On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

Accepted Manuscript On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions

Marcelo Rempel Ebert, Laila Fitriana, Fumihiko Hirosawa

PII: DOI: Reference:

S0022-247X(15)00605-8 http://dx.doi.org/10.1016/j.jmaa.2015.06.051 YJMAA 19604

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

9 March 2015

Please cite this article in press as: M.R. Ebert et al., On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2015.06.051

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the energy estimates of the wave equation with time dependent propagation speed asymptotically monotone functions Marcelo Rempel Ebert a , Laila Fitriana b , Fumihiko Hirosawa c,∗,1 a

Department of Computing and Mathematics, Universidade de S˜ ao Paulo (FFCLRP) Av. dos Bandeirantes 3900, CEP 14040-901, Ribeir˜ ao Preto, SP, Brazil

b

Graduate School of Science and Engineering, Yamaguchi University, 753-8512, Japan; Department of Mathematics Education, Sebelas Maret University Jl Ir Sutami 36 A Surakarta, Jawa Tenqah 57126, Indonesia

c

Department of Mathematical Sciences, Yamaguchi University, Yamaguchi 753-8512, Japan

Abstract We consider the energy estimates for the wave equation with time dependent propagation speed. It is known that the asymptotic behavior of the energy is determined by the interactions of the properties of the propagation speed: smoothness, oscillation and the difference from an auxiliary function. The main purpose of the article is that if the propagation speed behaves asymptotically as a monotone decreasing function, then we can extend the preceding results to allow faster oscillating coefficients. Moreover, we prove that the regularity of the initial data in the Gevrey class can essentially contribute for the energy estimate.

Keywords: Wave equations; Energy estimates; Time dependent coefficients

1

Introduction

Let us consider the following Cauchy problem of the wave equation with time dependent propagation speed:   ∂t2 − a(t)2 Δ u = 0, (t, x) ∈ R+ × Rn , (1.1) (u(0, x), (∂t u)(0, x)) = (u0 (x), u1 (x)), x ∈ Rn , where R+ = [0, ∞), a(t) ∈ C m (R+ ) with m ≥ 2 satisfy a(t) > 0 and supt {a(t)} < ∞. Here the total energy of (1.1) at t is defined by E(t) = E(t; u0 , u1 ) :=

 1 a(t)2 ∇u(t, ·)2 + ∂t u(t, ·)2 , 2

where  ·  denotes the usual L2 norm in Rn . If the propagation speed a(t) is a constant, then the energy conservation E(t) ≡ E(0) is valid. On the other hand, the energy conservation does not hold in general for variable propagation speed. However, the following equivalence between E(t) and E(0), which is called the generalized energy conservation: (GEC) C −1 E(0) ≤ E(t) ≤ CE(0) with a constant C > 1, can be expected even though the propagation speed is not a constant. For instance, if inf t {a(t)} > 0 and a (t) ∈ L1 (R+ ), then (GEC) is trivial by the classical energy method, which is derived by the inequality E  (t) ≤ C|a (t)|E(t) and Gronwall’s inequality. On the other hand, the classical energy method is useless for the proof of (GEC) if a (t) ∈ L1 (R+ ). Actually, the L1 property of a (t) is not enough to decide ∗ Corresponding

author. E-mail addresses: ebert@ffclrp.usp.br (M. R. Ebert), [email protected] (L. Fitriana), [email protected] (F. Hirosawa). 1 Supported by JSPS KAKENHI Grant Number 26400170.

1

whether (GEC) is valid or not because both cases are possible if a (t) ∈ L1 (R+ ); thus we introduce additional properties of a(t). Let us suppose that (1.2) inf {a(t)} > 0. t

For α ∈ [0, 1] and β ∈ R we introduce the following conditions:  t |a(s) − a∞ | ds ≤ C0 (1 + t)α

(1.3)

0

   (k)  a (t) ≤ Ck (1 + t)−βk (k = 1, . . . , m)

and

(1.4)

for some constants a∞ , C1 , . . . , Cm ; we shall denote universal positive constants by C and Ck with k = 0, 1, . . . without making any confusion. Here we remark the followings: • If α = 1, then (1.3) is trivial for any constant a∞ . On the other hand, a∞ is uniquely determined if (1.3) holds for α < 1. • If (1.4) holds for β > 1, then (GEC) is trivial because a (t) ∈ L1 (R+ ). • (1.3) and (1.4) impose more restricted conditions as α smaller and β larger, respectively. (1.3) is called the stabilization property, which describes an order of difference between the variable and constant propagation speeds, and (1.4) describes an order of oscillation and the smoothness of a(t). Under the assumptions above, we have the following result: Theorem 1.1 ([7]). Suppose that (1.2), (1.3) and (1.4) are valid. If α, β and m satisfy β ≥ βm := α +

1−α , m

(1.5)

then (GEC) is established. If β < α, then (GEC) does not hold in general. By Theorem 1.1 we see that (GEC) is determined by the interaction of the stabilization, the oscillation and the smoothness properties of a(t). For instance, β can be taken smaller as α smaller and m larger. That is, faster oscillation can be admitted for (GEC) if a(t) is smoother and strongly stabilized by a constant a∞ in the sense of (1.3). Remark 1.2. In the case of α = 1, that is, without the stabilization property, Theorem 1.1 is proved by [13]. Here we underline that (1.5) with α = 1 provides the optimal condition for (GEC). Moreover, βm is independent of m for α = 1. Remark 1.3. If α < 1, then Theorem 1.1 cannot give any answer whether (GEC) holds or not for α ≤ β < βm . We also see that limm→∞ βm = α, hence (1.5) provides almost optimal condition as m → ∞. In [9], a limit case m → ∞ is studied to consider a(t) in the Gevrey class, which is a subclass of C ∞ .

(GEC)

β<α No (iii)

Table 1: α ≥ 0 α ≤ β < β m βm ≤ β < 1 ? Yes (iii)

β=1 Yes (ii)

1<β Yes (i)

(i) Trivial by classical energy method. (ii) Theorem 1.1 [13] (α=1). (iii) Theorem 1.1 [7] (0 ≤ α < 1, m ≥ 2).

Taking into account Theorem 1.1 and the remarks above, the main purposes of this article are extending Theorem 1.1 from the following points of view: (P1) Introduce a new stabilization property in order to distinguish the class of functions a(t) satisfying (1.3) with α = 0. (P2) Remove the assumption (1.2) and consider non-strictly hyperbolic models.

2

(P3) Consider the following estimate for the boundedness of the energy: E(t) ≤ C,

(BE)

which is a weaker estimate than (GEC), where C may depend not only the initial energy E(0) but also the other properties of the initial data. Then, we expect that the restrictions to the coefficient for (BE) will be weakened than the restrictions for (GEC). At the end of this section we introduce the following notations. Let f, g : Ω → R be two strictly positive functions. We denote f  g if f (y) ≤ Cg(y) for all y ∈ Ω. Moreover, we denote f  g if both of the estimates f  g and g  f hold.

2

Main theorems

The stabilization property (1.3) is the most restricted for α = 0, and all the functions a(t) satisfying a(t) − a∞ ∈ L1 (R+ )

(2.1)

with some constants a∞ satisfy (1.3) for α = 0. Let us extend the stabilization property for α < 0 in order to classify the functions a(t) satisfying (2.1) as follows:  ∞ |a(s) − a∞ | ds ≤ C0 (1 + t)α . (2.2) t

Then Theorem 1.1 can be extended naturally as the following corollary of Theorems 2.4 and 2.7, which will be introduced below. Corollary 2.1. Suppose that (1.2), (1.4), (2.1) and (2.2) are valid. If α, β and m satisfy (1.5), then (GEC) is established. If β < α, then (GEC) does not hold in general. In Theorem 1.1, the minimum choice of β for (GEC) is β = 1/m; thus β must be positive. However, β can be taken as a non-positive number for (GEC) by Corollary 2.1 if (2.2) holds for α ≤ −1/(m − 1). In the next example, we cannot conclude (GEC) from Theorem 1.1, but we can do it by using Corollary 2.1. Example 2.2. Let κ < 0, m ≥ 2 and χ ∈ C m (R) be a 1-periodic function satisfying 0 ≤ χ(τ ) ≤ 1 and χ(τ ) ≡ 0 near τ = 0.

(2.3)

We define a(t) by a(t) := 1 + (1 + j)−1+κ χ



  (1 + j)−κ (t − j) , t ∈ [j, j + 1) for j = 0, 1, · · · ,

(2.4)

where [·] denotes Gauss’ symbol. Then for any t ∈ [j, j + 1) (j = 0, 1, · · · ) we have the following estimates: 

 1−κ 1−κ  (k)  max (2.5) a (t)  (1 + j)−k(κ+ k )  (1 + j)−k(κ+ m ) (k = 1, · · · , m), t∈[j,j+1]



∞

|a(s) − 1| ds 

t

and

∞ 

(1 + l)−1+κ  (1 + j)κ  (1 + t)

κ

(2.6)

l=j

 0

t

|a(s) − 1| ds 

j 

0

(1 + l)−1+κ  1 = (1 + t) .

(2.7)

l=0

Indeed, (GEC) is not concluded by Theorem 1.1 with (2.5) and (2.7), because the condition (1.5): β = κ + (1 − κ)/m ≥ 1/m(= α + (1 − α)/m |α=0 ) does not hold. On the other hand, by Corollary 2.1 with (2.5) and (2.6) we have (GEC) because of the equalities β = κ + (1 − κ)/m = α + (1 − α)/m |α=κ . Here we note that a (t) ∈ L1 (R+ ); indeed we have 

t 0



|a (s)| ds 

j 

(1 + l)−1 → ∞ (j → ∞).

l=1

3

The energy estimate (BE) is trivial if (GEC) holds; however, the converse is not always true if the constant C of (BE) depends not only on the initial energy E(0) but also the other properties of the initial data. Here we introduce the Gevrey class for the initial data in order to realize the assertion (P3). For ν > 1 and s ≥ 0 we define the Gevrey class γsν by    2   1   γsν := exp 2ρ|ξ| ν |ξ|2s fˆ(ξ) dξ < ∞ , (2.8) f ∈ H˙ s ; Rn

ρ>0

where fˆ(ξ) denotes the Fourier transformation of f (x) with respect to x. Here we note that f ∈ γsν if and only if there exist positive constants ρ and C such that ∂ h f H˙ s ≤ Cρ|h| |h|!ν for any h ∈ (N ∪ {0})n , where h = (h1 , · · · , hn ) and |h| = h1 + · · · + hn (see [15] for more detail of the Gevrey classes). Therefore, if (u0 , u1 ) ∈ γ1ν × γ0ν , then E(0) < ∞, and the converse is not true in general. The following corollary, which follows from Theorems 2.4 and 2.7, is a concrete answer to (P1): Corollary 2.3. Suppose that (1.2), (1.4), (2.1) and (2.2) are valid. If (u0 , u1 ) ∈ γ1ν × γ0ν , α, β and m satisfy β > β˜m :=

1−α αν + , ν−1 m

(2.9)

then (BE) is established. If β < αν/(ν − 1), then (BE) does not hold in general. Noting β˜m < βm , if (u0 , u1 ) ∈ γ1ν × γ0ν , then we observe from Corollaries 2.1 and 2.3 the followings: • If β˜m < β < βm , then Corollary 2.1 cannot give any answer for (GEC) and (BE), but Corollary 2.3 implies that (BE) is valid. • If β˜m < β < α, then (GEC) is not true by Corollary 2.1, but (BE) is valid by Corollary 2.3. More precisely, the conclusions of Corollaries 2.1 and 2.3 are summarized as the following tables:

αν ν−1

(GEC) (BE)

αν β < ν−1 No (iv) No (v)

αν ν−1

(GEC) (BE)

αν β < ν−1 No (iv) No (v)

Table 2: α < 0 ≤ β ≤ β˜m β˜m < β < α No (iv) No (iv) ? Yes (v)

≤β<α No (iv) ?

α ≤ β ≤ β˜m ? ?

α ≤ β < βm ? Yes (v)

βm ≤ β Yes (iv) Yes (iv)(v)

β˜m < β < βm ? Yes (v)

βm ≤ β Yes (iv) Yes (iv)(v)

(iv) Corollary 2.1. (v) Corollary 2.3.

Let us consider the generalization of Corollaries 2.1 and 2.3. If limt→∞ a(t) = ∞, then the energy estimates for (1.1) (not only L2 -L2 type but also Lp -Lq type) were studied in [14, 16] without considering any stabilization property. A generalization of the stabilization property (1.3) for unbounded propagation speed a(t) was introduced in [10, 12] by  t |a(s) − λ(s)| ds ≤ Ξ(t), (2.10) 0

where λ(t) and Ξ(t) are monotone increasing functions satisfying λ(t)  a(t). In this way, for monotone decreasing functions λ(t) and Ξ(t) satisfying λ(t) ∈ C 1 (R+ ), a(t) − λ(t) ∈ L1 (R+ ), a0 λ(t) ≤ a(t) ≤ a1 λ(t) for some positive constants a0 and a1 , we introduce the following generalization of (2.2):  ∞ |a(s) − λ(s)| ds ≤ Ξ(t). t

4

(2.11)

(2.12)

Moreover, for a non-negative function η(t) we introduce the following condition:    (k)  a (t) ≤ αk λ(t)η(t)k (k = 1, . . . , m)

(2.13)

as a generalization of (1.4). Then one of our main theorems is given as follows: Theorem 2.4. Suppose that (2.11), (2.12) and (2.13) are valid. If there exist positive constants K0 and K1 such that η(t)Ξ(t) ≤ K0 (2.14) λ(t) and





t

Ξ(t)m−1

λ(s) 0

η(s) λ(s)

m ds ≤ K1 ,

(2.15)

then the following estimates are established: C −1 λ(t)2 E(0) ≤ E(t) ≤ CE(0).

(GEC )

˜ 0 and K ˜ 1 such that In particular, if (u0 , u1 ) ∈ γ1ν × γ0ν and there exist positive constants K η(t)Ξ(t)κ ˜0 ≤K λ(t) and

 Ξ(t)



t

κm−1

λ(s) 0

for a κ satisfying κ>

η(s) λ(s)

m

(2.16)

˜1 ≤K

(2.17)

ν , ν−1

(2.18)

then (BE) is established. Remark 2.5. Generally, we cannot expect (GEC) but (GEC ) if (1.2) is not valid, that is, limt→∞ a(t) = 0. (See Concluding remark.) Remark 2.6. Corollaries 2.1 and 2.3 are special cases of Theorem 2.4 for η(t) = (1 + t)−β , Ξ(t) = C0 (1 + t)α and λ(t) = a∞ > 0. Then we see that the estimates (2.14) and (2.16) correspond to β ≥ α and β > αν/(ν − 1), respectively. In the next result about optimality, we shall restrict ourselves to consider the following conditions: Ξ(t) ≤ C0 (1 + t)α , η(t) ≤ (1 + t)−β , λ(t) ≥ (1 + t)−δ (α < 0, δ ≥ 0).

(2.19)

Here we can suppose that the following inequality holds: α < −δ + 1.

(2.20)

Otherwise, (2.12) is a trivial condition by (2.11). Then, (2.14) and (2.15) are valid for β ≥ β∞,δ := α + δ and β ≥ βm,δ := α + δ +

1−α−δ , m

respectively. Noting βm,δ > β∞,δ and limm→∞ βm,δ = β∞,δ , the condition (2.15) approaches (2.14) as m → ∞. Analogously, (2.16) and (2.17) are valid for β > β˜∞,δ :=

1−α−δ αν αν + δ and β > β˜m,δ := +δ+ , ν−1 ν−1 m

respectively. Moreover, we see that β˜m,δ > β˜∞,δ and limm→∞ β˜m,δ = β˜∞,δ . The following theorem ensures the optimality of the conditions (2.14) and (2.16), or the conditions (2.15) and (2.17) approach the optimal ones as m → ∞.

5

Theorem 2.7. Let n = 1 and ν > 1. For any β ∈ R satisfying β < min{0, β∞,δ },

(2.21)

∞ ∞ there exist {aj (t)}∞ j=1 and {(uj,0 , uj,1 )}j=1 such that aj (t) ∈ C (R+ ), (2.11), (2.12) and (2.13) hold with aj (t) = a(t) for some λ(t), η(t) and Ξ(t) satisfying (2.19), (2.20),

E(0; uj,0 , uj,1 ) ≤ 1

(2.22)

lim sup E(t; uj,0 , uj,1 ) = ∞.

(2.23)



β < min 0, β˜∞,δ ,

(2.24)

and j→∞ t>0

Moreover, if

then (2.23) is established though the following estimate is valid:

     1 2 2 2 2 ν sup aj (0) |ξ| |ˆ exp 2ρ|ξ| uj,0 (ξ)| + |ˆ uj,1 (ξ)| dξ ≤ 1 j

(2.25)

R

for a positive constant ρ. Remark 2.8. If β ≥ βm,δ , then (GEC ) is valid by Theorem 2.4; thus (2.23) is not realized since (2.22) holds. Analogously, if β > β˜m,δ , then (2.23) is not true since (2.25) holds. It is open problems whether (GEC ) and (BE) hold or not if β∞,δ ≤ β < βm,δ and β˜∞,δ ≤ β ≤ β˜m,δ , respectively. No (GEC )

d β∞,δ

 t (GEC ) -

No (BE)

d β˜∞,δ

βm,δ

d (BE)

-

β˜m,δ

Example 2.9. Let γ, δ, ε and κ be constants satisfying γ > 1, ε ≥ 0, κ < 0 and max{0, −1 + κ + γ} ≤ δ ≤ γ. Denoting ρj := [(1 + j)−κ−ε ]/[(1 + j)−κ ], we define a(t) by  (1 + t)−δ + (1 + j)−γ χ ([(1 + j)−κ ] (t − j)) t ∈ [j, j + ρj ), a(t) := (2.26) t ∈ [j + ρj , j + 1) (1 + t)−δ for j = 0, 1, · · · , where χ ∈ C m (R) with m ≥ 2 is defined by (2.3). Setting λ(t) = (1 + t)−δ , for t ∈ [j, j + 1) we have 

 γ−δ  (k)  max a (t)  (1 + t)−δ−k + (1 + j)−γ−kκ  (1 + t)−δ−k(κ+ m ) (k = 1, · · · , m), t∈[j,j+1]

and

 t

∞

|a(s) − λ(s)| ds 

∞ 

(1 + l)−γ−ε  (1 + j)−γ−ε+1  (1 + t)

−γ−ε+1

.

l=j

Therefore, (2.26) is an example satisfying (2.19) with α = −γ − ε + 1 and β = κ + (γ − δ)/m. It is common to introduce the Gevrey class on the well-posedness for the Cauchy problem with singular propagation speed, where a(t) is singular if it is non-Lipschitz continuous or having zeros. In [4], which is a pioneer work for this kind of problem, it is proved that if a(t) > 0 and a(t) ∈ C σ (R+ ) with σ ∈ (0, 1), where C σ (R+ ) denotes the class of H¨older continuous functions on R+ , then (1.1) is well-posed in the Gevrey class of order ν with ν < 1/(1 − σ). After that, the relations between various types of singularities of a(t) and the Gevrey order ν for the well-posedness of (1.1) was studied in many papers, for instance [1, 3, 5, 8, 11]. In particular, a sort of stabilization properties corresponding to (1.3) and (2.10) are introduced in [1, 8, 11]. As a consequence of these results, for any fixed T > 0 there exists a constant CT such that the energy estimate E(t) ≤ CT is valid for any t ∈ [0, T ] since (u0 , u1 ) ∈ γ1ν × γ0ν , where ν is determined by the corresponding singularities of a(t). However, CT is not bounded with respect to T , thus the estimate is different essentially from (BE). The estimate (BE) for singular propagation speed is studied in [2, 6]. However, their arguments are essentially based on the previous works, and our results cannot be derived from them. Indeed, the assumption a(t) ∈ C m (R+ ) with m ≥ 2 is essentially used for the proof. As mentioned above, the essential contribution of the Gevrey class for the energy estimate (BE) has never discussed in the previous works, thus our main theorems propose a new point of view on the research for (1.1).

6

3

Proof of Theorem 2.4

Denoting v(t, ξ) := u ˆ(t, ξ), (1.1) is reduced to the following problem:   ∂t2 + a(t)2 |ξ|2 v = 0, (t, ξ) ∈ R+ × Rn , u0 (ξ), u ˆ1 (ξ)), ξ ∈ Rn . (v(0, ξ), vt (0, ξ)) = (ˆ

(3.1)

We define the energy density function E(t, ξ), and E0 (t, ξ) by E(t, ξ) :=

 1 a(t)2 |ξ|2 |v(t, ξ)|2 + |∂t v(t, ξ)|2 , 2

E0 (t, ξ) :=

 1 λ(t)2 |ξ|2 |v(t, ξ)|2 + |∂t v(t, ξ)|2 . 2

and

Then, by (2.11) we have

(3.2)

a20 E0 (t, ξ) ≤ E(t, ξ) ≤ a21 E0 (t, ξ).

(3.3)

Noting (2.11), we have the following estimates: 

 λ(t)2 − a(t)2 |ξ|

{λ(t)|ξ|v(t, ξ)vt (t, ξ)} ∂t E0 (t, ξ) =λ (t)λ(t)|ξ| |v(t, ξ)| + λ(t) ≤(1 + a1 )|a(t) − λ(t)||ξ|E0 (t, ξ) 

2

and

 ∂t E0 (t, ξ) ≥

2

 2λ (t) − (1 + a1 )|a(t) − λ(t)||ξ| E0 (t, ξ). λ(t)

Therefore, by Gronwall’s inequality and (2.12) we have 

λ(t) λ(τ0 )

2 exp (−(1 + a1 )Ξ(τ0 )|ξ|) E0 (τ0 , ξ) ≤ E0 (t, ξ) ≤ exp ((1 + a1 )Ξ(τ0 )|ξ|) E0 (τ0 , ξ)

(3.4)

for any 0 ≤ τ0 < t. For a large constant N to be chosen later let us define tξ by tξ := min {t ∈ R+ ; Ξ(t)|ξ| ≤ N } .

(3.5)

Moreover, for a positive constant ρ, which provides the estimate    1 exp 2ρ|ξ| ν E(0, ξ) dξ < ∞ Rn

for (u0 , u1 ) ∈ γ1ν × γ0ν , we define t˜ξ by

t˜ξ := min t ∈ R+ ; Ξ(t)|ξ| ≤ N +

1 ρ |ξ| ν 1 + a1

 .

(3.6)

Then we define the pseudo-differential zones ZΨ and Z˜Ψ by

and

ZΨ := {(t, ξ) ∈ R+ × Rn ; t ≥ tξ }

(3.7)

  Z˜Ψ := (t, ξ) ∈ R+ × Rn ; t ≥ t˜ξ ,

(3.8)

respectively. Thus we have the following lemma by (3.3) and (3.4):

7

Lemma 3.1. The following estimates are established: 1 λ(t)2 E(tξ , ξ) ≤ E(t, ξ) ≤ K2 E(tξ , ξ) in ZΨ K2 λ(tξ )2 and

  1 E(t, ξ) ≤ K2 exp ρ|ξ| ν E(t˜ξ , ξ) in Z˜Ψ ,

(3.9)

(3.10)

2 N (1+a1 ) where K2 = a−2 0 a1 e

We define the hyperbolic zones ZH and Z˜H satisfying ZH ∪ ZΨ = Z˜H ∪ Z˜Ψ = R+ × Rnξ by

and

ZH := {(t, ξ) ∈ R+ × Rn ; t ≤ tξ }

(3.11)

  Z˜H := (t, ξ) ∈ R+ × Rn ; t ≤ t˜ξ .

(3.12)

Let (t, ξ) ∈ ZH . The equation of (3.1) is reduced to the following first order system: ∂t V1 = A1 V1 , where

(3.13)





 vt + ia(t)|ξ|v , vt − ia(t)|ξ|v

V1 =

b1 = b1 = − We denote

a (t) 2a(t)

φ1 b1

A1 =

 b1 , φ1

a (t) + ia(t)|ξ|. 2a(t)

and φ1 =

(3.14)

 d log a(t), φ1 = {φ1 } = a(t)|ξ|, dt  λ1 = φ1 + i φ21 − |b1 |2

φ1 = {φ1 } =

and λ1 − φ1 φ1 θ1 = = −i b1 b1



 1−

|b1 |2 1− 2 φ1

 .

Here we note that {λ1 , λ1 } and {t (1, θ1 ), t (θ1 , 1)} are the eigenvalues of A1 and their corresponding eigenvectors, respectively. Therefore, if |θ1 | < 1, then A1 is diagonalized by the diagonalizer Θ1 := (t (1, θ1 ) t (θ1 , 1)) as follows:   λ1 0 A Θ = . Θ−1 1 1 1 0 λ1 −1 −1 We define V2 := Θ−1 1 V1 . Noting Θ1 (∂t Θ1 ) = −(∂t Θ1 )Θ1 , V2 is a solution to the following system:

∂t V2 = A2 V2 , where



0 λ1

λ1 0

A2 =

b2 = −

 −

Θ−1 1 (∂t Θ1 )

(3.15)  =

φ2 b2

 b2 , φ2

(θ1 )t , φ2 = φ2 + iφ2 , 1 − |θ1 |2

φ2 =φ1 − ∂t log and φ2 =



1 − |θ1 |2

   φ21 − |b1 |2 −  θ1 b2 .

Generally, we have the following lemma:

8

Lemma 3.2. Let Vk be a solution to the following system: ∂t Vk = Ak Vk ,



Ak =

and the matrix Θk be defined by  Θk =

 θk , 1

1 θk

φk θk = −i bk

φk bk

 bk , φk 

 1−

|bk |2 1− 2 φk

 ,

where φk = {φk } and φk = {φk }. If |θk | < 1, then Vk+1 = Θ−1 k Vk is a solution to the following system: ∂t Vk+1 = Ak+1 Vk+1 , where

 Ak+1 =

and φk+1 = φk − ∂t log



φk+1 bk+1

 bk+1 , φk+1

bk+1 = −

(θk )t 1 − |θk |2

   1 − |θk |2 + i −φk + φ2k − |bk |2 − {θk bk+1 } .

(3.16)

Proof. The proof is straightforward. Denoting Vm = t (Vm,1 , Vm,2 ), by Lemma 3.2 we have ∂t |Vm |2 =2 (Am Vm , Vm )C2 = 2φm |Vm |2 + 4 {bm Vm,1 Vm,2 } ≤2 (φm + |bm |) |Vm |2 and ∂t |Vm |2 ≥ 2 (φm − |bm |) |Vm |2 , it follows that

⎧    ⎨≤ exp 2 t (φm (s, ξ) + |bm (s, ξ)|) ds |Vm (0, ξ)|2 , 0    |Vm (t, ξ)| ⎩≥ exp 2 t (φm (s, ξ) − |bm (s, ξ)|) ds |Vm (0, ξ)|2 0 2

(3.17)

in ZH and Z˜H . Let us introduce some symbol classes restricted in ZH in order to show |θk | < 1 and estimate the right hand sides of (3.17). For integers p ≥ 0, q and r the symbol class S (p) {q, r} is the set of functions satisfying  k  ∂t f (t, ξ) ≤ Ck (λ(t)|ξ|)q η(t)r+k (3.18) for k = 0, . . . , p in ZH . Then the following properties of the symbol classes are established: Lemma 3.3. (S1) If f ∈ S (p) {q, r} for p ≥ 1, then ∂t f ∈ S (p−1) {q, r + 1}. (S2) If f ∈ S (p) {q, r}, then f ∈ S (p) {q + 1, r − 1}. (S3) If f ∈ S (p) {−q, q} for q ≥ 1, then N q |f | ≤ C0 K0q . (S4) If fj ∈ S (pj ) {q, r} (j = 1, 2), then f1 + f2 ∈ S (min{p1 ,p2 }) {q, r}. (S5) If fj ∈ S (pj ) {qj , rj } (j = 1, 2), then f1 f2 ∈ S (min{p1 ,p2 }) {q1 + q2 , r1 + r2 }. Proof. (S1) and (S4) are trivial from the definition of the symbol class. Noting (2.14) and the estimate Ξ(t)|ξ| ≥ N in ZH , (S2) and (S3) are also trivial. (S5) is immediately proved by Leibniz rule as follows: k    k   k  j   k−j  q +q ∂t (f1 f2 ) ≤ ∂t f1  ∂t f2   (λ(t)|ξ|) 1 2 η(t)r1 +r2 +k j j=0

for any k = 0, . . . , min{p1 , p2 }.

9

Moreover, we have the following properties: Lemma 3.4. The following properties are established: (S6) If f ∈ S (p) {−q, q} for q ≥ 1, then there exists N0 > 0 and g ∈ S (p) {−q, q} such that 2(1 − f (1 + g) for any N ≥ N0 .

√

1 − f) =

(S7) If f1 ∈ S (p1 ) {q1 , r1 } and f2 ∈ S (p2 ) {−q2 , q2 } for q2 ≥ 1, then there exists N0 > 0 such that f1 /(1 − f2 ) ∈ S (min{p1 ,p2 }) {q1 , r1 } for any N ≥ N0 . Proof. Let us prove (S6). By (S3) and choosing N0 large, we can suppose that |f | < 1 for any N ≥ N0 . Then, by Taylor expansion we have 1 ∞ 1 ∞  1     2 (−f )j = 1 − f − f 2 (−f )l , 1 − f = 1 + 2 (−f ) + 1 j 2 l + 1 j=2 l=1

thus g is given by g=2

∞   l=1



1 2

l+1

(−f )l .

Let us estimate ∂tk f l focusing in the dependence of l. We define q  η(t) ζ = ζ(t, ξ) := λ(t)|ξ| and note the following representation due to Faa’ di Bruno’s formula: ∂tk f l =k!

k  h=1

k   l! (∂tj f )hj f l−h . (l − h)! h !j!hj j=1 j h∈Λh,k

where Λh,k := {h = (h1 , · · · , hk ) ∈ (N ∪ {0}) ; h1 + · · · + hk = h, 1h1 + · · · + khk = k}. Noting the following estimate with a positive constant Cq : k

k  k     h j  j hj h Cq ζη j = (Cq ζ) η k , ∂t f  ≤ j=1

j=1

we have k   (Cq ζ)h η k  k l ∂t f  ≤k! lh (Cp ζ)l−h . hj !j!hj h∈Λh,k

h=1

≤η k (Cp ζ)l lk k!

k 



1

h=1 h∈Λh,k

≤C˜p η k (Cp ζ)l lk , k  where C˜p = max1≤k≤p k! h=1 h∈Λh,k 1. By choosing N0 large enough, we have (Cp ζ)l−1 lk ≤ (1/2)l for any  1  2 N ≥ N0 . Consequently, noting | l+1 | ≤ 1/2, we have   ∞  1  ∞  l    k l  k  1 k 2 ∂t f  ≤ C˜p Cp ζη 1 + ∂t g  ≤2 l+1 2 l=1

l=1

≤2C˜p Cp (λ(t)|ξ|)−q η(t)q+k for any k = 0, · · · , p. Thus the proof of (S6) is concluded. (S7) can be proved on the analogy of the proof of (S7) to the expansion ∞  f1 = f1 f2l 1 − f2 l=0

for |f2 | < 1 since N  1.

10

By using the properties (S1)-(S7) we shall prove the following lemma: Lemma 3.5. There exists a positive constant N such that |θk | ≤ 1/2 for k = 1, . . . m and bm ∈ S (0) {−m+1, m}. Proof. Firstly we show the following estimates:    1 (k+1)  η(t)k+1   (k = 0, . . . , m − 1),    a(t)  λ(t)

(3.19)

j

d where (1/a)(j) = dt j (1/a). If k = 0, then (3.19) is trivial. Suppose that (3.19) is valid for k = 1, . . . , j. Noting the representations:

dj+1 0 = j+1 dt



1 a· a



  (j+1)   (j−l+1) j+1  1 j + 1 (l) 1 a =a + , l a a l=1

we have        j+1  (j−l+1)   1 (j+1)  1  j + 1  (l)   1    η j+1 , a    ≤   a  a   a l λ l=1

thus (3.19) is valid for any k = 0, . . . , m − 1. Therefore, for f ∈ S (p) {q, r} we have        k   (j)    k f  f (k)  k  (k−j)   1  −1 ∂t ≤ + |ξ| f     (λ|ξ|)q−1 η r+k .  φ1   a  j a|ξ| j=1 Thus we have the following property: (S8) If f ∈ S (p) {q, r}, then f /φ1 ∈ S (p) {q − 1, r}. By applying (S6) for f = |b1 |2 /φ21 ∈ S (m−1) {−2, 2}, which follows from (S5) and (S8), there exists g1 ∈ S (m−1) {−2, 2} such that    b1 φ1 |b1 |2 θ1 = −i 1− 1− 2 =− (1 + g1 ) ∈ S (m−1) {−1, 1}. (3.20) φ1 2φ1 b1 Therefore, by (S3) we have |θ1 | ≤ 1/2 for any sufficiently large N . Moreover, by (S1) and (S7) we have b2 ∈ S (m−2) {−1, 2}.

(3.21)

By (3.20), (3.21), (S4)-(S6) and (S8) we have     θ 1 b2 |b1 |2 1− 1− 2 + ∈ S (m−2) {−2, 2}. φ1 φ1 Therefore, by (3.21), (S5), (S7) and (S8) we have b2 b2 = φ2 φ1 and thus obtain

 1− 1−



1 1−

|b1 |2 φ21

+

{θ1 b2 } φ1

θ2 ∈ S (m−2) {−2, 2}

 ∈ S (m−2) {−2, 2},

(3.22)

(3.23)

by using the same arguments to show (3.20). Moreover, by (S3) we have |θ2 | ≤ 1/2 for any sufficiently large N . On the analogy of the arguments above, we can prove bk ∈ S (m−k) {−k + 1, k}, θk ∈ S (m−k) {−k, k} and |θk | ≤ 1/2 for k = 3, . . . , m.

11

The properties (S1)-(S8) except for (S3) are valid in Z˜H under the assumption (2.16), and the estimate of (S3) is changed into N qν/(ν−1) |f |  1. By (2.15), (2.17), Lemma 3.5 and the representation (3.16), there exists N  1 and Cm such that m   t  t η(s) 2|bm (s, ξ)| ds ≤2Cm |ξ|−m+1 λ(s) ds λ(s) 0 0 m    m−1 t η(s) ≤2Cm N −1 Ξ(t) λ(s) ds λ(s) 0 ≤2Cm K1 N −m+1

and 



t 0

2φm (s, ξ) ds =

0



t

∂s

log a(s) −

m−1 

   2 ds log 1 − |θk (s, ξ)|

k=1  m−1  1 − |θk (0, ξ)|2 k=1 = log m−1 a(0) k=1 (1 − |θk (t, ξ)|2 ) ⎧    ⎨≤ log 4 m−1 a1 λ(t) 3 a λ(0) 0    ⎩≥ log 3 m−1 a0 λ(t) 4 a1 λ(0)

a(t)

in ZH . Therefore, by (3.17) we have   m−1 ≤ 43 2  m−1 |Vm (t, ξ)| ≥ 34

a1 a0 a0 a1

 λ(t)  exp 2Cm K1 N −m+1 λ(0) |Vm (0, ξ)|2 ,   λ(t) exp −2Cm K1 N −m+1 λ(0) |Vm (0, ξ)|2 .

By Lemmas 3.2 and 3.5, we have   |Vk |2 = |Θk Vk+1 |2 = 1 + |θk |2 |Vk+1 |2 + 4 {θk Vk+1,1 Vk+1,2 }  2 ≤ (1 + |θk |) |Vk+1 |2 ≤ 94 |Vk+1 |2 , 2 ≥ (1 − |θk |) |Vk+1 |2 ≥ 14 |Vk+1 |2 , it follows that

 m−1  m−1 9 1 2 2 |Vm | ≤ |V1 | ≤ |Vm |2 . 4 4

uniformly in ZH . Consequently, noting   |V1 (t, ξ)|2 = 2 a(t)2 |ξ|2 |v(t, ξ)|2 + |vt (t, ξ)|2 = 4E(t, ξ), we have  m−1 1 9 |Vm (t, ξ)|2 4 4  λ(t)  3m−1 a1 |Vm (0, ξ)|2 ≤ exp 2Cm K1 N −m+1 4a0 λ(0)   12m−1 a1 exp 2Cm K1 N −m+1 λ(t) |V1 (0, ξ)|2 ≤ 4a0 λ(0)   12m−1 a1 exp 2Cm K1 N −m+1 λ(t) E(0, ξ). = a0 λ(0)

E(t, ξ) ≤

By the same way, we have E(t, ξ) ≥

12m−1 a

a0 λ(t) E(0, ξ). −m+1 ) λ(0) 1 exp (2Cm K1 N

12

Let us consider the estimate in Z˜H . We define τ˜ by

ν νm νm −1 − ν−1 ˜ 1 (1 + a1 ) ν−1 τ˜ := max t > 0 ; 2Cm K ρ Ξ(t)−m(κ− ν−1 ) . By (2.11)-(2.13) we have ∂t E(t, ξ) ≤

2α1 η(t) 2α1 η˜1 E(t, ξ) ≤ E(t, ξ) a0 a0

for 0 ≤ t ≤ τ˜, where η˜1 = max0≤t≤˜τ {η(t)}, it follows that   2α1 η˜1 E(0, ξ). E(t, ξ) ≤ exp a0

(3.24)

Let τ˜ ≤ t ≤ t˜ξ . Noting the estimate |ξ| ≥ (ρ−1 (1 + a1 )Ξ(t))−ν/(ν−1) and (2.17), we have m   t  t η(s) 2|bm (s, ξ)| ds ≤2Cm |ξ|−m+1 λ(s) ds λ(s) τ˜ 0 ˜ 1 Ξ(t)−κm+1 |ξ|−m+1 ≤2Cm K  νm −1  ν 1 1 + a1 ν−1 ˜ ≤2Cm K1 Ξ(t)−m(κ− ν−1 ) |ξ| ν ρ 1

≤ρ|ξ| ν

by (3.24). Then, by the same way in the estimates of ZH we have  m−1 1 9 |Vm (t, ξ)|2 E(t, ξ) ≤ 4 4   1 3m−1 a1 λ(t) exp ρ|ξ| ν |Vm (˜ ≤ τ , ξ)|2 4a0 λ(˜ τ)   1 12m−1 a1 ≤ exp ρ|ξ| ν E(˜ τ , ξ) a0     1 2α1 η˜1 12m−1 a1 exp ρ|ξ| ν E(0, ξ). exp ≤ a0 a0 Summarizing the estimates above we have the following lemma: Lemma 3.6. The following estimates are established: K3−1

λ(t) λ(t) E(0, ξ) ≤ E(t, ξ) ≤ K3 E(0, ξ) in ZH λ(0) λ(0)

  ˜ 3 exp ρ|ξ| ν1 E(0, ξ) in Z˜H , E(t, ξ) ≤ K   −m+1 ˜ 3 = 12m−1 a−1 a1 exp(2a−1 α1 η˜1 ). and K where K3 = 12m−1 a−1 0 a1 exp 2Cm K1 N 0 0

(3.25)

and

(3.26)

Proof of Theorem 2.4. If (t, ξ) ∈ ZH , then by Lemma 3.6 we have 1 λ(t)2 E(0, ξ) ≤ E(t, ξ) ≤ K3 E(0, ξ). K3 λ(0)2 If (t, ξ) ∈ {(t, ξ) ; |ξ| ≤ N/Ξ(0)} ⊂ ZΨ , and thus tξ = 0 by (3.5), then by Lemma 3.1 we have 1 λ(t)2 E(0, ξ) ≤ E(t, ξ) ≤ K2 E(0, ξ). K2 λ(0)2 By Lemma 3.6 we have 1 λ(tξ ) λ(tξ ) E(0, ξ) ≤ E(tξ , ξ) ≤ K3 E(0, ξ). K3 λ(0) λ(0) 13

(3.27)

Therefore, if (t, ξ) ∈ ZΨ ∩ {(t, ξ) ; |ξ| ≥ N/Ξ(0)}, then by Lemma 3.1 and (3.27) we have E(t, ξ) ≤ K2 K3

λ(tξ ) E(0, ξ) ≤ K2 K3 E(0, ξ) λ(0)

and E(t, ξ) ≥

λ(t)2 λ(t)2 1 E(0, ξ) ≥ E(0, ξ). K2 K3 λ(0)λ(tξ ) K2 K3 λ(0)2

Summarizing the estimates above and Parseval’s theorem, we have λ(t)2 E(0) ≤ E(t) ≤ K2 K3 E(0). K2 K3 λ(0)2 By the same way we conclude (BE) as follows:  ˜3 E(t) ≤ K2 K

Rn

  1 exp 2ρ|ξ| ν E(0, ξ) dξ < ∞

since (3.2) holds. Thus the proof of Theorem 2.4 is concluded.

4

Proof of Theorem 2.7

We shall prove Theorem 2.7 by constructing concrete examples of the series of the coefficients {aj (t)} and the initial data {(uj,0 , uj,1 )} making use of the ideas in [4, 7]. Let ϕ ∈ C ∞ (R) be a 2π-periodic function satisfying  2π ϕ(τ ) cos2 τ dτ = π. (4.1) ϕ ≥ 0, ϕ(τ ) ≡ 0 near τ = 0 and 0

We define the 2π-periodic function ψ(τ ) by ψ(τ ) = ψ(τ ; ε) := 1 + 4εϕ(τ ) sin(2τ ) − 2εϕ (τ ) cos2 τ − 4ε2 ϕ(τ )2 cos4 τ, where ε is a positive constant providing

3 1  ≤ ψ(τ ; ε) ≤ . 2 2

(4.2)

(4.3)

Then we have the following lemma: Lemma 4.1. The solution to the initial value problem w + ψ(τ ; ε)w = 0, w(0) = 1, w (0) = 0 is represented by

  w(τ ) = w(τ ; ε) = exp 2ε

τ

 ϕ(s) cos2 s ds cos τ.

(4.4)

(4.5)

0

Proof. The proof is straightforward. For α, β, δ ∈ R satisfying α < 0, β < 0, δ ≥ 0, (2.20) and (2.21) we define tj by 1

tj := (2πj) α−β+δ .

(4.6)

Here we note that {tj }∞ j=0 is a strictly increasing sequence satisfying limj→∞ tj = ∞. Moreover, for j = 1, 2, . . . we define λj , ξj and ρj by −β+δ and ρj := tα+δ . λj := t−δ j , ξj := tj j For χ ∈ C ∞ (R) satisfying χ ≤ 0, χ(k) (τ ) ≡ 0 near τ ∈ {0, 1} for any k ∈ N, χ(τ ) = 1 for τ ≤ 0 and χ(τ ) = 0 for τ ≥ 1, we define Λ ∈ C ∞ (R+ ) by   t − tl for t ∈ [tl , tl+1 ) and l = 0, 1, . . . , (4.7) Λ(t) := λl+1 + (λl − λl+1 )χ tl+1 − tl 14

where λ0 = λ1 . Then we see that Λ (t) ≤ 0, Λ(k) (t) = 0 for t ∈ [0, t1 ) and   λ − λ      (k)  l l+1 −α+β−2δ−(1−α+β−δ)k ≤ Ck t l ≤ Ck t−δ−βk Λ (t) ≤ max χ(k) (τ ) l k 0≤τ ≤1 (tl+1 − tl ) for t ∈ [tl , tl+1 ), l = 1, 2, . . .. Let us define aj (t) and λ(t) by ⎧ ⎪ ⎨Λ(t),  aj (t) := λj ψ (λj ξj (t − tj )), ⎪ ⎩ λj , 

and

Λ(t), λj ,

λ(t) :=

t ∈ [0, tj ), t ∈ [tj , tj + ρj ), t ∈ [tj + ρj , ∞),

t ∈ [0, tj ), t ∈ [tj , ∞).

(4.8)

(4.9)

Then we see that aj (t) ∈ C ∞ (R+ ) and (2.11) are valid. Moreover, noting the estimates    (k)  −δ−βk aj (t) ≤ Ck λj (λj ξj )k = Ck tj for t ∈ [tj , tj + ρj ), we have ⎧ ⎪ ⎪0,   ⎪ ⎨C t−δ−βk ,  (k)  k l aj (t) ≤ −δ−βk ⎪ C , k tj ⎪ ⎪ ⎩ 0,

t ∈ [0, t1 ), t ∈ [tl , tl+1 ), l = 1, . . . , j − 1, t ∈ [tj , tj + ρj ), t ∈ [tj + ρj , ∞)

for k = 1, 2, . . .. It follows that a(t) = aj (t) satisfy (2.12) and (2.13) uniformly with respect to j for  t +ρ j j |aj (s) − λj | ds  λj ρj = tα j , t ∈ [0, tj + ρj ), tj Ξ(t) = 0, t ∈ [tj + ρj , ∞), and

⎧ 0, ⎪ ⎪ ⎪ ⎨t−β , l η(t) = ⎪ t−β ⎪ j , ⎪ ⎩ 0,

t ∈ [0, t1 ), t ∈ [tl , tl+1 ), l = 1, . . . , j − 1, t ∈ [tj , tj + ρj ), t ∈ [tj + ρj , ∞).

Now we consider the following Cauchy problems:  ∂t2 uj − aj (t)2 ∂x2 uj = 0, (t, x) ∈ (0, ∞) × R, (uj (0, x), (∂t uj )(0, x)) = (uj,0 (x), uj,1 (x)), x ∈ R,

(4.10)

and the corresponding energy of (4.10) 1 Ej (t) = Ej (t; uj,0 , uj,1 ) := 2



 R

 aj (t)2 |∂x uj (t, x)|2 + |∂t uj (t, x)|2 dx

(4.11)

for j = 1, 2, . . .. Then we shall prove the following proposition, which implies the first part of the conclusion of Theorem 2.7: Proposition 4.2. Suppose that (2.20) and (2.21) are valid. There exists a series of initial data {(uj,0 , uj,1 )}∞ j=1 such that Ej (0) ≤ 1 and lim sup{Ej (t)} = ∞. (4.12) j→∞ t>0

15

Remark 4.3. If α − β + δ < 0, then (2.14) and (2.15) are valid for m ≥ (α + δ − 1)/(α − β + δ) uniformly with respect to j. Therefore, there exists a positive constant C independent of j such that Ej (tj + ρj ) ≤ CEj (0) ≤ C by Theorem 2.4. On the other hand, if α − β + δ > 0, that is, (2.21) holds, then we have η(tj )Ξ(tj )  tα−β+δ → ∞ (j → ∞) j λ(tj ) and

 Ξ(tj + ρj )m−1



tj +ρj

λ(s) 0

η(s) λ(s)

m

m(α−β+δ)

ds  tj

→ ∞ (j → ∞),

that is, (2.14) and (2.15) do not hold. Therefore, Proposition 4.2 implies that the conditions (2.14) and (2.15) cannot be removed. Proof of Proposition 4.2. Noting aj (t) = 0 on [tj + ρj , ∞), we have Ej (t) = Ej (tj + ρj ) for t ∈ [tj + ρj , ∞). Moreover, we have  2Λ (t) Ej (t) Ej (t) = Λ (t)Λ(t) |∂x uj (t, x)|2 dx ≥ Λ(t) R for t ∈ (0, tj ], it follows that Ej (t) ≥

Λ(t)2 Ej (0) ≥ λ2j Ej (0). Λ(0)2

(4.13)

∞ We shall find a series of initial data {(uj,0 , uj,1 )}∞ j=1 providing limj→∞ Ej (tj + ρj ) = ∞. Let {ζj }j=1 be a positive sequence, which will be defined in (4.23). By Lemma 4.1, the solution of

yj + aj (t)2 ξj2 yj = 0, yj (tj ) = is represented by

ζj , yj (tj ) = 0 λj

   λj ξj (t−tj ) ζj 2 yj (t) = exp 2ε ϕ(s) cos s ds cos(λj ξj (t − tj )). λj 0

(4.14)

By (4.6) we have λj ξj ρj = 2πj; hence we have yj (tj + ρj ) =

max

t∈[tj ,tj +ρj )

{|yj (t)|} =

ζj exp (2πεj) . λj

(4.15)

For a positive monotone decreasing sequence {σj }∞ j=1 satisfying σj ≤ ξj , which will be defined in (4.23), we define Xj (ξ) by   σ σ  ζj , ξ ∈ ξj − 2j , ξj + 2j ,  Xj (ξ) := σ σ  0, ξ ∈ ξj − 2j , ξj + 2j .  Here we note that R Xj (ξ)2 dξ = σj ζj2 . Denoting u ˆj (t, ξ) = vj (t, ξ), the equation of (4.10) is reduced to ∂t2 vj + aj (t)2 ξ 2 vj = 0.

(4.16)

(uj,0 (x), uj,1 (x)) = (ˇ vj (0, x), (∂t vˇj ) (0, x)) ,

(4.17)

Then we set the initial data of (4.10) by

where vˇ denotes the inverse partial Fourier transformation, and vj (0, x) is the solution of (4.16) at t = 0 with the initial data at t = tj :   yj (tj ) (vj (tj , ξ), (∂t vj ) (tj , ξ)) = Xj (ξ), 0 . (4.18) ζj We define zj (t, ξ) and Zj (t, ξ) by zj (t, ξ) := vj (t, ξ) −

16

yj (t) Xj (ξ) ζj

(4.19)

and Zj (t, ξ) := Noting (4.15),

 1 aj (t)2 ξ 2 |zj (t, ξ)|2 + |∂t zj (t, ξ)|2 . 2

  yj (t) ∂t2 zj (t, ξ) = −aj (t)2 ξ 2 zj (t, ξ) − aj (t)2 ξ 2 − ξj2 Xj (ξ) ζj

(4.20)

(4.21)

and the inequalities  2  ξ − ξj2 2 Xj (ξ)2 ≤ we have

 ξj +

σj  2 − ξj2 2

2

Xj (ξ)2 ≤

"   a2j ξ 2 − ξj2 yj − ∂ t zj Xj ζj

25 2 2 σ ξ Xj (ξ)2 , 16 j j

(4.22)



∂t Zj =aj aj ξ 2 |zj |2 2C1 λ2j ξj ≤ aj



1 2 2 2 1 a ξ |zj | + |∂t zj |2 2 j 2



1 aj + 2C1 λ2j ξj 2



2   a2j ξ 2 − ξj2 yj Xj ζj

25a5j σj2 ξj yj2 2 X 64C1 λ2j ζj2 j ≤4C1 λj ξj Zj + C˜1 λj σj2 ξj exp (4πεj) Xj2 ≤4C1 λj ξj Zj +

by (4.3), where C˜1 = 6075/2048C1 . Thus, noting Zj (tj , ξ) = 0, we have Zj (tj + ρj , ξ) ≤ We set σj and ζj by

C˜1 σj2 exp (4πεj + 8πC1 j) Xj (ξ)2 . 4C1 

 σj :=

C 1 ξj exp (−4πC1 j) and ζj := C˜1 4

8λ2j . 9ξj2 σj

Noting the inequalities (4.13) and |f + g|2 ≥ |f |2 /2 − 2|g|2 , we have σj ≤ ξj ,  1 Ej (tj + ρj ) ≥ λ2j ξ 2 |vj (tj + ρj , ξ)|2 dξ 2 R  2    1 2 2  yj (tj + ρj ) = λj ξ  Xj (ξ) + zj (tj + ρj , ξ) dξ 2 ζ j R   λ2j yj (tj + ρj )2 2 2 ≥ ξ X (ξ) dξ − 2 Zj (tj + ρj , ξ) dξ j 4ζj2 R R   ξj2 exp (4πεj) C˜1 σj2 − exp (4πεj + 8πC1 j) Xj (ξ)2 dξ ≥ 16 2C1 R λ2j exp (4πεj) ξj2 σj ζj2 exp (4πεj) = 32 36 →∞ (j → ∞) =

and 1 1 Ej (0) ≤ 2 Ej (tj ) = 2 λj 2λj

 R

ξ 2 Xj (ξ)2 ≤

9ξj2 σj ζj2 = 1. 8λ2j

Thus the proof of Proposition 4.2 is concluded. On the analogy of the proof of Proposition 4.2, we can prove the following proposition:

17

(4.23)

Proposition 4.4. Let ν > 1. Suppose that (2.24) is valid. There exist a positive constant ρ and a series of initial data {(uj,0 , uj,1 )}∞ j=1 such that     1 exp 2ρ|ξ| ν aj (0)2 |∂x uj,0 (x)|2 + |uj,1 (x)|2 dx ≤ 1 (4.24) R

and lim sup{Ej (t)} = ∞.

j→∞ t>0

(4.25)

Proof. Thanks to α < 0, we note that (2.21) is valid since (2.24) holds. Let vj (t, ξ) be a solution to (4.16), and denote  1 aj (t)2 |ξ|2 |vj (t, ξ)|2 + |∂t vj (t, ξ)|2 . Ej (t, ξ) := 2 We set the initial data of (4.10) by (4.17) with     1 yj (tj ) (vj (tj , ξ), (∂t vj ) (tj , ξ)) = Xj (ξ) exp −ρ|ξ| ν , 0 . ζj We may proceed as we did to derive (4.13), obtaining  ≥ λ12 Ej (0, ξ) j Ej (t, ξ) = Ej (tj + ρj , ξ)

t ∈ (0, tj ), t ∈ [tj + ρj , ∞).

We define z˜j (t, ξ) and Z˜j (t, ξ) by z˜j (t, ξ) := vj (t, ξ) − and

Then we have

  1 yj (t) Xj (ξ) exp −ρ|ξ| ν ζj

 1 aj (t)2 ξ 2 |˜ zj (t, ξ)|2 + |∂t z˜j (t, ξ)|2 . Z˜j (t, ξ) := 2   1 ∂t Z˜j ≤4C1 λj ξj Z˜j + C˜1 λj σj2 ξj exp (4πεj) Xj (ξ)2 exp −2ρ|ξ| ν ,

it follows that    ξj2 exp (4πεj) 1 Xj (ξ)2 exp −2ρ|ξ| ν dξ 32  R    ν1 2 1 ξj σj ζj2 3 ν exp 4πεj − 2 ρξj ≥ 32 2     ν1 −β+δ −β+δ λ2j 3 (2π) ν(α−β+δ) ρj ν(α−β+δ) = exp 4πεj − 2 36 2

Ej (tj + ρj ) ≥

→∞ (j → ∞) since (2.24) holds. Moreover, we have        1 1 1 1 exp 2ρ|ξ| ν Ej (0, ξ) dx ≤ 2 exp 2ρ|ξ| ν Ej (tj , ξ) dx = 2 ξ 2 Xj (ξ)2 dx ≤ 1. λ 2λ R j R j R Thus the proof of the proposition is concluded.

18

(4.26)

(4.27)

5

Concluding remark

In this paper, we focused on the estimates of the total energy E(t), which is given by the sum of the the elastic energy EE (t) and the kinetic energy EK (t), where EE (t) := a(t)2 ∇u(t, ·)2 /2 and EK (t) := ∂t u(t, ·)2 /2. If the propagation speed a(t) is a positive constant, then the “equipartition of the energy”: lim

t→∞

EE (t) =1 EK (t)

is valid. For variable propagation speed a(t) satisfying a(t)  1 and a suitable control on the oscillations and the stabilizations so that (GEC) is valid, we expect the following estimates: EE (t) 1 EK (t)

(5.1)

for any large t, which can be called the “generalized equipartition of the energy”. On the other hand, if limt→∞ a(t) = 0, then not the estimate (5.1) but limt→∞ EE (t)/EK (t) = 0 can be established though (GEC ) holds. Actually, we cannot distinguish the estimates of EE (t) and EK (t) from E(t) by using just the method in this paper. However, for a ∈ L1 (R+ ), we can do it if we consider more precisely the estimate of E(t, ξ) in ZΨ , and we shall study such a precise estimates of the energy in a forthcoming paper.

Acknowledgements The first author thanks Faculty of Science of Yamaguchi University for the warm hospitality and assistance during his one month February/2014 research visit.

References [1] M. Cicognani, F. Hirosawa, On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients, Math. Scand. 102 (2) (2008) 283–304. [2] F. Colombini, Energy estimates at infinity for hyperbolic equations with oscillating coefficients, J. Differential Equations 231 (2) (2006) 598–610. [3] F. Colombini, D. Del Santo, T. Kinoshita, Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2) (2002) 327–358. [4] F. Colombini, E. De Giorgi, S. Spagnolo, Sur les ´equations hyperboliques avec des coefficients qui ne d´ependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (3) (1979) 511–559. [5] F. Colombini, E. Jannelli, S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (2) (1983) 291–312. [6] F. Hirosawa, Energy decay for a degenerate hyperbolic equation with a dissipative term, Publ. Res. Inst. Math. Sci. 35 (3) (1999) 391–406. [7] F. Hirosawa, On the asymptotic behavior of the energy for the wave equations with time depending coefficients, Math. Ann. 339 (4) (2007) 819-839. [8] F. Hirosawa, On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions, Math. Nachr. 283 (12) (2010) 1771–1794. [9] F. Hirosawa, Energy estimates for wave equations with time dependent propagation speeds in the Gevrey class, J. Differential Equations 248 (12) (2010) 2972-2993. [10] F. Hirosawa, T. Inooka, T. D. Pham, On the global solvability for semilinear wave equations with smooth time dependent propagation speeds. Progress in Partial Differential Equations: Asymptotic Profiles, Regularity and Well-Posedness, 153–181, Springer Proceedings in Mathematics & Statistics, 44, Springer, Heidelberg, 2013.

19

[11] F. Hirosawa, H. Ishida, On second order weakly hyperbolic equations and the ultradifferentiable classes, J. Differential Equations 255 (7) (2013) 1437–1468. [12] F. Hirosawa, J. Wirth, Generalised energy conservation law for the wave equations with variable propagation speed, J. Math. Anal. Appl. 358 (1) (2009) 56–74. [13] M. Reissig, J. Smith, Lp -Lq estimate for wave equation with bounded time dependent coeffient, Hokkaido Math. J. 34 (3) (2005) 541–586. [14] M. Reissig, K. Yagdjian, Lp − Lq decay estimates for the solutions of strictly hyperbolic equations of Math. Nachr. 214 (2000) 71–104. [15] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. [16] K. Yagdjian, Global existence in the Cauchy problem for nonlinear wave equations with variable speed of propagation, Operator Theory: Advances and Applications, Vol. 159, 301–385, Birkh¨auser, 2005. [17] T. Yamazaki, Unique existence of evolution equations of hyperbolic type with countably many singular of degenerate points, J. Differential Equations 77 (1) (1989) 38–72.

20