One nonlocal problem of determination of the temperature and density of heat sources

We consider one family of problems simulating the determination of the temperature and density of heat sources from given values of the initial and final temperature. The mathematical statement of these problems leads to the inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. A specific feature of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. We prove the unique existence of a generalized solution to the mentioned problem.