A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:372

Article

Duan, Lian²  Liu, Shuangqian³  Zhao, Huijiang¹ 

[1] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Peoples R China

[2] Jiaxing Univ, Coll Math & Informat Sci, Jiaxing 314001, Zhejiang, Peoples R China

[3] Jinan Univ, Dept Math, Guangzhou 510632, Guangdong, Peoples R China

关键词: Cahn-Hilliard equation;    Optimal temporal decay estimates;    Sobolev's inequality;   

DOI  :  10.1016/j.jmaa.2010.06.009

来源: Elsevier

【 摘 要 】

This paper is concerned with the optimal temporal decay estimates on the solutions of the Cauchy problem of the Cahn-Hilliard equation. It is shown in Liu. Wang and Zhao (2007) [11] that such a Cauchy problem admits a unique global smooth solution u(t,x) provided that the smooth nonlinear function phi(u) satisfies a local growth condition. Furthermore if phi(u) satisfies a somewhat stronger local growth condition, the optimal temporal decay estimates on u(t,x) are also obtained in Liu, Wang and Zhao (2007) [11]. Thus a natural question is how to deduce the optimal temporal decay estimates on u(t, x) only under the local growth condition which is sufficient to guarantee the global solvability of the corresponding Cauchy problem and the main purpose of this paper is devoted to this problem. Our analysis is motivated by the technique developed recently in Ukai, Yang and Zhao (2006) [15] with a slight modification. (C) 2010 Elsevier Inc. All rights reserved.

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