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

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 $F(\xi )=\pm |\xi {|}^2\xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.