150 M. Reissig
4. Hirosawa’s counter-example
To end the proof of Theorem 8 we cite a result from [7] which explains that very fast oscillations have a deteriorating influence on C
∞
well-posedness. T
HEOREM
9. [see [7]] Let ω : 0, 12] → 0, ∞ be a continuous, decreasing
function satisfying lim ωs = ∞ for s → +0 and ωs2 ≤ c ωs for all s ∈
0, 12]. Then there exists a function a ∈ C
∞
R \ {0} ∩ C
R with the following
properties: • 12 ≤ at ≤ 32 for all t ∈ R;
• there exists a suitable positive T and to each p a positive constant C
p
such that |a
p
t | ≤ C
p
ω t
1 t
ln 1
t
p
for all t ∈ 0, T
; • there exist two functions ϕ and ψ from C
∞
R such that the Cauchy problem
u
t t
− atu
x x
= 0, u0, x = ϕx, u
t
0, x = ψx, has no solution in
C [0, r , D
′
R for all r 0.
The coefficient a = at possesses the regularity a ∈ C
∞
R \ {0}. To attack
the open problem 3 it is valuable to have a counter-example from [14] with lower regularity a
∈ C
2
R \ {0}. To understand this counter-example let us devote to the
Cauchy problem u
ss
− b ln
1 s
q 2
△ u = 0 , s, x ∈ 0, 1] × R
n
, 13
u1, x = ϕx, u
s
1, x = ψx,
x ∈ R
n
. Then the results of [14] imply the next statement.
T
HEOREM
10. Let us suppose that b = bs is a positive, 1-periodic, non-constant
function belonging to C
2
. If q 2, then there exist data ϕ, ψ ∈ C
∞
R
n
such that 13 has no solution in C
2
[0, 1], D
′
R
n
. Proof. We divide the proof into several steps.
Due to the cone of dependence property it is sufficient to prove H
∞
well-posedness. We will show that there exist positive real numbers s
ξ
= s|ξ| tending to 0 as |ξ| tends to infinity and data ϕ, ψ
∈ H
∞
R
n
such that with suitable positive constants C
1
, C
2
, and C
3
, |ξ| | ˆus
ξ
, ξ | + | ˆu
s
s
ξ
, ξ | ≥ C
1
|ξ|
1 2
expC
2
ln C
3
|ξ|
γ
. Here 1 γ q
− 1. This estimate violates H
∞
well-posedness of the Cauchy problem 13.
The assumption b ∈ C
2
guarantees that a unique solution u ∈
C
2
0, T ], H
∞
R
n
exists.
Hyperbolic equations with non-Lipschitz coeffi cients 151
Step 1. Derivation of an auxiliary Cauchy problem After partial Fourier transformation we get from 13
v
ss
+ b ln
1 s
q 2
|ξ|
2
v = 0 ,
s, ξ ∈ 0, 1] × R
n
, v
1, ξ = ˆϕξ , v
s
1, ξ = ˆ
ψ ξ , ξ
∈ R
n
, where vs, ξ
= ˆus, ξ. Let us define w = wt, ξ := τ t
1 2
v st, ξ , where t
= t s :
= ln
1 s
q
, τ = τ t := −
dt ds
st and s = st denotes the inverse function to
t = t s. Then w is a solution to the Cauchy problem
w
t t
+ bt
2
λ t, ξ w
= 0 , t, ξ
∈ [t 1, ∞ × R
n
, w
t 1, ξ = τ t 1
1 2
ˆϕξ , w
t
t 1, ξ = τ t 1
−
1 2
1 2
τ
t
t 1 ˆϕξ − ˆ
ψ ξ , where λ
= λt, ξ = λ
1
t, ξ + λ
2
t, and λ
1
t, ξ =
|ξ|
2
τ t
2
, λ
2
t =
θ t
bt
2
τ t
2
, θ = τ
′2
− 2τ
′′
τ . Simple calculations show that τ t
= q t
q−1 q
expt
1 q
and θ t ≈ − exp2t
1 q
. Hence, lim
t →∞
λ
2
t = 0. Let λ
be a positive real number, and let us define t
ξ
= t
ξ
λ by the
definition λt
ξ
, ξ = λ
. It follows from previous calculations that lim
|ξ |→∞
t
ξ
= ∞. Using the mean value theorem we can prove the following result.
L
EMMA
7. There exist positive constants C and δ such that |λ
1
t, ξ − λ
1
t − d, ξ| ≤ C d
τ
′
t τ
t λ
1
t, ξ , |λ
2
t − λ
2
t − d| ≤ C
τ
′
t τ
t for any 0
≤ d ≤ δ
τ t
τ
′
t
. In particular, we have |λt
ξ
, ξ − λt
ξ
− d, ξ| ≤ Cd τ
′
t
ξ
τ t
ξ
λ t
ξ
, ξ , 1
≤ d ≤ δ τ
t
ξ
τ
′
t
ξ
. We have the hope that properties of solutions of w
t t
+ bt
2
λ t, ξ w
= 0 are not “far away” from properties of solutions of w
t t
+ bt
2
λ t
ξ
, ξ w = 0. For this reason
let us study the ordinary differential equation w
t t
+ λ bt
2
w = 0.
Step 2. Application of Floquet’s theory We are interested in the fundamental solution X
= X t, t as the solution to the
Cauchy problem d
dt X
= −λ
bt
2
1 X , X t
, t
= 1
1 .
14 It is clear that X t
+ 1, t is independent of t
∈ N.
152 M. Reissig
L
EMMA
8 F
LOQUET
’
S THEORY
. Let b = bt ∈ C
2
, 1-periodic, positive and non-constant. Then there exists a positive real number λ
such that λ belongs to an
interval of instability for w
t t
+ λ bt
2
w = 0, that is, X t
+ 1, t has eigenvalues µ
and µ
−1
satisfying |µ
| 1. Let us define for t
ξ
∈ N the matrix X t
ξ
+ 1, t
ξ
= x
11
x
12
x
21
x
22
. According to Lemma 8 the eigenvalues of this matrix are µ
and µ
−1
. We suppose |x
11
− µ | ≥
1 2
|µ − µ
−1
| . 15
Then we have |x
22
− µ
−1
| ≥
1 2
|µ − µ
−1
|, too. Step 3. A family of auxiliary problems
For every non-negative integer n we shall consider the equation w
t t
+ λt
ξ
− n + t, ξbt
ξ
+ t
2
w = 0.
16 It can be written as a first-order system which has the fundamental matrix X
n
= X
n
t, t solving the Cauchy problem
d
t
X = A
n
X , X t ,
t = I
17 A
n
= A
n
t, ξ =
−λt
ξ
− n + t, ξbt
ξ
+ t
2
1 .
L
EMMA
9. There exist positive constants C and δ such that max
t
2
, t
1
∈[0,1]
kX
n
t
2
, t
1
k ≤ e
C λ
for 0 ≤ n ≤ δ
τ t
ξ
τ
′
t
ξ
and t
ξ
large. Proof. The fundamental matrix X
n
has the following representation: X
n
t
2
, t
1
= I +
∞
X
j =1
t
2
Z
t
1
A
n
r
1
, ξ
r
1
Z
t
1
A
n
r
2
, ξ · · ·
r
j −1
Z
t
1
A
n
r
j
, ξ dr
j
· · · dr
1
. By Lemma 7 we have
max
t
2
, t
1
∈[0,1]
kX
n
t
2
, t
1
k ≤ exp1 + b
2 1
λ
1
t
ξ
− n, ξ + sup
t 1 ≤t
|λ
2
t |
= exp1 + b
2 1
λ
1
t
ξ
− n, ξ − λ
1
t
ξ
, ξ + λ
− λ
2
t
ξ
+ sup
t 1 ≤t
|λ
2
t |
≤ e
C λ
Hyperbolic equations with non-Lipschitz coeffi cients 153
for large t
ξ
, ≤ n ≤ δ
τ t
ξ
τ
′
t
ξ
, where b
1
= max
[0,1]
bt. L
EMMA
10. Let η = ηt be a function satisfying
lim
t →∞
η t
τ
′
t τ
t = 0 .
18 Then there exist constants C and δ such that
kX
n
1, 0 − X t
ξ
+ 1, t
ξ
k ≤ C λ
η t
ξ τ
′
t
ξ
τ t
ξ
for 0 ≤ n ≤ δ ηt
ξ
. Consequently, kX
n
1, 0 − X t
ξ
+ 1, t
ξ
k ≤ ε for any given ε 0, sufficiently large t
ξ
∈ N and 0 ≤ n ≤ δ ηt
ξ
. Proof. Using the representation of X
n
1, 0 and of X t
ξ
+ 1, t
ξ
, then the application of Lemma 7 to
kX
n
1, 0 − X t
ξ
+ 1, t
ξ
k gives kX
n
1, 0 − X t
ξ
+ 1, t
ξ
k ≤ C λ n
+ 1 τ
′
t
ξ
τ t
ξ
expC λ n
+ 1 τ
′
t
ξ
τ t
ξ
≤ C λ δ η
t
ξ
+ 1 τ
′
t
ξ
τ t
ξ
expC λ δ η
t
ξ
+ 1 τ
′
t
ξ
τ t
ξ
→ 0 for t
ξ
→ ∞ and 1 ≤ n ≤ δ ηt
ξ
. Repeating the proofs of Lemmas 9 and 10 gives the following result.
L
EMMA
11. There exist positive constants C and δ such that kX
n +1
1, 0 − X
n
1, 0 k ≤ C λ
τ
′
t
ξ
− n τ
t
ξ
− n for 1
≤ n ≤ δ ηt
ξ
and large ξ . We will later choose η
= ηt ∼ t
α
with α ∈
1 2
,
q −1
q
. That the interval is non-empty follows from the assumptions of our theorem. If we denote X
n
1, 0 =
x
11
n x
12
n x
21
n x
22
n , then the statements of Lemmas 8 and 10 imply
• |µ
n
− µ | ≤ ε, where µ
n
and µ
−1 n
are the eigenvalues of X
n
1, 0; • |µ
n
| ≥ 1 + ε for ε ≤ |µ | − 12;
• |x
11
n − µ
n
| ≥
1 4
|µ − µ
−1
| , |x
22
n − µ
−1 n
| ≥
1 4
|µ − µ
−1
|. From Lemma 11 we conclude
• |x
i j
n + 1 − x
i j
n | ≤ C λ
τ
′
t
ξ
−n τ
t
ξ
−n
. This implies
154 M. Reissig
• |µ
n +1
− µ
n
| ≤ C λ
τ
′
t
ξ
−n τ
t
ξ
−n
. Step 4. An energy estimate from below
L
EMMA
12. Let n satisfy 0
≤ n ≤ δ ηt
ξ
≤ n + 1. Then there exist positive
constants C and C
1
such that the solution w = wt, ξ to
w
t t
+ bt
2
λ t, ξ w
= 0, w
t
ξ
− n − 1, ξ = 1 , w
t
t
ξ
− n − 1, ξ =
x
12
n µ
n
− x
11
n satisfies
|wt
ξ
, ξ | + |w
t
t
ξ
, ξ | ≥ C
expC
1
η t
ξ
19 for large ξ and η
= ηt fulfilling 18. Proof. The function w
= wt
ξ
− n + t, ξ satisfies 16 with n = n
. It follows that
d dt
w t
ξ
, ξ w
t
ξ
, ξ = X
1
1, 0X
2
1, 0 · · ·
· · · X
n −1
1, 0X
n
1, 0
d dt
w t
ξ
− n , ξ
w t
ξ
− n , ξ
. The matrix
B
n
=
x
12
n µ
n
−x
11
n
1 1
x
21
n µ
− 1
n
−x
22
n
is a diagonalizer for X
n
1, 0, that is, X
n
1, 0B
n
= B
n
diag µ
n
, µ
−1 n
. Since det X
n
1, 0 = 1 and trace of X
n
1, 0 is µ
n
+ µ
−1 n
we get det B
n
=
µ
n
−µ
− 1
n
µ
− 1
n
−x
22
n
. Us- ing the properties of µ
n
from the previous step we conclude | det B
n
| ≥ C 0 for all 0 n
≤ δ ηt
ξ
. Moreover, by Lemma 9 we have |x
i j
n | ≤ C , kB
n
k + kB
−1 n
k ≤ C for all 0 n
≤ δ ηt
ξ
. All constants C are independent of n. These estimates lead to kB
−1 n
−1
B
n
− I k = kB
−1 n
−1
B
n
− B
n −1
k ≤ C λ τ
′
t
ξ
− n τ
t
ξ
− n 20
Hyperbolic equations with non-Lipschitz coeffi cients 155
for large t
ξ
. If we denote G
n
: = B
−1 n
−1
B
n
− I , then we can write X
1
1, 0X
2
1, 0 · · · X
n −1
1, 0X
n
1, 0 = B
1
µ
1
µ
−1 1
B
−1 1
B
2
µ
2
µ
−1 2
B
−1 2
B
3
· · · B
−1 n
−1
B
n
µ
n
µ
−1 n
B
−1 n
= B
1
µ
1
µ
−1 1
I + G
2
µ
2
µ
−1 2
I + G
3
· · · I + G
n
µ
n
µ
−1 n
B
−1 n
. We shall show that the 1, 1 element y
11
of the matrix µ
1
µ
−1 1
I + G
2
µ
2
µ
−1 2
I + G
3
· · · · · · I + G
n
µ
n
µ
−1 n
can be estimated with suitable positive constants C and C
1
by C expC
1
η t
ξ
. It is evident from 20 that
|y
11
−
n
Y
n =1
µ
n
| ≤ C
n
Y
n =1
|µ
n
|
n
X
n =1
τ
′
t
ξ
− n τ
t
ξ
− n for large t
ξ
. We have
n
X
n =1
τ
′
t
ξ
− n τ
t
ξ
− n ≤
δ η t
ξ
Z τ
′
t
ξ
− t − 1 τ
t
ξ
− t − 1 dt
≤ ln τ
t
ξ
− 1 τ
t
ξ
− δ ηt
ξ
− 1 ≤ ln
1 − δ ηt
ξ
τ
′
t
ξ
− 1 τ
t
ξ
− 1
−1
→ 0 as t
ξ
→ ∞ . Hence, we can find a positive real ν such that
|y
11
| ≥ 1 − ν
n
Y
n =1
|µ
n
| ≥ 1 − νµ − ε
n
≥ 1 − νµ − ε
δ η t
ξ
−1
. The vector of data on t
= t
ξ
− n is an eigenvector of B
n
. Thus the estimate for y
11
holds for the vector d
t
w t
ξ
, ξ , w t
ξ
, ξ
T
too. This proves the energy estimate from below of the lemma.
156 M. Reissig
Step 5. Conclusion After choosing s
ξ
= st
ξ
= exp−t
1q ξ
for large t
ξ
and taking account of w
t
t, ξ =
1 2
τ
t
tτ t
−
1 2
v st, ξ
+ τ t
1 2
v
s
st, ξ we obtain |wt s, ξ| + |w
t
t s, ξ |
≤ τ t s
1 2
1 +
τ
t
t s 2τ t s
|vs, ξ| + τ t s
−
1 2
|v
s
s, ξ |
≤ 2τ t s
1 2
|vs, ξ| + τ t s
−
1 2
|v
s
s, ξ |
for large ξ . Finally, we use τ t s ∼ |ξ|. This follows from the definition λt
ξ
, ξ =
λ and lim
t
ξ
→∞
λ
2
t
ξ
= 0. Thus we have shown |ξ| | ˆus
ξ
, ξ | + | ˆu
s
s
ξ
, ξ | ≥ C
1
|ξ|
1 2
expC
2
η t
ξ
. The function ηt
= t
α
satisfies 18 if α
q −1
q
. The function t
ξ
behaves as ln |ξ|
q
. Together these relations give
|ξ| | ˆus
ξ
, ξ | + | ˆu
s
s
ξ
, ξ | ≥ C
1
|ξ|
1 2
expC
2
ln |ξ|
qα
≥ C
1
|ξ|
1 2
expC
2
ln |ξ|
γ
, where
γ ∈ 1, q − 1 .
From this inequality we conclude the statement of Theorem 10.
R
EMARK
6. The idea to apply Floquet’s theory to construct a counter-example goes back to [25] to study C
∞
well-posedness for weakly hyperbolic equations. This idea was employed in connection to L
p
− L
q
decay estimates for solutions of wave equations with time-dependent coefficients in [24]. The merit of [14] is the application
of Floquet’s theory to strictly hyperbolic Cauchy problems with non-Lipschitz coef- ficients. We underline that the assumed regularity b
∈ C
2
comes from statements of Floquet’s theory itself. An attempt to consider non-Lipschitz theory, weakly hyperbolic
theory and theory of L
p
− L
q
decay estimates for solutions of wave equations with a time-dependent coefficient is presented in [23].
5. How to weaken C
Kategori