Inverse Problem for the Sobolev Type Equation of Higher Order

Alyona Zamyshlyaeva,Aleksandr Lut

doi:10.3390/math9141647

Abstract

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with the Cauchy and overdetermination conditions. This problem was reduced to two equivalent problems in the aggregate: regular and singular. For these problems, the theory of polynomially bounded operator pencils is used. The unknown coefficient of the original equation is restored using the method of successive approximations. The main result of this work is a theorem on the unique solvability of the original problem. This study continues and generalizes the authors’ previous research in this area. All the obtained results can be applied to the mathematical modeling of various processes and phenomena that fit the problem under study.

Highlights

Let U, F, Y be Banach spaces, operators A, B0, B1, ..., Bn−1 ∈ L(U ; F ), i.e., linear and continuous operators defined on U and acting to F, ker A = {0}, C ∈ L(U ; Y ), given functions χ : [0, T] → L(Y; F ), f : [0, T] → F, Ψ : [0, T] → Y
The paper [7] contains a condition for the existence of a weak, local, timely solution to the Cauchy problem for a model Sobolev type equation
We study the unique solvability of the regular problem by reducing it to an equivalent problem of the first order and achieving the necessary smoothness for the required function q using the method of successive approximations

Summary

Introduction

The authors have obtained the result when studying the inverse problem, but only in the case of the second-order, Sobolev type equation [1]. The paper [7] contains a condition for the existence of a weak, local, timely solution to the Cauchy problem for a model Sobolev type equation. In the study of the direct problem for a higher-order, Sobolev type equation, the phase space method was used [10]. We study the unique solvability of the regular problem by reducing it to an equivalent problem of the first order and achieving the necessary smoothness for the required function q using the method of successive approximations. The significance of the obtained results is given in both the development of the studied theory and their practical application

Preliminary Information

Reduction of the Initial Inverse Problem

Solution of the Regular Inverse Problem

Solvability of the Original Inverse Problem

Discussion