Divergence form nonlinear nonsmooth parabolic equations with locally arbitrary growth conditions and nonlinear maximal regularity

Abstract

This is a generalization of our prior work on the compact fixed point theory for the elliptic Rosseland-type equations. We obtain the maximum principle without the technical Steklov techniques. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions instead of the common global restricted growth conditions. Its physical meaning is: the absolute temperature should be positive and bounded. There exists a fixed point for the linearized map (compact and continuous in $L^2$) in a closed convex set. We also consider the nonlinear maximal regularity in Sobolev space.