Existence of Rosseland equation

Abstract

The global boundness, existence and uniqueness are presented for the kind of Rosseland equation with a small parameter. This problem comes from conduction-radiation coupled heat transfer in the composites; it's with coefficients of high order growth and mixed boundary conditions. A linearized map is constructed by fixing the function variables in the coefficients and the right-hand side. The solution to the linearized problem is uniformly bounded based on De Giorgi iteration; it is bounded in the H\"older space from a Sobolev-Campanato estimate. This linearized map is compact and continuous so that there exists a fixed point. All of these estimates are independent of the small parameter. At the end, the uniqueness of the solution holds if there is a big zero-order term and the solution's gradient is bounded. This existence theorem can be extended to the nonlinear parabolic problem.