On one nonlocal inverse boundary problem for the Benney – Luke equation with integral conditions

An inverse boundary value problem for the Benney-Luke equation with periodic and integral condition is investigated. The definition of a classical solution of the problem is introduced. The goal of this paper is to determine the unknown coefficient and to solve the problem of interest. The problem is considered in a rectangular domain. To investigate the solvability of the inverse problem, we perform a conversion from the original problem to some auxiliary inverse problem with trivial boundary conditions. By the contraction mapping principle we prove the existence and uniqueness of solutions of the auxiliary problem. Then we make a conversion to the stated problem again and, as a result, we obtain the solvability of the inverse problem.


Introduction
There are many cases where the needs of the practice bring about the problems of determining coefficients or the right hand side of differential equations from some knowledge of its solutions. Such problems are called inverse boundary value problems of mathematical physics. Inverse boundary value problems arise in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control in industry etc., which makes them an active field of contemporary mathematics. Inverse problems for various types of have been studied in many papers.
The foundations of the theory and practice of studying inverse problems of mathematical physics were laid and developed in pioneering works of the outstanding scientists A. N. Tikhonov [1], M. M. Lavrent'ev [2], V. K. Ivanov [3] and V. G. Romanov [4]. At present, there are many works devoted to the study of inverse problems for partial differential equations [5][6][7][8][9][10].
Many problems of gas dynamics, theory of elasticity, theory plates and shells is reduced to the consideration of differential equations in high-order partial derivatives. Of great interest from the point of view of applications are differential equations of the fourth order. Partial differential equations of the Benneo -Luke type have applications in mathematical physics [11].
In this paper, we prove existence and uniqueness of the solution to an inverse boundary value problem for the Benney -Luke equation with integral conditions.

Problem statement and its reduction to an equivalent problem
Consider for the Benney -Luke equation [11] AMCSM_2018 IOP Conf. Series: Journal of Physics: Conf. Series 1203 (2019) 012100 IOP Publishing doi: 10.1088/1742-6596/1203/1/012100 2 u tt (x, t)−u xx (x, t)+αu xxxx (x, t)−βu xxtt (x, t) = a(t)u(x, t)+b(t)g(x, t)+f (x, t) (x, t) ∈ D T (1) in the domain D T = {(x, t) : 0 ≤ x ≤ 1, 0 ≤ t ≤ T } an inverse boundary problem with the nonlocal initial conditions the periodic conditions the non-local integral condition and with the additional condition where h i (t) (i = 1, 2) are the given functions, and u(x, t), a(t), b(t) are the required functions. We introduce the following set of functions: Definition For investigating problem (1)-(5), firstly we consider the following problem: where a(t), p(t) ∈ C[0, T ] is the given functions, y = y(t) is the unknown function, and if y = y(t) is the solution of problem (6), (7) then y(t) is continuous on [0, T ] together with all its derivatives contained in equation (6) and satisfying conditions (6), (7) in the ordinary sense. Analogously [8], the following lemma was proved.
where R is a constant. Then problem (6), (7) has only a trivial solution.
The function u(x, t), as an element of the space B 5 2,T , is continuous and has continuous derivatives u x (x, t), u xx (x, t), u xxx (x, t), u xxxx (x, t) in D T .
It can be shown that has continuous derivatives u t (x, t), u tt (x, t) in D T . It is easy to verify that the equation (1) and conditions (2), (3), (6), (7) are satisfied in the ordinary sense. Consequently, {u(x, t), a(t), b(t)} is a solution to the problem (1)-(3), (6), (7), and by Lemma 3 it is unique in the ball.
By Lemma 1 the unique solvability of the initial problem (1)-(5) follows from the theorem. The theorem is proved.
Theorem 2. Let all the conditions of Theorem 1 be fulfilled and