A note on the optimal temporal decay estimates of solutions to the Cahn–Hilliard equation
A note on the optimal temporal decay estimates of solutions to the Cahn–Hilliard equation
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DOI：
10.1016/j.jmaa.2010.06.009
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发表时间：
2010-12
影响因子：
1.3
通讯作者：
Lian Duan;Shuangqiang Liu;Huijiang Zhao
中科院分区：
文献类型：
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作者：
Lian Duan;Shuangqiang Liu;Huijiang Zhao
本文讨论Cahn-Hilliard方程柯西问题解的最优时间衰减估计。这类柯西问题存在唯一的全局光滑解u(t，x)，只要光滑的非线性函数φ(U)满足局部增长条件。此外，如果φ(U)满足某种较强的局部增长条件，则关于u(t，x)的最优时间衰减估计也在Liu，Wang和赵(2007)[11]中得到。因此，一个自然的问题是如何在保证相应柯西问题整体可解的局部增长条件下，得到u(t，x)的最优时间衰减估计，本文的主要目的就是研究这一问题。我们的分析是受到Ukai，Yang和赵(2006)[15]最近发展的技术的启发，并进行了轻微的修改。
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 φ(u) satisfies a local growth condition. Furthermore if φ(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.