Abstract. Our aim in this paper is to study the existence of local (in time) solutions for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms. This equation was proposed in view of applications to binary image inpainting. We also give some numerical simulations which show the efficiency of the model.
In this paper, we propose a model for multi-color image inpainting composed of n colors. In particular, as in the binary model, i.e., the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation [4], we add a fidelity term to the corresponding Cahn-Hilliard system. We are interested in the study of the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem. The main difficulty here is that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter c, as in the Cahn-Hilliard system. Instead, we prove that the spatial average of c is dissipative. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.
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