On quasilinear parabolic evolution equations in weighted $L_p$-spaces

TL;DR

This paper develops a geometric framework for quasilinear parabolic equations in weighted $L_p$-spaces, establishing existence, uniqueness, and long-term behavior of solutions using maximal regularity techniques.

Contribution

It introduces a new geometric approach for analyzing quasilinear parabolic problems in weighted $L_p$-spaces, including applications to free boundary problems.

Findings

Proved existence and uniqueness of solutions.

Analyzed long-time behavior and $ ext{ω}$-limit sets.

Applied techniques to free boundary value problems.

Abstract

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted $L_p$-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the $\om$-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) $L_p$-spaces.