Long time $\mathcal H^s_{\alpha}$ stability of a classical scheme for Cahn-Hilliard equation with polynomial nonlinearity

In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and $\mathcal H^s_\alpha$ ($s=1,2,3$) spaces independent of the initial data and time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes.The uniform $\mathcal H^3_\alpha$ bound is obtained on basis of the uniform $H^1$ estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.