Inverse Problems of Determination of the Time-Dependent Coefficient of a Parabolic Equation with Involution and Antiperiodicity Conditions

By the method of separation of variables, we construct the solution of the investigated problem with an unknown coefficient in the differential equation. We study the properties of the spectral problem for the second-order differential equation with involution. The dependence of the spectrum of the operator of the problem, its multiplicity, and the structure of the system of root functions and particular solutions of the problem on the involutive part of the equation is analyzed. We establish the conditions for the existence and uniqueness of the solution to the inverse problem. To determine the unknown coefficient, we find and solve the Volterra integral equation of the second kind.