On quasilinear parabolic equations and continuous maximal regularity

TL;DR

This paper develops a theoretical framework for quasilinear parabolic equations with singular lower-order terms, establishing well-posedness, stability, and global existence results, and applies these to surface diffusion flow.

Contribution

It introduces new methods for analyzing quasilinear parabolic problems with singular structures and extends stability principles to critical initial spaces.

Findings

Proved well-posedness and Lipschitz continuity of semiflows.

Established global existence of solutions under certain conditions.

Extended stability principles to critical spaces.

Abstract

We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.