Direct and inverse problems for a nonlocal heat equation under generalized- periodic boundary conditions

This paper studies direct and inverse source problems for a nonlocal heat equation with an involution in the spatial variable under generalized periodic-type boundary conditions. These conditions couple the traces of the solution and its derivatives at the endpoints of the interval and lead to a non-self-adjoint spatial operator, for which the classical separation-of-variables approach may fail due to the lack of the Riesz basis property of root functions. To overcome this difficulty, we develop a functional decomposition method that represents the solution as a specially chosen sum of two auxiliary components. This decomposition reduces the original nonlocal problem to the sequential solution of two classical heat equations with standard self-adjoint boundary conditions (a Dirichlet problem for the odd component and a Robin problem for the auxiliary even component). Using this reduction, we prove existence, uniqueness, and stability of strong Sobolev solutions for the direct initial-boundary value problem for all admissible real values of the model parameters, without imposing any arithmetic restrictions on the associated spectral ratio. We also investigate the inverse problem of recovering a stationary (space-dependent) source from a final-time observation and establish well-posedness and stability estimates for the reconstructed source and the corresponding solution.