NONLINEAR INVERSE PROBLEMS WITH INTEGRAL OVERDETERMINATION FOR NONSTATIONARY DIFFERENTIAL EQUATIONS OF HIGH ORDER

Abstract

Nonlinear inverse coecient problems for nonstationary higher order dierential equations of pseudohyperbolic type are the object of research. More precisely, we study the problems of determining both the solution of the corresponding equation, an unknown coecient at the solution or at the time derivative of the solution in the equation. A distinctive feature of these problems is the fact that the unknown coecient is a function of time only. Integral overdetermination is used as an additional condition. We prove the existence theorems of regular solutions (those solutions that have all generalized derivatives in the sense of S.L. Sobolev). The technique of the proof relies on the transition from the original inverse problem to a new direct problem for an auxiliary integral-dierential equation, and then on the proof of solvability of the latter and construction of some solution of the original inverse problem from a solution of the auxiliary problem.