An Inverse Problem for a Parabolic Equation with Involution

In this paper, we investigate the solvability of an inverse problem with the Dirichlet condition for a nonlocal analog of a parabolic equation, which generalizes the classical parabolic equation in two spatial variables. Theorems on existence and uniqueness of solution to the considered problem are proved. The inverse problem is solved by using the variable separation method. To solve this problem, we use the method of separation of variables, the application of which leads to the study of the spectral problem for the Laplace equation with involution. Also, we have shown the method for reducing this problem to the known spectral problem for the classical Laplace equation (without involution). The completeness and basis property of the eigenfunctions of the obtained spectral problem are proved. This made it possible to represent the solution of the inverse problem in the form of a sum of an absolutely and uniformly converging series, expanded in terms of the obtained eigenfunctions.