Determination of an Unknown Coefficient in the Third Order Pseudoparabolic Equation with Non-Self-Adjoint Boundary Conditions
The solvability of the inverse boundary problem with an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved.
Contemporary problems of natural sciences lead to the need for statement and investigation of the qualitative new problems. As an example we can consider a class of nonlocal problems for the partial differential equations. Researching such kind of problems aroused both theoretical interest and practical necessity and they are still studied actively today. The problems with both nonlocal boundary and initial conditions had previously been studied by many scientists. Classes of nonlocal problems with integral terms in boundary conditions are of great importance in the theory of heat conductivity, thermoelasticity, chemical engineering, underground water flow, population dynamics, and plasma physics.
The questions of solvability of the nonlocal problems with integral terms in boundary conditions had been studied by Samarskii . Auxiliary information for investigation of the solution of such kind of problems can be found in [2–7]. Inverse problems with integral condition of override for pseudoparabolic type of equations had been studied in [8–10].
Existence and uniqueness of the solution of an inverse boundary value problem for the third order pseudoparabolic equation with the integral condition of override is proved in the present paper.
2. Statement of the Problem and Reducing It to Equivalent
Problem 1. Let us consider inverse boundary problem for the equation
in the domain with initial condition
Neumann boundary condition and the additional condition where , are the given numbers, , , are the given functions, and and are the unknown functions.
Definition 2. The classical solution of problems (1)–(5) is the pair of the functions and possessing the following properties:(a) is continuous in together with all its derivatives contained in (1);(b) is continuous on ;(c)all the conditions of (1)–(5) are satisfied in the ordinary sense.
The following lemma takes place.
Lemma 3. Suppose that , , , , , , and .
Then the problem of finding the classical solution of problem (1)–(5) is equivalent to the problem of defining the functions and , possessing the properties (a) and (b) of definition of the classical solution of problem (1)–(5), from relations (1)–(4), and
Proof . Let be a classical solution of problem (1)–(5). Differentiating (5) and taking into account that we will get
From (1) we have
Taking into account conditions (5) and (7) in (8), the fulfillment of condition (6) takes place.
Now suppose that is a solution of problem (1)–(4), (6). Then from (6) and (8), taking into account (4), we find Further, by (2) and , we obtain
Since problems (9) and (10) has only a trivial solution, then ; that is, the condition (5) is fulfilled.
3. Auxiliary Facts
Now, in order to investigate problem (1)–(4), (6) we introduce known facts. Consider the following spectral problem: where is a parameter. Boundary value problem (11) is not self-adjoint . The problem
will be a conjugated problem.
Let us denote the system of eigen and adjoint functions of problem (11) in the following way: where
The system of function (13) forms a Riesz basis in the space .
Let us choose the system of eigen and adjoint functions of the conjugated problem as follows:
From this it follows that for systems (13) and (15) the biorthogonality condition in is satisfied. Here, is the Kronecker symbol.
Then the arbitrary function is expanded in biorthogonal series: and the following estimate is true: where
Under the assumptions
the following estimates hold:
Further, under the assumptions
the validity of the estimates is proved:
Similarly, under the assumptions
the estimations hold:
In order to investigate problem (1)–(4), (6), consider the following spaces.(1)Denote by  the set of all the functions of the form considered in , where each of the functions is continuous on and The norm in this set is determined as follows: (2)Denote by the space of the vector functions with the norm It is known that and are Banach spaces.
4. Investigation of the Existence and Uniqueness of the Classical Solution of the Inverse Boundary Value Problem
Since the system (13) is a Riesz basis in and system (13), (15) is a system of biorthogonal functions in , it is obvious that for each solution of problem (1)–(4), (6) its first component has the following form: where Moreover, and are defined by relations (13) and (15), respectively.
Then applying the formal scheme of the Fourier method, from (1) and (2), we have where
Solving the problem (34)–(37), we find the following: After substituting expressions into (32) for determining the component of the solution of problem (1)–(4), (6) we get
Now, taking into account (32), from (6), we get
Further, from (35), taking into account (40), we have
To find the second component of the solution of problem (1)–(4), (6) after substituting expression (44) into (43), we get
Thus, the solution of problem (1)–(4), (6) is reduced to the solution of system (42) and (45) with respect to the unknown functions and .
Now, proceeding from the definition of the classical solution of problem (1)–(4), (6) the following lemma is proved.
Lemma 4. If is any solution of problem (1)–(4), (6), then the function satisfies the system (39)-(41) in .
Remark 5. From Lemma 4 it follows that, to prove the uniqueness of the solution of problem (1)–(4), (6), it suffices to prove the uniqueness of the solution of system (42), (45).
Now, in the space consider the operator where and , , , and equal the right sides of (39), (40), (41), and (45), respectively.
It is easy to see that By means of simple transformations we find
Suppose that the data of problem (1)–(4), (6) satisfy the following conditions:(1), , , , ;(2), , ;(3), , , .
Then taking into account (21)–(27), from (50), we get where
So, the following theorem can be proved.
Theorem 6. Let conditions be satisfied, and Then problem (1)–(4), (6) has a unique solution in the ball of the space .
Remark 7. Inequality (53) is satisfied for sufficiently small values at .
Proof. Consider the equation
in the space , where and the components of the operator are defined from the right sides of (42), (45). Consider the operator in the ball from .
Similar to (51), for any the following estimates hold:
Then taking into account (53) in (55), it follows that the operator acts in the ball and is contractive. Therefore, in the ball the operator has a unique fixed point that is a unique solution of (54) in the ball ; that is, it is a unique solution of system (42), (45) in the ball .
The function as an element of the space is continuous and has continuous derivatives and in .
From (35), (36) it is easy to see that
From the last relation it is obvious that , , and are continuous in .
It is easy to verify that (1) and conditions (2), (3), (4), and (6) are satisfied in the ordinary sense.
Consequently, is a solution of problem (1)–(4), (6) and by Lemma 4 this solution is unique in the ball . The theorem is proved.
By means of Lemma 3 the uniqueness of the solution of the initial problem (1)–(5) follows from Theorem 6 and the following theorem takes place.
Theorem 8. Suppose that all the conditions of Theorem 6 are satisfied and
Then problem (1)–(5) has a unique classical solution in the ball of the space .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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