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

The decay behavior of a class of equation wt=-∇⋅(wn⁢∇⁡Δ⁢w+α⁢wn-1⁢Δ⁢w⁢∇⁡w+β⁢wn-2⁢|∇⁡w|2⁢∇⁡w) is considered under the Neumann boundary condition. The equation can be viewed as a generalization of the thin film equation wt+(wn⁢wx⁢x⁢x)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 L1-norm in the multi-dimensional case, and decays algebraically in L∞-norm in the one-dimensional case. The critical step solving the problem is to construct appropriate dissipative entropies.