Decay estimates of a tangential derivative to the light cone for the wave equation and their application
aa r X i v : . [ m a t h . A P ] A ug DECAY ESTIMATES OF A TANGENTIAL DERIVATIVE TOTHE LIGHT CONE FOR THE WAVE EQUATION AND THEIRAPPLICATION
SOICHIRO KATAYAMA AND HIDEO KUBO
Abstract.
We consider wave equations in three space dimensions and obtainnew weighted L ∞ - L ∞ estimates for a tangential derivative to the light cone.As an application, we give a new proof of the global existence theorem, whichwas originally proved by Klainerman and Christodoulou, for systems of nonlin-ear wave equations under the null condition. Our new proof has the advantageof using neither the scaling nor the Lorentz boost operators. Introduction
Solutions to the Cauchy problem for nonlinear wave equations with quadraticnonlinearity in three space dimensions may blow up in finite time no matter howsmall initial data are, and we have to impose some special condition on the nonlin-earity to get global solutions. The null condition is one of such conditions and isassociated with the null forms Q and Q ab , which are given by Q ( v, w ; c ) =( ∂ t v )( ∂ t w ) − c ( ∇ x v ) · ( ∇ x w ) , (1.1) Q ab ( v, w ) =( ∂ a v )( ∂ b w ) − ( ∂ b v )( ∂ a w ) (0 ≤ a < b ≤ v = v ( t, x ) and w = w ( t, x ), where c is a positive constant corresponding to thepropagation speed, ∂ = ∂ t = ∂/∂t , and ∂ j = ∂/∂x j ( j = 1 , , c > (cid:3) c u i = F i ( u, ∂u, ∇ x ∂u ) in (0 , ∞ ) × R (1 ≤ i ≤ m )with initial data(1.4) u = εf and ∂ t u = εg at t = 0,where (cid:3) c = ∂ t − c ∆ x , u = ( u j ), ∂u = ( ∂ a u j ), and ∇ x ∂u = ( ∂ k ∂ a u j ) with1 ≤ j ≤ m , 1 ≤ k ≤
3, and 0 ≤ a ≤
3, while ε is a positive parameter. Let F = ( F i ) ≤ i ≤ m be quadratic around the origin in its arguments and the system bequasi-linear. In other words, we assume that each F i has the form(1.5) F i ( u, ∂u, ∇ x ∂u ) = X ≤ j ≤ m ≤ k ≤ , ≤ a ≤ c ijka ( u, ∂u ) ∂ k ∂ a u j + d i ( u, ∂u ) , Mathematics Subject Classification.
Key words and phrases.
Nonlinear wave equation; null condition; global existence.The first and the second author were partially supported by Grant-in-Aid for Young Scientists(B) (No. 16740094), MEXT, and by Grant-in-Aid for Science Research (No.17540157), JSPS,respectively.Published in SIAM Journal on Mathematical Analysis Vol. (2008), no. 6, 1851–1862. where c ijka ( u, ∂u ) = O ( | u | + | ∂u | ) and d i ( u, ∂u ) = O ( | u | + | ∂u | ) around ( u, ∂u ) =(0 , c ijkℓ = c ijℓk for 1 ≤ i, j ≤ m and1 ≤ k, ℓ ≤
3. In addition, we always assume the symmetry condition c ijka = c jika for 1 ≤ i, j ≤ m , 1 ≤ k ≤
3, and 0 ≤ a ≤ . Then it is well known that the null condition (for the above system (1.3)) is satisfiedif and only if the quadratic terms of F i (1 ≤ i ≤ m ) can be written as linearcombinations of the null forms Q ( u j , ∂ α u k ; c ) and Q ab ( u j , ∂ α u k ) with 1 ≤ j, k ≤ m , 0 ≤ a < b ≤
3, and | α | ≤
1, where ∂ α = ∂ α ∂ α ∂ α ∂ α for a multi-index α = ( α , α , α , α ) (refer to [3] and [14] for the precise description of the nullcondition). Klainerman [14] and Christodoulou [3] proved the following globalexistence theorem independently by different methods. Theorem 1.1 (Klainerman [14], Christodoulou [3]) . Suppose that the null condi-tion is satisfied. Then, for any f , g ∈ C ∞ ( R ; R m ) , there exists a positive con-stant ε such that the Cauchy problem (1.3)–(1.4) admits a unique global solution u ∈ C ∞ ([0 , ∞ ) × R ; R m ) for any ε ∈ (0 , ε ] . Christodoulou used the so-called conformal method which is based on Penrose’sconformal compactification of Minkowski space. On the other hand, Klainermanused the vector field method and showed the above theorem by deriving some decayestimates in the original coordinates. In Klainerman’s proof, he introduced vectorfields L c,j = x j c ∂ t + ct∂ j (1 ≤ j ≤ , Ω ij = x i ∂ j − x j ∂ i (1 ≤ i < j ≤ , which are the generators of the Lorentz group, and the scaling operator S = t∂ t + x · ∇ x . These vector fields play an important role in getting Klainerman’s weighted L - L ∞ estimates for wave equations (see also H¨ormander [5]). In addition, using them, wecan see that an extra decay factor is expected from the null forms. For example,we have Q ( v, w ; c ) = 1 t + r (cid:26) ( ∂ t v ) (cid:0) Sw + cL c,r w (cid:1) − c X j =1 ( L c,j v )( ∂ j w )(1.6) − c ( Sv )( ∂ r w ) + c X j = k ω k (Ω jk v )( ∂ j w ) (cid:27) , where r = | x | , ω = ( ω , ω , ω ) = x/r , ∂ r = P j =1 ω j ∂ j , L c,r = P j =1 ω j L c,j , andΩ ij = − Ω ji for 1 ≤ j < i ≤ L c,j depend on thepropagation speed c , and they are unfavorable when we consider the multiplespeed case. Thus, the vector field method without the Lorentz boost fields wasdeveloped by many authors (see Kovalyov [17, 18], Klainerman and Sideris [16],Yokoyama [25], Kubota and Yokoyama [19], Sideris and Tu [23], Sogge [24], Hi-dano [4], Katayama [9, 11], and Katayama and Yokoyama [13], for example). Inplace of (1.6), the following identity was used in the above works relating to the ECAY ESTIMATES OF A TANGENTIAL DERIVATIVE 3 null condition for the multiple speed case: Q ( v, w ; c ) = 1 t ( Sv + ( ct − r ) ∂ r v )( Sw − ( ct + r ) ∂ r w )(1.7) + ct { ( Sv )( ∂ r w ) − ( ∂ r v )( Sw ) } + c r X j = k ω k ( ∂ j v )(Ω jk w ) , whose variant was introduced by Hoshiga and Kubo [6]. Equation (1.7) leads to agood estimate in the region r > δt with some small δ >
0, because r is equivalentto t + r in this region. Note that the operator S is still used in (1.7), and this isthe only reason why S was adopted in [9, 19, 25], because these works are basedon variants of L ∞ - L ∞ estimates due to John [7] and Kovalyov [17], where only ∂ a and Ω ij are used (see Lemma 3.2 below).Our aim here is to get rid of not only L c,j , but also S from the estimate ofthe null forms, and prove Theorem 1.1 using only ∂ a and Ω jk . Though the usageof the scaling operator S has not caused any serious difficulty in the study of theCauchy problem for nonlinear wave equations so far, we believe that it is worthwhiledeveloping a simple approach with a smaller set of vector fields. For this purpose,we make use of the identity Q ( v, w ; c ) = 12 (cid:8) ( D + ,c v )( D − ,c w ) + ( D − ,c v )( D + ,c w ) (cid:9) (1.8) + c r X j = k ω k ( ∂ j v )(Ω jk w ) , where D ± ,c = ∂ t ± c∂ r . Note that this identity was already used implicitly to obtainidentities like (1.7) (see [23], for example). In view of (1.8), what we need to treatthe null forms is an enhanced decay estimate for the tangential derivative D + ,c tothe light cone. We can say that, in the previous works, this enhanced decay hasbeen observed through D + ,c = 1 t (cid:0) S + ( ct − r ) ∂ r (cid:1) or D + ,c = 1 ct + r (cid:0) cS + cL c,r (cid:1) with the help of S or also L c,r = P j =1 ω j L c,j .In this paper, we take a different approach. We will establish the enhanceddecay of D + ,c u for the solution u to the wave equation directly. We formulate it asa weighted L ∞ - L ∞ estimate in Theorem 2.1 below, which is our main ingredientin this paper. The point is that such an estimate can be derived by using only ∂ a and Ω ij . This type of approach to D + ,c goes back to the work of John [8].2. The Main Result
Before stating our result precisely, we introduce several notations. We put Z = { Z a } ≤ a ≤ = { ( ∂ a ) ≤ a ≤ , (Ω jk ) ≤ j 0, a nonnegative integer s , and a smooth function G = G ( t, x ), whereΛ c ( t, x ) = { ( τ, y ) ∈ [0 , t ] × R ; | y − x | ≤ c ( t − τ ) } . We also define(2.4) B ρ,s [ φ, ψ ; c ]( t, x ) = sup y ∈ Λ ′ c ( t,x ) h| y |i ρ (cid:0) | φ ( y ) | s +1 + | ψ ( y ) | s (cid:1) for ρ ≥ 0, a nonnegative integer s , and smooth functions φ and ψ on R , whereΛ ′ c ( t, x ) = { y ∈ R ; | y − x | ≤ ct } .The following theorem is our main result. Theorem 2.1. Assume ≤ κ ≤ and µ > . (i) Let u be the solution to (cid:3) c u = G in (0 , ∞ ) × R with initial data u = ∂ t u = 0 at t = 0 . Then there exists a positive constant C , depending on κ and µ , such that h| x |i h t + | x |i h ct − | x |i κ − { log(2 + t + | x | ) } − | D + ,c u ( t, x ) | (2.5) ≤ CA κ,µ, [ G ; c ]( t, x ) for ( t, x ) ∈ (0 , ∞ ) × R with x = 0 , where A κ,µ, is given by ( ) .Moreover, if < κ < , then for any δ > , there exists a constantC, depending on κ , µ , and δ , such that (2.6) h t + | x |i h ct − | x |i κ − | D + ,c u ( t, x ) | ≤ CA κ,µ, [ G ; c ]( t, x ) for ( t, x ) ∈ (0 , ∞ ) × R satisfying | x | > δt . (ii) Let u ∗ be the solution to (cid:3) c u ∗ = 0 in (0 , ∞ ) × R with initial data u ∗ = φ and ∂ t u ∗ = ψ at t = 0 . Then we have (2.7) h| x |i h t + | x |i h ct − | x |i κ − | D + ,c u ∗ ( t, x ) | ≤ CB κ + µ +1 , [ φ, ψ ; c ]( t, x ) for ( t, x ) ∈ (0 , ∞ ) × R with x = 0 , where B κ + µ +1 , is given by ( ) .Remark . (1) Similar estimates for radially symmetric solutions are obtained byKatayama [11].(2) Suppose that A κ,µ, [ G ; c ]( t, x ) is bounded on [0 , ∞ ) × R for some κ ∈ [1 , 2) and µ > u solves (cid:3) c u = G with zero initial data. Then, from Lemma 3.2below, we see that u and ∂u decay like h t i − Ψ κ − ( t ) along the light cone ct = | x | ,where Ψ ρ ( t ) = log(2 + t ) if ρ = 0, and Ψ ρ ( t ) = 1 if ρ > 0. Compared with thisdecay rate, we find from (2.5) and (2.6) that D + ,c u gains extra decay of h t i − andbehaves like h t i − Ψ κ − ( t ) along the light cone. ECAY ESTIMATES OF A TANGENTIAL DERIVATIVE 5 (3) For tangential derivatives T c,j = ( x j / | x | ) ∂ t + c∂ j (1 ≤ j ≤ (cid:18)Z t Z R (cid:0) (cid:12)(cid:12) cτ − | x | (cid:12)(cid:12)(cid:1) − ρ | T c,j u ( τ, x ) | dxdτ (cid:19) / with ρ > k ∂u (0 , · ) k L ( R ) + R t k (cid:3) c u ( τ, · ) k L ( R ) dτ (see [1], for ex-ample). Observe that T c,j is closely connected to D + ,c . In fact, we have D + ,c = P j =1 ( x j / | x | ) T c,j . Though Alinhac’s estimate does not need S and means enhanceddecay of tangential derivatives implicitly, it seems difficult to recover a pointwise de-cay estimate from his weighted space-time estimate. On the other hand, Sideris andThomases [22] obtained the estimate for (cid:13)(cid:13)(cid:0) (cid:12)(cid:12) ct + | · | (cid:12)(cid:12)(cid:1) T c,j u ( t, · ) (cid:13)(cid:13) L ( R ) ; how-ever, S is used in their estimate.(4) The exterior problem for systems of nonlinear wave equations with the singleor multiple speed(s) is also widely studied (see Metcalfe, Nakamura, and Sogge [20]and Metcalfe and Sogge [21] and the references cited therein). In the exterior do-mains, because of their unbounded coefficients on the boundary, the Lorentz boostsare unlikely to be applicable even for the single speed case. This is another reasonwhy the vector field method without the Lorentz boosts is widely studied. In ad-dition, S also causes a technical difficulty in the exterior problems. We will discussthe exterior problem in a subsequent paper, and we will not go into further detailshere.We will prove Theorem 2.1 in the next section, after stating some known weighted L ∞ - L ∞ estimates for wave equations. Though we can apply our theorem to exclude S from the proof of the multiple speed version of Theorem 1.1 in [9, 19, 25], weconcentrate on the single speed case for simplicity, and we will give a new proof,without using S and L c,j , of Theorem 1.1 in section 4 as an application of our maintheorem.Throughout this paper, various positive constants, which may change line byline, are denoted just by the same letter C .3. Proof of Theorem 2.1 For c > φ = φ ( x ), and ψ = ψ ( x ), we write U ∗ c [ φ, ψ ] for the solution u to thehomogeneous wave equation (cid:3) c u = 0 in (0 , ∞ ) × R with initial data u = φ and ∂ t u = ψ at t = 0. Similarly, for c > G = G ( t, x ), we write U c [ G ] for thesolution u to the inhomogeneous wave equation (cid:3) c u = G in (0 , ∞ ) × R with initialdata u = ∂ t u = 0 at t = 0.For U ∗ c [ φ, ψ ] we have the following. Lemma 3.1. Let c > . Then, for κ > , we have h t + | x |i h ct − | x |i κ − | U ∗ c [ φ, ψ ]( t, x ) | (3.1) ≤ C sup y ∈ Λ ′ c ( t,x ) h| y |i κ ( h| y |i | φ ( y ) | + | y | | ψ ( y ) | ) for ( t, x ) ∈ [0 , ∞ ) × R . For the proof, see Katayama and Yokoyama [13, Lemma 3.1] (see also Asakura [2]and Kubota and Yokoyama [19]).After the pioneering work of John [7], a wide variety of weighted L ∞ - L ∞ esti-mates for U c [ G ] and ∂U c [ G ] have been obtained (see [2, 9, 10, 12, 13, 17, 18, 19, 25]). S. KATAYAMA AND H. KUBO Here we restrict our attention to what will be used directly in our proofs of Theo-rems 1.1 and 2.1. Lemma 3.2. Let c > . Define Φ ρ ( t, r ) = (cid:26) log (cid:0) h t + r i h t − r i − (cid:1) if ρ = 0 , h t − r i − ρ if ρ > , (3.2) Ψ ρ ( t ) = (cid:26) log(2 + t ) if ρ = 0 , if ρ > . (3.3) Assume κ ≥ and µ > . Then we have h t + | x |i Φ κ − ( ct, | x | ) − | U c [ G ]( t, x ) | ≤ CA κ,µ, [ G ; c ]( t, x ) , (3.4) h| x |i h ct − | x |i κ Ψ κ − ( t ) − | ∂U c [ G ]( t, x ) | ≤ CA κ,µ, [ G ; c ]( t, x )(3.5) for ( t, x ) ∈ [0 , ∞ ) × R , where A κ,µ,s [ G ; c ] is given by ( ) .Proof. For the proof of (3.4), see Katayama and Yokoyama [13, equation (3.6) inLemma 3.2, and section 8] for κ > κ = 1.Next we consider (3.5) with κ > 1. From Lemma 8.2 in [13], we find that(3.5) with ∂U c [ G ] replaced by U c [ ∂G ] is true. Now (3.5) follows immediately fromLemma 3.1, because we have ∂ a U c [ G ] = U c [ ∂ a G ] + δ a U ∗ c [0 , G (0 , · )] for 0 ≤ a ≤ δ ab , and h| y |i κ +1 | y | | G (0 , y ) | ≤ CA κ,µ, [ G ; c ]( t ) (note thatwe have w (0 , r ) = h r i ). Equation (3.5) for the case κ = 1 can be treated similarly(see [19] and [9]). (cid:3) Note that we will use (3.5) in the proof of Theorem 1.1 but not in that ofTheorem 2.1.Now we are in a position to prove Theorem 2.1. Suppose that all the assumptionsin Theorem 2.1 are fulfilled. Without loss of generality, we may assume c = 1.For simplicity of exposition, we write D ± for D ± , = ∂ t ± ∂ r . Similarly, U ∗ [ φ, ψ ], U [ G ], A ρ,µ,s ( t, x ), and B ρ,s ( t, x ) denote U ∗ [ φ, ψ ], U [ G ], A ρ,µ,s [ G ; 1]( t, x ), and B ρ,s [ φ, ψ ; 1]( t, x ), respectively.First we prove (2.5). Assume 0 < r = | x | ≤ 1. We have | D + u | ≤ | ∂ t u | + |∇ x u | ≤ X ≤ a ≤ | U [ ∂ a G ] | + | U ∗ [0 , G (0 , · )] | . From (3.4) in Lemma 3.2, we get(3.6) h t + r i Φ κ − ( t, r ) − | U [ ∂ a G ]( t, x ) | ≤ CA κ,µ, ( t, x ) , while Lemma 3.1 leads to h t + r i h t − r i κ | U ∗ [0 , G (0 , · )]( t, x ) | ≤ C sup y ∈ Λ ′ ( t,x ) | y | h| y |i κ +1 | G (0 , y ) |≤ CA κ,µ, ( t, x ) . Thus we obtain (2.5) for 0 < | x | ≤ v ( t, r, ω ) = ru ( t, rω ) for r > ω ∈ S . Then we have(3.7) D − D + v ( t, r, ω ) = rG ( t, rω ) + 1 r X ≤ j Let r = | x | ≥ ≤ κ ≤ 2. From (3.4), we get1 r X ≤ j 1. This completes the proof of (2.5).To prove (2.6), we first note that h t + r i ≤ C h r i for r > δt . Let 1 < κ < 2. Bythe first line of (3.8), we have(3.11) 1 r X ≤ j 1. Lemma 3.1 also implies1 r X ≤ j 1. Set v ∗ ( t, r, ω ) = ru ∗ ( t, rω ) for r ≥ ω ∈ S . For r ≥ | D + v ∗ ( t, r, ω ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( D + v ∗ )(0 , t + r, ω ) + Z t ( D − D + v ∗ )( τ, t + r − τ, ω ) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C h t + r i − κ B κ +1 , ( t, rω )+ C h t + r i − κ B κ + µ +1 , ( t, rω ) Z t h t + r − τ i − − µ dτ + C h t + r i − κ B κ + µ +1 , ( t, rω ) Z t h t + r − τ i − − µ dτ ≤ C h t + r i − κ B κ + µ +1 , ( t, rω ) , which ends up with h r i h t + r i h t − r i κ − | D + u ∗ ( t, x ) | ≤ CB κ + µ +1 , ( t, x )for r = | x | ≥ 1. This completes the proof of (2.7). (cid:3) Proof of Theorem 1.1 As an application of Theorem 2.1, we give a new proof of Theorem 1.1. First wederive estimates for the null forms. Lemma 4.1. Let c be a positive constant, and v = ( v , . . . , v M ) . Suppose that Q is one of the null forms. Then, for a nonnegative integer s , there exists a positiveconstant C s , depending only on c and s , such that | Q ( v j , v k ) | s ≤ C s (cid:26) | ∂v | [ s/ X | α |≤ s | D + ,c Z α v | + | ∂v | s X | α |≤ [ s/ | D + ,c Z α v | + 1 r (cid:0) | ∂v | [ s/ | v | s +1 + | v | [ s/ | ∂v | s (cid:1)(cid:27) . Proof. The case Q = Q and s = 0 follows immediately from (1.8). We can obtainsimilar identities for other null forms by using( ∂ t , ∇ x ) = (cid:18) , − x cr (cid:19) D − ,c + (cid:18) , x cr (cid:19) D + ,c − (cid:16) , xr ∧ Ω (cid:17) with Ω = (Ω , − Ω , Ω ) (see (5.2) in Sideris and Tu [23, Lemma 5.1]), and wecan show the desired estimate for s = 0. Since Z α Q ( v j , v k ) can be written in termsof Q ( Z β v j , Z γ v k ; c ) and Q ab ( Z β v j , Z γ v k ) (0 ≤ a < b ≤ 3) with | β | + | γ | ≤ | α | , thedesired estimate for general s follows immediately. (cid:3) ECAY ESTIMATES OF A TANGENTIAL DERIVATIVE 9 Now we are going to prove Theorem 1.1. Without loss of generality, we mayassume c = 1. Assume that the assumptions in Theorem 1.1 are fulfilled. Let u bethe solution to (1.3)–(1.4) on [0 , T ) × R , and we set e ρ,k ( t, x ) = h t + | x |i h t − | x |i ρ | u ( t, x ) | k +2 + h| x |i h t − | x |i ρ +1 | ∂u ( t, x ) | k +1 + χ ( t, x ) h t + | x |i h t − | x |i ρ X | α |≤ k | D + , Z α u ( t, x ) | for ρ > k , where χ ( t, x ) = 1 if | x | > (1 + t ) / 2, while χ ( t, x ) = 0 if | x | ≤ (1 + t ) / 2. We fix ρ ∈ (1 / , 1) and s ≥ 8, and assume that(4.1) sup ≤ t 0) and small ε ( > M ε ≤ 1. Our goal hereis to get (4.1) with M replaced by M/ 2. Once such an estimate is established,it is well known that we can obtain Theorem 1.1 by the so-called bootstrap (orcontinuity) argument.In the following we always assume M is large enough, and ε is sufficiently small.For simplicity of exposition, we will not write dependence of nonlinearities on theunknowns explicitly. Namely we abbreviate F ( u, ∂u, ∇ x ∂u )( t, x ) as F ( t, x ), andso on.First we evaluate the energy. For any nonnegative integer k ≤ s , (4.1) implies(4.2) | F (2) ( t, x ) | k ≤ CM ε h| x |i − h t − | x |i − − ρ | ∂u ( t, x ) | k +1 , where F (2) denotes the quadratic terms of F . Put H = F − F (2) , and Zu =( Z u, . . . , Z u ). Since we have(4.3) h r i − h t − r i − ≤ C h t + r i − for any ( t, r ) ∈ [0 , ∞ ) × [0 , ∞ ),and since h| x |i − | Zu | ≤ C | ∂u | , from (4.1) we obtain | H ( t, x ) | k ≤ C (cid:0) | u | + | ( u, ∂u ) | k/ ( | Zu | k − + | ∂u | k +1 ) (cid:1) (4.4) ≤ CM ε h t + | x |i − h t − | x |i − ρ + CM ε h t + | x |i − h t − | x |i − ρ | ∂u ( t, x ) | k +1 for any nonnegative integer k ≤ s . Similarly to (4.2) and (4.4), using (4.3), weobtain(4.5) | F i,α ( t, x ) | ≤ CM ε (1 + t ) − | ∂u ( t, x ) | s + CM ε h t + | x |i − h t − | x |i − ρ for | α | ≤ s , where F i,α = Z α F i − X j,k,a c ijka ∂ k ∂ a ( Z α u j )with c ijka coming from (1.5). It is easy to see that(4.6) k h t + | · |i − h t − | · |i − ρ k L ( R ) ≤ C (1 + t ) − for ρ > / 2. Therefore, from (4.5), we obtain k F i,α ( t, · ) k L ≤ CM ε (1 + t ) − k ∂u ( t, · ) k s + CM ε (1 + t ) − for | α | ≤ s . We also have X j,k,a | c ijka ( t, x ) | ≤ CM ε (1 + t ) − . Now, applying the energy inequality for the systems of perturbed wave equations (cid:3) ( Z α u i ) − P j,k,a c ijka ∂ k ∂ a ( Z α u j ) = F i,α , we find ddt k ∂u ( t, · ) k s ≤ CM ε (1 + t ) − k ∂u ( t, · ) k s + CM ε (1 + t ) − , and the Gronwall lemma leads to(4.7) k ∂u ( t, · ) k s ≤ C ( ε + M ε )(1 + t ) C Mε ≤ CM ε (1 + t ) C Mε with an appropriate positive constant C which is independent of M (note that theenergy inequality for the systems of perturbed wave equations is available becauseof the symmetry condition).In the following, we repeatedly use Theorem 2.1 and Lemmas 3.1 and 3.2 withthe choice of N = 1 and c = 1(= c ). In other words, from now on we put w ( t, r ) = min (cid:8) h r i , h t − r i (cid:9) . Note that we have(4.8) h r i − h t − r i − ≤ C h t + r i − w ( t, r ) − , which is more precise than (4.3).By (4.7) and the Sobolev-type inequality h| x |i | v ( t, x ) | ≤ C k v ( t, · ) k , whose proof can be found in Klainerman [15], we see that(4.9) h| x |i | ∂u ( t, x ) | s − ≤ CM ε (1 + t ) C Mε . Using (4.8) and (4.9), from (4.2) and (4.4) with k = 2 s − 3, we obtain | F ( t, x ) | s − ≤ CM ε h r i − h t + | x |i − w ( t, | x | ) − ρ (1 + t ) C Mε , which implies(4.10) A ν, ρ − , s − [ F ; 1]( t, x ) ≤ CM ε h t + | x |i C Mε + ν , where ν is a positive constant to be fixed later (note that we have h τ + | y |i ≤h t + | x |i for ( τ, y ) ∈ Λ ( t, x )). Since 2 ρ > ν > 1, by Lemmas 3.1 and 3.2with Theorem 2.1, we obtain e , s − ( t, x ) ≤ e ν, s − ( t, x ) ≤ Cε + CM ε h t + | x |i C Mε + ν (4.11) ≤ CM ε h t + | x |i C Mε + ν . Finally, we are going to estimate e ρ,s ( t, x ). By (4.11) and (4.2) with k = 2 s − | F (2) ( t, x ) | s − ≤ CM ε h t + | x |i − − ρ + C Mε + ν h| x |i − for ( t, x ) satisfying | x | ≤ ( t + 1) / 2. On the other hand, (4.1), (4.11), and Lemma 4.1imply | F (2) ( t, x ) | s − ≤ CM ε h t + | x |i − C Mε + ν h t − | x |i − − ρ for ( t, x ) satisfying | x | ≥ ( t + 1) / 2. Summing up, we obtain(4.12) | F (2) ( t, x ) | s − ≤ CM ε h| x |i − h t + | x |i − C Mε + ν w ( t, | x | ) − − ρ . By the first line of (4.4) with k = 2 s − 6, using (4.1) and (4.11), we get(4.13) | H ( t, x ) | s − ≤ CM ε h| x |i − h t + | x |i − C Mε + ν w ( t, | x | ) − ρ . Equations (4.12) and (4.13) yield(4.14) | F ( t, x ) | s − ≤ CM ε h| x |i − h t + | x |i − C Mε + ν w ( t, | x | ) − ρ . ECAY ESTIMATES OF A TANGENTIAL DERIVATIVE 11 Now we fix some ν satisfying 0 < ν < − ρ , and assume that ε is sufficientlysmall to satisfy − C M ε + ν ≤ − − ρ . Then from (4.14) we find that(4.15) A ρ, ρ − , s − [ F ; 1]( t, x ) ≤ CM ε . Since we have s + 2 ≤ s − 6, 1 + ρ > 1, and 2 ρ > 1, from Theorem 2.1, Lemmas 3.1and 3.2, we obtain(4.16) e ρ,s ( t, x ) ≤ C (cid:0) ε + M ε (cid:1) for ( t, x ) ∈ [0 , T ) × R , with an appropriate positive constant C which is indepen-dent of M . Finally, if M is large enough to satisfy 4 C ≤ M , and ε is small enoughto satisfy C M ε ≤ / 4, by (4.16) we obtain(4.17) sup ≤ t S. Alinhac , Remarks on energy inequalities for wave and Maxwell equations on a curvedback ground , Math. Ann., 329 (2004), pp. 707–722.[2] F. Asakura , Existence of a global solution to a semi-linear wave equation with slowly de-creasing initial data in three space dimensions , Comm. Partial Differential Equations, 11(1986), pp. 1459–1487.[3] D. Christodoulou , Global solutions of nonlinear hyperbolic equations for small initial data ,Comm. Pure Appl. Math., 39 (1986), pp. 267–282.[4] K. Hidano , The global existence theorem for quasi-linear wave equations with multiple speeds ,Hokkaido Math. J., 33 (2004), pp. 607–636.[5] L. H¨ormander , L , L ∞ estimates for the wave operator , in Analyse Math´ematique et Appli-cations, Contributions en l’Honneur de J. L. Lions, Gauthier–Villars, Paris, 1988, pp. 211–234.[6] A. Hoshiga and H. Kubo , Global small amplitude solutions of nonlinear hyperbolic systemswith a critical exponent under the null condition , SIAM J. Math. Anal., 31 (2000), pp. 486–513.[7] F. John , Blow-up of solutions of nonlinear wave equations in three space dimensions ,Manuscripta Math., 28 (1979), pp. 235–268.[8] F. John , Lower bounds for the life span of solutions of nonlinear wave equations in threespace dimensions , Comm. Pure Appl. Math., 36 (1983), pp. 1–35.[9] S. Katayama , Global and almost-global existence for systems of nonlinear wave equationswith different propagation speeds , Differential Integral Equations, 17 (2004), pp. 1043–1078.[10] S. Katayama , Global existence for systems of wave equations with nonresonant nonlinearitiesand null forms , J. Differential Equations, 209 (2005), pp. 140–171.[11] S. Katayama , Lifespan for radially symmetric solutions to systems of semilinear wave equa-tions with multiple speeds , Osaka J. Math, to appear.[12] S. Katayama and A. Matsumura , Sharp lower bound for the lifespan of the systems ofsemilinear wave equations with multiple speeds , J. Math. Kyoto Univ., 45 (2005), pp. 391–403.[13] S. Katayama and K. Yokoyama , Global small amplitude solutions to systems of nonlinearwave equations with multiple speeds , Osaka J. Math., 43 (2006), pp. 283–326.[14] S. Klainerman , The null condition and global existence to nonlinear wave equations , inNonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lecturesin Appl. Math. 23, AMS, Providence, RI, 1986, pp. 293–326.[15] S. Klainerman , Remarks on the global Sobolev inequalities in the Minkowski space R n +1 ,Comm. Pure Appl. Math., 40 (1987), pp. 111–117.[16] S. Klainerman and T. C. Sideris , On almost global existence for nonrelativistic wave equa-tions in D , Comm. Pure Appl. Math., 49 (1996), pp. 307–321.[17] M. Kovalyov , Long-time behavior of solutions of a system of nonlinear wave equations ,Comm. Partial Differential Equations, 12 (1987), pp. 471–501. [18] M. Kovalyov , Resonance-type behaviour in a system of nonlinear wave equations , J. Differ-ential Equations, 77 (1989), pp. 73–83.[19] K. Kubota and K. Yokoyama , Global existence of classical solutions to systems of nonlin-ear wave equations with different speeds of propagation , Japan. J. Math. (N.S.), 27 (2001),pp. 113–202.[20] J. Metcalfe, M. Nakamura, and C. D. Sogge , Global existence of quasilinear, nonrel-ativistic wave equations satisfying the null condition , Japan. J. Math. (N.S.), 31 (2005),pp. 391–472.[21] J. Metcalfe and C. D. Sogge , Global existence of null-form wave equations in exteriordomains , Math. Z., 256 (2007), pp. 521–549.[22] T. C. Sideris and B. Thomases , Local energy decay for solutions of multidimensionalisotropic symmetric hyperbolic systems , J. Hyperbolic Differ. Equ., 3 (2006), pp. 673–690.[23] T. C. Sideris and S.-Y. Tu , Global existence for systems of nonlinear wave equations in Dwith multiple speeds , SIAM J. Math. Anal., 33 (2001), pp. 477–488.[24] C. D. Sogge , Global existence for nonlinear wave equations with multiple speeds , in HarmonicAnalysis at Mount Holyoke, W. Beckner, A. Nagel, A. Seeger, and H. F. Smith, eds., Contemp.Math. 320, AMS, Providence, RI, 2003, pp. 353–366.[25] K. Yokoyama , Global existence of classical solutions to systems of wave equations withcritical nonlinearity in three space dimensions , J. Math. Soc. Japan, 52 (2000), pp. 609–632. Department of Mathematics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan E-mail address : [email protected] Department of Mathematics, Graduate School of Science, Osaka University, Toy-onaka, Osaka 560-0043, Japan E-mail address ::