Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in four and more spatial dimensions

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we prove the small data global well-posedness and scattering at the critical space in $d \ge 4$ by applying the radial Strichartz estimates and $U^2, V^2$ type spaces. On the other hand, if the initial datum is non-radial, then we prove the local well-posedness at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$ by applying the $U^2, V^2$ type spaces.

Then, (1.2) is globally well-posed in H s (R d ) ×Ḣ s (R d ). We consider both the radial case and the non-radial case. First, we consider the radial case. In the radial case, the Strichartz estimates hold for a more wider range of (q, r). More precisely, see Propostions 2.11, 2.12. On the other hand, we have to recover a half derivative loss to derive the key bilinear estimates at the critial space. Thanks to c > 0 and c = 1, if |ξ| ≫ |ξ ′ |, then it holds that Here, ξ, ξ ′ denote frequency for the wave equation, Klein-Gordon equation respectively and τ ± c|ξ| (resp. τ ′ ± ξ ′ , τ − τ ′ ± ξ − ξ ′ ) denote the symbol of the linear part for the wave equation (resp. the Klein-Gordon equation). From (1.3) and by applying the U 2 , V 2 type spaces, then we can recover the derivative loss. Therefore, we can obtain the bilinear estimates at the critical space by applying the radial Strichartz estimates and U 2 , V 2 type spaces. Next, we consider d = 4 and the nonradial case. When d ≤ 4, the Lorentz regularity s l is an important index as well as the scaling regularity for the well-posedness for the wave equation. When d = 4 with quadratic nonlinearity, the Lorentz regularity s l = 1/4. On the other hand, s c = 0, so we need to consider s ≥ s l = 1/4. When d = 4, we obtain local well-posedness at s = s l = 1/4 by applying U 2 , V 2 type spaces. Finally, we consider d ≥ 5 and the non-radial case. Since s c ≥ s l when d ≥ 5, we expect the local well-posedness with s = s c . However, we only obtain the local well-posedness with s = s c + 1/(d + 1).
It seems difficult to prove the bilinear estimate with s = s c . The reason is as below.
We observe the first equation of (1.2). We regard the nonlinearity as n ± (ω −1 1 u ± ). Here, we consider the following cases. The case |ξ| |ξ ′ | and the case |ξ| ≫ |ξ ′ |, where ξ, ξ ′ denote the frequency of n ± , u ± respectively. For the case |ξ| |ξ ′ |, the nonlinearity does not have the derivative loss, so we can derive the bilinear estimate at the critical space only by applying the Strichartz estimates. However, for the case |ξ| ≫ |ξ ′ |, we need to recover a half derivative loss by (1.3). Here, there are three cases in (1.3). The cases (a) M ′ = τ ± c|ξ| , (b) M ′ = |τ ′ ± ξ ′ | and (c) M ′ = |τ − τ ′ ± ξ − ξ ′ |. For the case (a) or (c), we apply (1.3) for n ± and the Strichartz estimates for ω −1 1 u ± . Then we can obtain the bilinear estimate at the critical space. Whereas for (b), we apply the Strichartz estimates for n ± and apply (1.3) for ω −1 1 u ± . In this case, we cannot prove the bilinear estimate at the critical space. As a result, we have to impose more regularity.
In section 2, we prepare some notations and lemmas with respect to U p , V p , in section 3, we prove the bilinear estimates and in section 4, we prove the main result.

Acknowledgement
The author appreciate Professor K. Tsugawa for giving useful advice. Also, the author would like to thank S. Kinoshita for telling the author about the radial Strichartz estimates for the wave equation.

Notations and Preliminary Lemmas
In this section, we prepare some lemmas, propositions and notations to prove the main theorem. A B means that there exists C > 0 such that A ≤ CB. Also, A ∼ B means A B and B A. Let u = u(t, x). F t u, F x u denote the Fourier transform of u in time, space, respectively. F t, x u = u denotes the Fourier transform of u in space and time. Let Z be the set of finite partitions −∞ < t 0 < t 1 < · · · < t K = ∞ and let Z 0 be the set of finite partitions −∞ < t 0 < t 1 < · · · < t K ≤ ∞.
1, we call the function a : R → L 2 x given by a U p -atom. Furthermore, we define the atomic space λ j a j , λ j ∈ C, a j : U p -atom .
(iv) The closed subspace U p c of all continuous functions in U p is a Banach space.
The definitions of V p and V p − , see the erratum [9].
Let N = 2 n (n ∈ Z) be dyadic number. P N and P <1 denote For dyadic number N, M, Here summation over N means that summation over n ∈ Z. Similarly, we define Definition 4. For the Klein-Gordon equation, we defineẎ s K ± ,Ż s K ± , Y s K ± , Z s K ± as the closure of all u ∈ C(R; H s x (R d )) such that Similarly, for the wave equation, we defineẎ s W ±c ,Ż s W ±c , Y s W ±c , Z s W ±c by replacing K ± with W ±c in the above norms.
Definition 5. For a Hilbert space H and a Banach space X ⊂ C(R; H), we define We denote the Duhamel term for the Klein-Gordon equation and the wave equation respectively.
If 0 < c < 1, then the right hand side of (2.2) is bounded by If c > 1, then the right hand side of (2.2) is bounded by The following proposition is in [8] (Theorem 2.8 and Proposition 2.10).
Proposition 2.4. u ∈ V 1 − ⊂ U 2 be absolutely continuous on compact intervals and v ∈ V 2 . Then, u U 2 = sup Corollary 2.5. Let A = K ± or W ±c and u ∈ V 1 −, A ⊂ U 2 A be absolutely continuous on compact intervals and v ∈ V 2 A . Then, (2.3) Proof. By scaling, we only prove (2.3) for M = 1. We will show (2.
By the unitarity of K ± , we have For some Schwartz function φ, it holds that Hence by the Young inequality and the Hölder inequality, we have (2.6) Collecting (2.5)-(2.6), we obtain (2.4).
be a n-linear operator. Assume that for some 1 ≤ p, q ≤ ∞, it holds that See Proposition 2.19 in [8] for the proof of the above proposition.
For the proof of Proposition 2.9, see [22]. Combining Proposition 2.2, Proposition 2.8, Proposition 2.9 and Proposition 2.7, we have the following proposition.
is radial function, then it holds that (2.9) If N < 1 and f ∈ L 2 x (R d ) is radial function, then it holds that See (3.13) in [7] for the proof of Proposition 2.12.
Then, for all radial function Proof. When q = r, (2.11) follows from (2.9). Interpolating L q t,x with L ∞ t L 2 x , we obtain (2.11).
Proof. We only prove for A = K ± since we can prove similarly for A = W ±c . By L 2 x orthogonality, we have Proposition 2.17. It holds that The same estimates hold by replacing the Klein-Gordon operator K ± by the wave operator W ±c .
For the proof of Lemma 2.18, see [13].
By Corollary 2.5, we have We apply Corollary 2.5 to have For d = 4 and s = 1/4, by (3.4), For the estimate of J 2 , we take M = εN 1 for sufficiently small ε > 0. Then, from Lemma 2.3, we have For the estimate of F 1 , we apply Corollary 2.5 to have the right-hand side of (3.5) is bounded by For the estimate of F 2 , we apply Corollary 2.5 to have For d = 4, s = 1/4, we apply Lemma 2.19 (iv), , then the right-hand side of (3.7) is bounded by (3.8) For the estimate for F 3 , we apply Corollary 2.5 to have For d = 4, s = 1/4, we apply Lemma 2.19 (iv), , then the right-hand side of (3.9) is bounded by (3.10) Collecting (3.6), (3.8) and (3.10), we obtain J 2 T 1/2 n 2Ẏ . By Corollary 2.5 and the triangle inequality to have (3.12) From (3.12), the right-hand side of (3.11) is bounded by Hence, · l 2 l 1 · l 1 l 2 and the Cauchy-Schwarz inequality to have We prove (3.2). By Corollary 2.5, we only need to estimate K i (i = 1, 2, 3):

I. KATO
First, we estimate K 1 . Put K 1 = K 1,1 + K 1,2 where For d = 4, s = 1/4, by the same manner as the estimate for Lemma 2.19 (i) and Hence, We take M = εN 2 for sufficiently small ε > 0. Then, from Lemma 2.3, we have Therefore, Hence, it follows that (3.14) (3.16) By Lemma 2.18, (3.17) By the same manner as the estimate for Lemma 2.19 (iv), i = 5, for d = 4, s = 1/4, we find By Lemma 2.18, By the same manner as the estimate for Lemma 2.19 (iv), i = 6, for d = 4, s = 1/4, we find (3.20) By Lemma 2.18, (3.21) By the same manner as the estimate for Lemma 2.19 (iv), i = 4, for d = 4, s = 1/4, we find By symmetry, the estimate for K 2 is obtained by the same manner as the estimate for K 1 . Hence, we estimate K 3 . By the triangle inequality, Lemma 2.19 (i) and the Cauchy-Schwarz inequality, we have If d = 4, s = 1/4, then we apply Lemma 2.19 (i) and the Cauchy-Schwarz inequality, the right-hand side of (3.23) is bounded by Next, we prove , the right-hand side of (3.5) is bounded by (3.24) From Lemma 2.19 (iv), , the right-hand side of (3.7) is bounded by , the right-hand side of (3.9) is bounded by (3.26) . By the same manner as the estimate for Lemma 2.19 (iii), we obtain From (3.27), the right-hand side of (3.11) is bounded by Hence, · l 2 l 1 · l 1 l 2 and the Cauchy-Schwarz inequality to have We prove (3.2) for d ≥ 5, s = s ′ = (d 2 − 3d − 2)/2(d + 1) by the same manner as the proof for d = 4, s = 1/4. By the Hölder inequality to have (3.28) By Proposition 2.10, N 1 ≪ N 3 1 and discarding ω −1 1 to have From (3.13), (3.28), (3.29), (2.17) and (2.21), we obtain By the same manner as the estimate for Lemma 2.19 (iv), i = 5, we see From (3.14), (3.17) and (3.30), we have By the same manner as the estimate for Lemma 2.19 (iv), i = 6, we see From (3.15), (3.19) and (3.31), we have By the same manner as the estimate for Lemma 2.19 (iv), i = 4, we see (3.32) From (3.16), (3.21) and (3.32), we have By symmetry, the estimate for K 2 is obtained by the same manner as the estimate for K 1 . We apply Lemma 2.19 (i) and the Cauchy-Schwarz inequality, the right-hand side of (3.23) is bounded by Thus, we obtain K 1/2 3 Finally, we prove and Lemma 2.19 (ii), we obtain From Lemma 2.19 (iv), , the right-hand side of (3.5) is bounded by , the right-hand side of (3.7) is bounded by , the right-hand side of (3.9) is bounded by From (3.36), the right-hand side of (3.11) is bounded by Hence, · l 2 l 1 · l 1 l 2 and the Cauchy-Schwarz inequality to have We prove (3.2) for d ≥ 4, s = s c = d/2 − 2 and spherically symmetric functions (u, v, n) by the same manner as the proof of d = 4, s = 1/4. By the Hölder inequality to have Discarding ω −1 1 , then N 1 ≪ N 3 1 and the same manner as (2.23), we find (3.38) Collecting (3.13), (3.37), (3.38), (2.24), (2.25) and N 2 ∼ N 3 1, we obtain By the same manner as the estimate for Lemma 2.19 (iv), i = 5, we obtain From (3.14), (3.17) and (3.39), we have By the same manner as the estimate for Lemma 2.19 (iv), i = 6, we obtain From (3.15), (3.19) and (3.40), we have By the same manner as the estimate for Lemma 2.19 (iv), i = 4, we obtain From (3.16), (3.21) and (3.41), we have By symmetry, the estimate for K 2 is obtained by the same manner as the estimate for K 1 . For d = 4, from (3.23), Lemma 2.19 (i) and the Cauchy-Schwarz inequality to have Hence, for d = 4, we obtain For d > 4, from (3.23) and Lemma 2.19 (i), we have (3.44) For d > 8 and N 2 < 1, it holds that N 2 1. Hence, by (3.43) to have (3.46) Collecting (3.42), (3.44)-(3.46), we obtain K
) be radial, then for all 0 < T < ∞, there exists a unique spherically symmetric solution of (4.2) on [0, T ] such that .
Finally, we prove Proposition 4.2. The proof is the same manner as the proof for Thus, Hence, there exists f ± := lim t→±∞ ∇ x s K ± (−t)u ± (t) in L 2 x (R d ). Then put u ± := ∇ x −s f ± , we obtain ∇ x s K ± (−t)u ± (t) − f ± L 2 x = u ± (t) − K ± (t)u ±∞ H s x → 0 as t → ±∞. The scattering result for the wave equation is obtained similarly.