Existence of Rosseland equation

TL;DR

This paper proves the existence, boundedness, and uniqueness of solutions for the Rosseland equation, a nonlinear PDE modeling heat transfer in composites, using fixed point and iterative methods.

Contribution

It establishes the global existence and uniqueness of solutions for the Rosseland equation with high-order growth coefficients and mixed boundary conditions, extending to nonlinear parabolic problems.

Findings

Solutions are uniformly bounded independently of the small parameter.

Existence of solutions is proven via fixed point methods.

Uniqueness holds under certain conditions on the solution's gradient.

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.