# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020277

## On the Cahn-Hilliard equation with mass source for biological applications

 1 Lebanese International University, School of Arts and Sciences, Department of Mathematics and Physics, Bekaa campus, Lebanon 2 Lebanese University, Faculty of Sciences, Department of Mathematics, Houch el Oumara, Zahle, Lebanon 3 Politehnica University of Bucharest, Splaiul Independentei 313, 060042, Bucharest, Romania

* Corresponding author

Received  December 2019 Revised  September 2020 Published  December 2020

This article deals with some generalizations of the Cahn–Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn–Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn–Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.

Citation: Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020277
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(a) Initial datum at $t = 0$. (b) Solution after $2000$ iterations. (c) Solution after $3000$ iterations. (d) Solution after $4000$ iterations. (e) Solution after $5000$ iterations. (f) Solution after $6000$ iterations. (g) Solution after $8000$ iterations. (h) Solution after $9000$ iterations
(a) initial image. (b) mask. (c) Inpainting result
(a) original image. (b) mask. (c) Inpainting result
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