Mathematical analysis of a fourth-order degenerate equation with a nonlinear second-order term

Article type: Research Article

Authors: Kong, Linghua^a | Liang, Bo^{b; *} | Li, Ying^b | Wang, Ying^b | Wu, Xiaoqin^b

Affiliations: [a] School of Information Engineering, Dalian Ocean University, Dalian, Liaoning, China | [b] School of Science, Dalian Jiaotong University, Dalian, Liaoning, China

Correspondence: [*] Corresponding author: Bo Liang, School of Science, Dalian Jiaotong University, Dalian, Liaoning 116028, China. E-mail: cnliangbo@163.com.

Abstract: A fourth-order partial differential model in this paper is called the thin-film equation in the field of fluid theory. The existence of the weak solution is obtained by solving two approximation problems. In order to perform the limit for small parameters in the approximation problems, the Galerkin method and the entropy functional method are both used. By means of some classical compactness results we can give the existence of nonnegative weak solutions. Finally, by redefining the entropy functional, the long-time exponential decay is given in the sense of L1-norm.

Keywords: Fourth-order degenerate equation, existence, long-time behavior, entropy functional

DOI: 10.3233/JCM-226854

Journal: Journal of Computational Methods in Sciences and Engineering, vol. 23, no. 6, pp. 3271-3281, 2023