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

TL;DR

This paper extends fixed point theory to divergence form nonlinear nonsmooth parabolic equations with locally arbitrary growth, establishing maximum principles and nonlinear maximal regularity without Steklov techniques, inspired by heat transfer models.

Contribution

It introduces a novel approach to handle locally arbitrary growth conditions in nonlinear parabolic equations, avoiding Steklov techniques and proving fixed point existence and maximal regularity.

Findings

Established maximum principle under local growth conditions

Proved existence of fixed points for the linearized map

Achieved nonlinear maximal regularity in Sobolev spaces

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.