Well-Posedness of the Cauchy Problem for Hyperbolic Equations with Non-Lipschitz Coefficients

and Applied Analysis 3 Here the coefficients ak t satisfy different conditions of type 1.6 and 1.7 , so that qk and pk corresponding to different k are different. The smoothness of the solution depending on smoothness on initial data with respect to each variable xk depends not only on lk but also on qk and pk. 2. Statement of the Problem and Results We considered the Cauchy problem 1.8 . Suppose that ak t and bα t satisfy the following conditions: ak t ≥ a > 0, t ∈ 0, T , k 1, 2, . . . , n, 2.1 tk |ȧk t | ≤ c, t ∈ 0, T , k 1, 2, . . . , n, 2.2 bα t ∈ L∞ 0, T , |α : l| ≤ 1. 2.3 In order to formulate the basic results we introduce some denotation. Let H be some Hilbert space. ByW 2 R ,H we will denote a functional space with the norm ‖u‖Wλ,L 2 Rm,H ⎡ ⎣ ∫ Rm ( 1 m ∑ k 1 ηk k )∥∥û ( η )∥∥2 Hdη ⎤ ⎦ 1/2 , 2.4 where L L1, . . . , Lm , Lj ∈ N, j 1, 2, . . . , m, λ ≥ 0, and û η Fx u η ; Fx is a Fourier transformation with respect to variable x ∈ R. For s ≥ 1 by γ β R,H we will denote a functional space with the norm ‖u‖γs,L β R m,H ⎡ ⎣ ∫ exp ⎧ ⎨ ⎩ ∣∣∣∣ m ∑ k 1 η2 k ∣∣∣∣ 1/s ⎫ ⎬ ⎭ ∥û ( η )∥∥2 Hdη ⎤ ⎦ 1/2 . 2.5 DenoteW 2 R ,R W 2 R m , γ β R ,R γ β R m , C∞ R;H ⋂ λ≥0 W 2 R ;H , γ s R;H ⋂ β≥0 γ s β R ;H . 2.6 If L 1, . . . , 1 then W 2 R ,H H R;H , γ β R ,H γ β R ,H , and γ β R,R γ s β , where γ s β is the Geverey space of order s see 12, 13 . If λ ∈ H then W 2 R ,H is Hilbert-valued anisotropic Sobolev space W λL1,...,λLm 2 R ;H . For the read valued functions the anisotropic Sobolev spaces are stated in 14 . The basic results led in 14 are also valid for abstract-valued functions. 4 Abstract and Applied Analysis We introduce also the following denotation: x′ x1, . . . , xn1 , x ′′ xn1 1, . . . , xn , ξ′ ξ1, . . . , ξn1 , ξ ′′ ξn1 1, . . . , ξn , l′ l1, . . . , ln1 , l ′′ ln1 1, . . . , ln , |ξ| n ∑ k 1 ξk k , ∣ξ′ ∣∣ l′ n1 ∑ k 1 ξk k , ∣ξ′′ ∣∣ l′′ n ∑ k n1 1 ξk k , n2 n − n1. 2.7 The main results are the following theorems. Theorem 2.1. Let the conditions 2.1 – 2.3 be satisfied, where qk ∈ 0, 1 , for k 1, 2, . . . , n1, 2.8 qk 1, for k n1 1, . . . , n. 2.9 Then for any λ′ ≥ 0, λ′′ ≥ 0 the energy estimates E ( t, λ′, λ′′ ) ≤ME0, λ′, λ′′ λ0 ) , 2.10 hold, whereM and λ0 are some constants indepent of t ∈ 0, T , E ( t, λ′, λ′′ λ ) ∫ Rn ( 1 ∣ξ′ ∣∣ l′ λ′1 ∣ξ′′ ∣∣ l′′ )λ′′ λ|v̇ t, ξ | 1 |ξ|l |v t, ξ | ] dξ, λ ≥ 0, v̇ t, ξ ∂v t, ξ ∂t . 2.11 Theorem 2.2. Let the conditions 2.1 – 2.3 be satisfied, where qk ∈ 0, 1 , for k 1, 2, . . . , n1, 2.12 qk q > 1, for k n1 1, . . . , n. 2.13 Additionally, let the conditions t|ak t | ≤ c, t ∈ 0, T , for k n1 1, . . . , n. 2.14 be satisfied, where p ∈ 0, 1 ∩ 0, q − 1 . Then for any β > 0, λ′ ≥ 0, and 1 ≤ s < q − p / q − 1 the energy estimates, Et, β, s, λ′ ≤ME0, β δ, s, λ′, 2.15 Abstract and Applied Analysis 5 hold, whereM and δ are some constants independent of t ∈ 0, T , Et, β, s, λ′ ∫ Rn exp { β ∣ξ′′ ∣∣1/s l′′ }( 1 ∣ξ′ ∣∣ l′ )λ′|v̇ t, ξ | 1 |ξ|l |v t, ξ | ] dξ. 2.16and Applied Analysis 5 hold, whereM and δ are some constants independent of t ∈ 0, T , Et, β, s, λ′ ∫ Rn exp { β ∣ξ′′ ∣∣1/s l′′ }( 1 ∣ξ′ ∣∣ l′ )λ′|v̇ t, ξ | 1 |ξ|l |v t, ξ | ] dξ. 2.16 Remark 2.3. It is clear by our notation that E ( t, λ′, λ′′ ) ≤ ‖u̇ t, · ‖ W ′′ ,l′′ 2 ( R n2 x′′ ;W λ′ 1,l′ 2 R n1 x′ ) ‖u t, · ‖ W ′′ ,l′′ 2 ( R n2 x′′ ;W λ′ 1,l′ 2 R n1 x′ ) ‖u t, · ‖ W ′′ 1,l′′ 2 ( R n2 x′′ ;W λ′ ,l′ 2 R n1 x′ ) ≤ 2Eλ′, λ′′, t, 2.17 and we can write Et, β, s, λ′ ‖u t, · ‖ γ ′′ β ( R n2 x′′ ;W λ′ ,l′ 2 R n1 x′ ). 2.18 Remark 2.4. It is possible to replace the conditions a1 t , . . . , an1 t ∈ C1 0, T and 2.8 or 2.12 by Lipschitz conditions. The following theorems are obtained from Theorems 2.1 and 2.2. Theorem 2.5. Let condition 2.1 – 2.9 be satisfied. Then for any s ≥ 0, u0 ∈ C∞ R2 x′′ ;W 1,l ′ 2 R n1 x′ , u1 ∈ C∞ R2 x′′ ;W ′ 2 R n1 x′ the problem 1.1 , 1.2 admits a unique solution u ∈ C ( 0, T ;C∞ ( R2 x′′ ;W s 1,l′ 2 ( R1 x′ ))) ∩ C1 ( 0, T ;C∞ ( R2 x′′ ;W s,l′ 2 ( R1 x′ ))) . 2.19 Theorem 2.6. Let conditions 2.1 – 2.3 and 2.12 – 2.14 be satisfied. Then for any s′ ≥ 0, 1 ≤ s′′ < q− p / q− 1 , u0 ∈ γs R2 x′′ ;W ′ 1,l′ 2 R n1 x′ , u1 ∈ γ ′′ R2 x′′ ;W s′,l′ 2 R n1 x′ the problem 1.1 , 1.2 admits a unique solution u ∈ C ( 0, T ; γ ′′( R2 x′′ ;W s′ 1,l′ 2 ( R1 x′ ))) ∩ C1 ( 0, T ; γ ′′( R2 x′′ ;W s′,l′ 2 ( R1 x′ ))) . 2.20 In particular it follows from Theorem 2.1 that if the conditions 2.1 – 2.3 are satisfied, then the problem 1.1 , 1.2 is well-posed in C∞ R , and if the conditions 2.1 – 2.3 and 2.12 – 2.14 are satisfied then the problem 1.1 , 1.2 is well-posed in the Geverey class γ s . 3. Proof of Theorems At first we reduce some auxiliary statements. We denote v t, ξ Fx u t, ξ and define the weighted energetic function in the following way: Φ t Φ ( t, ξ, λ′, λ′′, β, r ) [ |v̇ t, ξ | 1 ∣ξ′∣l′ d ( t, ξ′′ |v t, ξ | ] ·H t, ξ , 3.1 6 Abstract and Applied Analysis where H t, ξ H ( t, ξ, λ′, λ′′, β, r ) ( 1 ∣ξ′ ∣∣ l′ λ1 ∣ξ′′ ∣∣ l′′ )λ′′


Introduction
Let us consider the Cauchy problem for a second-order hyperbolic equation: where the matrix a ij t is real and symmetric for all t ∈ 0, T , ü u tt .Suppose that 1.1 is strictly hyperbolic, that is, there exists λ 0 > 0 such that a t, ξ ≡ n i,j 1 where ω |τ| monotonically decreasing tends to zero, and log |τ| • ω |τ| tends to infinity, then there exists δ > 0 such that, for all u 0 ∈ H s R n , u 1 ∈ H s−1 R n the problem 1.1 , 1.2 has a unique solution u ∈ C 0, T ; H s−δ R n ∩ C 1 0, T , H s−1−δ R n this behavior goes under the name of loss of derivatives .
In the paper 5 it is considered the case when a i,j t 0, i / j, a part of coefficients belongs to the class LL ω 0, T , and another part of coefficients satisfies the Lipschitz condition.It is proved that the loss of derivatives occurs in those variables x k for which appropriate coefficient a kk t belongs to the class LL ω 0, T .
It is interesting to investigate the Cauchy problem for 1.1 , with singular coefficients.Many interesting results have been obtained in this direction.For example, in the paper 6 it is supposed that for each ξ ∈ R n \ {0}a t, ξ ∈ C 1 0, T and where q ≥ 1, c > 0. It is proved that if q 1, the problem 1.1 , 1.2 is well-posed in C ∞ R n .If q > 1 and where p ∈ 0, 1 ∩ 0, q − 1 , then the problem 1.1 , 1.2 is well-posed in the Geverey class γ s R n , s < q − p / q − 1 see 6 .If the coefficients a ij t satisfy only Holder conditions of order α < 1 then in 3 it is established that the problem 1.1 , 1.2 is γ s well-posed for all s < 1/ 1 − α .In this direction see also the results obtained in the papers 6-13 .
In this paper we consider the Cauchy problem for a higher-order hyperbolic equation with anisotropic elliptic part: where Here the coefficients a k t satisfy different conditions of type 1.6 and 1.7 , so that q k and p k corresponding to different k are different.The smoothness of the solution depending on smoothness on initial data with respect to each variable x k depends not only on l k but also on q k and p k .

Statement of the Problem and Results
We considered the Cauchy problem 1.8 .Suppose that a k t and b α t satisfy the following conditions:

2.3
In order to formulate the basic results we introduce some denotation.Let H be some Hilbert space.By W λ,L 2 R m , H we will denote a functional space with the norm where L L 1 , . . ., L m , L j ∈ N, j 1, 2, . . ., m, λ ≥ 0, and u η F x u η ; F x is a Fourier transformation with respect to variable x ∈ R n .
For s ≥ 1 by γ s,L β R m , H we will denote a functional space with the norm

2.7
The main results are the following theorems.

2.9
Then for any λ ≥ 0, λ ≥ 0 the energy estimates hold, where M and λ 0 are some constants indepent of t ∈ 0, T ,

2.18
Remark 2.4.It is possible to replace the conditions a 1 t , . . ., a n 1 t ∈ C 1 0, T and 2.8 or 2.12 by Lipschitz conditions.
The following theorems are obtained from Theorems 2.1 and 2.2.
Theorem 2.5.Let condition 2.1 -2.9 be satisfied.Then for any s ≥ 0, x the problem 1.1 , 1.2 admits a unique solution

2.20
In particular it follows from Theorem 2.1 that if the conditions 2.1 -2.3 are satisfied, then the problem 1.1 , 1.2 is well-posed in C ∞ R n , and if the conditions 2.1 -2.3 and 2.12 -2.14 are satisfied then the problem 1.1 , 1.2 is well-posed in the Geverey class γ s .

Proof of Theorems
At first we reduce some auxiliary statements.
We denote v t, ξ F x u t, ξ and define the weighted energetic function in the following way:

3.2
The following auxiliary lemmas are proved similar to the paper 6 .The proofs of the lemmas are in appendix.

3.6
On the other hand from 1.8 we have

3.15
On the other hand

3.19
On the other hand from the definition of Φ and E we have

Abstract and Applied Analysis
On the other hand from the definition of φ and E we have Φ t, ξ, λ , 0, β, r dξ ≥ c 11 E t, λ , s, β .

3.25
Proof of Theorem 2.5.For any ξ ∈ R n the problem 3.7 , 3.8 has a unique solution v t, ξ ∈ C 1 0, T see 15, Chapter I .
, then for any s ≥ 0, λ ≥ 0, where c s,λ > 0 is some constant dependent on s ≥ 0 and λ ≥ 0. Taking into account Theorem 2.1 it follows from 3.20 that

3.28
It follows from 3.28 that

3.29
By the expression of u t, x it follows that the function u t, x is the solution of problem 1.8 .
The uniqueness of the solution is proved by standard method.
The proof of Theorem 2.6 is carried out in the similar way.

A. Proof of Lemmas
Proof of Lemma 3.1.Let q k 1, k n 1 1, . . ., n.Then from 2.2 we have A. A.4 If t|ξ | l > 1, then using A.1 we get A.5 Consequently if q 1, the statement of the lemma follows from A.2 -A.5 .
Let q k > 1, k n A.8 Thus if q k > 1, k n 1 1, . . ., n then the statement of the lemma follows from A.2 , A.6 , and A.8 .
The lemma is proved.
Abstract and Applied Analysis 13 Now let us consider the case q k > 1, k n 1 1, . . ., n.In this case r q

A.12
As r q − 1 s, and s < q − p / q − 1 , it follows that 1 − 1 − p /s < 1/s and q − 1 /r 1/s.Then according to the Young inequality there exists such δ > 0 that Thus, by A.9 -A.13 the following inequality is valid: