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

Abstract

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.