Stationary problem of radiative heat transfer with Cauchy boundary conditions

A stationary problem of radiative-conductive heat transfer in a three-dimensional domain is studied in the ${{P}_{1}}$-approximation of the radiative transfer equation. A formulation is considered in which the boundary conditions for the radiation intensity are not specified but an additional boundary condition for the temperature field is imposed. Nonlocal solvability of the problem is established, and it is shown that the solution set is homeomorphic to a finite-dimensional compact. A condition for the uniqueness of the solution is presented.