Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting

In this article, we are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of a generalization of the Cahn-Hilliard equation with a fidelity term (integrated over Ω\D instead of the entire domain Ω, D ⊂⊂ Ω). Such a model has, in particular, applications in image inpainting. The difficulty here is that we no longer have the conservation of mass, i.e. of the spatial average of the order parameter u, as in the Cahn-Hilliard equation. Instead, we prove that the spatial average of u is dissipative. We finally give some numerical simulations which confirm previous ones on the efficiency of the model.