June  2020, 10(2): 305-331. doi: 10.3934/mcrf.2019040

Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme

Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125 Milano, Italy

Received  February 2019 Revised  May 2019 Published  June 2020 Early access  August 2019

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long-time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, we assume some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide the existence of optimal controls and a necessary condition for optimality is addressed.

Citation: Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control and Related Fields, 2020, 10 (2) : 305-331. doi: 10.3934/mcrf.2019040
References:
[1]

A. AgostiP. F. AntoniettiP. CiarlettaM. Grasselli and M. Verani, A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.  doi: 10.1002/mma.4548.

[2]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597.  doi: 10.1016/0022-247X(81)90125-6.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, , Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leyden, 1976.

[4]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973.

[5]

C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., (2019), 1–49. doi: 10.1007/s00245-019-09562-5.

[6]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.

[7]

P. ColliM. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24.  doi: 10.1007/s00245-014-9250-8.

[8]

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

[9]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.

[10]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.

[11]

P. ColliG. GilardiE. Rocca and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108.  doi: 10.1016/j.nonrwa.2015.05.002.

[12]

P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[13]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modeling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[14]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.

[15]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Opt., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.

[16]

P. ColliG. Gilardi and J. Sprekels, Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition, Nonlinear Anal., 170 (2018), 171-196.  doi: 10.1016/j.na.2018.01.003.

[17]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.

[18]

P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A 'deep quench' approach, J. Convex Anal., 26 (2019), 485–514. Available from: http://www.heldermann.de/JCA/JCA26/JCA262/jca26024.htm.

[19]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[20]

V. CristiniX. R. LiJ. 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.

[21] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Leiden, 2010.  doi: 10.1017/CBO9780511781452.
[22]

M. DaiE. FeireislE. RoccaG. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.  doi: 10.1088/1361-6544/aa6063.

[23]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, preprint, arXiv: 1903.00333.

[24]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019). doi: 10.1007/s00526-019-1579-z.

[25]

M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998-6036.  doi: 10.1016/j.jde.2018.10.045.

[26]

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.

[27]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821-856.  doi: 10.4310/CMS.2018.v16.n3.a11.

[28]

S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 217–254. doi: 10.1007/978-3-319-64489-9_9.

[29]

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.

[30]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.

[31]

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.

[32]

H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, Trends in applications of mathematics to mechanics, Springer INdAM Ser., Springer, Cham, 27 (2018), 243–264. doi: 10.1007/978-3-319-75940-1_12.

[33]

H. GarckeK. F. LamR. Nürnberg 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.

[34]

H. GarckeK. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495-544.  doi: 10.1007/s00245-017-9414-4.

[35]

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.

[36]

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.

[37]

A. Hawkins-DaarudS. PrudhommeK. G. van der Zee and J. T. Oden, Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, J. Math. Biol., 67 (2013), 1457-1485.  doi: 10.1007/s00285-012-0595-9.

[38]

A. Hawkins-DaruudK. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng., 28 (2012), 3-24.  doi: 10.1002/cnm.1467.

[39]

D. HilhorstJ. KampmannT. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011-1043.  doi: 10.1142/S0218202515500268.

[40]

S. Kurima, Asymptotic analysis for Cahn-Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci., 42 (2019), 2431-2454.  doi: 10.1002/mma.5520.

[41]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Mathematical Monographs, 23. American mathematical society, Providence, R. I. 1968.

[42]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Grundlehren der Mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[43]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[44]

A. MiranvilleE. Rocca and and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.

[45]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[46]

J. T. OdenA. 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.

[47]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, preprint, arXiv: 1903.04930.

[48]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, preprint, arXiv: 1811.08626.

[49]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9538-1.

[50]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65–96. Available from: https://doi.org/10.1007/BF01762360.

[51]

J. Sprekels and H. Wu, Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources, Appl. Math. Optim., (2019), 1–42. doi: 10.1007/s00245-019-09555-4.

[52]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[53]

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.

[54]

X. WuG. J. van Zwieten and K. G. van der Zee, Stabilized second-order splitting schemes for Cahnf-Hilliard models with applications to diffuse–interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180-203.  doi: 10.1002/cnm.2597.

show all references

References:
[1]

A. AgostiP. F. AntoniettiP. CiarlettaM. Grasselli and M. Verani, A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626.  doi: 10.1002/mma.4548.

[2]

V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597.  doi: 10.1016/0022-247X(81)90125-6.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, , Editura Academiei Republicii Socialiste România, Bucharest, Noordhoff International Publishing, Leyden, 1976.

[4]

H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973.

[5]

C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., (2019), 1–49. doi: 10.1007/s00245-019-09562-5.

[6]

P. ColliM. H. Farshbaf-ShakerG. Gilardi and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), 2696-2721.  doi: 10.1137/140984749.

[7]

P. ColliM. H. Farshbaf-Shaker and J. Sprekels, A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles, Appl. Math. Optim., 71 (2015), 1-24.  doi: 10.1007/s00245-014-9250-8.

[8]

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

[9]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112.  doi: 10.3934/mcrf.2016.6.95.

[10]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a conserved phase field system with a possibly singular potential, Evol. Equ. Control Theory, 7 (2018), 95-116.  doi: 10.3934/eect.2018006.

[11]

P. ColliG. GilardiE. Rocca and J. Sprekels, Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108.  doi: 10.1016/j.nonrwa.2015.05.002.

[12]

P. ColliG. GilardiE. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.  doi: 10.1088/1361-6544/aa6e5f.

[13]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modeling tumor growth, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[14]

P. ColliG. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994.  doi: 10.1016/j.jmaa.2014.05.008.

[15]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Opt., 73 (2016), 195-225.  doi: 10.1007/s00245-015-9299-z.

[16]

P. ColliG. Gilardi and J. Sprekels, Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition, Nonlinear Anal., 170 (2018), 171-196.  doi: 10.1016/j.na.2018.01.003.

[17]

P. ColliG. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691.  doi: 10.1137/17M1146786.

[18]

P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A 'deep quench' approach, J. Convex Anal., 26 (2019), 485–514. Available from: http://www.heldermann.de/JCA/JCA26/JCA262/jca26024.htm.

[19]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[20]

V. CristiniX. R. LiJ. 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.

[21] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Leiden, 2010.  doi: 10.1017/CBO9780511781452.
[22]

M. DaiE. FeireislE. RoccaG. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.  doi: 10.1088/1361-6544/aa6063.

[23]

M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, preprint, arXiv: 1903.00333.

[24]

M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calc. Var. Partial Differential Equations, 58 (2019). doi: 10.1007/s00526-019-1579-z.

[25]

M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998-6036.  doi: 10.1016/j.jde.2018.10.045.

[26]

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.

[27]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Comm. Math. Sci., 16 (2018), 821-856.  doi: 10.4310/CMS.2018.v16.n3.a11.

[28]

S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Ser., Springer, Cham, 22 (2017), 217–254. doi: 10.1007/978-3-319-64489-9_9.

[29]

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.

[30]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.

[31]

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.

[32]

H. Garcke and K. F. Lam, On a Cahn-Hilliard-Darcy system for tumour growth with solution dependent source terms, Trends in applications of mathematics to mechanics, Springer INdAM Ser., Springer, Cham, 27 (2018), 243–264. doi: 10.1007/978-3-319-75940-1_12.

[33]

H. GarckeK. F. LamR. Nürnberg 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.

[34]

H. GarckeK. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495-544.  doi: 10.1007/s00245-017-9414-4.

[35]

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.

[36]

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.

[37]

A. Hawkins-DaarudS. PrudhommeK. G. van der Zee and J. T. Oden, Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth, J. Math. Biol., 67 (2013), 1457-1485.  doi: 10.1007/s00285-012-0595-9.

[38]

A. Hawkins-DaruudK. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng., 28 (2012), 3-24.  doi: 10.1002/cnm.1467.

[39]

D. HilhorstJ. KampmannT. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011-1043.  doi: 10.1142/S0218202515500268.

[40]

S. Kurima, Asymptotic analysis for Cahn-Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci., 42 (2019), 2431-2454.  doi: 10.1002/mma.5520.

[41]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Mathematical Monographs, 23. American mathematical society, Providence, R. I. 1968.

[42]

J.-L. Lions, Équations Différentielles Opérationnelles et Problèmes aux Limites, Grundlehren der Mathematischen Wissenschaften, Bd. 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[43]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[44]

A. MiranvilleE. Rocca and and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.

[45]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[46]

J. T. OdenA. 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.

[47]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, preprint, arXiv: 1903.04930.

[48]

A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach, preprint, arXiv: 1811.08626.

[49]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., (2018), 1–33. doi: 10.1007/s00245-018-9538-1.

[50]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1986), 65–96. Available from: https://doi.org/10.1007/BF01762360.

[51]

J. Sprekels and H. Wu, Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources, Appl. Math. Optim., (2019), 1–42. doi: 10.1007/s00245-019-09555-4.

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F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

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