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In this paper, we study the diffusive limit of the steady state radiativeheat transfer system for non-homogeneous Dirichlet boundary conditions in abounded domain with flat boundaries. A composite approximate solution isconstructed using asymptotic analysis taking into account of the boundarylayers. The convergence to the approximate solution in the diffusive limit isproved using a Banach fixed point theorem. The major difficulty lies on thenonlinear coupling between elliptic and kinetic transport equations. Toovercome this problem, a spectral assumption ensuring the linear stability ofthe boundary layers is proposed. Moreover, a combined $L^2$-$L^\infty$ estimateand the Banach fixed point theorem are used to obtain the convergence proof.This results extend our previous work \cite{ghattassi2020diffusive} for thewell-prepared boundary data case to the ill-prepared case when boundary layerexists.

Diffusive limits of the steady state radiative heat transfer system: Boundary layers