- 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
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 ; , , , .
Let , and let be densely in and continuously imbedded into it, .
For each , the function is continuously differentiable, .
For each and is a linear closed operator in whose domain is ; acts boundedly from to ; is strongly continuously differentiable.
The linear operators , that act from to , bounded, where is interpolation space between and of order (see ).
For each , the linear operators , that act from to , are bounded; is strongly continuously differentiable , ; .
The linear operators , from into , act boundedly , ; .
Let the linear manifold be dense in , and let linear manifold be dense in .
(Green's Identity). For arbitrary and , the following identity is valid:
from to is twice continuously differentiable and (1.1)-(1.2) are satisfied.
Let conditions (i)–(xi) are satisfied, then the problem (1.1)-(1.2) has a unique solution.
is the solution of problem (1.16), then , and is the solution of problem (1.1)-(1.2).
where , .
We denote space with inner product (1.19) by .
We will prove later the following auxiliary results.
and the function is continuously differentiable, where = .
is a symmetric operator in for each .
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 .
To complete the proof of the theorem, we need to show that and .
Thus, . The theorem is proved.
1.2. Proof of Auxiliary Results
Validity of Statement 1.3 follows from condition (i), the Statement 1.4 from condition (vii).
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
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]).
(10) ; , ,
(50) are nonlinear functionals acting from
where , and —is defined as in (3.3),
(60) , , where
By applying Theorem 2.1, we obtain the following result.
Let us denote , , , , , , , , where , .
From differentiability of the functions , , and , it follows that the condition (i) is satisfied.
Let us define the following operators:
, for all other ,
, for all other ,
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.
is dense .
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, .
where , ; , , .
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.
where , , ; and
and , satisfy the conditions – .
where , , .
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