WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 24, 2025
On the Nonlocal Cahn-hilliard Inpainting Model with Singular Nonlinearity
Authors: , , , ,
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Abstract: 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.
Keywords:
Nonlocal Cahn-Hilliard, Well-possednes, Singular potential, Image Inpainting, Separation
property, Galerkin algorithm
Pages: 726-742
DOI: 10.37394/23206.2025.24.72