NONE 20260116132343154 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Hussein Fakih Department of Mathematics and Physics Lebanese International University (LIU) LEBANON Sami Hammoud Department of Mathematics and Physics Lebanese International University (LIU) LEBANON Salwa Manosur Department of Mathematics and Physics Lebanese International University (LIU) LEBANON Ragheb Mghames Department of Mathematics and Physics Lebanese International University (LIU) LEBANON Wafa Saoud Department of Preparatory Classes Saint-Joseph University of Beirut Beirut, LEBANON In this paper, we consider the nonlocal Cahn-Hilliard equation with a fidelity term and singular nonlinearities, which has numerous applications in biology and image inpainting. First, we prove the existence of a unique solution. We approximate the singular nonlinearities by regular nonlinearities, from which we derive important priori-estimates for the solution of the approximating problem. We then establish the existence of a solution to the approximating problem using the Faedo-Galerkin method. Additionally, we demonstrate that the solution of the Cahn-Hilliard equation in question is of higher regular by applying Sobolev embedding theorems. Finally, we discuss the important separation property of the solution that allows us to obtain a global (in time) unique solution of the problem. 12 31 2025 12 31 2025 726 72 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.72 https://wseas.com/journals/mathematics/2025/b465106-038(2025).pdf N. Abukhdeir, D. Vlachos, M. Katsoulakis and M. Plexousakis, Long–time integration methods for mesoscopic models of pattern–forming systems. J. Comput. Phys. 18 (2013) 2211-2238. P.W. Bates and J. Han, The Dirchlet boundary for a nonlocal Cahn–Hilliard equation, J. Math. Anal. App .311 (2005), 289–312. P.W. Bates and J. Han, The Neumann boundary for a nonlocal Cahn–Hilliard equation, J. Diff. Equ. 212 (2005), 235–277. A.L. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model, Simul. 6 (2007) 913–936. M. Bahiana, and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A 41 (1990), 6763-6771 J.W. Cahn, On spindol decomposition, Acta Metallurgica 9 (1961), 795–801. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications. Discrete Contin Dyn.Syst. Ser. B 19 (2014), 2013–2026. L. Cherfils, H. Fakih, and A. Miranville, A complex version of the Cahn-Hilliard equation for grayscale image inpainting, Multiscale Modeling and Simulation 15 (1) 575-605. L. Cherfils, H. Fakih, M. Grasselli and A. Miranville, A convergent convex spliting scheme for a nonloacl Cahn–Hilliard–Oona type equation with a transport term, Math. Sc. App. (2018) 225-250. D. Cohen and J.M. Murray, A generalized diffusion model for growth and dispersion in a population. J. Math. Biol. 12 (1981) 237–248. F. Della Porta M. Grasselli, Convective nonlocal Cahn-Hilliard equations with reaction terms. Disc. Cont. Dyn. Syst. B 20 (2015) 1529-1553. I.C. Dolcetta, S.F. Vita and R. March, Area preserving curve shortening flows: from phase separation to image processing, Interfaces Free Bound. 4 (2002) 325-343. S. Frigeri, M. Grasselli,and E. Rocca, A diffuse interfance model for two phase incompressible flows with nonlocal interactions and nonconstant mobility, preprint arXiv: 13036446v2 (2014), 1-44. G. Giacomin and J.L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett. 76 (1996) 1094-1096. G. Giacomin and J.L. Lebowitz, Exact Interface description of phase segregation in model alloys with long range interactions, Math. Rev. Lett. 58 (1998) 1707-1729. H. Gajewski and K. Zacharias, On nonlocal phase separation model, J. Math. Anal. Appl. 286 (2003) 11-31. D. Giglbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Verlag, Berlin, (1977). P. Krejci, E. Rocca , and J. Sprekels, A nonlocal phase field model with nonconstant specific heat, Interfaces Free Bound. 9 (2007), 285-306. S. Melchionna and E. Rocca, On a nonlocal Cahn-Hilliard equation with a reaction term, Adv. Math. Sci. Appl. 24 (2014), 461-497. A. Miranville, Asymptotic behaviour of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal. 92 (2013), 1308-1321.