GLOBAL EXISTENCE FOR SYSTEMS OF NONLINEAR WAVE AND KLEIN-GORDON EQUATIONS WITH COMPACTLY SUPPORTED INITIAL DATA

. We consider the Cauchy problem for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions. The author previously proved the small data global existence for rapidly decreasing data under a certain condition on nonlinearity. In this paper, we show that we can weaken the condition, provided that the initial data are compactly supported.

(1) is called a nonlinear wave equation when m j = 0, and a nonlinear Klein-Gordon equation when m j > 0. For simplicity, we assume that there is N 0 ∈ {0, 1, . . . , N } such that m j > 0 for 1 ≤ j ≤ N 0 , and m j = 0 for N 0 + 1 ≤ j ≤ N (we understand this relation as m j > 0 for all j if N 0 = N , and m j = 0 for all j if N 0 = 0). We set

SOICHIRO KATAYAMA
For a while, we assume that f = (f j ) 1≤j≤N and g = (g j ) 1≤j≤N are smooth and decay sufficiently fast at spatial infinity, but are not necessarily of compact support. The parameter ε is always supposed to be sufficiently small.
For systems of nonlinear Klein-Gordon equations (namely the case where N 0 = N ), we do not need any further assumption to get the small data global existence (see Klainerman [15] and Shatah [22]). On the other hand, for systems of nonlinear wave equations (namely the case where N 0 = 0), we need an additional condition to obtain the small data global existence because the blow-up of the solution in finite time may occur even for small initial data; for example, the blow-up occurs when N = 1 and F = (∂ t u) 2 (see John [8]). Klainerman [16] and Christodoulou [3] showed the small data global existence when the null condition is satisfied: We say that the null condition is satisfied if we have with ω 0 = −1. Here and hereafter F q = F q j 1≤j≤N denotes the quadratic part of F , that is to say F q (u, ∂u) := lim λ→0 λ −2 F (λu, λ∂u).
The left-hand side of (5) means that X and ω a Y are substituted in place of u and ∂ a u, respectively. The null condition for the wave equation is closely related to the null forms Q ab 's are sometimes called the strong null forms. In fact, the null condition is satisfied if and only if each F q j is a linear combination of Q 0 (u k , u l ) and Q ab (u k , u l ) with 0 ≤ a, b ≤ 3 and 1 ≤ k, l ≤ N .
Before we proceed further, we introduce some notation. For 1 ≤ j ≤ N , we write with F kl j [u k , u l ] := |α|,|β|≤1 where c klαβ j are real constants satisfying c kkαβ j = c kkβα j . For 1 ≤ j ≤ N we set Then F q j is decomposed into three parts: Similarly, for * = q, K, KW, W, we put . For the case of coupled systems of nonlinear wave and Klein-Gordon equations (namely, the case where 1 ≤ N 0 < N ), Georgiev [4] proved the small data global existence under the strong null condition, that is to say, We can show that the strong null condition is satisfied if and only if each F q j is a linear combination of the strong null forms Q ab (u k , u l ) with 0 ≤ a, b ≤ 3 and 1 ≤ k, l ≤ N .
The author [10] extremely relaxed the condition for the small data global existence in the following way. We assume the null condition for the interaction between wave components in the wave equations. To be more precise, we assume with ω 0 = −1.
We also assume that other quadratic parts are independent of the wave component w itself 1 .
To be more precise, we assume the followings: . Instead of (A1) and (A2), which restrict the dependence on w, we have an alternative condition which is an additional restriction on the form of F W : (B) There are functions G j,a = G j,a (u) such that holds for any C 1 -function u.
Under the conditions (N), (A1), and (A2), or under the conditions (N) and (B), the author proved the small data global existence for (1)-(2) 2 . Observe that this result under the conditions (N), (A1), and (A2) naturally covers the known results for the Klein-Gordon equations and the wave equations in [15], [22], [16], and [3]. See also LeFloch-Ma [20] for an alternative proof of the small data global existence under the conditions (N), (A1), and (A2), where much less regularity for initial data is necessary, though the compactness of the support of initial data is required.
In this paper, we would like to show that the conditions (N) and (A1) are sufficient for the small data global existence, if we assume the compactness of the support of the initial data; in other words, F W K can depend also on w itself under the compactness assumption.
Here we say that u = (u j ) 1≤j≤N is asymptotically free, if there are ( whileḢ 1 and H 1 denote the homogeneous and inhomogeneous Sobolev spaces, respectively. More precisely, is the set of L 2 -functions whose first derivatives (in the sense of distributions) also belong to L 2 (R 3 ).
We emphasize that in the previous result of [10], the compactness of the support was not required. In Theorem 1.1 here, the condition (A2) is removed, but the compactness of the support is essentially used in the proof. We do not know whether Theorem 1.1 holds true or not for the initial data with non-compact support.
Our theorem can be extended to the quasi-linear systems. See Section 5 below for the details.

2.
Preliminaries. For z ∈ R d , we define z = 1 + |z| 2 . In what follows, C stands for positive constants whose actual values may change line by line.
First we recall Hardy's inequality.
Integrating this inequality over S 2 , we obtain the desired result.
The next Hardy-type inequality is due to Lindblad [21].
for any smooth function φ = φ(t, x) satisfying By replacing t − |x| with (2R + t − |x|) on the left-hand side of (12), this lemma is proved by performing integration by parts in a similar manner to the proof of Hardy's inequality. This replacement requires the support condition.
The following corollary is a key estimate for the treatment of nonlinear terms w k (∂ a w l ) in the energy estimate, and this is one of the points where we need the compactness of the support of the initial data in our theorem.
Proof. We have Therefore, using Lemmas 2.1 and 2.2, we obtain This completes the proof.
We use the so-called vector field method to obtain decay estimates of solutions. We introduce and we set For a multi-index α = (α 1 , . . . , α 10 ), we write Γ α = Γ α1 1 · · · Γ α10 10 . For a smooth function φ = φ(t, x) and a non-negative integer s, we define It is easy to check the following: Hence, for any multi-index α, we get We also have for 1 ≤ j, k ≤ 10 and 0 ≤ a ≤ 3 with appropriate constants c jkl and d jab . As a consequence, for any non-negative integer s, there is a positive constant C s such that we have for any smooth function φ.
For the decay estimate of solutions to Klein-Gordon equations, we use the following estimate due to Georgiev [5]: 2]. Let m > 0, and v be a smooth solution to Then there exists a positive constant C = C(m) such that we have for (t, x) ∈ (0, ∞) × R 3 , provided that the right-hand side of (15) is finite.
where (a) + = a for a > 0, and (a) + = 0 for a < 0. In Section 4 below, we will use Lemma 2.4 in combination with (16), which requires the support condition for Φ. But this usage of (16) is just for the simplification of the calculation, and not essential.
We use the weighted L ∞ -L ∞ estimates for wave equations. To state the estimates, we define Lemma 2.5. Let w be a smooth solution to , and we have used the standard notation of multi-indices.
For the proof, see Asakura [1] (see also [14] for the above expression). The next estimates for the inhomogeneous wave equations are due to Kubota-Yokoyama [19] (see also [12] for the expression below): Lemma 2.6. Let w be a smooth solution to Suppose that ρ ≥ 0, κ ≥ 0, and ν > 0. Then there exists a positive constant C = C(ρ, κ, ν) such that Because of the terms in F K W , whose decay rate is t + |x| −3 at best, we need to choose κ = 0 when we apply Lemma 2.6 in the proof of Theorem 1.1, but then, as w is included in F W K , the factor W 0 (t, |x|) in (18) causes some trouble in the energy estimates. It is known that some logarithmic factor like W 0 (t, |x|) is unremovable from (18) with κ = 0 in general (see the author [11]); however, as shown in Lemma 2.7 below, we can remove W 0 (t, |x|) if we assume a certain support condition on Ψ. This is another point where we need the compactness of the support of initial data in our proof. Lemma 2.7. Let w be as in Lemma 2.6. Suppose that there is a positive constant R such that Then, for ρ ≥ 0 and ν > 0, there is a positive constant Proof. Since the right-hand side of (20) becomes greater as ν becomes greater, we may assume 0 < ν < 1.
The following Sobolev type inequality will be used to combine decay estimates with the energy estimates (see Klainerman [17] for the proof): Lemma 2.8. There is a positive constant C such that we have for any smooth function ϕ on R 3 , provided that the right-hand side of (23) is finite.

3.
Algebraic normal forms and the null condition. In this section, we summarize the method of algebraic normal forms. Roughly speaking, the method of algebraic normal forms enables us to remove the undesirable nonlinear terms in the decay estimates through a certain algebraic transformations of the unknowns. See Sunagawa [23] and Katayama-Ozawa-Sunagawa [13] for the application of the algebraic normal form technique to systems of nonlinear Klein-Gordon equations with mass resonance in one and two space dimensions. Similar ideas were previously used in Bachelot [2], the author [9], Kosecki [18], and Tsutsumi [24] for example, but the method here is more systematic.

It follows from simple calculation and (29) that
Using these formulas, we can show the following key lemma in the method of algebraic normal forms: Lemma 3.1. Let u be the solution to (1). We assume that |α|, |β| ≤ 1, and 1 ≤ j, k, l ≤ N . If A jkl is invertible, then there exist two constants c jkl and d jkl such that, writing for any non-negative integer s, where C s is a positive constant.
Proof. By (30), we have If we replace k u k and l u l with F k (u, ∂u) and F l (u, ∂u), respectively, in (31) and (32), we obtain the desired estimate for |R αβ jkl | s with the help of (28). When we derive decay estimates for local solutions, Lemma 3.1 can be used to replace terms like (∂ α u k )(∂ β u l ) in F j with harmless terms, provided that det A jkl = 0. For example, if m j > 0 and m k = m l = 0, we have det A jkl = m 4 j > 0. Thus we can replace F W K with harmless terms in the decay estimates below. When m j = 0, we have det A jkl = (m 2 k − m 2 l ) 2 . Hence, if m j = 0 and m k = m l , we can replace (∂ α u k )(∂ β u l ) in F q W with harmless terms. Especially, F KW W can be replaced. When m j = 0 and m k = m l , we cannot apply Lemma 3.1 in general, since det A jkl = 0. However, it turns out that the null form Q 0 (u k , u l ) in F W W can be similarly replaced with better terms. Lemma 3.2. Let u be the solution to (1).
then, for any non-negative integer s we have with a positive constant C s depending only on s.
Proof. Since we have we can show the result in a similar fashion to the proof of Lemma 3.1.

4.
Proof of the main theorem. Let u be a solution to (1)- (2). If f (x) = g(x) = 0 for |x| ≥ R with some R > 0, then we have u(t, x) = 0 for |x| ≥ t + R and t ≥ 0, which is known as finite speed of propagation. This enables us to use Lemmas 2.3 and 2.7 in the following proof. Let c klαβ j be from (9). In the proof of the theorem, without loss of generality, we may assume the following condition in addition to (N) and (A1): (A3) F W K (w, ∂w) does not contain terms like w k w l for N 0 + 1 ≤ k, l ≤ N . Indeed, we can reduce the system to a new one satisfying (A3) by a kind of algebraic normal form as follows. If we set , and if we define u = ( u j ) 1≤j≤N , v = ( v j ) 1≤j≤N0 , and w = ( w j ) N0+1≤j≤N by then we see that for 1 ≤ j ≤ N 0 , and F j u, ∂ u := F j (u, ∂u) for N 0 + 1 ≤ j ≤ N , with the substitution u = (v, w) = ( v + p( w), w) on the right-hand sides. We can easily check that F KW u, ∂ u = F KW u, ∂ u and F W W w, ∂ w = F W W w, ∂ w . Hence the conditions (N) and (A1) are preserved by this reduction, while we can see that F W K does not contain w k w l (= w k w l ) for N 0 + 1 ≤ k, l ≤ N . In conclusion, by the reduction (33), we can obtain a system satisfying the condition (A3), without affecting the assumptions (N) and (A1).
Recall that | · | s and · s are defined by (13). We put for any smooth function φ = φ(t, x).
For a smooth solution u = (v, w) to (1) where λ ∈ (0, 1/100), and I is an integer with I ≥ 15. Observe that there is a positive constant C such that where W − is given by (17). Note also that we have Let M be a sufficiently large constant, and let ε be sufficiently small compared to M , so that we have M ε ≤ 1 and M 2 ε ≤ 1. We are going to prove that E[u](T ) ≤ M ε implies E[u](T ) ≤ M ε/2, provided that ε is sufficiently small. Then, by the so-called bootstrap argument, we obtain a priori estimates for u, and we can show the global existence of the solution u.
In what follows, C stands for positive constants which are independent of M , ε, and T .
Step 2: Rough decay estimates. By Lemma 2.4 and (38), we obtain in view of Remark 1. For ρ ≥ 0, ν > 0, a non-negative integer s, and a smooth function φ = φ(t, x), we put Suppose that φ(t, x) = 0 for |x| ≥ t + R and t ≥ 0 with some R > 0. Then, for a positive integer s, Lemmas 2.5 and 2.7 yield for ρ ≥ 0 and ν > 0, provided that φ(0) and (∂ t φ)(0) are compactly supported, and their amplitude is of order ε. By Lemma 2.6, we also have We put We also use the notation By (N) and (A1), F q W is independent of w itself, and we get By Lemma 2.8 and (39), we get Therefore we get which leads to It follows from (41) that and if we choose sufficiently small ε satisfying CM 2 ε 2 ≤ 1/2, we get By (42), we also have Step 3: Better decay estimates through the method of algebraic normal forms. We use the method of the algebraic normal forms to replace some part of F q W with harmless terms. F KW W can be treated by Lemma 3.1; we can use Lemma 3.2 to replace Q 0 (w j , w k ) in F W W , and only the strong null forms are left in F W W . Because of the enhanced decay estimate (28), we can consider the strong null forms as harmless terms. In this way, we can find P W = P j (u, ∂u, ∂ 2 u) N0+1≤j≤N , whose components are homogeneous polynomials of degree 2 in (u, ∂u, ∂ 2 u), such that for s ≤ 2I, which, together with (43) and (44), yields From (40), (45), and (46), we get Now, it follows from (47) that On the other hand, (40) implies For ρ ≥ 0 and s ≤ 2I − 7, (43) and (48) lead to Therefore, we get If ε is sufficiently small to satisfy CM 2 ε 2 ≤ 1/2, we obtain Here we switch to the estimate for v. We use the method of algebraic normal forms to replace F W K with harmless terms. By Lemma 3.1, we can find P K = P j (w, ∂w, ∂ 2 w) 1≤j≤N0 , whose components are homogeneous polynomials of degree 2 in (w, ∂w, ∂ 2 w), such that I+1 |(u, ∂u)| 2I−7 + t + r −1 |∂u| I |∂u| 2I−7 . It follows from (53) that . Other terms in R j can be easily treated by (39), and we get R j 2I−9 ≤ CM ε 2 t λ−(3/2) . By (39), we also have (39) and (40). Since F h j enjoys the same estimate as R j , we obtain and Lemma 2.4 yields By (53), we have t + |x| 3/2 |P K | 2I−13 ≤ C t + |x| 3/2 |w| 2 2I−11 ≤ CM ε 2 t + |x| 2λ−1/2 , and we obtain t + |x| 3/2 |v(t, x)| 2I−13 ≤ Cε.
Now we go back to the estimate of w. (55) implies (cf. (50)). By (49), (56), and (51) with ρ = 0 and s = 2I − 14, we obtain As before, because of (52), it follows from (41) and (57) that provided that ε is sufficiently small. Since we have From (57) and (58), we get from which we get (w j − P j )(t) L 2 ≤ Cε t −1−λ . Hence, with the help of (54), we have From this, we see that there is a free solution u + j to ( + m 2 j )u + j = 0 such that (u j − P j )(t) − u + j (t) E,mj → 0 as t → ∞, where · E,mj is given by (11). Since it is easy to get P j (t) E,mj → 0 as t → ∞, we find that u j (t) − u + j (t) E,mj → 0 as t → ∞.
Finally, in the case where we use the reduction (33), the above argument implies u j (t) − u + j (t) E,mj as t → ∞. Since it is easily checked that p j ( w) E,mj ≤ Cε 2 (1 + t) −1/2 → 0 as t → ∞, we obtain u j (t) − u + j (t) E,mj → 0 as desired. This completes the proof for the asymptotic behavior.

5.
Remarks on the quasi-linear case. In fact, the following quasi-linear system is considered in almost all the previous works mentioned in the introduction: for j = 1, 2, . . . , N , where F j has the form G ab jk (u, ∂u)(∂ a ∂ b u k ) + H j (u, ∂u) with some G ab jk and H j satisfying G ab jk (u, ∂u) = O(|u| + |∂u|) and H j (u, ∂u) = O(|u| 2 + |∂u| 2 ) near (u, ∂u) = (0, 0). If we assume the symmetry condition G ab jk (u, ∂u) = G ab kj (u, ∂u) = G ba jk (u, ∂u) for 1 ≤ j, k ≤ N and 0 ≤ a, b ≤ 3, and if we make some apparent modifications in the assumptions, such as addition of ∂ 2 u to the arguments of the nonlinearity, and replacement of the null condition for the semilinear case with that for the quasilinear case, then the global existence results in [15,22,16,3,4], and [10] hold true also for (60).
Remark 2. We note that terms like w k (∂ a ∂ b w l ) seem to be allowed in F W K if we only look at (A3) for the quasi-linear system; however such terms are excluded by (61) because entry of such terms causes entry of terms like w k (∂ a ∂ b v l ) in F KW W , which is excluded by (A1). Therefore the quadratic part of G ab jk does not depend on w itself under the assumption of modified Theorem 1.1 for (60).