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A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions
Advances in Difference Equations volume 2010, Article number: 101959 (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.
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
Let (
;
;
;
) be Hilbert Spaces. Consider the following abstract initial-boundary value problem:


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
;
,
,
,
.
We will investigate this problem under the following conditions.
-
(i)
Let
, and let
be densely in
and continuously imbedded into it,
.
In the Hilbert space , it was defined the system of the inner products
, which generate uniform equivalent norms, that is,

For each , the function
is continuously differentiable,
.
In the Hilbert space , it was defined the system of the inner products
, which generate uniform equivalent norms, that is,

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)
The linear operators
, from
into
, act boundedly
,
;
.
Let us introduce the following designations:

From condition (v), it follows that the space with the norm

is a subspace of

-
(vi)
Let the linear manifold
be dense in
, and let linear manifold
be dense in
.
-
(vii)
(Green's Identity). For arbitrary
and
, the following identity is valid:

-
(viii)
For all
, the following inequality is fulfilled:
(1.9)
where .
-
(ix)
For each
, an operator pencil
(1.10)
which acts boundedly from to
, has a regular point
, where

-
(x)
(1.12)
-
(xi)
,

Definition 1.1.
The function is called a solution of problem (1.1)-(1.2) if the function
from
to
is continuous, and the function

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.
We define the operator in the Hilbert space
in the following way:

Then the problem (1.1)-(1.2) is represented as the Cauchy problem

where ,

It is obvious that if is the solution of problem (1.1)-(1.2), then
is the solution of the problem (1.16). On the contrary, if

is the solution of problem (1.16), then ,
and
is the solution of problem (1.1)-(1.2).
Let us define the system of inner product in Hilbert space in the following way:

where ,
.
We denote space with inner product (1.19) by
.
We will prove later the following auxiliary results.
Statement 1.3.
There exists such , that

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]).
Taking into account (viii) and Statement 1.3, we get

that is, is a lower semibounded selfadjoint operator in
.
Thus, the operator is selfadjoint and positive definite, where
.
Problem (1.16) can be rewritten as

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
.
By conditions of the theorem ;
,
and
are bounded operators from
to
. Therefore,

On the other hand, and
, therefore,
. Consequently,

From the definition of interpolation spaces (see [3, chapter 1], [7, chapter 1]), we get the following inclusion:

By virtue of definition, the powers of positive selfadjoint operator (see [8, chapter 2], [7, chapter 1]), we have that and

Assume that , then

By virtue of conditions (ii), (viii), (1.26), and (1.27), we get

Let . By virtue of condition (vi),
is dense in
; therefore, there exists a sequence
, such that
and

Hence it follows, that

Then from (1.28) and (1.30) it follows that is fundamental in
, that is,

where ,….
Thus, there exists such that

On the other hand, , therefore,

Hence,

where . From this, by virtue of (1.29),
, that is,

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).
Proof.
Consider in Hilbert space the equation

where .
Equation (1.36) is equivalent to the following system of differential-operator equations:

By virtue of (ix), problem (1.37) has a solution for some
. Thus, for each
,

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
Consider the following initial boundary value problem:

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

satisfy the local Lipschitz conditions in the following sense: for arbitrary ,

where ,

Theorem 2.1.
Let conditions (i)–(x) and (xi′) be satisfied, then there exists , such that the problem (2.1) has a unique solution

Additionally, if

where , then
. Otherwise, there exists
, such that

In the Hilbert space , the problem (2.1) is represented as the Cauchy problem

where ,

From (), it follows that, for arbitrary
,

where .
Thus, the nonlinear operator satisfies the condition of local solvability of the Cauchy problem for the quasilinear hyperbolic equations in Hilbert space (see [6, 9]). Taking this into account, the problem (2.8) has a unique solution

3. Initial Boundary Value Problem with Non Stationary Boundary and Transmission Condition for the Impulsive Semilinear Hyperbolic Equations
Let . We consider in the domain
the following mixed problem

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]).
Assume that the following conditions are held:
-
(10)
;
,
,
-
(20)
,
-
(30)
,
-
(40)
are nonlinear functionals acting from
(3.2)to
and for arbitrary
the following inequality holds
(3.3)where
,
(3.4) -
(50)
are nonlinear functionals acting from
(3.5)to
and for arbitrary
the following inequality holds
(3.6)where
, and
—is defined as in (3.3),
-
(60)
,
, where
(3.7)By applying Theorem 2.1, we obtain the following result.
Theorem 3.1.
Let conditions (10)–(60) be held, then there exists a , such that the problem (3.1) has a unique solution
, where

Proof.
Let us denote ,
,
,
,  
,
,
,  
, where
,
.
In space and
are defined the following inner products:

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:
,
,
,
.
It is easy to verify that linear operators and the nonlinear operators
,
satisfy the conditions of Theorem 2.1, and the problem (3.1) is represented as an abstract initial boundary-value problem in the following way:

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 linear manifolds and
are defined in the following way:

We also define the spaces

Statement 3.2.
is dense in

Proof.
Assume that . Consider the following functions:

From definitions of ,
, we can see that

Let . Consider the function

It is obvious that . On the other hand,
, where
is a space of infinitely differentiable finite functions. Therefore, for an arbitrary
, there exist the functions
,
, such that

By denoting from (3.17), we get

where .
Thus,

The following statement is proved in the same way.
Statement 3.3.
is dense
.
Now, we prove that the condition (vi) holds.
Let , then

Similary, we obtain the following identity:

Thus, by virtue of (3.20)-(3.21), the condition (vi) holds.
From (3.20) or (3.21), putting , we also obtain the identity

that is, condition (viii) is satisfied, .
Now, we verify fulfillment of condition (ix). To that end, we consider the mixed problem


where ,
;
,
,
.
Let be the extend of function
to
. We consider the system of the differential equations

Hence, we have

where is a Fourier transformation of the function
. From (3.26), we obtain
, then functions
satisfy (3.25), and their constrictions on (
) satisfy the (3.23). It is clear that
. Considering linearity of the problem (3.23), (3.24), the solution can be represented in the form

where is a solution of the following problem:


where ,

A general solution of a system (3.28) is found in the following form:

Then, for determination of , from (3.29), we get the following system of the algebraic equations:

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.
Let

where ,
,
;
and
and
,
satisfy the conditions
–
.
,
,
,
where ,
,
.
Theorem 3.4.
Let conditions be held and initial data satisfy the condition
, then the problem (3.1) has a unique solution
, where

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Aliev, A.B., Mamedova, U.M. A Mixed Problem for Quasilinear Impulsive Hyperbolic Equations with Non Stationary Boundary and Transmission Conditions. Adv Differ Equ 2010, 101959 (2010). https://doi.org/10.1155/2010/101959
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DOI: https://doi.org/10.1155/2010/101959