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August  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  May 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 and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, SpringerVerlag, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[3]

S. BosiaM. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13 (2015), 1541-1567.  doi: 10.4310/CMS.2015.v13.n6.a9.  Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans le Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[5]

H. BrezisM. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Commun. Pure Appl. Math., 23 (1970), 123-144.  doi: 10.1002/cpa.3160230107.  Google Scholar

[6]

Y. ChenS. M. WiseV. B. Shenoy and 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.  doi: 10.1002/cnm.2624.  Google Scholar

[7]

P. ColliS. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444.  doi: 10.1016/j.jmaa.2011.08.008.  Google Scholar

[8]

P. ColliG. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field model related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar

[9]

H. B. FrieboesF. JinY. -L. ChuangS. M. WiseJ. S. Lowengrub and V. Cristini, Threedimensional multispecies nonlinear tumor growth -Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278.  doi: 10.1016/j.jtbi.2010.02.036.  Google Scholar

[10]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.  Google Scholar

[11]

S. FrigeriM. Grasselli and E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.  doi: 10.1088/0951-7715/28/5/1257.  Google Scholar

[12]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360.  doi: 10.3934/math.2016.3.318.  Google Scholar

[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. Google Scholar

[14]

H. Garcke and 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.  doi: 10.1017/s0956792516000292.  Google Scholar

[15]

H. GarckeK. F. LamE. 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

[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.   Google Scholar

[17]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[18]

A. Hawkins-DaarudK. 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

[19]

J. JiangH. Wu and 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.  doi: 10.1016/j.jde.2015.04.009.  Google Scholar

[20]

J. S. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.  Google Scholar

[21]

D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, Romania, 1978.  Google Scholar

[22]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Providence, R. I. , 1997.  Google Scholar

[23]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth -Ⅰ: Model and numerical method, J. Theor. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.  Google Scholar

[24]

E. Zeidler, Nonlinear Functional Analysis and its Applications. Part Ⅱ/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0981-2.  Google Scholar

[25]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.  Google Scholar

[2]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, SpringerVerlag, New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[3]

S. BosiaM. Conti and M. Grasselli, On the Cahn-Hilliard-Brinkman system, Commun. Math. Sci., 13 (2015), 1541-1567.  doi: 10.4310/CMS.2015.v13.n6.a9.  Google Scholar

[4]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans le Espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[5]

H. BrezisM. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Commun. Pure Appl. Math., 23 (1970), 123-144.  doi: 10.1002/cpa.3160230107.  Google Scholar

[6]

Y. ChenS. M. WiseV. B. Shenoy and 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.  doi: 10.1002/cnm.2624.  Google Scholar

[7]

P. ColliS. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, J. Math. Anal. Appl., 386 (2012), 428-444.  doi: 10.1016/j.jmaa.2011.08.008.  Google Scholar

[8]

P. ColliG. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field model related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.  doi: 10.3934/dcds.2015.35.2423.  Google Scholar

[9]

H. B. FrieboesF. JinY. -L. ChuangS. M. WiseJ. S. Lowengrub and V. Cristini, Threedimensional multispecies nonlinear tumor growth -Ⅱ: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254-1278.  doi: 10.1016/j.jtbi.2010.02.036.  Google Scholar

[10]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.  Google Scholar

[11]

S. FrigeriM. Grasselli and E. Rocca, A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity, 28 (2015), 1257-1293.  doi: 10.1088/0951-7715/28/5/1257.  Google Scholar

[12]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360.  doi: 10.3934/math.2016.3.318.  Google Scholar

[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. Google Scholar

[14]

H. Garcke and 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.  doi: 10.1017/s0956792516000292.  Google Scholar

[15]

H. GarckeK. F. LamE. 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

[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.   Google Scholar

[17]

G. GilardiA. Miranville and G. Schimperna, On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.  Google Scholar

[18]

A. Hawkins-DaarudK. 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

[19]

J. JiangH. Wu and 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.  doi: 10.1016/j.jde.2015.04.009.  Google Scholar

[20]

J. S. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.  Google Scholar

[21]

D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Editura Academiei, Romania, 1978.  Google Scholar

[22]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, AMS, Providence, R. I. , 1997.  Google Scholar

[23]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth -Ⅰ: Model and numerical method, J. Theor. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.  Google Scholar

[24]

E. Zeidler, Nonlinear Functional Analysis and its Applications. Part Ⅱ/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0981-2.  Google Scholar

[25]

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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