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

In this paper, we investigate the large-times behavior of weak solutions to the fourth-order degenerate parabolic equation ut = −(unuxxx)x modeling the evolution of thin films. In particular, for all n > 0, we prove exponential decay of u(x, t) towards its mean value (1/Ω) ∫Ω u dx in L1-norm for long times and we give the explicit (n-dependent) rate of decay. The result is based on classical entropy estimates, and on detailed lower bounds for the entropy production.