Time rescaling and asymptotic behavior of some fourth-order degenerate diffusion equations (Q1609133)

The authors investigate the large-time behavior of weak solutions of the degenerate parabolic fourth-order equation \(u'=-(|u|^nu_{xxx})_x\) modeling the evolution of thin films. The main result says that, for all \(n>0\), \(u\) decays exponentially as \(t\to\infty\) towards its mean value \((|\Omega|)^{-1} \int_\Omega u\) (i.e. the limiting stationary state) in the \(L^1\)-norm. Explicit decay rates (depending on \(n\)) are presented.