  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, bring additional difficulty in mathematical analysis, compared
to the well-developed theory for Milne problem of linear transport equation.
With the help of the monotonicity property of the second order ODE, we prove
the existence of the nonlinear Milne problem on a finite interval using
monotonic convergence theorems. Then the solution is extended to the half-space
using a uniform weighted estimate and the compactness method. Moreover, the
solutions are proved to converge to constants as $x\to \infty$. Therefore, the
linear stability analysis is used to study the uniqueness of the nonlinear
Milne problem. The existence and uniqueness for the linearized system is
established under a spectral assumption on the solution of the nonlinear
problem. The spectral assumption is shown to be satisfied when the boundary
data is close to the well-prepared case by using a generalized Hardy's
inequality. The uniqueness of the solution to the half-line nonlinear Milne
problem is established in a neighborhood of solutions satisfying a spectral
assumption and the energy estimate.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.
