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

Abstract

The decay behavior of a class of equation $w_t=-\nabla \cdot \left(w^n\nabla \mathrm{\Delta }w+\alpha w^{n-1}\mathrm{\Delta }w\nabla w+\beta w^{n-2}{|\nabla w|}^2\nabla w\right)$ is considered under the Neumann boundary condition. The equation can be viewed as a generalization of the thin film equation $w_t+{\left(w^n{w}_{xxx}\right)}_x=0$ , 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 $L^{\mathrm{\infty }}$ -norm in the one-dimensional case. The critical step solving the problem is to construct appropriate dissipative entropies.