About the influence of oscillations 379
3. Wave equations with slow oscillations in the time-dependent coefficient 3.1. Classification of oscillations
D
EFINITION
1. Let us suppose that there exists a real β ∈ [0, 1] such that the following condition is satisfied:
|d
t
bt | ≤ C λ
t 3
t ln 3t
β
for large t ,
11 where 3t :=
R
t
λ s ds. If β ∈ [0, 1, β = 1, respectively, we call the oscillations slow
oscillations, fast oscillations, respectively. If 11 is not satisfied for β = 1, then we call the oscillations very fast oscillations.
R
EMARK
3. Very fast oscillations may destroy L
p
−L
q
decay estimates. These oscillations give us an exact description of a fairly wide class of equations in which the oscillating part
dominates the increasing one. In [6] it is shown that one can prove in this case a statement similar to Theorem 1.
R
EMARK
4. The case of fast oscillations is studied in [7]. We could derive L
p
− L
q
decay estimates only for large dimension n. Moreover, the behaviour of b and λ and its first two
derivatives has an influence on the decay rate. The goal of the following considerations is to show that for slow oscillations β ∈ [0, 1 in
11 we have L
p
− L
q
decay estimates similar to Strichartz decay estimate 1 • for any dimension n ≥ 2,
• with a decay rate which coincides with the classical decay rate, • with the decay function 1 + 3t,
• without an essential influence of the oscillating part.
3.2. Main result and philosophy of our approach
Let us study u
t t
− λt
2
bt
2
△ u = 0 , u0, x = ϕx ,
u
t
0, x = ψx , 12
under the following assumptions for the positive coefficient λt
2
bt
2
: • it holds
3 t → ∞ as t → ∞ ;
13 • there exist positive constants c
, c
1
and c such that c
λ t
3 t
≤ λ
′
t λ
t ≤ c
1
λ t
3 t
≤ cln 3t
c
for large t ;
14 • there exist positive constants c
k
such that for all k = 2, 3, . . . it holds d
k t
λ t
≤ c
k
λt 3
t
k
λ t
for large t ;
15
380 M. Reissig – K. Yagdjian
• with two positive constants d and d
1
we have d
≤ b
2
t ≤ d
1
for t ∈ [0, ∞ ;
16 • there exist positive constants d
k
such that for all k = 2, 3, . . . it holds d
k t
bt ≤
d
k
λt 3
t ln 3t
β k
for large t .
17 T
HEOREM
2 M
AIN RESULT
. Assume that the conditions 13 to 17 are satisfied with β ∈ [0, 1. Then for every ε 0 there exists a constant C
ε
such that the decay estimate ku
t
t, ·k
L
q
+ kλt∇
x
ut, ·k
L
q
≤ C
ε
1 + 3t
1+ε−
n−1 2
1 p
−
1 q
kϕk
W
L+1 p
+ kψk
W
L p
holds for the solution u = ut, x to 12. Here L = h
n
1 p
−
1 q
i + 1, 1 p 2,
1 p
+
1 q
= 1. Let us explain the philosophy of our approach. By F, F
− 1
we denote the Fourier transform, inverse Fourier transform with respect to x, respectively. Applying F to 12 we get with v =
Fu the Cauchy problem v
t t
+ λt
2
bt
2
|ξ |
2
v = 0 , v
0, ξ = Fϕ , v
t
0, ξ = Fψ . 18
Setting V = V
1
, V
2 T
:= λt|ξ |v, D
t
v , D
t
:= ddt, the differential equation can be trans- formed to the system of first order
D
t
V − λ
t |ξ | λ
t bt
2
|ξ | V −
D
t
λ λ
1 V = 0 .
19 Our main object is the fundamental solution X = X t, τ, ξ ∈ C
∞
[τ, ∞ ×
n
of 19, that is the solution of
D
t
X − λ
t |ξ | λ
t bt
2
|ξ | X −
D
t
λ λ
1 X
= 0 ,
20 X τ, τ, ξ
= I ,
21 with τ ≥ 0. We prove that X = X t, 0, ξ can be represented in the form
X t, 0, ξ = X
+
t, 0, ξ exp i
Z
t
λ sbs ds|ξ |
+ X
−
t, 0, ξ exp −i
Z
t
λ sbs ds|ξ |
, 22
where X
+
and X
−
have connections to symbol classes. Using this representation we obtain the solution of 12 in the form
ut, x =
F
− 1
λ0 λ
t X
11
t, 0, ξ Fϕ + 1
λ t |ξ |
X
12
t, 0, ξ Fψ ,
23 D
t
ut, x =
F
− 1
λ 0|ξ |X
21
t, 0, ξ Fϕ + X
22
t, 0, ξ Fψ ,
24 where X
j k
are the elements of X . For these Fourier multipliers L
p
− L
q
decay estimates are derived in Section 3.5.
We intend to obtain the representation 22 in t, ξ ∈ [0, ∞ ×
n
\ {0} by splitting this set
into two zones.
About the influence of oscillations 381
D
EFINITION
2. We define the pseudodifferential zone Z
pd
β, N by
Z
pd
β, N :=
n t, ξ ∈ [0, ∞ ×
n
\ {0} : e
4
+ 3t |ξ| ≤ N ln e
4
+ 3t
β
o ,
the hyperbolic zone Z
hyp
β, N by
Z
hyp
β, N :=
n t, ξ ∈ [0, ∞ ×
n
\ {0} : e
4
+ 3t |ξ| ≥ N ln e
4
+ 3t
β
o .
The positive constant N will be chosen later. For |ξ | ∈ 0, p
], p = 4
β
N e
4
, we define the function t
ξ
= t |ξ | as the solution of e
4
+ 3t
ξ
|ξ| = N ln e
4
+ 3t
ξ β
. L
EMMA
3. The derivatives ∂
k |
ξ |
t
ξ
can be estimated in the following way: ∂
k |
ξ |
t
| ξ |
≤ C
k
|ξ |
− k
e
4
+ 3t
ξ
λ t
ξ
for all ξ ∈
n
, |ξ | ∈ 0, p ] .
3.3. The fundamental solution in Z
