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June  2015, 35(6): 2423-2442. doi: 10.3934/dcds.2015.35.2423

On a Cahn-Hilliard type phase field system related to tumor growth

Pierluigi Colli 1, , Gianni Gilardi 1, and  Danielle Hilhorst 2,

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia
2. 

Laboratoire de Mathématiques, CNRS et Université de Paris-Sud, 91405 Orsay

Received  January 2014 Revised  April 2014 Published  December 2014

The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.
Keywords: Phase field model, viscous Cahn-Hilliard equations, tumor growth, asymptotic analysis., long-time behavior, well-posedness.
Mathematics Subject Classification: 35B20, 35K20, 35K35, 35R3.
Citation: Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[2]

H. Brezis, Opérateurs Maximaux Monotones Et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.  Google Scholar

[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.  Google Scholar

[4]

V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, N. Bellomo, M. Chaplain, and E. De Angelis, eds., Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2008), 113-181.  Google Scholar

[5]

V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763. doi: 10.1007/s00285-008-0215-x.  Google Scholar

[6]

V. Cristini and J. S. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511781452.  Google Scholar

[7]

V. Cristini, J. S. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.  Google Scholar

[8]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414. doi: 10.1006/jdeq.1996.0101.  Google Scholar

[10]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[11]

S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, preprint arXiv:1405.3446 [math.AP] (2014), 1-27. Google Scholar

[12]

A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24. doi: 10.1002/cnm.1467.  Google Scholar

[13]

D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, preprint (2013), 1-28, to appear in Math. Models Methods Appl. Sci.. Google Scholar

[14]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91. doi: 10.1088/0951-7715/23/1/R01.  Google Scholar

[16]

J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517. doi: 10.1142/S0218202510004313.  Google Scholar

[17]

R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[18]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[19]

S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

show all references

References:
[1]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.  Google Scholar

[2]

H. Brezis, Opérateurs Maximaux Monotones Et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.  Google Scholar

[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.  Google Scholar

[4]

V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, N. Bellomo, M. Chaplain, and E. De Angelis, eds., Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2008), 113-181.  Google Scholar

[5]

V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763. doi: 10.1007/s00285-008-0215-x.  Google Scholar

[6]

V. Cristini and J. S. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511781452.  Google Scholar

[7]

V. Cristini, J. S. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.  Google Scholar

[8]

R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414. doi: 10.1006/jdeq.1996.0101.  Google Scholar

[10]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[11]

S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, preprint arXiv:1405.3446 [math.AP] (2014), 1-27. Google Scholar

[12]

A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24. doi: 10.1002/cnm.1467.  Google Scholar

[13]

D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, preprint (2013), 1-28, to appear in Math. Models Methods Appl. Sci.. Google Scholar

[14]

J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[15]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91. doi: 10.1088/0951-7715/23/1/R01.  Google Scholar

[16]

J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517. doi: 10.1142/S0218202510004313.  Google Scholar

[17]

R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[18]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[19]

S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203492222.  Google Scholar

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