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

- Jie Xin
^{1}Email author and - Xiuyan Sha
^{1}

**2009**:489061

https://doi.org/10.1155/2009/489061

© J. Xin and X. Sha. 2009

**Received: **27 June 2008

**Accepted: **29 April 2009

**Published: **5 May 2009

## Abstract

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.

## 1. Introduction

This paper is concerned with the exterior problem for hyperbolic system of second order. Let be a closed domain with smooth boundary in and let the origin belong to . Consider the following exterior problem for the hyperbolic system of second order:

for all symmetric matrixes , where , .

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.

## 2. Existence of the Exterior Problem for Hyperbolic System of Second Order

By (1.2) and Korn inequality (cf. [4, 5]), we have

Lemma 2.1.

where It is obvious that is a densely defined operator.

Lemma 2.2.

holds.

Proof.

Corollary 2.3.

Proof.

The estimate of the resolvent operator is the following.

Lemma 2.4.

Proof.

of (2.10). Therefore, is a surjection.

Corollary 2.5.

Proof.

Definition 2.6.

*Banach*space. A family of infinitesimal generators of semigroups on is called stable if there are constants and (called the stability constants) such that

Lemma 2.7.

Lemma 2.8.

has a unique classical solution such that

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 and , then there exists one and only one solution of (1.3) such that .

Proof.

where . From Lemma 2.4, for any , . Then by Lemma 2.7, is a stable family. Obviously, is continuously differentiable in . 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.

Proof.

Let , . By Proposition 2.9, there exists a solution of problem (1.3) such that . Let denote the forgoing three components of , then is the solution of problem (1.1) and satisfies (2.27).

## 3. 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.

Proposition 3.1.

where is a constant which depends on .

Proof.

This completes the proof of (3.2).

Theorem 3.2.

Proof.

Therefore is the solution of problem (1.1) and satisfies (3.19).

We now prove (3.18) by induction. When , (3.18) follows from (3.2). For , suppose that (3.18) holds for . We show that it still holds for .

and (3.18).

## 4. Application to Elastrodynamics

where stands for the elastic tensor.

The system (4.3) is the special case of the system (1.1). So by the existence Theorem 3.2, we derive the existence of solutions for the initial-boundary problem to the elastrodynamic system (4.3) outside a domain.

## Declarations

### Acknowledgments

Projects 10626046 supported by NSFC and 20070410487 supported by China Postdoctoral Science Foundation. The authors would like to thank Professor Tatsien Li and Professor Tiehu Qin for helpful discussions and suggestions.

## Authors’ Affiliations

## References

- Ikawa M:
**Mixed problems for hyperbolic equations of second order.***Journal of the Mathematical Society of Japan*1968,**20:**580–608. 10.2969/jmsj/02040580MathSciNetView ArticleMATHGoogle Scholar - Hörmander L:
*Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications*.*Volume 26*. Springer, Berlin, Germany; 1997:viii+289.MATHGoogle Scholar - Dafermos CM, Hrusa WJ:
**Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics.***Archive for Rational Mechanics and Analysis*1985,**87**(3):267–292.MathSciNetView ArticleMATHGoogle Scholar - Ciarlet PG:
*Mathematical Elasticity. Volume I: Three-Dimensional Elasticity, Studies in Mathematics and Its Applications*.*Volume 20*. North-Holland, Amsterdam, The Netherlands; 1988:xlii+451.Google Scholar - Li T-T, Qin T:
*Physics and Partial Differential Equations. Vol. I*. Higher Educational Press, Beijing, China; 1997.Google Scholar - Pazy A:
*Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences*.*Volume 44*. Springer, New York, NY, USA; 1983:viii+279.View ArticleGoogle Scholar

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