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.

Our aim in this article is to propose a generalization of the Bertozzi--Esedoglu--Gillette--Cahn--Hilliard equation, introduced for binary image inpainting, for grayscale image inpainting. In particular, we consider the solution to the corresponding Cahn--Hilliard inpainting model as a complex valued function. We are interested in the study of the well-posedness and of the asymptotic behavior, in terms of finite-dimensional attractors, of the associated dynamical system. We have to face two major difficulties here. The first one comes from the fact that we no longer have the conservation of mass, i.e., of the spatial average of the order parameter $u$, contrary to the classical Cahn--Hilliard equation. The second one is due to the estimates on the nonlinear terms, combined with the fact that the order parameter $u$ is complex valued. We finally give numerical simulations which confirm and extend previous ones on the efficiency of the binary model.

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 $\Omega\backslash D$ instead of the entire domain $\Omega$, $D \subset \subset \Omega$). 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.

In this paper, we are interested in the study of the solution to a generalization of the Cahn–Hilliard equation endowed with Neumann boundary conditions. This model has, in particular, applications in biology and in chemistry. We show that the solutions blow up in finite time or exist globally in time. We further prove that the relevant, from a biological and a chemical point of view, solutions converge to a constant as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.