Existence of Solutions for Hyperbolic System of Second Order Outside a Domain

Jie Xin and Xiuyan Sha School of Mathematics and Information, Ludong University, Yantai, Shandong 264025, China Correspondence should be addressed to Jie Xin, fdxinjie@sina.com Received 27 June 2008; Accepted 29 April 2009 Recommended by Robert Bob Gilbert We study the mixed initial-boundary value problem for hyperbolic system of second order outside a closed domain. The existence of solutions to this problem is proved and the estimate for the regularity of solutions is given. The application of the existence theorem to elastrodynamics is discussed. Copyright q 2009 J. Xin and X. Sha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
This paper is concerned with the exterior problem for hyperbolic system of second order. Let K be a closed domain with smooth boundary in R 3 and let the origin belong to K. Consider the following exterior problem for the hyperbolic system of second order: where a ijkl t, x ∈ C 2 B 0, ∞ × R 3 \ K and b b 1 , b 2 , b 3 . We assume that a ijkl t, x satisfies 3 j,k,l 1 a ijkl t, x e ij e kl ≥ α|E| 2 , α > 0 , 1 i,j 1 e 2 ij , t, x ∈ R × R 3 \ K.
Let v ∂ t u. The system 1.1 can be written as an evolution system in the form

1.4
Ikawa considered in 1 the mixed problem of a hyperbolic equation of second-order. The existence theorem is known for the obstacle free problem in 2 . Dafermos and Hrusa proved in 3 the local existence of the Dirichlet problem for the hyperbolic system inside a domain by energy method.
In this paper, we deal with the exterior problem for the second order hyperbolic system. In Section 2, we show the existence of the exterior problem for the problem 1.1 by the semigroup theory. In Section 3, we prove the regularity for the solutions of the exterior problem 1.1 and give the estimate for the regularity of solutions. In Section 4, we discuss the application of the existence theorem to elastrodynamics.

Existence of the Exterior Problem for Hyperbolic System of Second Order
Note that H t

Journal of Inequalities and Applications 3
Then H t is a Hilbert space with the inner product defined as above. We define the operator without loss of generality, we still write this operator as A t in H t by It is obvious that A t is a densely defined operator.

Lemma 2.2.
There exists a constant c > 0 such that for any U ∈ D,

2.5
Corollary 2.3. For all real λ such that |λ| > 2c, the estimate holds for any U ∈ D.

2.7
The estimate of the resolvent operator λI − A t −1 is the following.

Lemma 2.4.
There exists a constant δ > 0 such that for all λ real and |λ| > δ, Proof. Consider the system in the second of 2.11 gives −a t λ 2 u λp q w ∈ L 2 R 3 \ K .

2.13
Journal of Inequalities and Applications 5 By the well-known variation method, there exists a solution u ∈ H 2 R 3 \ K ∩ H 1 0 R 3 \ K of the elliptic system 2.13 for any w ∈ L 2 R 3 \ K . Defining v by 2.12 , we have a solution 14 of 2.10 . Therefore, λI − A t is a surjection. From 2.6 , it follows that the existence of λI − A t −1 and the estimate Let δ 2c, we have 2.9 .
we define the following norm: Proof. From Lemma 2.4, is a bijective continuous mapping, then λ 0 I − A t is a closed operator. It implies that λ 0 I − A t −1 is also a closed operator. By Banach's closed graph theorem,

2.19
Definition 2.6. Let X be a Banach space. A family {A t } t∈ 0,T of infinitesimal generators of C 0 semigroups on X is called stable if there are constants M ≥ 1 and δ called the stability constants such that

Lemma 2.8. Let {A t } t∈ 0,T be a stable family of infinitesimal generators of C 0 semigroups S t s on the Banach space X such that D A t D is independent of t and for every
The proofs of Lemmas 2.7 and 2.8 are in 6 . The straightforward application of the semigroup theory to the system 1.3 gives the following proposition. Proposition 2.9. Given U 0 ∈ D and B t ∈ C 1 0, T , H 1 0 R 3 \ K × L 2 R 3 \ K , then there exists one and only one solution U t ∈ C 1 0, T ; Proof. Let X H t . For given t > 0, A t is an infinitesimal generator of C 0 semigroups S t s on X. For any U ∈ D, it is easy to know that S t s U H t ≤ e δs U H t .

2.23
Then for any U ∈ D, t 1 , t 2 > 0, we have Journal of Inequalities and Applications 7 namely,

2.26
where M ≥ 1. From Lemma 2.4, for any t ∈ 0, T , δ, ∞ ⊂ ρ A t . Then by Lemma 2.7, {A t } t∈ 0,T is a stable family. Obviously, A t U 0 is continuously differentiable in X. So Proposition 2.9 follows from Lemma 2.8.
From Proposition 2.9, we obtain the existence of solutions to the problem 1.1 .
Theorem 2.10. Given f, g ∈ D and b ∈ C 1 0, T ; L 2 R 3 \ K , then there exists one and only one solution u t, x of 1.1 such that

2.27
Proof. Let U 0 f, g T , B 0, b T . By Proposition 2.9, there exists a solution U t ∈ C 1 0, T ; H 1 0 R 3 \ K × L 2 R 3 \ K ∩ C 0, T ; D of problem 1.3 such that U 0 U 0 . Let u t, x denote the forgoing three components of U t , then u t, x is the solution of problem 1.1 and satisfies 2.27 .

Regularity of Solutions for the Exterior Problem
First, we show the energy inequalities for our problem. These inequalities play an important role in the proof of the regularity of solutions.
is a solution of problem 1.1 and that b t, x ∈ C 1 0, T ; L 2 R 3 \ K , then for any given t ∈ 0, T , we have

3.2
where C T is a constant which depends on T .
Proof. Put U t u, ∂ t u , then U t ∈ D and satisfies Without loss of generality, we assume that ∂ t u t, x ∈ C 0, T ;

3.10
Obviously, Also we have Inserting these estimates to the above inequality, we get 3.14 An application of Gronwall's inequality implies Namely,

3.16
This completes the proof of 3.2 .