Time-periodic problem for inhomogeneous heat equations in a two-layer domain with Dirichlet boundary and linear transmission conditions

Heat conduction problems in multilayered media are widely used to model processes in heterogeneous structures, including composites, semiconductors, biomaterials, and nanostructures. In this context, the study of models that describe the evolution of the temperature field under periodic forcing is relevant. In the case, the initial conditions lose their physical meaning and are naturally replaced by time-periodic conditions.

This work investigates a two-layer heat conduction problem for inhomogeneous equations with Dirichlet boundary conditions in the spatial variable and periodic conditions in time. The problem is formulated in the cylindrical domain, defined as the Cartesian product of an interval on the real line and the unit circle. At the interface between the two layers, transmission conditions generalize the classical continuity conditions of the temperature and the heat flux.

The solution is determined as a time-periodic regime, consistent with physical scenarios such as periodic heating or cooling. The existence and uniqueness of the solution in Sobolev spaces of time-periodic functions are proved. The analysis is based on the method of separation of variables, Fourier series expansions, the construction of Green’s functions, and estimates of the determinants related to the problem. A representative model problem is presented along with an approximate solution obtained using a truncated Fourier series expansion.