September  2014, 19(7): 2013-2026. doi: 10.3934/dcdsb.2014.19.2013

On a generalized Cahn-Hilliard equation with biological applications

1. 

Université de La Rochelle, Laboratoire Mathématiques, Image et Applications, Avenue Michel Crépeau, F-17042 La Rochelle Cedex, France

2. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex

3. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  March 2013 Revised  May 2013 Published  August 2014

In this paper, we are interested in the study of the asymptotic behavior of a generalization of the Cahn-Hilliard equation with a proliferation term and endowed with Neumann boundary conditions. Such a model has, in particular, applications in biology. We show that either the average of the local density of cells is bounded, in which case we have a global in time solution, or the solution blows up in finite time. We further prove that the relevant, from a biological point of view, solutions converge to $1$ as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.
Citation: Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Amsterdam, (1992).

[2]

J. W. Cahn, On spinodal decomposition,, Acta. Metall., 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102.

[4]

V. Chalupeckí, Numerical studies of Cahn-Hilliard equations and applications in image processing,, in Proceedings of Czech-Japanese Seminar in Applied Mathematics, (2004).

[5]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4.

[6]

L. Cherfils, M. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions,, Discrete Cont. Dyn. Systems, 27 (2010), 1511. doi: 10.3934/dcds.2010.27.1511.

[7]

D. Cohen and J. M. Murray, A generalized diffusion model for growth and dispersion in a population,, J. Math. Biol., 12 (1981), 237. doi: 10.1007/BF00276132.

[8]

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. doi: 10.4171/IFB/64.

[9]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, in Mathematical Models for Phase Change Problems, 88, (1989).

[10]

C. M. Elliott, D. A. French and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation,, Numer. Math., 54 (1989), 575. doi: 10.1007/BF01396363.

[11]

, FreeFem++ is freely, available at http://www.freefem.org/ff++., ().

[12]

M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term,, Math. Models Methods Appl. Sci., 20 (2010), 1363. doi: 10.1142/S0218202510004635.

[13]

S. Injrou and M. Pierre, Stable discretizations of the Cahn-Hilliard-Gurtin equations,, Discrete Cont. Dyn. Systems, 22 (2008), 1065. doi: 10.3934/dcds.2008.22.1065.

[14]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.051129.

[15]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.031902.

[16]

R. V. Kohn and F. Otto, Upper bounds for coarsening rates,, Commun. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4.

[17]

J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. Phys., 65 (1975), 53. doi: 10.1016/0003-4916(71)90162-X.

[18]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate,, Commun. Math. Phys., 195 (1998), 435. doi: 10.1007/s002200050397.

[19]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics,, Arch. Ration. Mech. Anal., 151 (2000), 187. doi: 10.1007/s002050050196.

[20]

A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term,, Appl. Anal., 92 (2013), 1308. doi: 10.1080/00036811.2012.671301.

[21]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0.

[22]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations,, Commun. Partial Diff. Eqns., 14 (1989), 245. doi: 10.1080/03605308908820597.

[23]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.

[24]

A. Novick-Cohen, The Cahn-Hilliard equation,, in Handbook of Differential Equations, 4 (2008), 201. doi: 10.1016/S1874-5717(08)00004-2.

[25]

A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films,, Rev. Mod. Phys., 69 (1997), 931. doi: 10.1103/RevModPhys.69.931.

[26]

M. Pierre, Habilitation Thesis,, Université de Poitiers, (1997).

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[28]

U. Thiele and E. Knobloch, Thin liquid films on a slightly inclined heated plate,, Phys. D, 190 (2004), 213. doi: 10.1016/j.physd.2003.09.048.

[29]

S. Tremaine, On the origin of irregular structure in Saturn's rings,, Astron. J., 125 (2003), 894. doi: 10.1086/345963.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Amsterdam, (1992).

[2]

J. W. Cahn, On spinodal decomposition,, Acta. Metall., 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1.

[3]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102.

[4]

V. Chalupeckí, Numerical studies of Cahn-Hilliard equations and applications in image processing,, in Proceedings of Czech-Japanese Seminar in Applied Mathematics, (2004).

[5]

L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials,, Milan J. Math., 79 (2011), 561. doi: 10.1007/s00032-011-0165-4.

[6]

L. Cherfils, M. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions,, Discrete Cont. Dyn. Systems, 27 (2010), 1511. doi: 10.3934/dcds.2010.27.1511.

[7]

D. Cohen and J. M. Murray, A generalized diffusion model for growth and dispersion in a population,, J. Math. Biol., 12 (1981), 237. doi: 10.1007/BF00276132.

[8]

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. doi: 10.4171/IFB/64.

[9]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, in Mathematical Models for Phase Change Problems, 88, (1989).

[10]

C. M. Elliott, D. A. French and F. A. Milner, A second order splitting method for the Cahn-Hilliard equation,, Numer. Math., 54 (1989), 575. doi: 10.1007/BF01396363.

[11]

, FreeFem++ is freely, available at http://www.freefem.org/ff++., ().

[12]

M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term,, Math. Models Methods Appl. Sci., 20 (2010), 1363. doi: 10.1142/S0218202510004635.

[13]

S. Injrou and M. Pierre, Stable discretizations of the Cahn-Hilliard-Gurtin equations,, Discrete Cont. Dyn. Systems, 22 (2008), 1065. doi: 10.3934/dcds.2008.22.1065.

[14]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.051129.

[15]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.031902.

[16]

R. V. Kohn and F. Otto, Upper bounds for coarsening rates,, Commun. Math. Phys., 229 (2002), 375. doi: 10.1007/s00220-002-0693-4.

[17]

J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. Phys., 65 (1975), 53. doi: 10.1016/0003-4916(71)90162-X.

[18]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate,, Commun. Math. Phys., 195 (1998), 435. doi: 10.1007/s002200050397.

[19]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics,, Arch. Ration. Mech. Anal., 151 (2000), 187. doi: 10.1007/s002050050196.

[20]

A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term,, Appl. Anal., 92 (2013), 1308. doi: 10.1080/00036811.2012.671301.

[21]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0.

[22]

B. Nicolaenko, B. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations,, Commun. Partial Diff. Eqns., 14 (1989), 245. doi: 10.1080/03605308908820597.

[23]

A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives,, Adv. Math. Sci. Appl., 8 (1998), 965.

[24]

A. Novick-Cohen, The Cahn-Hilliard equation,, in Handbook of Differential Equations, 4 (2008), 201. doi: 10.1016/S1874-5717(08)00004-2.

[25]

A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films,, Rev. Mod. Phys., 69 (1997), 931. doi: 10.1103/RevModPhys.69.931.

[26]

M. Pierre, Habilitation Thesis,, Université de Poitiers, (1997).

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[28]

U. Thiele and E. Knobloch, Thin liquid films on a slightly inclined heated plate,, Phys. D, 190 (2004), 213. doi: 10.1016/j.physd.2003.09.048.

[29]

S. Tremaine, On the origin of irregular structure in Saturn's rings,, Astron. J., 125 (2003), 894. doi: 10.1086/345963.

[1]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[2]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[3]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[4]

Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315

[5]

Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855

[6]

Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543

[7]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

[8]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[9]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

[10]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033

[11]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

[12]

Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859

[13]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[14]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

[15]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037

[16]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275

[17]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137

[18]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

[19]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251

[20]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (14)

[Back to Top]