On a Cahn–Hilliard–Oono model for image segmentation

Article type: Research Article

Authors: Li, Lu^{a; b; *}

Affiliations: [a] Laboratoire I3M et Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348, Equipe DACTIM-MIS, Site du Futuroscope – Téléport 2, 11 Boulevard Marie et Pierre Curie – Bâtiment H3 – TSA 61125, 86073 Poitiers Cedex 9, France | [b] CHU de Poitiers, 2 rue de la Milétrie, 86000 Poitiers, France

Correspondence: [*] Corresponding author. E-mail: Lu.Li@math.univ-poitiers.fr.

Abstract: This paper studies firstly the well-posedness and the asymptotic behavior of a Cahn–Hilliard–Oono type model, with cubic nonlinear terms, which is proposed for image segmentation. In particular, the existences of the global attractor and the exponential attractor have been proved, and it shows that the fractal dimension of the global attractor will tend to infinity as α→0. The difficulty here is that we no longer have the conservation of mass. Furthermore, this model with logarithmic nonlinear terms has been studied as well. One difficulty here is to make sure that the logarithmic terms can pass to the limit under the standard Galerkin scheme. Another difficulty is to prove additional regularities on the solutions which is essential to prove a strict separation from the pure states 0 and 1 in one and two space dimensions. It eventually shows that the dimension of the global attractor is finite by proving the existence of the exponential attractor.

Keywords: Image segmentation, Cahn–Hilliard–Oono equation, well-posedness, global attractor, exponential attractor, strict separation

DOI: 10.3233/ASY-221801

Journal: Asymptotic Analysis, vol. 132, no. 3-4, pp. 519-548, 2023