The coefficient inverse problem for a pseudoparabolic equation of the third order

In this paper, we consider the coefficient inverse problem for a third-order pseudoparabolic equation, which represents mathematical model for the movement of moisture and salts in soils. Such non-classical equations are also called Sobolev-type equations. At present, the study of direct and inverse problems for a pseudoparabolic equation is readily developing due to the needs of modeling and controlling processes in hydrodynamics, mechanics, thermal physics and continuum mechanics. At the same time, the investigation of coefficient inverse problems is also important, since they are used in solving problems of planning the development of oil fields, in particular, in determining the filtration parameters of fields, in creating new types of measuring equipment, in solving environmental monitoring problems, etc. Thus both trend directions such as pseudoparabolic equations and coefficient inverse problems are relevant due to the abundance of various applications where such non-classical objects arise. In this work, the Galerkin method is used to prove the existence of the solution for the inverse coefficient problem and obtained sufficient conditions for the blow up of its solution in a finite time in a bounded domain. Moreover, authors developed the algorithm for the numerical solution of the given problem by using the finite difference method. In addition, computational experiments were carried out illustrating the theoretical calculations obtained in the work.