- Research Article
- Open Access
A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions
© Akbar B. Aliev and Ulviya M. Mamedova. 2010
- Received: 10 March 2010
- Accepted: 26 October 2010
- Published: 21 November 2010
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.
- 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  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 .
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
The linear operators , that act from to , bounded, where is interpolation space between and of order (see ).
Let conditions (i)–(xi) are satisfied, then the problem (1.1)-(1.2) has a unique solution.
We will prove later the following auxiliary results.
is symmetric and , for some ; therefore, for each is a selfadjoint operator in (see [4, chapter x]).
1.2. Proof of Auxiliary Results
Validity of Statement 1.3 follows from condition (i), the Statement 1.4 from condition (vii).
2. Abstract Model of Initial Boundary Value Problem with Non Stationary Boundary and Transmission Conditions for the Impulsive Semilinear Hyperbolic Equations
(xi′)Suppose that the nonlinear operators
3. Initial Boundary Value Problem with Non Stationary Boundary and Transmission Condition for the Impulsive Semilinear Hyperbolic Equations
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]).
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 ; ; ; ; .
The following statement is proved in the same way.
Now, we prove that the condition (vi) holds.
Thus, by virtue of (3.20)-(3.21), the condition (vi) holds.
Now, let us consider a class of nonlinear equations, for which the large solvability theorem takes place.
- 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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