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

Article type: Research Article

Authors: Kong, Linghua^a | Zhu, Yongbo^b | Liang, Bo^{c; *} | Wang, Ying^b

Affiliations: [a] School of Information Engineering, Dalian Ocean University, Dalian, Liaoning, China | [b] School of Science, Dalian Jiaotong University, Dalian, Liaoning, China | [c] School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui, China

Correspondence: [*] Corresponding author: Bo Liang, School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui, China. E-mail: cnliangbo@163.com.

Abstract: 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.

Keywords: Fourth-order, large-time behavior, exponential decay, dissipative entropy

DOI: 10.3233/JCM-247227

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 24, no. 4-5, pp. 2085-2094, 2024