Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions. We prove that for a given asymptotic profile, there exists a solution to (NLKG) which converges to given asymptotic profile as t to infinity. Here the asymptotic profile is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper and smooth modification of phase correction by Ginibre-Ozawa.


Introduction
This paper is devoted to the study of the final state problem for the nonlinear Klein-Gordon equation with a critical nonlinearity in three space dimensions: (1.1) ( + 1)u = λ|u| 2/3 u t ∈ R, x ∈ R 3 , u − u ap → 0 in L 2 as t → +∞, where = ∂ 2 t −∆ is d'Alembertian, u : R×R 3 → R is an unknown function, u ap : R × R 3 → R is a given function, and λ is a non-zero real constant. The aim of this paper is to find a proper choice of the function u ap so that the equation (1.1) admits a nontrivial solution. In other words, we want to determine a "right" asymptotic behavior which actually takes place. This is a continuation of our previous study of the two dimensional case in [23].
Let us briefly review the known results on the global existence and long time behavior of solution to the more general nonlinear Klein-Gordon equation where p > 1 and λ ∈ R\{0}. Since the point-wise decay of solution to the linear Klein-Gordon equation is O(t −d/2 ) as t → ∞, the linear scattering theory indicates that the power p = 1 + 2/d will be a borderline between the short and long range scattering theories. This formal observation was firstly justified by Glassey [7], Matsumura [24] and Georgiev and Yordanov [5] for p 1+2/d. More precisely, they proved that solutions to (1.2) do not scatter to the solution to the linear Klein-Gordon equation if 1 < p 1+2/d.
Later, Georgiev and Lecente [4] obtained a point-wise decay estimates for small solutions to the (1.2) for p > 1+2/d with d = 1, 2, 3 by using the vector field approach by Klainerman [16]. Furhtermore, Hayashi and Naumkin [10] proved that small solutions to (1.2) scatter to the solution to the linear Klein-Gordon equation if p > 1 + 2/d and d = 1, 2. Notice that it is an still open problem for the asymptotic behavior of small solution to (1.2) when p is close to 1 + 2/d and n 3. See [9,16,28,29,30,32] for the small data scattering when n 3 and p is large. For the critical case p = 1 + 2/d and d = 1, Georgiev and Yordanov [5] studied the point-wise decay of a solution to the initial value problem. Delort [1] obtained an asymptotic profile of a global solution to the equation. His proof is based on hyperbolic coordinates and the compactness of the support of the initial data was assumed. See also Lindblad and Soffer [18] for the alternative proof of [1]. The compact support assumption in [1] was later removed by Hayashi and Naumkin in [8] by using the vector field approach.
Recently, the authors [23] consider (1.2) with p = 1 + 2/d and d = 2 and specify an asymptotic profile u ap that allows a unique solution u which converges to u ap as t → ∞. The asymptotic profile u ap has the same form as in the d = 1 case. Namely, it is the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. The key ingredient is to extract a resonance term by means of Fourier series expansion of the nonlinearity. In this paper, we consider (1.1), that is, a similar final value problem for (1.2) with p = 1 + 2/d and d = 3. Because the power becomes a fractional number, the argument in the two dimensional case [23] is not directly applicable. To deal with the nonlinearity, we use the argument in Ginibre-Ozawa [6].
Let us introduce the asymptotic profile u ap which we work with. To this end, we first recall that the leading term of a solution to the linear Klein-Gordon equation is given by is the characteristic function supported on Ω ⊂ R 1+3 , and ρ 0 and β ∈ [0, 2π) are given by the relation , see [11] for instance.
For given final state (φ 0 , φ 1 ), we define the asymptotic profile u ap by , where the phase correction term is given by Remark that the coefficient comes from the first Fourier-cosine coefficient of a 2π-periodic function | cos θ| 2/3 cos θ. The final state (φ 0 , φ 1 ) is taken from the function space Y defined by The main result in this paper is as follows.     [25], Katayama [12], Sunagawa [33] for one dimensional cubic case and Ozawa, Tsutaya and Tsutsumi [27], Delort, Fang and Xue [2], Kawahara and Sunagawa [14], Katayama, Ozawa and Sunagawa [13] for the two dimensional quadratic case.
The rest of the paper is organized as follows. In Section 2, we exhibit the outline of the proof of Theorem 1. In this section, we give an outline of the proof of Theorem 1.1.

2.1.
On the solvability of the final state problem. We first remark that solvability of (1.1) is reduced to the appropriateness of the choice of the asymptotic behavior u ap .
Let A(t, x) be a given asymptotic profile of a solution to (1.1). We show that if A(t, x) is well-chosen then we obtain a solution which asymptotically behaves like A(t, x). Let N (u) = λ|u| 2/3 u.
for some T 0 3, then there exist T T 0 and a unique solution u ∈ C([T, ∞); L 2 x ) for the equation (1.1) satisfying for the same γ.
The proposition will be proven in Section 3.

2.2.
Choice of an appropriate asymptotic profile. An easy choice is A = u ap . However, it does not work well. Hence, we choose a suitable A that satisfies the assumptions (2.1) and (2.2) and Then, the solution obtained by means of Proposition 2.1 from A possesses the desired asymptotics. The obstacle in three-dimensional case lies in the fact that the phase correction term Ψ, given in (1.4), has the fractional power term ρ 2/3 . The power comes from the nonlinearity. Notice that because of the fractional power, we may not estimate ( + 1)u ap , in general. To overcome the difficulty, we exploit the argument in Ginibre-Ozawa [6]. We introduce a modified phase corrector and an auxiliary approximate solution We will see the error from the modification is acceptable. Starting from the modified asymptotic profile u ap , we construct the profile A as in the two dimensional case [23]. This is the idea of the proof. For readers' convenience, we recall the construction of A.
x as t → ∞, we have to, at least, find the main parts of it and cancel them out, otherwise (2.2) fails. In [23], a Fourier series expansion is introduced for this purpose. Here, we split where c n is the Fourier-cosine coefficient of | cos θ| 2 3 cos θ. We employ the following estimate on the coefficient. 21]). Let c n := 1 π π −π | cos θ| 2 3 cos θ cos nθdθ for n 0. Then, c n = 0 for even n and for odd n. In particular, Thanks to the choice of the phase function Ψ, the resonance part N r is close to ( + 1) u ap (See Proposition 4.5). To cancel out N nr , we introduce ).
It will turn out that the non-resonance part N nr is successfully canceled out by v ap (See Proposition 4.6).

Remark 2.3. This kind of approximation was introduced in Hörmander [11]
for the Klein-Gordon equation with polynomial nonlinearity in (u, u). See also [26,31] for the nonlinear Schrödinger equation with polynomial nonlinearity in (u, u).
Based on the above observation, we will show the following proposition.
Together with Proposition 2.1, this proposition implies Theorem 1.1. Section 4 is devoted to the proof of the above proposition.
Remark 2.5. Let us consider a generalization of Theorem 1.1 to any realvalued nonlinearity satisfying F (λu) = λ 5/3 F (u) for any λ > 0 and u ∈ R. Notice that this class of nonlinearity is written as F (u) = λ 1 |u| 2 3 u + λ 2 |u| 5 3 . Theorem 1.1 corresponds to the case where λ 2 = 0. By means of the following lemma, we see that the nonlinearity λ 2 |u| 5/3 does not contain a resonant part, and so that we can treat the above general nonlinearity by the same argument.

Proof of Proposition 2.1
In this section, we prove Proposition 2.1. The proof is essentially the same as in the two dimensional case [23]. The following inhomogeneous Strichartz estimates associated with the Klein-Gordon equation is crucial for the proof. Let Lemma 3.1. Let 2 q 6 and 2/p + 3/q = 3/2. Then we have x ) . Proof. The above inequalities follow from combination of the L p -L q estimate for the solution to the Klein-Gordon equation by [19] with the duality argument by [35] for the non-endpoint case q = 6 and the argument by [15] for the endpoint case q = 6. Since the proof is now standard, we omit the detail.
Proof of Proposition 2.1. We introduce .
For R > 0 and T > 0, we define The function space X T is a Banach space with the norm · X T and X T (ρ) is a complete metric space with the · X T -metric. We put v = u − A. Then the equation (1.1) is equivalent to The associate integral equation to the equation (3.2) is where G is given by (3.1). It suffices to show the existence of a unique solution v to the equation (3.3) in X T for suitable η > 0 and T T 0 . We prove this assertion by the contraction argument. Define the nonlinear operator Φ by Φv : for v ∈ X T (R). We show that Φ is a contraction map on X T (ρ) if R > 0, T T 0 , and η > 0 are suitably chosen. Let v ∈ X T (R) and t T . By the assumptions and Lemma 3.1, we see where M is an upper bound on the right hand side of (2.2). Therefore we obtain In the same way as above, for v 1 , v 2 ∈ X T (R), we can show (3.5) We first fix R so that C 1 M R/2. Then, using the fact that γ > 3/4, we are able to choose a sufficiently large T > 0 and a sufficiently small η > 0 such that

Proof of Proposition 2.4
In this section, we prove Proposition 2.4. Since (2.1) is trivial, we prove (2.4) and (2.2) in Sections 4.2 and 4.3, respectively, after preparing preliminary estimates in Section 4.1. Hereafter we always restrict our attention to the region |x| < t and t 3.

Preliminaries.
We collect preliminary estimates.
Proof. It follows by direct calculation.
Recall that ρ(s, µ) = ρ(µ) 2 + s −1 µ −3 . An elementary inequality will be useful. The following will be used to estimate the error comes from the phase modification.
Lemma 4.2. For s 3, we have the following inequality: where the implicit constant is independent of ρ.
Proof. By a direct calculation, we obtain for every µ ∈ R 3 . Thus we obtain the desired inequality. Now, we turn to the estimate of the derivatives of the modified phase part Ψ(s, ρ) = −(λc 1 /2) ρ(s, µ) 2/3 .
where the implicit constants are independent of ρ.
where the implicit constants are independent of ρ.
Proof. The first two inequalities (4.11) and (4.12) follow from (4.5). To obtain the inequality (4.13), we use where δ jk is the Kronecker delta. The inequality (4.14) is a consequence of This completes the proof of Lemma 4.4.

Proof.
Since for t 3, we deduce from (4.4) and Lemma 4.2 that Further, we see from (4.5) that Similarly, we have Indeed, in view of (4.2), the leading term with respect to t appears only when the derivative ∇ x hits e in µ −1 t . Furthermore, in that case, ∇ x e in µ −1 t = −inµe i µ −1 t and so the estimate is essentially the same. Combining these estimates, we conclude that A satisfies (2.4) as long as γ < 5/6.
The third term of the right hand side is estimated as Hence, we estimate the first and the second terms, in what follows. holds.
Proof. To show the inequality (4.15), we begin with the computation of the linear part: We split (4.16) Moreover, since and (4.20) Re J k .
To estimate the right hand side of (4.21), we show several elementary lemmas.