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

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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

Abstract
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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 and 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.
[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.
[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. 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.
[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.
[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.
[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.
[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.
[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.
[10]	C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.
[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.
[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.
[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..
[14]	J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[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.
[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.
[17]	R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997.
[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.
[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.

show all references

References:

[1]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.
[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.
[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. 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.
[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.
[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.
[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.
[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.
[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.
[10]	C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.
[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.
[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.
[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..
[14]	J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[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.
[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.
[17]	R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997.
[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.
[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.

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