Nonlocal unique solvability of a steady-state problem of complex heat transfer (Q327061)

The authors study the solvability of the system: \(-a \Delta\theta +v\cdot\nabla\theta +b\kappa_a |\theta|\theta^3=b\kappa_a\varphi\), \( -\alpha\Delta\varphi +\kappa_a\varphi =\kappa_a |\theta|\theta^3\) in a bounded domain \(\Omega\subset \mathbb{R}^3\), which models radiative, conductive, and convective heat transfer in the \(P_1\) diffusion approximation of the radiative transfer equation. The system is supplemented with boundary conditions: \(\theta|_{\partial\Omega}=\Theta_0\), \(\alpha\partial_n\varphi +\beta (\varphi -\Theta^4_0)|_{\partial\Omega} =0\). Using the fixed-point iteration and the Carlemann estimates for Poisson's equation, the authors prove the existence, boundedness and uniqueness of weak solutions to this system without assuming the smallness of the input data, thus extending the authors' related result in [Zh. Vychisl. Mat. Mat. Fiz. 54, No. 4, 711--719 (2014; Zbl 1313.80005); translation in Comput. Math. Math. Phys. 54, No. 4, 719--726 (2014)].