# American Institue of Mathematical Sciences

• Previous Article
The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators
• DCDS Home
• This Issue
• Next Article
Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one
2017, 37(8): 4277-4308. doi: 10.3934/dcds.2017183

## Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis

 Fakultät für Mathematik Universität Regensburg 93040 Regensburg, Germany

Received  April 2016 Revised  March 2017 Published  April 2017

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.
Citation: Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183
##### References:
 [1] H. W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. [2] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, SpringerVerlag, New York, 2010. [3] S. Bosia, M. Conti, M. Grasselli, On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13 (2015), 1541-1567. [4] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans le Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] H. Brezis, M. G. Crandall, A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Commun. Pure Appl. Math., 23 (1970), 123-144. [6] Y. Chen, S. M. Wise, V. B. Shenoy, J. S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 726-754. [7] P. Colli, S. Frigeri, M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444. [8] P. Colli, G. Gilardi, D. Hilhorst, On a Cahn-Hilliard type phase field model related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442. [9] H. B. Frieboes, F. Jin, Y. -L. Chuang, S. M. Wise, J. S. Lowengrub, V. Cristini, Threedimensional multispecies nonlinear tumor growth -Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278. [10] S. Frigeri, M. Grasselli, E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243. [11] S. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293. [12] H. Garcke, K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. [13] H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, preprint, arXiv: 1611. 00234. [14] H. Garcke, K. F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European J. Appl. Math., 28 (2017), 284-316. [15] H. Garcke, K. F. Lam, E. Sitka, V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148. [16] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161. [17] G. Gilardi, A. Miranville, G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. [18] A. Hawkins-Daarud, K. G. van der Zee, J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24. [19] J. Jiang, H. Wu, S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equ., 259 (2015), 3032-3077. [20] J. S. Lowengrub, E. Titi, K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734. [21] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, Romania, 1978. [22] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Providence, R. I. , 1997. [23] S. M. Wise, J. S. Lowengrub, H. B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth -Ⅰ: Model and numerical method, J. Theor. Biol., 253 (2008), 524-543. [24] E. Zeidler, Nonlinear Functional Analysis and its Applications. Part Ⅱ/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. [25] W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989.

show all references

##### References:
 [1] H. W. Alt, S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. [2] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, SpringerVerlag, New York, 2010. [3] S. Bosia, M. Conti, M. Grasselli, On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13 (2015), 1541-1567. [4] H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans le Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] H. Brezis, M. G. Crandall, A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Commun. Pure Appl. Math., 23 (1970), 123-144. [6] Y. Chen, S. M. Wise, V. B. Shenoy, J. S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 726-754. [7] P. Colli, S. Frigeri, M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444. [8] P. Colli, G. Gilardi, D. Hilhorst, On a Cahn-Hilliard type phase field model related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442. [9] H. B. Frieboes, F. Jin, Y. -L. Chuang, S. M. Wise, J. S. Lowengrub, V. Cristini, Threedimensional multispecies nonlinear tumor growth -Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278. [10] S. Frigeri, M. Grasselli, E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243. [11] S. Frigeri, M. Grasselli, E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293. [12] H. Garcke, K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. [13] H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, preprint, arXiv: 1611. 00234. [14] H. Garcke, K. F. Lam, Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport, European J. Appl. Math., 28 (2017), 284-316. [15] H. Garcke, K. F. Lam, E. Sitka, V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148. [16] G. Gilardi, On a conserved phase field model with irregular potentials and dynamic boundary conditions, Istit. Lombardo Accad. Sci. Lett. Rend. A, 141 (2007), 129-161. [17] G. Gilardi, A. Miranville, G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. [18] A. Hawkins-Daarud, K. G. van der Zee, J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24. [19] J. Jiang, H. Wu, S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equ., 259 (2015), 3032-3077. [20] J. S. Lowengrub, E. Titi, K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734. [21] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, Romania, 1978. [22] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Providence, R. I. , 1997. [23] S. M. Wise, J. S. Lowengrub, H. B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth -Ⅰ: Model and numerical method, J. Theor. Biol., 253 (2008), 524-543. [24] E. Zeidler, Nonlinear Functional Analysis and its Applications. Part Ⅱ/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. [25] W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989.
 [1] Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275 [2] Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn--Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002 [3] Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423 [4] Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881 [5] Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683 [6] Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511 [7] Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625 [8] Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 [9] Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763 [10] Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929 [11] Maurizio Grasselli, Alain Miranville, Giulio Schimperna. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 67-98. doi: 10.3934/dcds.2010.28.67 [12] Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581 [13] Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041 [14] Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573 [15] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [16] Gisèle Ruiz Goldstein, Alain Miranville. A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 387-400. doi: 10.3934/dcdss.2013.6.387 [17] Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419 [18] Rainer Brunnhuber, Barbara Kaltenbacher. Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4515-4535. doi: 10.3934/dcds.2014.34.4515 [19] Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297 [20] Pavel Krejčí, Elisabetta Rocca. Well-posedness of an extended model for water-ice phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 439-460. doi: 10.3934/dcdss.2013.6.439

2016 Impact Factor: 1.099