# American Institute of Mathematical Sciences

• Previous Article
Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line
• CPAA Home
• This Issue
• Next Article
Response solutions to harmonic oscillators beyond multi–dimensional Brjuno frequency
February  2021, 20(2): 495-510. 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, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277
##### References:

show all references

##### References:
(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
 [1] Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001 [2] Amanda E. Diegel. A C0 interior penalty method for the Cahn-Hilliard equation. Electronic Research Archive, , () : -. doi: 10.3934/era.2021030 [3] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393 [4] Matthias Ebenbeck, Harald Garcke, Robert Nürnberg. Cahn–Hilliard–Brinkman systems for tumour growth. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021034 [5] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006 [6] Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022 [7] Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057 [8] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 [9] Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 [10] Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227 [11] Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005 [12] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 [13] Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $n$-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018 [14] Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021018 [15] Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021032 [16] Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021 [17] Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 [18] De-han Chen, Daijun jiang. Convergence rates of Tikhonov regularization for recovering growth rates in a Lotka-Volterra competition model with diffusion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021023 [19] Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005 [20] Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402

2019 Impact Factor: 1.105