Nutrient control for a viscous Cahn–Hilliard–Keller–Segel model with logistic source describing tumor growth

Gianni Gilardi; Andrea Signori; Jürgen Sprekels

doi:10.3934/dcdss.2023123

Article Contents

2023, Volume 16, Issue 12: 3552-3572. Doi: 10.3934/dcdss.2023123

This issue Previous Article A convergence result for a Stefan problem with phase relaxation Next Article Well-posedness and optimal control for a viscous Cahn–Hilliard–Oono system with dynamic boundary conditions

Nutrient control for a viscous Cahn–Hilliard–Keller–Segel model with logistic source describing tumor growth

1.
Dipartimento di Matematica "F. Casorati", Università di Pavia and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, I-27100 Pavia, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, I-20133 Milano, Italy
3.
Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
4.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany

^*Corresponding author: Gianni Gilardi
^*Corresponding author: Gianni Gilardi

Dedicated to Pierluigi Colli on the occasion of his 65^th birthday, with friendship and admiration

Received: February 2023

Revised: May 2023

Early access: June 2023

Published: December 2023

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Abstract

In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn–Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller–Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.

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Mathematics Subject Classification: Primary: 35K55, 49J20, 49K20; Secondary: 35K61, 49J50.

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References

[1]	A. Agosti, A. Giotta Lucifero and S. Luzzi, An image-informed Cahn–Hilliard Keller–Segel multiphase field model for tumor growth with angiogenesis, Appl. Math. Comput., 445 (2023), Paper No. 127834, 33 pp. doi: 10.1016/j.amc.2023.127834.
[2]	A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-899. doi: 10.1006/bulm.1998.0042.
[3]	P. Colli, A. Signori and J. Sprekels, Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials, Appl. Math. Optim., 83 (2021), 2017-2049. doi: 10.1007/s00245-019-09618-6.
[4]	P. Colli, A. Signori and J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 73, 46 pp. doi: 10.1051/cocv/2021072.
[5]	P. Colli, A. Signori and J. Sprekels, Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities, J. Optim. Theory Appl., 194 (2022), 25-58. doi: 10.1007/s10957-022-02000-7.
[6]	M. Ebenbeck and H. Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differ. Equ., 266 (2019), 5998-6036. doi: 10.1016/j.jde.2018.10.045.
[7]	J. Folkman, Tumor angiogenesis, Adv. Cancer Res., 43 (1985), 175-203.
[8]	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.
[9]	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.
[10]	D. Horstmann, From 1970 until now: The Keller–Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verei., 106 (2004), 51-69.
[11]	P. Knopf and A. Signori, Existence of weak solutions to multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman models for stratified tumor growth with chemotaxis and general source terms, Comm. Partial Differential Equations, 47 (2022), 233-278. doi: 10.1080/03605302.2021.1966803.
[12]	E. Rocca, L. Scarpa and A. Signori, Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis, Math. Models Methods Appl. Sci., 31 (2021), 2643-2694. doi: 10.1142/S0218202521500585.
[13]	E. Rocca, G. Schimperna and A. Signori, On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth, J. Differ. Equ., 343 (2023), 530-578. doi: 10.1016/j.jde.2022.10.026.
[14]	L. Scarpa and A. Signori, On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport, Nonlinearity, 34 (2021), 3199-3250. doi: 10.1088/1361-6544/abe75d.
[15]	A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549. doi: 10.1007/s00245-018-9538-1.
[16]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.
[17]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.
[18]	M. Winkler, Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793. doi: 10.3934/dcdsb.2017135.