On nonlinear elastic waves in 2-D

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On nonlinear elastic waves in 2-D Dongbing Zha Department of Mathematics and Institute for Nonlinear Sciences, Donghua University, Shanghai 201620, PR China Received 20 March 2019; accepted 24 August 2019

Abstract For the Cauchy problem of 2-D nonlinear elastic waves of isotropic, homogeneous and hyperelastic materials satisfying null conditions, assuming the smallness of H 3 (R2 ) × H 2 (R2 ) norm and the boundedness of H 4 (R2 ) × H 3 (R2 ) norm for radially symmetric initial data, we show the global existence of classical solutions. Similar results for exterior domain problems are also proved. © 2019 Elsevier Inc. All rights reserved. MSC: 35L52; 35Q74 Keywords: Nonlinear elastic waves; 2-D; Null conditions; Global existence

1. Introduction and main results For isotropic, homogeneous and hyperelastic materials, the motion for the displacement u = u(t, y) satisfies the following second-order quasilinear hyperbolic system: ∂t2 u − c22 u − (c12 − c22 )∇∇ · u = N (∇u, ∇ 2 u).

(1.1)

Some physical backgrounds of nonlinear elastic waves can be found in [4]. Here the main concern for us is the problem of long time existence of classical solutions for (1.1), which can trace back to Fritz John’s pioneering works on elastodynamics (see [22]). In the 3-D case, for the Cauchy problem of (1.1), John showed that local classical solutions in general will develop singularities for radial and small initial data [13], and they almost globally E-mail address: [email protected]. https://doi.org/10.1016/j.jde.2019.08.044 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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exist for small data with compact support [14]. See also simplified proofs in [23] and [48], some improved result in [11], and a lower bound estimate in [25]. Agemi [1] and Sideris [39] proved independently that for certain classes of materials satisfying null conditions, there exist global classical solutions with small initial data. The exterior domain analogues of the almost global existence result and the global existence result were obtained in [32] and [38], respectively. We also note that for incompressible elastodynamics in 3-D, global existence of small classical solutions was established in [40,41]. In fact, the long time existence theory for nonlinear elastic waves largely follows the paradigm of nonlinear wave equations. For the Cauchy problem of 3-D scalar quasilinear wave equations or quasilinear wave systems with single wave speed, the pioneering works are [3,15,20,21]. The multiple speeds case was treated in [23,24,42,46]. For the 3-D exterior domain problem, some classical references can be found in [17,19,33,35,36]. Because of the slow time decay of the linear system, the 2-D case is much more difficult. For the Cauchy problem of 2-D quasilinear wave equations with quadratic and cubic nonlinear terms satisfying null conditions, global existence of small classical solutions was proved in Alinhac [2]. We refer the reader to [16,26,27,43] for the exterior domain problem, in which the orders of nonlinear terms are all cubic or higher. For 2-D nonlinear elastic waves, in [47] we suggest null conditions on quadratic and cubic nonlinear terms based on the variational structure of nonlinear elastic waves. In [47], for the Cauchy problem of (1.1) with radially symmetric initial data, we show the global existence of classical solution under the null condition. Our main observation is that in the radially symmetric case 2-D nonlinear elastic waves can be reduced to a 2-D quasilinear wave system satisfying the null condition with single wave speed c1 . Thus we can apply the global existence result in [2]. The non-radially symmetric case is still open. We note that for incompressible elastodynamics in 2-D, global existence result has been established in [28], then a different approach was given in [45]. Because the former proof of global existence in [47] heavily relies upon the method of Alinhac [2], smallness of radial data is assumed in a weighted H N+1 (R2 ) × H N (R2 ) norm with a relatively large N ∈ N. In this paper, we will suggest an alternative approach for the global existence of classical solutions to 2-D Cauchy problem of nonlinear elastic waves satisfying null conditions. In this approach, we can only assume the smallness of H 3 (R2 ) × H 2 (R2 ) norm and the boundedness of H 4 (R2 ) × H 3 (R2 ) norm for radially symmetric initial data. Note that no weight in the norm is necessary. Another advantage of this approach is that it is well adapted to exterior domain problem, which will be also treated in this paper. To the best of the author’s knowledge, in the literature there is no global existence result for exterior domain problems of 2-D quasilinear wave equations with quadratic nonlinearity satisfying null conditions. We now more precisely describe the problem we consider in this paper. First we have the Lagrangian  L (u) =

1 |ut |2 − W (∇u) dydt, 2

(1.2)

where W is the stored energy function which is used to characterize the potential energy density. Denote W (∇u) = l2 (∇u) + l3 (∇u) + l4 (∇u) + O(|∇u|5 ),

(1.3)

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where li (∇u) (i = 2, 3, 4) represents the homogeneous i-th order part of W (∇u) with respect to ∇u. By the frame indifference and isotropy assumptions of the materials under consideration, after some computation, we can show that (see [47]) l2 (∇u) =

c12 − c22 c2 (∇ · u)2 + 2 |∇u|2 − c22 Q12 (u1 , u2 ), 2 2

(1.4)

l3 (∇u) = d1 (∇ · u)3 + d2 (∇ · u)(∇ ⊥ · u)2 + d3 (∇ · u)Q12 (u1 , u2 ), ⊥

⊥

l4 (∇u) = e1 (∇ · u) + e2 (∇ · u) + e3 (∇ · u) (∇ · u) 4

4

2

2

+ e4 Q212 (u1 , u2 ) 2

+ e5 (∇ · u)2 Q12 (u1 , u2 ) + e6 (∇ ⊥ · u)2 Q12 (u1 , u ).

(1.5)

(1.6)

Here ∇ ⊥ = (∂y2 , −∂y1 ), the null form Q12 (u1 , u2 ) = ∂1 u1 ∂2 u2 − ∂1 u2 ∂2 u1 , the material constants c1 and c2 (c1 > c2 > 0) correspond to the speed of pressure wave and shear wave respectively, and ci (i = 1, 2), di (i = 1, 2, 3), ei (i = 1, 2, · · · , 6) are some constants which only depend on the stored energy function W . From (1.2)–(1.6), by the Hamilton’s principle we get the nonlinear elastic wave equations in 2-D: ∂t2 u − c22 u − (c12 − c22 )∇∇ · u = N (∇u, ∇ 2 u).

(1.7)

N (∇u, ∇ 2 u) = N2 (∇u, ∇ 2 u) + N3 (∇u, ∇ 2 u),

(1.8)

Here1

N2 stands for the quadratic nonlinear term and N3 stands for the cubic nonlinear term. We have   N2 (∇u, ∇ 2 u) = 3d1 ∇(∇ · u)2 + d2 ∇(∇ ⊥ · u)2 + 2∇ ⊥ (∇ · u ∇ ⊥ · u) + Q(u, ∇u),

(1.9)

N3 (∇u, ∇ 2 u) = 4e1 ∇(∇ · u)3 + 4e2 ∇ ⊥ (∇ ⊥ · u)3   + 2e3 ∇((∇ · u)(∇ ⊥ · u)2 ) + ∇ ⊥ ((∇ ⊥ · u)(∇ · u)2 )  ∇u), + Q(u,

(1.10)

where   Q(u, ∇u) = d3 ∇Q12 (u1 , u2 ) + d3 Q12 (∇ · u, u2 ), Q12 (u1 , ∇ · u) ,       ∇u) = 2e4 Q12 Q12 (u1 , u2 ), u2 , Q12 u1 , Q12 (u1 , u2 ) Q(u,   + 2e5 ∇ (∇ · u)Q12 (u1 , u2 )      + e5 Q12 (∇ · u)2 , u2 , Q12 u1 , (∇ · u)2

(1.11)

1 We truncate at fourth order in u, because the higher order corrections have no influence on the existence of small solutions.

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  + 2e6 ∇ ⊥ (∇ ⊥ · u)Q12 (u1 , u2 )      + e6 Q12 (∇ ⊥ · u)2 , u2 , Q12 u1 , (∇ ⊥ · u)2 .

(1.12)

We call d1 = 0 and e1 = e2 = 0 the first null condition and the second null condition, respectively (see [47]). Consider the following Cauchy problem of 2-D nonlinear elastic waves: ∂t2 u − c22 u − (c12 − c22 )∇∇ · u = N (∇u, ∇ 2 u), t > 0, y ∈ R2 ,

(1.13)

t = 0 : u = u0 , ut = u1 , y ∈ R2 .

(1.14)

We have the following Theorem 1.1. Consider the Cauchy problem (1.13)–(1.14). Assume that d1 = 0, e1 = 0, and the initial data (u0 , u1 ) are radially symmetric. Then there is a constant ε0 > 0 such that for any 0 < ε < ε0 , if u0 H 3 (R2 ) + u1 H 2 (R2 ) ≤ ε

(1.15)

M0 := u0 H 4 (R2 ) + u1 H 3 (R2 ) < +∞,

(1.16)

and

then (1.13)–(1.14) admits a unique global classical solution u. Remark 1.1. The value of e2 plays no role in the radially symmetric case. Now we will consider the exterior domain problem. Denote R2∗ = R2 \B1 (0), where B1 (0) stands for the unit ball centered at original point in R2 . Consider the following initial-boundary value problem of 2-D nonlinear elastic waves: ∂t2 u − c22 u − (c12 − c22 )∇∇ · u = N (∇u, ∇ 2 u), t > 0, y ∈ R2∗ ,

(1.17)

u|∂B1 (0) = 0, t > 0,

(1.18)

t = 0 : u = u0 , ut = u1 , y ∈ R2∗ .

(1.19)

To solve (1.17)–(1.19), the data must be assumed to satisfy the relevant compatibility conditions. Setting Jk u = {∂yα u : 0 ≤ |α| ≤ k}, we know that for a fixed m and a formal H m solution u of (1.17)–(1.19), we can write ∂tk u(0, ·) = ψk (Jk u0 , Jk−1 u1 ), 0 ≤ k ≤ m, in which the compatibility functions ψk (0 ≤ k ≤ m) depend on the nonlinearity, Jk u0 and Jk−1 u1 . For (u0 , u1 ) ∈ H m × H m−1 , the compatibility condition simply requires that ψk vanish on ∂B1 (0) for 0 ≤ k ≤ m − 1. If this condition holds, we say that the compatibility conditions of order m − 1 are satisfied. For some further descriptions, see Keel et al. [18].

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We have the following Theorem 1.2. Consider the initial-boundary value problem (1.17)–(1.19). Assume that d1 = 0, e1 = 0, and the initial data (u0 , u1 ) are radially symmetric and satisfy the compatibility conditions of order 3. Then there is a constant ε0 > 0 such that for any 0 < ε < ε0 , if u0 H 3 (R2∗ ) + u1 H 2 (R2∗ ) ≤ ε

(1.20)

M0 = u0 H 4 (R2∗ ) + u1 H 3 (R2∗ ) < +∞,

(1.21)

and

then (1.17)–(1.19) admits a unique global classical solution u. An outline of this paper is as follows. In the next section, we will reduce Theorem 1.1 to Proposition 2.1 concerning a Cauchy problem for some 4-D scalar quasilinear wave equation. Then we will show Proposition 2.1 in Sect. 3. Finally, in Sect. 4 we give a brief proof for Theorem 1.2 following the proof of Theorem 1.1. 2. Proof of Theorem 1.1: reduction to Proposition 2.1 Assume that u is a global classical solution to the Cauchy problem (1.13)–(1.14). By the invariance of nonlinear elastic wave equation (1.13) under the simultaneous rotation, the radial symmetry of initial data (u0 , u1 ), and the uniqueness of classical solutions of the Cauchy problem (1.13)–(1.14), we know that the solution u is also radially symmetric, i.e., there exists a function φ : R+ × R+ −→ R such that u(t, y) = yφ(t, |y|).

(2.1)

By (1.7)–(1.12) and (2.1), via some computations it can be shown that φ = φ(t, r) satisfies 3 3 φtt − c12 (φrr + φr ) = c2 (φ, rφr )(φrr + φr ) + r −2 B(φ, rφr ), r r φ(0, r) = φ0 (r), φt (0, r) = φ1 (r),

(2.2) (2.3)

where c2 (α, β) = 2(6d1 + d3 )α + 6d1 β + 2(24e1 + e4 + 5e5 )α 2 + 6(8e1 + e5 )αβ + 12e1 β 2 ,   B(α, β) = β 2 d3 + 2(e4 + 2e5 )α + 2e5 β ,     u0 (y), u1 (y) = yφ0 (|y|), yφ1 (|y|) .

(2.4) (2.5) (2.6)

On the other hand, if φ = φ(t, r) is a classical solution to (2.2)–(2.3), then we can verify that u(t, y) = yφ(t, |y|) is a classical solution to (1.13)–(1.14). Thus our task is to solve (2.2)–(2.3).

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Note that the linear part of equation (2.2) is just the radial part of 4-D linear wave equation with wave speed c1 . Consider the following Cauchy problem for 4-D quasilinear wave equations ψtt − c12 ψ = c2 (ψ, x · ∇ψ)ψ +

B(ψ, x · ∇ψ) , t > 0, x ∈ R4 , |x|2

ψ(0, x) = ψ0 (x), ψt (0, x) = ψ1 (x), x ∈ R4 ,

(2.7) (2.8)

where 

   ψ0 (x), ψ1 (x) = φ0 (|x|), φ1 (|x|) ,

(2.9)

and in view of (2.4) and (2.5), we have c2 (ψ, x · ∇ψ) = 2(6d1 + d3 )ψ + 6d1 (x · ∇ψ) + 2(24e1 + e4 + 5e5 )ψ 2 + 6(8e1 + e5 )ψ(x · ∇ψ) + 12e1 (x · ∇ψ)2 ,

(2.10)

B(ψ, x · ∇ψ) = d3 |∇ψ|2 + 2(e4 + 2e5 )ψ|∇ψ|2 + 2e5 (x · ∇ψ)|∇ψ|2 . |x|2

(2.11)

For the 4-D Cauchy problem (2.7)–(2.8), we will establish the following Proposition 2.1. Consider the Cauchy problem (2.7)–(2.8). Assume that d1 = 0, e1 = 0 and the initial data (ψ0 , ψ1 ) are radially symmetric. Then there is a constant ε0 > 0 such that for any 0 < ε < ε0 , if ψ0 H 3 (R4 ) + ψ1 H 2 (R4 ) ≤ ε

(2.12)

M0 = ψ0 H 4 (R4 ) + ψ1 H 3 (R4 ) < +∞,

(2.13)

and

then (2.7)–(2.8) admits a unique global classical solution ψ. The proof of Proposition 2.1 will be given in the next section. Now we will prove Theorem 1.1 based on Proposition 2.1. Proof of Theorem 1.2. The first key step is to show the fact that the smallness condition (1.15) on (u0 , u1 ) can imply the corresponding one (2.12) on (ψ0 , ψ1 ) and the boundedness condition (1.16) on (u0 , u1 ) can imply the corresponding one (2.13) on (ψ0 , ψ1 ), based on relationships (2.6) and (2.9). This fact is a consequence of Lemma 5.1 in the Appendix. Then by Proposition 2.1, we know that Cauchy problem (2.7)–(2.8) admits a unique global classical solution ψ . By the rotation invariance of the equation (2.7), the radial symmetry of initial data (ψ0 , ψ1 ), and the uniqueness of classical solutions of (2.7)–(2.8), we know that the solution ψ is also radially symmetric, i.e., there exists a function ϕ : R+ × R+ −→ R such that ψ(t, x) = ϕ(t, |x|).

(2.14)

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By (2.7)–(2.9), we know that ϕ = ϕ(t, r) satisfies 3 3 ϕtt − c12 (ϕrr + ϕr ) = c2 (ϕ, rϕr )(ϕrr + ϕr ) + r −2 B(ϕ, rϕr ), r r ϕ(0, r) = φ0 (r), ϕt (0, r) = φ1 (r).

(2.15) (2.16)

Thus for the Cauchy problem (2.2)–(2.3), we have constructed a classical solution φ(t, r) = ϕ(t, r). Let u(t, y) = yφ(t, |y|). By the argument at the beginning of this section, we know that u is the unique global classical solution to (1.13)–(1.14). 3. Proof of Proposition 2.1 In this section, we will prove Proposition 2.1. In the following, for convenience we will set c1 = 1. For the Cauchy problem of 4-D quasilinear wave equations with the general form 2ψ = F1 (ψ, ∂ψ, ∂∇ψ),

(3.1)

where 2 = ∂t2 −  is the wave operator in R1+4 and F1 (λ) = O(|λ|2 )(|λ| 1), there are many results on the long time existence of small classical solutions [12,20,29,30,49] (see also [31] for low regularity solutions). Results on the exterior problem of (3.1) can be found in [5,7,34,37, 50]. In our case, the equation takes the form 2ψ = F2 (ψ, x · ∇ψ, ∇ψ, ψ),

(3.2)

where F2 can be determined by (2.7), (2.10) and (2.11). There are some new terms concerning x · ∇ψ in F2 in contrast with F1 . These terms will impact the long time behavior of classical solutions dramatically, since it will produce a growth factor |x|. In fact, we can understand that null conditions d1 = 0 and e1 = 0 are used to rule out the occurrence of the terms only containing x · ∇ψ in (2.10) at the linear and quadratic level respectively. In the reminder of this section, we will give the detail of the proof of Proposition 2.1. We first give some necessary preliminaries including decay estimates and energy type estimates. Then we will prove Proposition 2.1 by a bootstrap argument. In the following of this paper, for convenience, we will denote 4-D space gradient by ∇ = (∂x1 , · · · , ∂x4 ) and use ∂ = (∂t , ∇) to denote the full space-time gradient in R1+4 . Since we only consider radially symmetric solutions in Proposition 2.1, we can only use usual space gradient ∇ as the commuting vector fields. 3.1. Preliminaries Lemma 3.1. Let ψ be a radially symmetric function on R4 . Then we have ψL∞ (R4 ) ≤ C∇ψH 2 (R4 ) ,  1 a  |x| 2 ∇ ψ  ∞ 4 ≤ C∇∇ a ψ 1 4 , H (R ) L (R )

(3.3) (3.4)

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  |x|∇ a ψ 

L∞ (R4 )

≤ C∇∇ a ψL2 (R4 ) ,

(3.5)

L∞ (R4 )

≤ C∇ a ψH 1 (R4 ) .

(3.6)

 3 a  |x| 2 ∇ ψ 

Proof. (3.3) is just the usual 4-D Sobolev inequality concerning the control of L∞ norm. (3.6) is a consequence of the Sobolev embedding on the unit sphere on R4 : H 2 (S 3 ) → L∞ (S 3 ), the fundamental theorem of calculus, and the fact that ψ is radially symmetric, which makes the angular derivatives play no role. While (3.4) and (3.5) follows from some trace inequality (see (3.15) of [8]) and the fact that ψ is radially symmetric. 2 The following lemma is a special case of the well known Keel-Smith-Sogge type inequality (KSS for short). This kind of estimate was first established for the Cauchy problem of linear wave equations with constant coefficients in Keel et al. [17] via a constructive method. In the appendix of [44], Rodnianski suggested an energy method and gave a new proof. Then the energy method was further developed in [35,36] for perturbed linear wave equations outside of star shaped obstacles and in [48] for 3-D perturbed linear elastic wave equations. KSS type inequality including weight which is singular at the origin was first developed in [10] for the Cauchy problem of linear wave equations with constant coefficients, then in [9] for perturbed linear wave equations and in [11] for 3-D perturbed linear elastic wave equations. Lemma 3.2. Fix δ > 0, T > 0 and k ∈ N. Let ψ satisfy 2ψ = h(t, x)ψ + Q(t, x), t > 0, x ∈ R4

(3.7)

and denote ST = [0, T ] × R4 . Assume that h satisfies the smallness condition |h| 1. Then we have |a|≤k−1

∂∇ a ψL∞ + 2 t Lx (ST ) +

≤C







1

1

|x|− 4 x − 4 −δ ∂∇ a ψL2

t,x (ST )

|a|≤k−1 5

1

|x|− 4 x − 4 −δ ∇ a ψL2

|a|≤k−1

t,x (ST )

∂∇ a ψ(0, ·)L2 (R4 )

|a|≤k−1

+C

  (|∂h| + |x|− 12 x − 12 |h|)|∇∇ a ψ|(|∇∇ a ψ| + |x|−1 |∇ a ψ|)

|a|≤k−1

+C



  (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

|a|≤k−1 b+c=a b =0

+C

  (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ a Q|

|a|≤k−1

where · = (1 + | · |2 )1/2 . Proof. See Theorem 2.1 of [9].

2

L1t,x (ST )

,

L1t,x (ST )

L1t,x (ST )

(3.8)

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3.2. Proof of Proposition 2.1 In this section, we will prove Proposition 2.1. Assume that the radially symmetric function ψ is a local classical solution on [0, T ] to the Cauchy problem (2.7)–(2.8). Denote Ek (ψ(t)) =

|a|≤k−1

∂∇ a ψ(t, ·)2L2 (R4 )

(3.9)

and 

Ek (ψ(t)) =

|a|≤k−1

1

1

|x|− 4 x − 2 ∂∇ a ψ2L2

t,x (St )

+

|a|≤k−1

5



1

|x|− 4 x − 2 ∇ a ψ2L2

t,x (St )

. (3.10)

Let 1/2

1/2

ψXk (T ) = sup Ek (ψ(t)) + Ek (ψ(T )).

(3.11)

0≤t≤T

We will prove that there exist positive constants ε0 and A0 such that ψX3 (T ) ≤ A0 ε, ψX4 (T ) ≤ A0 M0 under the assumption ψX3 (T ) ≤ 2A0 ε, ψX4 (T ) ≤ 2A0 M0 for any T > 0, where 0 < ε < ε0 . In view of (2.7), (2.10) and (2.11), noting d1 = 0 and e1 = 0, in Lemma 3.2, we take h = 2d3 ψ + 2(e4 + 5e5 )ψ 2 + 6e5 ψ(x · ∇ψ),

(3.12)

Q = d3 |∇ψ|2 + 2(e4 + 2e5 )ψ|∇ψ|2 + 2e5 (x · ∇ψ)|∇ψ|2 .

(3.13)

Then we can get ψXk (T ) ≤ C



∂∇ a ψ(0, ·)L2 (R4 )

|a|≤k−1

+C

  (|∂h| + |x|− 12 x − 12 |h|)|∇∇ a ψ|(|∇∇ a ψ| + |x|−1 |∇ a ψ|)

|a|≤k−1

+C



  (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

|a|≤k−1 b+c=a b =0

+C

  (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ a Q|

|a|≤k−1

L1t,x (ST )

.

L1t,x (ST )

L1t,x (ST )

(3.14)

We first focus on the case k = 3. Without loss of generality, we only consider the case |a| = 2. In view of (3.12), we have pointwise estimates

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  |h| ≤ C |ψ| + |ψ|2 + |x||ψ∇ψ| ,   |∂h| ≤ C ||∂ψ| + |ψ∂ψ| + |x||∇ψ∂ψ| + |x||ψ∂∇ψ| ,   |∇ 2 h| ≤ C |∇ 2 ψ| + |∇ψ|2 + |ψ∇ 2 ψ| + |x||∇ψ∇ 2 ψ| + |x||ψ∇ 3 ψ| .

(3.15) (3.16) (3.17)

On the region |x| ≥ 1. We first estimate the right hand side of (3.14) on the region |x| ≥ 1. By (3.15) and (3.5), we have   |x|h

L∞ x (|x|≥1)

      ≤ C |x|ψ L∞ (R4 ) + C |x|2 ψ 2 L∞ (R4 ) + C |x|2 ψ∇ψ L∞ (R4 ) x x x      2          ≤ C |x|ψ L∞ (R4 ) + C |x|ψ L∞ (R4 ) + C |x|ψ L∞ (R4 ) |x|∇ψ L∞ (R4 ) x

x

x

x

≤ C∇ψL2 (R4 ) + C∇ψ2L2 (R4 ) + C∇ψL2 (R4 ) ∇ 2 ψL2 (R4 ) 1/2

≤ CE3 (ψ(t)) + CE3 (ψ(t)).

(3.18)

From (3.16), (3.5) and (3.6), we obtain  3  |x| 2 ∂h

L∞ x (|x|≥1)

 3  ≤ C |x| 2 ∂ψ 

 5  + C |x| 2 ψ∂ψ L∞ (R4 ) x  5   5     + C |x| 2 ∇ψ∂ψ L∞ (R4 ) + C |x| 2 ψ∂∇ψ L∞ (R4 ) x x  3   3         ≤ C |x| 2 ∂ψ L∞ (R4 ) + C |x| 2 ∂ψ L∞ (R4 ) |x|ψ L∞ (R4 ) x x x      3   3  + C |x| 2 ∂ψ L∞ (R4 ) |x|∇ψ L∞ (R4 ) + C |x| 2 ∂∇ψ L∞ (R4 ) |x|ψ L∞ (R4 ) 4 L∞ x (R )

x

x

x

x

≤ C∂ψH 1 (R4 ) + C∂ψH 1 (R4 ) ∇ψL2 (R4 ) + C∂ψH 1 (R4 ) ∇ 2 ψL2 (R4 ) + C∂∇ψH 1 (R4 ) ∇ψL2 (R4 ) 1/2

≤ CE3 (ψ(t)) + CE3 (ψ(t)).

(3.19)

Thus (3.18) and (3.19) imply   (|∂h| + |x|− 12 x − 12 |h|)|∇∇ a ψ|(|∇∇ a ψ| + |x|−1 |∇ a ψ|)

L1t,x (|x|≥1)

  1 1 ≤ C (|∂h| + |x|− 2 x − 2 |h|)|∇ 3 ψ|(|∇ 3 ψ| + |x|−1 |∇ 2 ψ|)L1  3  ≤ C |x| 2 ∂h

L∞ t,x (|x|≥1)

 1  + |x| 2 h

L∞ t,x (|x|≥1)

t,x (|x|≥1)



E3 (ψ(T ))

   3   ≤ C |x| 2 ∂hL∞ ((|x|≥1)) + |x|hL∞ ((|x|≥1)) E3 (ψ(T )) t,x t,x   1/2 ≤ C sup E3 (ψ(t)) + sup E3 (ψ(t)) E3 (ψ(T )) 0≤t≤T

0≤t≤T

≤ Cψ3X3 (T ) + Cψ4X3 (T ) .

(3.20)

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Now we will consider the third part on the right hand side of (3.14). For any |a| = 2, b + c = a, b = 0, if |b| = |c| = 1, it is obvious that   |∇h||∇ 3 ψ| ≤ C |∇ψ| + |ψ∇ψ| + |x||∇ψ|2 + |x||ψ∇ 2 ψ| |∇ 3 ψ|.

(3.21)

In view of (3.20), we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≥1)

  1 1 ≤ C (|∇ 3 ψ| + |x|− 2 x − 2 |∇ 2 ψ|)|∇h||∇ 3 ψ|L1  3  ≤ C |x| 2 ∇h

t,x (|x|≥1)

E3 (ψ(T )) ≤ Cψ3X3 (T ) L∞ t,x ((|x|≥1))

+ Cψ4X3 (T ) .

(3.22)

For the case |b| = 2, c = 0, by (3.17) it is obvious that   |∇ 2 h||∇ 2 ψ| ≤ C |∇ 2 ψ| + |∇ψ|2 + |ψ∇ 2 ψ| + |x||∇ψ||∇ 2 ψ| |∇ 2 ψ| + C|x||ψ∇ 2 ψ||∇ 3 ψ|.

(3.23)

Thus by (3.23), (3.5) and (3.6), we can get   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≥1)

  1 1 ≤ C (|∇ 3 ψ| + |x|− 2 x − 2 |∇ 2 ψ|)|∇ 2 h||∇ 2 ψ|  3  ≤ C |x| 2 ∇ 2 ψ 

L∞ t,x (ST )

 5  + |x| 2 ∇ψ∇ 2 ψ   3  ≤ C |x| 2 ∇ 2 ψ   3  + |x| 2 ∇ 2 ψ  ≤C



 3  + |x| 2 |∇ψ|2 L∞ (S t,x

L∞ t,x (ST )

L∞ t,x (ST )

L∞ t,x (ST )

 5  + |x| 2 ψ∇ 2 ψ 

L1t,x (|x|≥1)

T)

L∞ t,x (ST )



E3 (ψ(T ))

 1    + |x| 2 ∇ψ L∞ (S ) |x|∇ψ L∞ (S

  |x|∇ψ 

t,x

L∞ t,x (ST )

T

 3  + |x| 2 ∇ 2 ψ 

t,x

T)

L∞ t,x (ST )

  |x|ψ 

 L∞ t,x (ST )

E3 (ψ(T ))

 1/2 sup E3 (ψ(t)) + sup E3 (ψ(t)) E3 (ψ(T )) ≤ Cψ3X3 (T ) + Cψ4X3 (T ) .

0≤t≤T

(3.24)

0≤t≤T

Finally we consider the fourth part on the right hand side of (3.14). In view of (3.13), we have pointwise estimate   |∇ 2 Q| ≤ C |∇ψ| + |ψ∇ψ| + |x||∇ψ|2 |∇ 3 ψ|   + C |∇ 2 ψ| + |∇ψ|2 + |ψ∇ 2 ψ| + |x||∇ψ||∇ 2 ψ| |∇ 2 ψ|.

(3.25)

Note that all terms on the right hand side of (3.25) have already appeared in the right hand side of (3.21) and (3.23), which have been treated in the above. Thus we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ a Q|

L1t,x (|x|≥1)

≤ Cψ3X3 (T ) + Cψ4X3 (T ) . (3.26)

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On the region |x| ≤ 1. Now we will estimate the right hand side of (3.14) on the region |x| ≤ 1. By (3.15), (3.3) and (3.5), we have   hL∞ (R4 ) ≤ CψL∞ (R4 ) + Cψ 2 L∞ (R4 ) + C |x|ψ∇ψ L∞ (R4 ) x   2 ≤ CψL∞ (R4 ) + CψL∞ (R4 ) + CψL∞ (R4 ) |x|∇ψ L∞ (R4 ) x

≤ C∇ψH 2 (R4 ) + C∇ψ2H 2 (R4 )

+ C∇ψH 2 (R4 ) ∇ ψL2 (R4 ) 2

1/2

≤ CE3 (ψ(t)) + CE3 (ψ(t)).

(3.27)

By (3.16), (3.3), (3.4), (3.5) and (3.6), we get  1  |x| 2 ∂h

4 L∞ x (R )

 1   1  ≤ C |x| 2 ∂ψ L∞ (R4 ) + C |x| 2 ψ∂ψ L∞ (R4 ) x

x

 3  + C |x| 2 ∇ψ∂ψ 

4 L∞ x (R )

 3  + C |x| 2 ψ∂∇ψ 

4 L∞ x (R )

 1  1   ≤ C |x| 2 ∂ψ L∞ (R4 ) + CψL∞ (R4 ) |x| 2 ∂ψ L∞ (R4 )  1  + C |x| 2 ∂ψ 

x

4 L∞ x (R )

  |x|∇ψ 

x

4 L∞ x (R )

 3  + CψL∞ (R4 ) |x| 2 ∂∇ψ L∞ (R4 ) x

≤ C∂∇ψH 1 (R4 ) + C∇ψH 2 (R4 ) ∂∇ψH 1 (R4 ) + C∇ ψL2 (R4 ) ∂∇ψH 1 (R4 ) 2

1/2

≤ CE3 (ψ(t)) + CE3 (ψ(t)).

(3.28)

Thus (3.27) and (3.28) imply   (|∂h| + |x|− 12 x − 12 |h|)|∇∇ a ψ|(|∇∇ a ψ| + |x|−1 |∇ a ψ|)

L1t,x (|x|≤1)

  1 1 ≤ C (|∂h| + |x|− 2 x − 2 |h|)|∇ 3 ψ|(|∇ 3 ψ| + |x|−1 |∇ 2 ψ|)L1  1  ≤ C |x| 2 ∂h ≤C



sup 0≤t≤T

L∞ t,x (ST

t,x (|x|≤1)

  + hL∞ (S ) E3 (ψ(T )) ) t,x

1/2 E3 (ψ(t)) +

T

 sup E3 (ψ(t)) E3 (ψ(T )) ≤ Cψ3X3 (T ) + Cψ4X3 (T ) .

0≤t≤T

(3.29) Now we will consider the third part on the right hand side of (3.14). For any |a| = 2, b + c = a, b = 0, if |b| = |c| = 1, in view of (3.29), we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≤1)

  1 1 ≤ C (|∇ 3 ψ| + |x|− 2 x − 2 |∇ 2 ψ|)|∇h||∇ 3 ψ|L1  1  ≤ C |x| 2 ∇h

t,x (|x|≤1)

E3 (ψ(T )) ≤ Cψ3X3 (T ) L∞ t,x (ST )

+ Cψ4X3 (T ) .

(3.30)

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For the case |b| = 2, c = 0, by (3.23), (3.3), (3.4), (3.6) and the Sobolev embedding H 1 (B1 ) → L∞ (B1 ), we obtain   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≤1)

  1 1 ≤ C (|∇ 3 ψ| + |x|− 2 x − 2 |∇ 2 ψ|)|∇ 2 h||∇ 2 ψ|L1 (|x|≤1) t,x  1     1  1 ≤ C |x| 2 ∇ 2 ψ  ∞ 4 + |x| 2 |∇ψ|2  ∞ 4 + |x| 2 ψ∇ 2 ψ  Lt Lx (|x|≤1)

4 L∞ t Lx (|x|≤1)

Lt Lx (|x|≤1)

 3  + |x| 2 ∇ψ∇ 2 ψ 

4 L∞ t Lx (|x|≤1)

  |x|− 14 ∇ 2 ψ 

1/2 E (ψ(T )) L2t L4x (|x|≤1) 3

 3    1 1/2 + C |x| 2 ψ∇ 2 ψ L∞ (S ) |x|− 4 ∇ 3 ψ L2 (S ) E3 (ψ(T )) t,x T t,x T    1  ≤ C ∇ 2 ψ L∞ H 1 (|x|≤1) + |x| 2 ∇ψ L∞ (S ) ∇ψL∞ 1 t Hx (|x|≤1) t

+ ψL∞ t,x (ST

t,x

x

 2    ) ∇ ψ

1 L∞ t Hx (|x|≤1)

T

 3  + |x| 2 ∇ 2 ψ 

 3 2  |x| 2 ∇ ψ  ∞ E (ψ(T )) + CψL∞ t,x (ST ) L (S ) 3 ≤C



t,x

sup 0≤t≤T

1/2 E3 (ψ(t)) +

L∞ t,x (ST )

 ∇ψL∞ 1 (|x|≤1) E3 (ψ(T )) H t x

T

 sup E3 (ψ(t)) E3 (ψ(T )) ≤ Cψ3X3 (T ) + Cψ4X3 (T ) .

(3.31)

0≤t≤T

The fourth part on the right hand side of (3.14) has been essentially treated in the above and we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ a Q|

L1t,x (|x|≤1)

≤ Cψ3X3 (T ) + Cψ4X3 (T ) . (3.32)

Then we consider the case k = 4. We only treat the case |a| = 3. Noting (3.18) and (3.19), similar to (3.20), on the region |x| ≥ 1 the second part on the right hand side of (3.14) admits the bound 

 1/2 sup E3 (ψ(t)) + sup E3 (ψ(t)) E4 (ψ(T ))

0≤t≤T

0≤t≤T

 ≤ C ψX3 (T ) + ψ2X3 (T ) ψ2X4 (T ) . 

(3.33)

Noting (3.27) and (3.28), similar to (3.29), on the region |x| ≤ 1 the second part on the right hand side of (3.14) also admits the bound (3.33). For the third part on the right hand side of (3.14), for any |a| = 3, b + c = a, b = 0, if |b| = 1, |c| = 2, then corresponding terms have been treated in the second part and admit the bound (3.33). For the case |b| = 2, |c| = 1, by (3.17), it is obvious that   |∇ 2 h||∇ 3 ψ| ≤ C |∇ 2 ψ| + |∇ψ|2 + |ψ∇ 2 ψ| + |x||∇ψ||∇ 2 ψ| + |x||ψ∇ 3 ψ| |∇ 3 ψ|. (3.34)

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By (3.34), (3.5) and (3.6), we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≥1)

  1 1 ≤ C (|∇ 4 ψ| + |x|− 2 x − 2 |∇ 3 ψ|)|∇ 2 h||∇ 3 ψ|L1 (|x|≥1) t,x  3    5   3 ≤ C |x| 2 ∇ 2 ψ  ∞ + |x| 2 |∇ψ|2  ∞ + |x| 2 ψ∇ 2 ψ  Lt,x (ST )

Lt,x (ST )

 5  + |x| 2 ∇ψ∇ 2 ψ 

L∞ t,x (ST )

 3  ≤ C |x| 2 ∇ 2 ψ L∞ (S  3  + |x| 2 ∇ 2 ψ 

T)

t,x

L∞ t,x (ST )



 5  + |x| 2 ψ∇ 3 ψ 

L∞ t,x (ST )

L∞ t,x (ST )

1/2

1/2

E3 (ψ(T ))E4 (ψ(T ))

   1  + |x| 2 ∇ψ L∞ (S ) |x|∇ψ L∞ (S   |x|ψ 

t,x

L∞ t,x (ST )

T

t,x

 3  + |x| 2 ∇ 2 ψ 

T)

L∞ t,x (ST )

  |x|∇ψ 

  3    1/2 1/2 + |x| 2 ∇ 3 ψ L∞ (S ) |x|ψ L∞ (S ) E3 (ψ(T ))E4 (ψ(T )) ≤C ≤C

 

T

t,x

sup 0≤t≤T

t,x

1/2 E3 (ψ(t)) +

sup 0≤t≤T

ψX3 (T ) + ψ2X3 (T )

L∞ t,x (ST )

T

1/2 E3 (ψ(t))



 1/2 1/2 1/2 sup E4 (ψ(t)) E3 (ψ(T ))E4 (ψ(T ))

0≤t≤T

ψ2X4 (T ) .

(3.35)

By (3.34), (3.3), (3.4), (3.6) and the Sobolev embedding H 1 (B1 ) → L∞ (B1 ), we have   (|∇∇ a ψ| + |x|− 12 x − 12 |∇ a ψ|)|∇ b h||∇ c ψ|

L1t,x (|x|≤1)

  1 1 ≤ C (|∇ 4 ψ| + |x|− 2 x − 2 |∇ 3 ψ|)|∇ 2 h||∇ 3 ψ|L1 (|x|≤1) t,x  1     1  1 ≤ C |x| 2 ∇ 2 ψ  ∞ 4 + |x| 2 |∇ψ|2  ∞ 4 + |x| 2 ψ∇ 2 ψ  Lt Lx (|x|≤1)

4 L∞ t Lx (|x|≤1)

Lt Lx (|x|≤1)

 3  + |x| 2 ∇ψ∇ 2 ψ 

4 L∞ t Lx (|x|≤1)

  |x|− 14 ∇ 3 ψ 

1/2 E (ψ(T )) L2t L4x (|x|≤1) 4

  3   1 1/2 + C |x| 2 ψ∇ 3 ψ L∞ (S ) |x|− 4 ∇ 3 ψ L2 (S ) E4 (ψ(T )) t,x T t,x T    1  ≤ C ∇ 2 ψ L∞ H 1 (|x|≤1) + |x| 2 ∇ψ L∞ (S ) ∇ψL∞ 1 t Hx (|x|≤1) t

+ ψL∞ t,x (ST

t,x

x

 2    ) ∇ ψ

1 L∞ t Hx (|x|≤1)

T

 3  + |x| 2 ∇ 2 ψ 

L∞ t,x (ST )

 ∇ψL∞ E4 (ψ(T )) 1 t Hx (|x|≤1)

 3 3  1/2 1/2 |x| 2 ∇ ψ  ∞ + CψL∞ E (ψ(T ))E4 (ψ(T )) (S ) T t,x L (S ) 3 ≤C



t,x

sup 0≤t≤T

+ C sup 0≤t≤T

1/2 E3 (ψ(t)) + 1/2 E3 (ψ(t))

T

 sup E3 (ψ(t)) E4 (ψ(T )) 0≤t≤T 1/2

1/2

1/2

sup E4 (ψ(t))E3 (ψ(T ))E4 (ψ(T ))

0≤t≤T

  ≤ C ψX3 (T ) + ψ2X3 (T ) ψ2X4 (T ) .

(3.36)

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15

Thus in the case |b| = 2, |c| = 1, the third part on the right hand side of (3.14) also admits the bound 

 ψX3 (T ) + ψ2X3 (T ) ψ2X4 (T )

(3.37)

For the case |b| = 3, c = 0, in view of (3.12), we have  |∇ 3 h| ≤ C |∇ 3 ψ| + |∇ψ∇ 2 ψ| + |ψ∇ 3 ψ| + |x||∇ 2 ψ∇ 2 ψ|  + |x||∇ψ∇ 3 ψ| + |x||ψ∇ 4 ψ| .

(3.38)

Then we have pointwise estimate   |∇ 3 h||∇ 2 ψ| ≤ C|x||ψ∇ 2 ψ||∇ 4 ψ| + C |∇ 2 ψ| + |ψ∇ 2 ψ| + |x||∇ψ∇ 2 ψ| |∇ 3 ψ|   (3.39) + C |∇ψ∇ 2 ψ| + |x||∇ 2 ψ∇ 2 ψ| |∇ 2 ψ|. Thus all terms for the case |b| = 3, c = 0 have been treated above and the bound (3.37) can be also obtained. In conclusion, we have got ψ2X3 (T ) ≤ CE3 (ψ(0)) + Cψ3X3 (T ) + Cψ4X3 (T ) ≤ C0 ε 2 + 8C0 A3 ε 3 + 16C0 A4 ε 4

(3.40)

and   ψ2X4 (T ) ≤ CE4 (ψ(0)) + C ψX3 (T ) + ψ2X3 (T ) ψ2X4 (T ) ≤ C0 M02 + C0 (2Aε + 4A2 ε 2 )ψ2X4 (T ) .

(3.41)

Take A2 = 4C0 and ε0 > 0 sufficiently small such that 32C0 Aε0 + 64C0 A2 ε02 ≤ 1.

(3.42)

ψX3 (T ) ≤ Aε, ψX4 (T ) ≤ AM0 ,

(3.43)

Then for any 0 < ε < ε0 , we have

which completes the proof of Proposition Proposition 2.1. 4. Proof of Theorem 1.2 In this section, we will consider the exterior domain problem and prove Theorem 1.2. Denote R4∗ = R4 \B(0, 1), where B(0, 1) stands for the unit ball centered at original point in R4 . Consider the following 4-D initial-boundary value problem

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ψtt − c12 ψ = c2 (ψ, x · ∇ψ)ψ +

B(ψ, x · ∇ψ) , t > 0, x ∈ R4∗ , |x|2

ψ|∂B(0,1) = 0,

(4.1) (4.2)

ψ(0, x) = ψ0 (x), ψt (0, x) = ψ1 (x), x

∈ R4∗ ,

(4.3)

where 

   ψ0 (x), ψ1 (x) = φ0 (|x|), φ1 (|x|) , x ∈ R4∗     u0 (y), u1 (y) = yφ0 (|y|), yφ1 (|y|) , y ∈ R2∗ .

(4.4) (4.5)

Similarly to the Cauchy problem case, Theorem 1.2 can be reduced to the following Proposition 4.1. Consider the initial-boundary value problem (4.1)–(4.3). Assume that d1 = 0, e1 = 0 and the initial data (ψ0 , ψ1 ) are radially symmetric and satisfy the compatibility conditions of order 3. Then there is a constant ε0 > 0 such that for any 0 < ε < ε0 , if ψ0 H 3 (R4∗ ) + ψ1 H 2 (R4∗ ) ≤ ε

(4.6)

M0 = ψ0 H 4 (R4∗ ) + ψ1 H 3 (R4∗ ) < +∞,

(4.7)

and

then (4.1)–(4.3) admits a unique global classical solution ψ. In fact, in the exterior domain problem case, since we do not need to consider the singularity at the origin, we can easily show that (1.20) implies (4.6) and (1.21) implies (4.7) by direct computations. The remained task is to prove Proposition 4.1. Similarly to the Cauchy problem case, we first need some decay estimates and KSS type estimates. The proof of decay estimates in the following lemma is similar to the corresponding ones in the Cauchy problem case. Lemma 4.1. Let ψ be a radially symmetric function on R4∗ . Then we have   x ∂ a ψ 

L∞ (R4∗ )

≤ C∇∂ a ψL2 (R4∗ ) ,

(4.8)

L∞ (R4∗ )

≤ C∂ a ψH 1 (R4∗ ) .

(4.9)

 3 a  x 2 ∂ ψ 

Lemma 4.2. Fix δ > 0, T > 0 and k ∈ N. Let ψ satisfy

2ψ(t, x) = h(t, x)ψ + Q(t, x), t > 0, x ∈ R4∗ , ψ|∂B(0,1) = 0, t > 0

(4.10)

and denote ST∗ = [0, T ] × R4∗ . Assume that h satisfies the smallness condition |h| 1. Then we have

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|a|≤k−1

∂∂ a ψL∞ 2 ∗ + t Lx (S ) T



≤C



17

1

x − 2 −δ ∂∂ a ψL2

∗ t,x (ST )

|a|≤k−1

∂∂ a ψ(0, ·)L2 (R4 )

|a|≤k−1

  (|∂h| + x −1 |h|)|∂∂ a ψ|(|∂∂ a ψ| + x −1 |∂ a ψ|)

+C

|a|≤k−1



+C

  (|∂∂ a ψ| + x −1 |∂ a ψ|)|∂ b h||∂ c ψ|

L1t,x (ST∗ )

|a|≤k−1 b+c=a b =0

  (|∂∂ a ψ| + x −1 |∂ a ψ|)|∂ a Q|

+C

|a|≤k−1



+C

|a|≤k−2

L1t,x (ST∗ )

∂ a w2L2

∗ t,x (ST )

L1t,x (ST∗ )

(4.11)

.

Proof. See Lemma 3.2 and Lemma 5.2 of [35]. 2 Assume that the radially symmetric function ψ is a local classical solution on [0, T ] to the initial-boundary value problem (4.1)–(4.3). Denote Ek (ψ(t)) =

|a|≤k−1

∂∂ a ψ(t, ·)2L2 (R4 ) , Ek (ψ(t)) = ∗

|a|≤k−1

3

x − 4 ∂∂ a ψ2L2

∗ t,x (St )

. (4.12)

Let 1/2

1/2

ψXk (T ) = sup Ek (ψ(t)) + Ek (ψ(T )).

(4.13)

0≤t≤T

Once we have Lemma 4.1 and Lemma 4.2 at hand, the estimates for ψX3 (T ) and ψX4 (T ) are completely similar to the |x| ≥ 1 part of the proof of Proposition 2.1. We can get ψ2X3 (T ) ≤ CE3 (ψ(0)) + Cψ3X3 (T ) + Cψ4X3 (T )

(4.14)

  ψ2X4 (T ) ≤ CE4 (ψ(0)) + C ψX3 (T ) + ψ2X3 (T ) ψ2X4 (T ) .

(4.15)

and

Based on the above two estimates, similarly to the Cauchy problem case, we can prove that there exist positive constants ε0 and A0 such that ψX3 (T ) ≤ A0 ε, ψX4 (T ) ≤ A0 M0 under the assumption ψX3 (T ) ≤ 2A0 ε, ψX4 (T ) ≤ 2A0 M0 for any T > 0, where 0 < ε < ε0 . Thus we have proved Proposition 4.1.

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5. Appendix Denote the Fourier transformation on Rn by Fn . For a n-dimensional radial function (x) = f (|x|) for some function f on R+ , its Fourier transformation is also radial and we use the notation Fn ( )(ξ ) = Fn (f )(r),

(5.1)

where r = |ξ |. Between Fn+2 (f )(r) and Fn (f )(r), there is the following relationship (see (1) of [6]) Fn+2 (f )(r) = −

1 1 d Fn (f )(r). 2π r dr

(5.2)

Using (5.2), we can get the following Lemma 5.1. Let u(y) = yφ(|y|), y ∈ Rn ,

(5.3)

ψ(x) = φ(|x|), x ∈ Rn+2 .

(5.4)

ψH s (Rn+2 ) ∼ uH s (Rn ) ,

(5.5)

Then for any s ∈ R,

where ∼ means the equivalent relationship. Proof. By the definition of H s norm and (5.2), we have ψH s (Rn+2 ) = ξ s Fn+2 (ψ)(ξ )L2 (Rn+2 ) = ξ s Fn+2 (φ)(|ξ |)L2 (Rn+2 ) ξ

∼ ξ s ∼

n

1 d d Fn (φ)(|ξ |)L2 (Rn+2 ) ∼ ξ s Fn (φ)(|ξ |)L2 (Rn ) ξ ξ |ξ | dr dr

n     ξ s ∂ξi Fn (φ)(|ξ |) L2 (Rn ) ∼ ξ s Fn yi φ(|y|) (ξ )L2 (Rn ) ξ

i=1

∼

ξ

n

ξ

i=1

ξ s Fn (ui )(ξ )L2 (Rn ) ∼ uH s (Rn ) . ξ

2

(5.6)

i=1

Acknowledgments The author is supported by National Natural Science Foundation of China (No. 11801068), a grant of institute for nonlinear sciences of Donghua University, Shanghai Sailing Program (No. 17YF1400700) and Fundamental Research Funds for Central Universities (No. 17D110913).

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