Asymptotic behavior of solutions to a fourth-order degenerate parabolic equation

The decay behavior of a class of equation w(t) = -del center dot (w(n)del Delta w + alpha w(n-1) Delta w del w + beta w(n-2 vertical bar)del w(vertical bar 2)del w) is considered under the Neumann boundary condition. The equation can be viewed as a generalization of the thin film equation w(t) + (w(n) w(xxx))(x) = 0, which can be used to describe the movement of the skinny viscous layer of compressible fluid along the slope. We obtain that the solution decays exponentially in L-1-norm in the multi-dimensional case, and decays algebraically in L-infinity-norm in the one-dimensional case. The critical step solving the problem is to construct appropriate dissipative entropies.