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 [

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.

Let us consider inverse boundary problem for the equation

Neumann boundary condition

The classical solution of problems (

all the conditions of (

The following lemma takes place.

Suppose that

Then the problem of finding the classical solution of problem (

Let

From (

Taking into account conditions (

Now suppose that

Since problems (

Now, in order to investigate problem (

will be a conjugated problem.

Let us denote the system of eigen and adjoint functions of problem (

The system of function (

Let us choose the system of eigen and adjoint functions of the conjugated problem as follows:

From this it follows that for systems (

Then the arbitrary function

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 (

Denote by

Denote by

Since the system (

Then applying the formal scheme of the Fourier method, from (

Solving the problem (

Now, taking into account (

Further, from (

To find the second component

Thus, the solution of problem (

Now, proceeding from the definition of the classical solution of problem (

If

From Lemma

Now, in the space

It is easy to see that

Suppose that the data of problem (

Then taking into account (

So, the following theorem can be proved.

Let conditions

Inequality (

Consider the equation

Similar to (

Then taking into account (

The function

From (

From the last relation it is obvious that

It is easy to verify that (

Consequently,

By means of Lemma

Suppose that all the conditions of Theorem

Then problem (

The authors declare that there is no conflict of interests regarding the publication of this paper.