Well-posedness and stability for a class of fourth-order nonlinear parabolic equations

TL;DR

This paper investigates the well-posedness and stability of solutions for a class of fourth-order nonlinear parabolic equations, establishing existence, uniqueness, and long-term behavior under various initial data conditions.

Contribution

It provides new results on existence, uniqueness, and stability for solutions to a specific class of nonlinear fourth-order parabolic equations with cubic growth conditions.

Findings

Existence and uniqueness of solutions for small initial data in local BMO spaces.

Analysis of large-time behavior and stability of global solutions in the cubic case.

Stability results for solutions with small initial data in VMO spaces.

Abstract

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(\xi) = \pm \lvert \xi \rvert^2 \xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.