# American Institute of Mathematical Sciences

April  2021, 20(4): 1545-1557. doi: 10.3934/cpaa.2021032

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

 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  April 2021 Early access  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.

Citation: Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1545-1557. 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. [2] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [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. [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. [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. [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. [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. [8] H. Gomez, Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262. [9] E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77(2008), 7 pp. [10] L. Li, A. Miranville and R. Guillevin, Cahn-Hilliard models for glial cells, Appl. Math. Optim., to appear. [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. [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. [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. [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. [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. [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. [17] Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.

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##### 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. [2] J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. [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. [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. [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. [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. [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. [8] H. Gomez, Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262. [9] E. Khain and L. M. Sander, A generalized Cahn-Hilliard equation for biological applications, Phys. Rev. E, 77(2008), 7 pp. [10] L. Li, A. Miranville and R. Guillevin, Cahn-Hilliard models for glial cells, Appl. Math. Optim., to appear. [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. [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. [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. [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. [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. [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. [17] Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.
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