Well-posedness and long time behavior of a perturbed Cahn-Hilliard system with regular potentials (Q2857019)

The aim of this paper is to study the well-posedness and long time behaviour, in terms of finite-dimensional attractors, of a perturbed Cahn-Hilliard equation, NEWLINE\[NEWLINE\partial_tu + \Delta(\Delta u- f(u)) + \varepsilon(-\Delta u+f(u))=0. NEWLINE\]NEWLINE This equation differs from the usual Cahn-Hilliard equation by the presence of an additional Allen-Cahn-type term \(\varepsilon(-\Delta u+f(u))\).NEWLINENEWLINEThe main results are related to the existence of attractors. On one hand exponential attraction is shown, while on the other robustness of the attractors is studied as \(\varepsilon\) goes to zero.NEWLINENEWLINEThe existence of solutions is studied via spectral Galerkin-methods, where uniform a priori estimates are established.