- Research Article
- Open Access

# A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions

- Akbar B. Aliev
^{1}Email author and - Ulviya M. Mamedova
^{2}

**2010**:101959

https://doi.org/10.1155/2010/101959

© Akbar B. Aliev and Ulviya M. Mamedova. 2010

**Received: **10 March 2010

**Accepted: **26 October 2010

**Published: **21 November 2010

## Abstract

The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space is considers. We prove the existence and uniqueness of solution of this problem. The results of this investigation are applied to solvability of initial-boundary value problems for quasilinear impulsive hyperbolic equations with non-stationary transmission and boundary conditions.

## Keywords

- Hilbert Space
- Cauchy Problem
- Hyperbolic Equation
- Initial Boundary
- Nonlinear Operator

## 1. Abstract Model Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Linear Hyperbolic Equations

In paper [1] there is given an abstract scheme of investigation of mixed problems for hyperbolic equations with non stationary boundary conditions. In this direction, some results were obtained in [2].

In this paper, we offer the analogues abstract model of investigation of mixed problem with non stationary boundary and transmission conditions for impulsive linear and semilinear hyperbolic equations.

### 1.1. Statement of the Problem and Main Theorem

where , , , are the linear closed operators in ; are the linear operators from to ; are the linear operators from to ; are the linear operators from to ; , , , .

- (i)

For each , the function is continuously differentiable, .

- (ii)
For each and is a linear closed operator in whose domain is ; acts boundedly from to ; is strongly continuously differentiable.

- (iii)
The linear operators , that act from to , bounded, where is interpolation space between and of order (see [3]).

- (iv)
For each , the linear operators , that act from to , are bounded; is strongly continuously differentiable , ; .

- (v)

- (vi)
- (vii)

Definition 1.1.

from to is twice continuously differentiable and (1.1)-(1.2) are satisfied.

Theorem 1.2.

Let conditions (i)–(xi) are satisfied, then the problem (1.1)-(1.2) has a unique solution.

Proof.

is the solution of problem (1.16), then , and is the solution of problem (1.1)-(1.2).

We denote space with inner product (1.19) by .

We will prove later the following auxiliary results.

Statement 1.3.

and the function is continuously differentiable, where = .

Statement 1.4.

is a symmetric operator in for each .

Statement 1.5.

has a regular point for each in .

is symmetric and , for some ; therefore, for each is a selfadjoint operator in (see [4, chapter x]).

that is, is a lower semibounded selfadjoint operator in .

Thus, the operator is selfadjoint and positive definite, where .

It is known that if and , then the problem (1.22) has a unique solution (see [5, 6]).

To complete the proof of the theorem, we need to show that and .

### 1.2. Proof of Auxiliary Results

Validity of Statement 1.3 follows from condition (i), the Statement 1.4 from condition (vii).

Proof.

where is an identity operator in , that is, has a regular point.

## 2. Abstract Model of Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Semilinear Hyperbolic Equations

where , , , , , , , , and satisfy all conditions of Theorem 1.2.

Assume, that the nonlinear operators and satisfy the following conditions.

(xi′)*Suppose that the nonlinear operators*

Theorem 2.1.

## 3. Initial Boundary Value Problem with Non Stationary Boundary and Transmission Condition for the Impulsive Semilinear Hyperbolic Equations

where , , , are some functions, and are some functionals, which will be specified below, .

Recently, differential equations with impulses are great interest because of the needs of modern technology, where impulsive automatic control systems and impulsive computing systems are very important and intensively develop broadening the scope of their applications in technical problems, heterogeneous by their physical nature and functional purpose (see [10,chapter 1]).

Theorem 3.1.

^{0})–(6

^{0}) be held, then there exists a , such that the problem (3.1) has a unique solution , where

Proof.

Let us denote , , , , , , , , where , .

From differentiability of the functions , , and , it follows that the condition (i) is satisfied.

Let us define the following operators:

We also define the nonlinear operators as follows:

We will show that conditions of Theorem 2.1 are satisfied. Conditions (i)–(v) follow immediately from definitions of spaces and operators , and traces theorems (see [3, chapter 2]), where ; ; ; ; .

Statement 3.2.

Proof.

The following statement is proved in the same way.

Statement 3.3.

Now, we prove that the condition (vi) holds.

Thus, by virtue of (3.20)-(3.21), the condition (vi) holds.

that is, condition (viii) is satisfied, .

Let be a matrix of coefficients of system (3.32). From (3.32), it is clear that , where and as . Thus, for sufficiently large , is invertible and . Therefore, the system (3.32) has a unique solution.

Thus, for sufficiently positive large , the problem (3.23)-(3.24) has a unique solution .

Thus, the condition (ix) is satisfied. The fulfillment of other conditions follows from .

Now, let us consider a class of nonlinear equations, for which the large solvability theorem takes place.

and , satisfy the conditions – .

Theorem 3.4.

## Authors’ Affiliations

## References

- Yakubov Y:
**Hyperbolic differential-operator equations on a whole axis.***Abstract and Applied Analysis*2004, (2):99-113. 10.1155/S1085337504311103Google Scholar - Lancaster P, Shkalikov A, Ye Q:
**Strongly definitizable linear pencils in Hilbert space.***Integral Equations and Operator Theory*1993,**17**(3):338-360. 10.1007/BF01200290MATHMathSciNetView ArticleGoogle Scholar - Lions J-L, Magenes E:
*Problemes aux Limites non Homogenes et Applications*.*Volume 1*. Dunod, Paris, France; 1968.MATHGoogle Scholar - Reed M, Simon B:
*Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness*. Academic Press, New York, NY, USA; 1975:xv+361.MATHGoogle Scholar - Kato T:
**Linear evolution equations of "hyperbolic" type. II.***Journal of the Mathematical Society of Japan*1973,**25:**648-666. 10.2969/jmsj/02540648MATHMathSciNetView ArticleGoogle Scholar - Hughes TJR, Kato T, Marsden JE:
**Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity.***Archive for Rational Mechanics and Analysis*1977,**63**(3):273-294. 10.1007/BF00251584MATHMathSciNetView ArticleGoogle Scholar - Yakubov S, Yakubov Y:
*Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics*.*Volume 103*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xxvi+541.Google Scholar - Krein SQ:
*Linear Differential Equation in Banach Spaces*. Nauka, Moscow, Russia; 1967.Google Scholar - Aliev AB:
**Solvability "in the large" of the Cauchy problem for quasilinear equations of hyperbolic type.***Doklady Akademii Nauk SSSR*1978,**240**(2):249-252.MathSciNetGoogle Scholar - Samoĭlenko AM, Perestyuk NA:
*Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 14*. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar

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