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

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