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doi: 10.3934/cpaa.2021032

On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells

Lu Li a,b,

a. 

Université de Poitiers, Laboratoire I3M et Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Equipe DACTIM-MIS, 11 Boulevard Marie et Pierre Curie-Bâtiment H3-TSA 61125, 86073 Poitiers Cedex 9, France
b. 

CHU de Poitiers, 2 rue de la Milétrie, 86000 Poitiers, France

Received  October 2020 Revised  January 2021 Published  March 2021

Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a Cahn–Hilliard equation for the tumor density and a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering modified equations and taking logarithmic nonlinear terms in the Cahn–Hilliard equations. After that we show a local in time weak solution which is conditionally global in time.

Keywords: Hypoxic glioma cells, proliferative-to-invasive transition, Cahn–Hilliard equation, logarithmic nonlinear term, existence of solutions.
Mathematics Subject Classification: Primary: 35K55, 35B45; Secondary: 35Q92.
Citation: 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
References:
[1]

A. C. Aristotelous, O. A. Karakashian and S. M. Wise, Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn-Hilliard equation with a mass source, IMA J. Numer. Anal., 35 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[2]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.   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.   Google Scholar

[4]

L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[5]

M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliferative-to-invasive transition of hypoxic glioma cells, Nonlinear Anal., 189(2019), 17 pp. doi: 10.1016/j.na.2019.111572.  Google Scholar

[6]

H. Garcke, K. F. Lam, R. Nurnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525-577.  doi: 10.1142/S0218202518500148.  Google Scholar

[7]

H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar

[8]

H. Gomez, Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262.   Google Scholar

[9]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77(2008), 7 pp. Google Scholar

[10]

L. Li, A. Miranville and R. Guillevin, Cahn-Hilliard models for glial cells, Appl. Math. Optim., to appear. Google Scholar

[11]

L. Li, A. Miranville and R. Guillevin, A coupled Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Quart. Appl. Math., to appear. Google Scholar

[12]

L. Li, L. Cherfils, A. Miranville and R. Guillevin, A Cahn-Hilliard model with a proliferation term for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Math. Sci., to appear. doi: 10.1080/00036811.2012.671301.  Google Scholar

[13]

A. Miranville, Existence of solutions to a Cahn-Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.  doi: 10.1007/s00009-018-1284-8.  Google Scholar

[14]

A. Miranville, The Cahn-Hilliard equation: recent advances and applications, CBMS-NSF Regional Conference Series in Applied Mathematics 95, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. doi: 10.1137/1.9781611975925.  Google Scholar

[15]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differ. Equ., 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.  Google Scholar

[16]

A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Partial Differential Equations, Vol.(4), (eds. C.M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[17]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

show all references

References:
[1]

A. C. Aristotelous, O. A. Karakashian and S. M. Wise, Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn-Hilliard equation with a mass source, IMA J. Numer. Anal., 35 (2015), 1167-1198.  doi: 10.1093/imanum/dru035.  Google Scholar

[2]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.   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.   Google Scholar

[4]

L. Cherfils, A. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[5]

M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliferative-to-invasive transition of hypoxic glioma cells, Nonlinear Anal., 189(2019), 17 pp. doi: 10.1016/j.na.2019.111572.  Google Scholar

[6]

H. Garcke, K. F. Lam, R. Nurnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525-577.  doi: 10.1142/S0218202518500148.  Google Scholar

[7]

H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.  Google Scholar

[8]

H. Gomez, Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262.   Google Scholar

[9]

E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77(2008), 7 pp. Google Scholar

[10]

L. Li, A. Miranville and R. Guillevin, Cahn-Hilliard models for glial cells, Appl. Math. Optim., to appear. Google Scholar

[11]

L. Li, A. Miranville and R. Guillevin, A coupled Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Quart. Appl. Math., to appear. Google Scholar

[12]

L. Li, L. Cherfils, A. Miranville and R. Guillevin, A Cahn-Hilliard model with a proliferation term for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Math. Sci., to appear. doi: 10.1080/00036811.2012.671301.  Google Scholar

[13]

A. Miranville, Existence of solutions to a Cahn-Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.  doi: 10.1007/s00009-018-1284-8.  Google Scholar

[14]

A. Miranville, The Cahn-Hilliard equation: recent advances and applications, CBMS-NSF Regional Conference Series in Applied Mathematics 95, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2019. doi: 10.1137/1.9781611975925.  Google Scholar

[15]

A. Miranville, E. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differ. Equ., 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.  Google Scholar

[16]

A. Novick-Cohen, The Cahn-Hilliard equation, in Handbook of Differential Equations: Evolutionary Partial Differential Equations, Vol.(4), (eds. C.M. Dafermos and M. Pokorny), Elsevier, Amsterdam, 2008. doi: 10.1016/S1874-5717(08)00004-2.  Google Scholar

[17]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

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