1. Introduction and the Main Result

J. Partial Diff. Eqs. 17(2004), 97–121

Vol.17, No.2

c

International Academic Publishers

GLOBAL WELL-POSEDNESS FOR THE KLEIN–GORDON EQUATION BELOW THE ENERGY NORM* Miao Changxing (Institute of Applied Physics and Computational Mathematics PO Box 8009, Beijing 100088, China) (E-mail: miao [email protected])

Zhang Bo (School of Mathematical and Information Sciences, Coventry University Coventry CV1 5FB, UK) (E-mail: [email protected] )

Fang Daoyuan (Department of Mathematics, Zhejiang University Hangzhou, 310027, China) (E-mail: [email protected]) (Received Feb. 11, 2004)

Abstract We study global well-posedness below the energy norm of the Cauchy problem for the Klein-Gordon equation in Rn with n ≥ 3. By means of Bourgain’s method along with the endpoint Strichartz estimates of Keel and Tao, we prove the H s -global well-posedness with s < 1 of the Cauchy problem for the Klein-Gordon equation. This we do by establishing a series of nonlinear a priori estimates in the setting of Besov spaces. Key Words Klein-Gordon equations; Strichartz estimates; Besov spaces; wellposedness. 2000 MR Subject Classification 35L05, 35L15. Chinese Library Classification O175.27, O175.29.

1. Introduction and the Main Result Recently, a large amount of work has been devoted to the study of the Cauchy problem for the semilinear wave equation utt − 4u = −|u|ρ−1 u,

(t, x) ∈ R × Rn ,

n ≥ 3,

(1.1)

*The first author was partly supported by the NSF, Special Funds for Major State Basic Research Projects of China and NSF of China Academy of Engineering Physics.

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u(x, 0) = φ(x),

x ∈ Rn ,

(1.2)

ut (x, 0) = ψ(x),

x ∈ Rn

(1.3)

in energy space, that is, for initial data (φ, ψ) ∈ H 1 (Rn ) × L2 (Rn ), where ρ > 1 and H s (Rn ) = (1 − ∆)−s/2 L2 (Rn ) for s ∈ R. For example, the global well-posedness, the scattering theory as well as regularity of solutions to the Cauchy problem (1.1)-(1.3) have been established in, e.g. [1, 2], [3-7],[8, 9] and [10-12] for the case of sub-critical growth 1 < ρ < 1 + 4/(n − 2) or in, e.g. [7, 13] and [9, 14, 15] for the case of critical growth ρ = 1 + 4/(n − 2). However, certain questions remain open. For example, when ρ > 1 + 4/(n − 2), it is not yet clear whether or not there exists a global, regular, solution to the Cauchy problem (1.1)-(1.3) with arbitrary initial data. On the other hand, the local well-posedeness (as well as global well-posedness with small initial data in the critical growth case) in fractional Sobolev spaces has also been studied recently by many authors for the Cauchy problem of general semi-linear wave equations including (1.1)-(1.3) under minimal regularity assumptions on the initial data (see, e.g. [16], [17-20] and [9, 21, 22].However, very few authors have undertaken a study of global well-posedness below the energy norm of Cauchy problems with less regular initial data. In [23, 24], Bourgain established the global well-posedness of the Cauchy problem of nonlinear wave or dispersive wave equations for rough initial data (with infinite energy) for the first time. Bourgain’s method has been further developed to prove the global well-posedness below the energy norm of the Cauchy problem for the modified KdV equation in [25] and for the semi-linear wave equation (1.1)-(1.3) under minimal regularity assumptions on the data in the three-dimensional case in [26]. It should be pointed out that Keel and Tao have recently proposed a different approach to the study of both local and global well-posedness below the energy norm for the wave map equation [27, 28] and the Yang-Mills equation [29]. In this paper we consider the following Cauchy problem for the Klein–Gordon equation: utt − 4u + m2 u = −|u|ρ−1 u, u(x, 0) = φ(x), ut (x, 0) = ψ(x),

x ∈ Rn , x ∈ Rn ,

(t, x) ∈ R × Rn ,

n ≥ 3, m 6= 0,

(1.4) (1.5) (1.6)

and establish the global well-posedness below the energy norm for the Cauchy problem in the case of general spatial dimensions. To deal with the case of general spatial dimensions, we have to develop some a priori nonlinear estimates in Besov spaces. Our purpose is to give a unified method to deal

No.2

Global well-posedness for the Klein–Gordon equation below the energy norm

99

with nonlinear estimates in Besov spaces and then to establish the global well-posedness of the Cauchy problem (1.4)-(1.6) under minimal regularity assumptions on the realvalued initial data (φ, ψ). To this end, we first rewrite some Strichartz inequalities, including the endpoint Strichartz estimates established by Keel and Tao in [30] for solutions of the linear Klein-Gordon equation, and constructe suitable working-spaces to establish a series of nonlinear estimates in Besov spaces. The H s -global well-posedness with s < 1 is then proved for the Cauchy problem (1.4)-(1.6) in the case with n ≥ 3 by following Bourgain’s ideas (cf. [23, 24]) along with Kenig-Ponce-Vega’s technique [26]. It is necessary to point out that the way of deriving the a priori nonlinear estimates will be changed with different dimensions. Before stating our main result note first the local well-posedness result that for (φ, ψ) ∈ H s (Rn ) × H s−1 (Rn ) with H s (Rn ) = (I − ∆)−s/2 L2 (Rn ), where  s ∈ (ν(ρ), 1], if n = 3 and ρ = 2, s ∈ [ν(ρ), 1], if n > 3 or if n = 3 and ρ > 2, n+1 1 n+3  − , if k0 (n) ≤ ρ ≤ ,  ρ−1 n−1 ν(ρ) = n 4 2 n+3 n+2   − , if ≤ρ< 2 ρ−1 n−1 n−2 and k0 (n) =

(n + 1)2 , (n − 1)2 + 4

(1.7)

n ≥ 3,

the Cauchy problem (1.4)-(1.6) is locally well-posed in the time interval [0, T0 ) with T0 = T0 (kφkH s , kψkH s−1 ) (see [9, 19] for the case when s ∈ (ν(ρ), 1] and [22, 30] for the case when s = ν(ρ) and n ≥ 4). It should be pointed out that in the case with n = 3, the lower bound ν(ρ) given in (1.7) is optimal [18, 19] and that in the case with n ≥ 4, this local well-posedness result has been greatly extended in [22] by use of the endpoint Strichartz estimates. Define α(ρ) :=

2(ρ − 1)2 − [n + 2 − ρ(n − 2)][n + 1 − ρ(n − 1)] 2(ρ − 1)[n + 1 − ρ(n − 3)]

if n ≥ 4 and k0 (n) ≤ ρ
0 and (φ, ψ) ∈ H s (Rn ) ∩ Lρ+1 (Rn ) × H s−1 (Rn ), the Cauchy problem (1.4) − (1.6) has a unique solution u ∈
C [0, T ]; H s (Rn ) ∩ Lρ+1 (Rn ) . Moreover, ˙ u(t) = K(t)φ + K(t)ψ + z(t) with 1−s

sup kz(t)kH 1 ≤ CT 1−s−η , [0,T ) − 12

where K(t) = (m2 − 4) η=

1

sin[(m2 − 4) 2 t],

2(ρ − 1)2 (1 − s) nρ − (n + ρ − 1) + − ρs n + 2 − ρ(n − 2) 2

if n ≥ 4 and k0 (n) ≤ ρ < η=

n−1 n+3 or if n = 3 and 2 < ρ < , n−3 n−1 nρ − (n + 2) 2(ρ − 1)(1 − s) + − ρs n + 2 − ρ(n − 2) 2

n+3 n+2 ≤ρ< . n−1 n−2 Remark 1.1 (i) Let k1 (n) = 1 + 2/(n − 2). Then it is easy to verify that k0 (n) < k1 (n) in the case with n = 3, 4 and k0 (n) > k1 (n) in the case with n ≥ 5. Moreover, it 2 n+3 is clear that 1 + = when n = 3. It is also easy to see that n−2 n−1 if n = 3 and

n−1 n+2 = , n−3 n−2 n−1 n+2 < , n−3 n−2

n = 4, n ≥ 5.

(ii) When n ≥ 5, we only consider the case k0 (n) ≤ ρ < (n − 1)/(n − 3) for technical reason. (iii) When n = 3 and m = 0, our result recovers Kenig-Ponce-Vega’s result in [26] for semilinear wave equations. We conclude this section by introducing some notations. Denote by S(Rn ) and S 0 (Rn ) the Schwartz space and the Schwartz distribution functions space respectively. For 1 ≤ r ≤ ∞, Lr (Rn ) denotes the usual Lebesgue space on Rn with the norm k · kr . s For s ∈ R and 1 < r < ∞, let Lsr (Rn ) = (1−4)− 2 Lr (Rn ), the inhomogeneous Sobolev s space in terms of Bessel potentials, let L˙ sr (Rn ) = (−4)− 2 Lr (Rn ), the homogeneous Sobolev space in terms of Riesz potentials, and write H s (Rn ) = Ls2 (Rn ) and H˙ s (Rn ) = s (Rn ) the Besov space defined L˙ s2 (Rn ). For s ∈ R and 1 ≤ r, m ≤ ∞, denote by Br,m

No.2 Global well-posedness for the Klein–Gordon equation below the energy norm

101

m as the space of distributions u such that {2js kϕj ∗ ukr }∞ j=0 ∈ ` , where ∗ stands for s (Rn ) the the convolution and {ϕj } is a dyadic decomposition on Rn , and by B˙ r,m homogeneous Besov space defined as the space of distributions u modulo polynomials m n such that {2js kψj ∗ ukr }∞ j=0 ∈ ` , where {ψj } is a dyadic decomposition on R \{0}. For the detailed definitions of the above function spaces see, e.g. [31-33]. We shall omit Rn from spaces and norms. 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 Lq (I; X) the space of strongly measurable functions from I to X with ku(·); Xk ∈ Lq (I). Finally, for any q > 0, q 0 stands for the dual to q, i.e., 1/q + 1/q 0 = 1.

2. Space-time Estimates for the Linear Problem In this section we present some a priori estimates for the solution of the following Cauchy problem for the linear Klein-Gordon equation associated with the problem (1.4)-(1.6): Wtt − 4W + m2 W = f (x, t), (t, x) ∈ R × Rn ,

(2.1)

W (x, 0) = φ(x),

x ∈ Rn ,

(2.2)

Wt (x, 0) = ψ(x),

x ∈ Rn .

(2.3)

The solution, W (x, t), of the above Cauchy problem is given by Z t ˙ W (x, t) = K(t)φ(x) + K(t)ψ(x) + K(t − τ )f (x, τ )dτ, 0 − 21

where K(t) = (m2 − 4) define

1 2

˙ sin[(m2 − 4) t]. Let w(x, t) = K(t)φ(x) + K(t)ψ(x) and t

Z

K(t − τ )f (x, τ )dτ,

(Gf )(x, t) :=

(t, x) ∈ R × Rn .

0

Then w(x, t) is the solution of the Cauchy problem for the free Klein-Gordon equation. 0 We have the Lr –Lr estimate (see [1, 2, 34]): s ≤ Cm(t)kψkB s˜0 , kK(t)ψ(·)kBr,2

(2.4)

r ,2

where ( m(t) ≤

1

1

|t|−(n−1−σ)( 2 − r ) , |t| ≤ 1, 1 1 |t|−(n−1+σ)( 2 − r ) , |t| ≥ 1,

and (

2 ≤ r < ∞,

0 ≤ σ ≤ 1, 1 1 (n + 1 + σ)( − ) ≤ 1 + s˜ − s. 2 r

(2.5)

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It is certainly true that under the condition (2.5) we have either 1

1

s kK(t)ψ(·)kBr,2 ≤ C|t|−(n−1+σ)( 2 − r ) kψkB s˜0 ,

t 6= 0

(2.6)

t 6= 0.

(2.7)

r ,2

or 1

1

s ≤ C|t|−(n−1−σ)( 2 − r ) kψkB s˜0 , kK(t)ψ(·)kBr,2 r ,2

0

We present the classical Strichartz estimates corresponding to the Lr –Lr estimate (2.6); as for the case corresponding to (2.7) we only give a remark. To this end, given n, define, for 2 ≤ r ≤ ∞ and 0 ≤ σ ≤ 1, 2β(r, σ) γ(r, σ) 1 1 δ(r, σ) = = = − , n+σ n+1+σ n−1+σ 2 r

(2.8)

and let, for 0 < s < 1, 2 ≤ r ≤ ∞ and 0 ≤ σ ≤ 1, 1 1 (n + 1 + σ)( − ) = 2β(r, σ) = 1 + s˜ − s. 2 r

(2.9)

Then (2.6) can be rewritten as s ≤ C|t|−γ(r,σ) kψkB s˜0 , kK(t)ψ(·)kBr,2

t 6= 0.

(2.10)

r ,2

In particular, kK(t)ψ(·)kB 1−β(r,σ) ≤ C|t|−γ(r,σ) kψkB β(r,σ) , r,2

t 6= 0.

(2.11)

r 0 ,2

¯ Then, since (1 − 4)−ρ is Let I = R or I ⊂ R be the interval with 0 ∈ I. s into B s+ρ , and by the T T ∗ –method, the Hardy– a holomorphic mapping from Br,2 r,2 Littlewood–Sobelev inequality and the Sobolev embedding theorem, we have the following Strichartz estimates. Proposition 2.1 Let ρ1 , ρ2 , µ ∈ R, 2 ≤ q1 , q2 , r1 and r2 ≤ ∞ such that 2 ≤ min(γ(rj , σ), 1), n ≥ 3, j = 1, 2, qj (qj , rj , n, σ) 6= (2, ∞, 3, 0), j = 1, 2, 1 ρ1 + δ(r1 , σ) − = µ, q1 1 = 1 − µ. ρ2 + δ(r2 , σ) − q2 0≤

Set Y µ = H µ × H µ−1 . (i) For (φ, ψ) ∈ Y µ , (w, ∂t w) ∈ C(I; Y µ ) ∩ Lq1 (I; Brρ11,2 ) and −1 kw; Lq1 (I; Brρ11,2 )k + k∂t w; Lq1 (I; Brρ11,2 )k ≤ Ck(φ, ψ)kY µ .

(2.12)

No.2 Global well-posedness for the Klein–Gordon equation below the energy norm

103

ρ1 2 µ q1 (ii) For f ∈ Lq2 (I; Br−ρ 0 ,2 ), Gf ∈ C(I; Y ) ∩ L (I; Br1 ,2 ) and 0

2

0

2 kGf ; Lq1 (I; Brρ11,2 )k ≤ Ckf ; Lq2 (I; Br−ρ 0 ,2 )k.

(2.13)

2

(iii) For I = [0, T ) with 0 < T ≤ ∞, −1 )k kW ; Lq1 (I; Brρ11,2 )k + k∂t W ; Lq1 (I; Brρ11,2   0 2 ≤ C k(φ, ψ)kY µ + kf ; Lq2 (I; B˙ r−ρ 0 ,2 )k .

(2.14)

2

Remark 2.1 Proposition 2.1 was proved in [35, 19, 34] for the case (2/qj , γ(rj , σ)) 6= (1, 1), and in [30] for the endpoint (2/qj , γ(rj , σ)) = (1, 1) case when n ≥ 4. When n = 3 and σ 6= 0, the endpoint Strichartz estimate can also be shown to be valid by the method of Keel and Tao [30]. However, when n = 3 and σ = 0, the endpoint Strichartz estimate (i.e., the case (2/qj , γ(rj , 0)) 6= (1, 1)) fails for the wave equation (m = 0) for general data but still holds for radial data (see [17] for details). In the case when n = 3 and σ = 0, it remains an open question whether or not the endpoint Strichartz estimate (i.e., the case (2/qj , γ(rj , 0)) 6= (1, 1)) still holds for the linear Klein-Gordon equation. As immidiate consequences of Proposition 2.1 and the Sobolev embedding theorem we have the following corollaries, which need the following definition. Definition 2.1 Given n, we say that the exponent pair (q, r) is sharp admissible if q, r ≥ 2, (q, r, n, σ) 6= (2, ∞, 3, 0) and 2 1 1 = (n − 1 + σ)( − ) = γ(r, σ). q 2 r

(2.15)

It is easy to see that if (q, r) is a sharp admissible pair, then q is uniquely determined by r and σ and is usually denoted by q = q(r, σ). Corollary 2.2 (i) Let 0 ≤ σ ≤ 1 and let (q(r, σ), r) and (qj (rj , σ), rj ) (j = 1, 2) be sharp admissible pairs. If w is the solution of the Cauchy problem for the free Kleins−β(r,σ) Gordon equation with initial data (φ, ψ) ∈ H s × H s−1 , then w ∈ Lq(r,σ) (I; Br,2 )∩ s C(I; H ) and ˙ kwkLq(r,σ) (I;B s−β(r,σ) ) = kK(t)φ + K(t)ψkLq(r,σ) (I;B s−β(r,σ) ) r,2

r,2

≤ C (kφk 0

s+β(r2 ,σ)−1

(ii) Let f ∈ Lq2 (r2 ,σ) (I; Br0 ,2 2

Hs

+ kψkH s−1 ) .

(2.16) s−β(r1 ,σ)

). Then Gf ∈ Lq1 (r1 ,σ) (I; Br1 ,2

) and

kGf kLq1 (r1 ,σ) (I;B s−β(r1 ,σ) ) ≤ Ckf kLq20 (r2 ,σ) (I;B s+β(r2 ,σ)−1 ) .

(2.17)

kGf kLq(r,σ) (I;B s−β(r,σ) ) ≤ Ckf kLq0 (r,σ) (I;B s+β(r,σ))−1 ) .

(2.18)

r1 ,2

0 ,2 r2

In particular, r,2

r 0 ,2

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Corollary 2.3 Let w be the solution of the Cauchy problem for the free Kleinn+1+σ Gordon equation with initial data (φ, ψ) and let θ, σ satisfy that 0 ≤ θ ≤ < 2(n − 1 + σ) 1 in the case when n ≥ 3 and 0 ≤ σ ≤ 1, and 0 ≤ θ < 1 in the case when n = 3 and σ = 0. Then kwk

n+1+σ L (n−1+σ)θ

kwk

! I;B l−θ 2(n+1+σ) ,2 n+1+σ−4θ

n+1+σ

L (n−1+σ)θ (I;Lr )

≤ Cθ (kφ(x)kH l + kψ(x))kH l−1 ) ,

≤ Cθ (kφkH l + kψkH l−1 ) ,

(2.19) (2.20)

where l ≥ 1 and 2(n + 1 + σ) 2(n + 1 + σ)n ≤r≤ . n + 1 + σ − 4θ (n − 2l)(n + 1 + σ) − 2(n − 1 − σ)θ Corollary 2.4 Let w be the solution of the Cauchy problem for the free KleinGordon equation with initial data (φ, ψ). Then, for q, r ≥ 2 and 0 ≤ σ ≤ 1 with (q, r, n, σ) 6= (2, ∞, 3, 0), kwkLq (I;Lr ) ≤ C (kφkH l + kψkH l−1 ) ,

δ(r, σ) −

1 ≤ l, q

(2.21)

where l ≥ 1. Remark 2.2 (i) When σ ≡ 0, we write δ(r, 0) = δ(r), β(r, 0) = β(r) γ(r, 0) = γ(r). Then (2.8) becomes 2β(r) γ(r) 1 1 δ(r) = = = − , n n+1 n−1 2 r which corresponds to the Strichartz estimates for the linear wave equation. (ii) In [32], Marshall established the mixed estimates kwkLq (R;B a ) ≤ C (kφkH 1 + kψkL2 ) r,2 under the condition that 1 2 1 1 2  < < − ,  − 2 (n − 1)q r 2 nq   0 ≤ 1 ≤ 1, a < 1 + 1 − 1, q 2 2 r q

(2.22)

(2.23)

and kwkLq (R;Lr ) ≤ C (kφkH 1 + kψkL2 )

(2.24)

105

No.2 Global well-posedness for the Klein–Gordon equation below the energy norm

under the condition that 0≤

1 1 ≤ q 2

n−2 1 1 1 1 − ≤ < − . 2n nq r 2 nq

(2.25)

One easily sees that the condition (2.23) is equivalent to the condition 2 1 1   = (n − 1 + σ) · ( − ) 0 < σ < 1, q 2 r 1 1 n+1+σ 1 1  0 ≤ ≤ , a < 1 − ( − ), q 2 2 2 r

(2.26)

so (2.22) follows from (2.16). On the other hand, (2.24) follows from Proposition 2.1 and the Sobolev embedding theorem. (iii) If one uses (2.7) and defines ˜ σ) ˜ σ) δ(r, 2β(r, γ˜ (r, σ) 1 1 = = = − , n n+1+σ n−1−σ 2 r

0 ≤ σ ≤ 1,

(2.27)

then we have the estimates similar to those in Proposition 2.1 and Corollaries 2.2-2.4. In particular, we have kwk

  ˜ s−β(r,σ) Lq˜(r,σ) I;Br,2

≤ C (kφkH s + kψkH s−1 ) ,

kGf k

  ˜ s−β(r ,σ) Lq˜1 (r1 ,σ) I;Br ,2 1

≤ Ckf k

  ˜ s−β(r,σ) Lq˜(r,σ) I;Br,2

(2.29)

r2 ,2

1

kGf k

,  ˜ 0 s+β(r 2 ,σ)−1 Lq˜2 (r2 ,σ) I;B˙ 0

(2.28)

≤ Ckf k

 , ˜ s+β(r,σ))−1 0 Lq˜ (r,σ) I;Br0 ,2

(2.30)

where 0 < s < 1, 2 ≤ r < ∞, 0 ≤ σ ≤ 1, and 2 1 1 = (n − 1 − σ)( − ) = γ˜ (r, σ). q˜(r, σ) 2 r From (2.28)-(2.30) and the Sobolev embedding theorem it follows that for l ≥ 1, kwk

n+1+σ

L (n−1−σ)θ

! I;B l−θ 2(n+1+σ) n+1+σ−4θ

kwk

n+1+σ

L (n−1−σ)θ (I;Lr )

≤ Cθ (kφkH l + kψkH l−1 ) ,

(2.31)

,2

≤ Cθ (kφkH l + kψkH l−1 ) ,

(2.32)

where 2(n + 1 + σ)n 2(n + 1 + σ) ≤r≤ , n + 1 + σ − 4θ (n − 2l)(n + 1 + σ) − 2(n − 1 − σ)θ n+1+σ < 1 in the case when n ≥ 4 and 0 ≤ σ ≤ 1, 0 ≤ θ < 1 in the 2(n − 1 − σ) case when n = 3 and (n, σ) = (4, 1). The following results follow easily from the Sobolev imbedding theorem and the H¨ older inequality.

and 0 ≤ θ ≤

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Lemma 2.5 Let kj ≥ 0 and αj be multi-indices on Rn with |αj | ≤ kj (j = 1, 2, · · · , N ). For bj ≥ 0 and 1 ≤ p, rj < ∞, let   kj − |αj | 1 − , j = 1, 2, · · · , N. aj = bj rj n (i) If aj > 0 and

N X

aj = 1/p, then

j=1



N

N Y

Y α

b bj j

|D u | ≤ C kuj k jkj . j

L˙ rj

j=1

j=1

(2.33)

p

(ii) If N

X

aj ≤

aj >0

1 X bj , ≤ p rj j=1

then

Y

N Y

N

b bj αj

u | |D ≤ C kuj k jkj . j

Lrj

j=1

j=1

(2.34)

p

3. Nonlinear Eestimates In this section we establish some nonlinear a priori estimates for the solution of the Cauchy problem (1.4)-(1.6) which is necessary to prove Theorem 1.1. Our proof of Theorem 1.1 follows both Bourgain’s and Kenig-Ponce-Vega’s ideas [23, 26]. To this end, we split the initial data (φ(x), ψ(x)) up into two parts: the high frequency one and the low frequency one (regular part). Let ϕ ∈ Cc∞ (Rn ) such that  1, for |ξ| ≤ 1, ϕ(ξ) = 0, for |ξ| ≥ 2, and define φ(x) := φ1 (x) + φ2 (x), ψ(x) := ψ1 (x) + ψ2 (x), where   ˆ ˆ (φ1 (x), ψ1 (x)) = F −1 (ϕN (ξ)φ(ξ)), F −1 (ϕN (ξ)ψ(ξ)) , (φ2 (x), ψ2 (x)) = (φ(x) − φ1 (x), ψ(x) − ψ1 (x))

No.2 Global well-posedness for the Klein–Gordon equation below the energy norm

107

with ϕN (ξ) = ϕ(ξ/N ) and N > 0 being chosen later. Hereafter we denote by vˆ(ξ) the Fourier transform of v(x) and F −1 the inverse Fourier transform. One easily sees that φ1 , ψ1 ∈ S(Rn ) and Z

l 2

l 2

ˆ 2 dξ |(1 + |ξ| ) ϕN (ξ)φ| 2

k(I − 4) φ1 k2 =

1

2

Rn

Z

2

≤

(1 + |ξ| )

l − 2s 2

s 2

ˆ 2 dξ |(1 + |ξ| ) ϕN (ξ)φ| 2

1 2

Rn

≤ C(1 + N )l−s kφkH s ∼ (1 + N )l−s , l ≥ s, Z 1 2 l 2 2 2l ˆ 2 k(I − 4) φ2 k2 = |(1 + |ξ| ) (1 − ϕN (ξ))φ| dξ

(3.1)

Rn

Z

2

≤

(1 + |ξ| )

l − 2s 2

s 2

2

ˆ dξ |(1 + |ξ| ) (1 − ϕN (ξ))φ| 2

1

2

Rn

≤ C(1 + N )l−s kφkH s ∼ (1 + N )l−s ,

l ∈ [0, s].

(3.2)

Hereafter, fN ∼ gN means |fN | ≤ C|gN | for some positive constant independent of N. Similarly, we have k(I − 4)

l−1 2

ψ1 k2 ∼ (1 + N )l−s ,

l ≥ s,

(3.3)

k(I − 4)

l−1 2

ψ2 k2 ∼ (1 + N )l−s ,

l ∈ [0, s].

(3.4)

We first consider the Cauchy problem with the regular data (φ1 , ψ1 ): vtt − 4v + m2 v = −|v|ρ−1 v, (t, x) ∈ R × Rn , ρ ∈ (1, v(x, 0) = φ1 (x), vt (x, 0) = ψ1 (x),

n+2 ) n−2

x ∈ Rn ,

(3.5) (3.6)

x ∈ Rn ,

(3.7)

and its integral formulation ˙ v(x, t) = K(t)φ 1 (x) + K(t)ψ1 (x) −

t

Z

K(t − τ )|v|ρ−1 vdτ.

(3.8)

0

The solution v(x, t) to the above problem satisfies the conservation law, i.e.  1 2 2 ρ+1 (3.9) E(v(·, t), ∂t v(·, t)) := |∂t v| + |∇v| + m |v| + |v| (x, t) dx ρ+1 Rn = E(φ1 , ψ1 ), ∀t > 0, Z



2

2

2

2

which, together with (3.1), (3.3) and the Sobolev embedding theorem, implies that ρ+1

2 ∼ (1 + N )1−s , k∂t v(t)k2 , kv(t)kH 1 , kv(t)kρ+1

∀t > 0.

(3.10)

108

Miao Changxing, Zhang Bo and Fang Daoyuan

Vol.17

¯ define For l ≥ 0 and I ⊂ R with 0 ∈ I, ||| · |||l :=

k·k

sup +

||| · |||l :=

k · kLq (I;B l−β(r) ) ,

sup 2≤r≤

k·k

sup

n ≥ 4,

r,2

2(n−1) , γ(r)=2/q n−3

0≤θ

2(ρ − 1)2 + (n + 2 − ρ(n − 2)) · (nρ − n − ρ − 1) 2(ρ − 1)2 + 2(ρ − 1) (n + 2 − ρ(n − 2))

in the case when n ≥ 4 and k0 (n) ≤ ρ < (n − 1)/(n − 3). Taking N sufficiently large 1 (e.g. N = T 1−s−η with η being defined in Theorem 1.1) completes the proof of the theorem. Acknowledgement The first author (CM) thanks Professor G Ponce for providing the paper [26] before its publication.

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