American Institute of Mathematical Sciences

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 and 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), 795-801. 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), 258-267. 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, Czech Technical University in Prague. [5] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. 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), 1511-1533. 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), 237-249. 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), 325-343. 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, (ed. J. F. Rodrigues), International Series of Numerical Mathematics, 88, Birkhäuser, Basel, 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), 575-590. 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), 1363-1390. doi: 10.1142/S0218202510004635. [13] S. Injrou and M. Pierre, Stable discretizations of the Cahn-Hilliard-Gurtin equations, Discrete Cont. Dyn. Systems, 22 (2008), 1065-1080. 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), 051129. 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), 0319021. doi: 10.1103/PhysRevE.74.031902. [16] R. V. Kohn and F. Otto, Upper bounds for coarsening rates, Commun. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4. [17] J. S. Langer, Theory of spinodal decomposition in alloys, Ann. Phys., 65 (1975), 53-86. 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), 435-464. 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), 187-219. doi: 10.1007/s002050050196. [20] A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321. 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, Evolutionary Partial Differential Equations, 4, (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008, 103-200. 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), 245-297. doi: 10.1080/03605308908820597. [23] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. [24] A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, 4, (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008, 201-228. 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), 931-980. 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, Applied Mathematical Sciences, 68, Springer-Verlag New York, 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), 213-248. 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), 894-901. doi: 10.1086/345963.

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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), 795-801. 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), 258-267. 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, Czech Technical University in Prague. [5] L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596. 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), 1511-1533. 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), 237-249. 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), 325-343. 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, (ed. J. F. Rodrigues), International Series of Numerical Mathematics, 88, Birkhäuser, Basel, 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), 575-590. 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), 1363-1390. doi: 10.1142/S0218202510004635. [13] S. Injrou and M. Pierre, Stable discretizations of the Cahn-Hilliard-Gurtin equations, Discrete Cont. Dyn. Systems, 22 (2008), 1065-1080. 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), 051129. 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), 0319021. doi: 10.1103/PhysRevE.74.031902. [16] R. V. Kohn and F. Otto, Upper bounds for coarsening rates, Commun. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4. [17] J. S. Langer, Theory of spinodal decomposition in alloys, Ann. Phys., 65 (1975), 53-86. 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), 435-464. 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), 187-219. doi: 10.1007/s002050050196. [20] A. Miranville, Asymptotic behavior of a generalized Cahn-Hilliard equation with a proliferation term, Appl. Anal., 92 (2013), 1308-1321. 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, Evolutionary Partial Differential Equations, 4, (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008, 103-200. 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), 245-297. doi: 10.1080/03605308908820597. [23] A. Novick-Cohen, The Cahn-Hilliard equation: Mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. [24] A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations, Evolutionary Partial Differential Equations, 4, (eds. C. M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008, 201-228. 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), 931-980. 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, Applied Mathematical Sciences, 68, Springer-Verlag New York, 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), 213-248. 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), 894-901. doi: 10.1086/345963.
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