Stability of the Nonlinear Milne Problem for Radiative Heat Transfer System

This paper focuses on the nonlinear Milne problem of the radiative heat transfer system on the half-space. The nonlinear model is described by a second order ODE for temperature coupled to transport equation for radiative intensity. The nonlinearity of the fourth power Stefan–Boltzmann law of black body radiation, brings additional difficulty in mathematical analysis, compared to the well-developed theory for the Milne problem of the linear transport equation. To overcome this difficulty, the monotonicity properties of the second order ODE are used, together with the uniform estimate and compactness method, to prove the existence of the nonlinear Milne problem and to show the exponential decay of solutions. Moreover, the linear stability of the problem is established under a spectral assumption on its solutions, and the uniqueness of the nonlinear Milne problem is established in a neighborhood of solutions satisfying a spectral assumption or when the boundary conditions are close to the well-prepared case. The current work extends the study of Milne problem for linear transport equations and provides a comprehensive study on the nonlinear Milne problem of radiative heat transfer systems.