The maximum principle for the transport equation in the case of Compton scattering

The properties of solutions to the transport equation describing the Compton scattering of photons is investigated. The maximum and minimum principles are proved for this equation. According to them, the radiation density within a region cannot be greater than the maximum positive value of the incident radiation density and cannot be less than its minimum negative value. Furthermore, conditions under which a solution to the equation under consideration is constant are presented. The results of this work are obtained under the assumption that the properties of the medium change continuously with respect to the spatial and energy variables.