of free Klein-Gordon solutions { v n (t) = e it ∇ v n (0)} with uniformly bounded L 2 x (R d )-norm, we have a sequence of the free concentrating wave v j n (t, x) with v j n (t j n ) = T j n ϕ j , ϕ j ∈ L 2 (R d ) for j = 0, 1, · · · , k − 1, such that v n (t, Now for any free concentrating wave v j n , we undo the group action T j n to look for the Next let u j n be the nonlinear solution with the same initial data v j n (0) where τ j n = −t j n /h j n . In order to look for the nonlinear profile U j ∞ associated with the free concentrating wave v j n , we also need undo the group action. Define u j n (t, x) = T j n U j n (t − t j n )/h j n , then U j n satisfies the rescaled equation Extracting a subsequence, we may assume that there exist h j ∞ ∈ {0, 1} and τ j ∞ ∈ [−∞, +∞] for every j, such that as n → +∞ h j n → h j ∞ , and τ j n → τ j ∞ . Thus we have the limit equations as follows We remark that the unique existence of a local solution U j ∞ around t = τ j ∞ is known in all cases, including h j ∞ = 0 and τ j ∞ = ±∞. We say that U j ∞ on the maximal existence interval is the nonlinear profile corresponding to the free concentrating wave ( v j n ; t j n , x j n , h j n ). The nonlinear concentrating wave u j (n) corresponding to v j n is defined by When h j ∞ = 1, u j (n) solves (1.1). While h j ∞ = 0, then it solves (3.13) The existence time interval of u j (n) may be finite and even go to 0, however, we have as n → +∞.
Let u n be a sequence of solutions of (1.1) around t = 0, and let v n be the sequence of the free solutions with the same initial data. By Lemma 3.1, we have the linear profile decomposition for { v n } as follows Now we define the nonlinear profile decomposition as follows.
Definition 3.1 (Nonlinear profile decomposition). Let { v j n } n∈N be the free concentrating wave, and { u j (n) } n∈N be the sequence of the nonlinear concentrating wave corresponding to { v j n } n∈N . Then we define the nonlinear profile decomposition of u n by We will show that u <k (n) + ω k n is a good approximation for u n provided that each nonlinear profile has finite global Strichartz norm.
Next we define the Strichartz norms for the nonlinear profile decomposition. Recall that ST (I) and ST * (I) are the functions spaces on I × R d defined as above . And the Strichartz norm for the nonlinear profileÛ j ∞ is defined by (3.16) The following two lemmas derive from Lemma 3.1 and the perturbation lemma. The first lemma concerns the orthogonality in the Strichartz norms.
x (R) < +∞, ∀ j < k. Then, for any finite interval I, j < k, one has (3.19) where the implicit constants is independent of I and j. Furthermore, we have (3.20) lim Proof. Proof of (3.18): Case 1: h j ∞ = 1. It is easy to see that u j (n) is just a sequence of space-time translations ofÛ j ∞ in this case. And so (3.18) follows in this case.
Case 2: h j ∞ = 0. We drop the superscript j in the following. Using the definition of u (n) andÛ ∞ , we derive u (n) (t, x) = h sc n T n |∇| sc ∇ −sc n U ∞ (t − t n )/h n .
By Sobolev embeddingḂ 0 p,2 ⊂ L p with p 2 in the lower frequencies and scaling, one has for s := s c − 1 Therefore, we obtain by scaling , which concludes the proof of (3.18).
Proof of (3.19): We estimate the left hand side of (3.19) by .
For the case h j ∞ = 1. Define U j ∞,R , u j (n),R and u <k (n),R by where χ R (t, x) = χ(t/R, x/R) and χ(t, x) ∈ C ∞ c (R 1+d ) is the cut-off defined by Then we have .
On one hand, we know that as R → +∞. On the other hand, by (3.6), the similar orthogonality and approximation analysis as in [14], we obtain , and for the case h j .
Proof of (3.20): By the definition of u j (n) andÛ j ∞ , we know that Then we have .

(3.22)
Using (3.6) and the approximation argument in [14], we get (3.21) → 0 as n → ∞. In addition, by h j n → 0 as n → ∞, one has as n → ∞. Hence we obtain (3.20). And so we complete the proof of this lemma.
With this preliminaries in hand, we now show that u <k (n) + ω k n is a good approximation for u n provided that each nonlinear profile has finite global Strichartz norm. Lemma 3.3. Assume that u n is a sequence of local solutions of (1.1) around t = 0 obeying lim n→∞ (u n ,u n ) L ∞ t (In;H sc x ×H sc−1 x ) < +∞. Assume also that in its nonlinear profile decomposition (3.15), every nonlinear profile U j ∞ has finite global Strichartz and L 2 x norms; that is x (R) < +∞. Then u n is bounded for large n in the Strichartz and the H sc norms, i.e.
We only need to verify the conditions of Lemma 2.5. For this purpose, by (3.13), we derive that u <k (n) + ω k n satisfies that where the error term eq u <k (n) , ω k n is First, by the definition of the nonlinear concentrating wave u j (n) and (3.14), we have as n → +∞. This verifies the condition (2.17) by the Strichartz estimate. Next, by the linear profile decomposition in Lemma 3.1, we get by (3.7) Hence except for a finite set J ⊂ N, the H sc x × H sc−1 x -norm of u j (n) (0),u j (n) (0) with j ∈ J is smaller than the iteration threshold (the small data scattering, Theorem 2.1), and so u j This together with (3.18), (3.19), (3.23) and (3.25) yield that for any finite interval I This together with the Strichartz estimate for ω k n implies that sup k lim n→∞ u <k (n) + ω k n ST (I) < +∞.
By the similar argument as above, we obtain for large n Hence we verify the conditions (2.15) and (2.16). It remains to verify the condition (2.18). Using Lemma 3.1 and Lemma 3.2, we have as n → +∞. On the other hand, the linear part in eq u <k (n) , ω k n vanishes if h j ∞ = 1, and from (3.8), we know that it is controlled if h j ∞ = 0 by →0, as n → +∞, (3.27) by the continuity in t and Lebesgue domainted convergence theorem for bounded t, and by the scattering ofÛ j ∞ for t → ±∞, which follows from Û j ∞ ST j ∞ (R) < +∞, and again Lebesgue domainted convergence theorem.
Thus, eq u <k (n) , ω k n ST * (I) → 0, as n → +∞. Therefore, for k sufficiently close to K and n large enough, the true solution u n and the near solution u <k (n) + ω k n satisfy all the assumptions of the perturbation Lemma 2.5. Thus, we conclude this Theorem.

Concentration Compactness
Using the profile decomposition in the previous section and the perturbation theory, we argue in this section that if the scattering result does not hold, then there must exist a minimal solution with some good compactness properties. Moreover, there exists Proof. By the definition of E c , we can choose a sequence of solutions to (1.1): {u n (t) : I n × R → R} such that (4.2) sup t∈In (u n ,u n ) H sc ×H sc−1 → E c , and u n ST (In) → +∞, as n → +∞.
By Lemma 3.1, we have and by (3.7), we derive that there is only one profile, i.e. K = 1, and so for large n But Bulut has shown that there is no such solution in [5,6]. Therefore, h 0 n ≡ 1. And so there exist a sequence (t n , x n ) ∈ R × R d and φ ∈ L 2 (R d ) such that along some subsequence, If not, then there exist a sequence t n ∈ R which approaches the maximal existence time. Noting that Û 0 ∞ (t + t n ), ∂ tÛ 0 ∞ (t + t n ) satisfies (4.2), and then by the same argument as (4.5), we deduce that there exist another sequence (t ′ n , x ′ n ) ∈ R × R d and for some ψ ∈ L 2 so that as n → ∞. We write v := e it ∇ ψ. From Strichartz estimate, we know that for any ε > 0, there exist δ > 0 with I = [−δ, δ] so that where η 0 = η(d) is the threshold from the small data theory. This together with (4.6) shows that for sufficiently large n Hence, by the small data theory (Theorem 2.1), we derive that the solution U 0 ∞ exists on [t n − δ, t n + δ] for large n, which contradicts with the choice of t n . ThusÛ 0 ∞ is a global solution and it is just the desired critical element u c . Moreover, since (1.1) is symmetric in t, we may assume that u c ST (0,+∞) = +∞.
We call such u a forward critical element. Next we prove the precompactness of K. It is equivalent to show the precompactness of { u(t n )} in L 2 x for any t 1 , t 2 , · · · > 0. It is easy to prove this by the continuity in t when t n converges. Thus, we can suppose that t n → +∞. Applying the property of (4.5) to the sequence of solution u(t + t n ), we get another sequence (t ′ n , x ′ n ) ∈ R d and φ ∈ L 2 such that If t ′ n → −∞, then we obtain by triangle inequality Thus, by the small data theory, we can solve u for t < t n with large n globally, which contradicts with its forward criticality.
If t ′ n → +∞, then one has Hence, we can solve u for t < t n with large n with diminishing Strichartz norms. We give a contradiction since u = 0 by taking the limit. Thus, t ′ n is bounded, which shows that {t ′ n } is precompact, so is u(t n , x + x ′ n ) in L 2 x by (4.8).
As a direct consequence of the above proposition, we have We refer to the function x(t) as the spatial center function, and to C(η) as the compactness modules function.
We remark that the small data theory shows that the H sc x (R d )×H sc−1 x (R d ) norm of a blowup solution must remain bounded from below. The fact that this norm is nonlocal in odd space dimensions reduces the efficacy of this statement. Our next lemma gives a lower bound in a more suitable norm, and also a mild control of x(t). (1) ((∇ t,x u, u) nontrivially) We have and so by Sobolev embedding, In particular, taking t = t 2 , we get (4.15) |x−x(t 1 )| |t 2 −t 1 |+C(η) On the other hand, we have by Corollary 4.1 and Sobolev embedding, (4.16) |x−x(t 2 )| C(η) This together with (4.10) and (4.15) yield that This concludes the proof of (4.11). The next corollary shows that the potential energy of the critical element must concentrate. Proof. The bound from above follows from Sobolev's inequality and sup t∈R (u c ,u c ) H sc ×H sc−1 = E c < +∞. Suppose the bound from below is not true. Then there exist τ > 0 and a sequence t k such that Using the precompactness of K, we can extract a subsequence and assume that Let U : I × R d → R be the nonlinear strong solution of (1.1) with initial data (U 0 , U 1 ) at time t = 0. Then, E(U,U ) = E(u,u) = 0. By wellposedness and (4.19), we get Hence, we have U (t) = 0 for all t in (0, τ ) ∩ I, hence U t (t) = 0 for all such t. Consequently, E(u,u) = 0. This is a contradiction.

Extinction of the critical element
In this section, we prove that the critical solution constructed in Section 4 does not exist, thus ensuring that E c = +∞. This implies Theorem 1.1. Proof. We argue by contradiction. Assume there exists a solution u : R + ×R d → R such that the set K defined in Proposition 4.1 is precompact in H sc (R d )×H sc−1 (R d ). We will show that this scenario is inconsistent with the following Morawetz inequality [4,23,25] (5.1) |x| dxdt E(u, u t ).
On one hand, since the solution u has finite energy, the right-hand side in the Morawetz inequality is finite and so