A note on a fourth order degenerate parabolic equation in higher space dimensions (Q2458838)

The author deals with a problem which appears in the lubrication theory for thin viscous films. The following initial boundary value problem of fourth order degenerate parabolic equation in higher space dimensions is considered \[ \begin{cases} u_t+ \text{div}( | u| ^n\nabla \Delta u) =0 &\text{in }\Omega \times (0,T], \\ \frac{\partial u}{\partial \nu }=\frac \partial {\partial \nu }\Delta u=0&\text{on }\partial \Omega \times (0,T], \\ u(x,0) =u_0(x)&\text{in }\Omega,\end{cases} \] where \(\Omega \subset \mathbb{R}^N\)\, is an open and bounded domain, \(T>0\), \(\nu \) denotes the unit outer normal vector to \(\partial \Omega .\) Here the function \(u\) is the height of the film. The author is concerned with existence, positivity property and long-time behaviour of the weak solution of the problem (1) and the main result of the paper generalize a result of Dal Passo, Garke and Grün to the space dimension \(N=4.\)