Open Access
Issue
ESAIM: COCV
Volume 31, 2025
Article Number 85
Number of page(s) 37
DOI https://doi.org/10.1051/cocv/2025070
Published online 08 October 2025
  1. D. Horstmann, From 1970 until now: the Keller-Segel model in chemotaxis and its consequences. Jahresber. Deutsch. Math.-Verein. 106 (2004) 51-69. [Google Scholar]
  2. J. Dolbeault, and B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in R2. C. R. Math. 339 (2004) 611-616. [Google Scholar]
  3. G. Gilardi, A. Signori and J. Sprekels, Nutrient control for a viscous Cahn—Hilliard—Keller—Segel model with logistic source describing tumor growth. Discrete Contin. Dyn. Syst. Ser. S 16 (2023) 3552-3572. [Google Scholar]
  4. M.H. Hashim and A. J. Harfash, Finite element analysis of a Keller-Segel model with additional cross-diffusion and logistic source. Part I: Space convergence. Comput. Math. Appl. 89 (2021) 44-56. [Google Scholar]
  5. 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. [CrossRef] [Google Scholar]
  6. M. Winkler, Finite-time blow-up in the higher-dimensional parabolic—parabolic Keller-Segel system. J. Math. Pures Appl. 100 (2013) 748-767. [Google Scholar]
  7. A. Agosti, A. Giotta Lucífero and S. Luzzi, An image-informed Cahn-Hilliard Keller-Segel multiphase field model for tumor growth with angiogenesis. Appl. Math. Comput. 445 (2023) 127834. [Google Scholar]
  8. A. Agosti, C. Cattaneo, C. Giverso, D. Ambrosi and P. Ciarletta, A computational framework for the personalized clinical treatment of glioblastoma multiforme. ZAMM 98 (2018) 2307-2327. [Google Scholar]
  9. A. Agosti and A. Signori, Analysis of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model with chemotaxis and angiogenesis. J. Differ. Equ. 403 (2024) 308-367. [Google Scholar]
  10. P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth. Discrete Contin. Dyn. Syst. 35 (2015) 2423-2442. [Google Scholar]
  11. V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press (2010). [Google Scholar]
  12. 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. Sei. 26 (2016) 1095-1148. [Google Scholar]
  13. H. Garcke, K.F. Lam and A. Signori, On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects. Nonlinear Anal. Real World Appl. 57 (2021) 103192. [Google Scholar]
  14. P. Colli, G. Gilardi, A. Signori and J. Sprekels, Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials. Nonlinearity 36 (2023) 4470-4500. [CrossRef] [MathSciNet] [Google Scholar]
  15. M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis. J. Differ. Equ. 266 (2019) 5998-6036. [CrossRef] [Google Scholar]
  16. 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 Differ. Equ. 47 (2022) 233-278. [CrossRef] [Google Scholar]
  17. P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth. Nonlinearity 30 (2017) 2518-2546. [Google Scholar]
  18. 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. [Google Scholar]
  19. 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. [Google Scholar]
  20. M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation. Calc. Var. Partial Differ. Equ. 58 (2019) 1-31. [Google Scholar]
  21. H. Garcke, K.F. Lam and A. Signori, Sparse optimal control of a phase field tumour model with mechanical effects. SIAM J. Control Optim. 59 (2021) 1555-1580. [CrossRef] [MathSciNet] [Google Scholar]
  22. A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential. Appl. Math. Optim. 82 (2020) 517-549. [CrossRef] [MathSciNet] [Google Scholar]
  23. A. Signori, Penalisation of long treatment time and optimal control of a tumour growth model of Cahn-Hilliard type with singular potential. Discrete Contin. Dyn. Syst. 41 (2021) 2519-2542. [Google Scholar]
  24. L.C. Evans, Partial Differential Equations, Vol. 19. American Mathematical Society (2022). [Google Scholar]
  25. A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sei. 27 (2004) 545-582. [Google Scholar]
  26. R. Denk, M. Hieber and J. Prüss, Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257 (2007) 193-224. [Google Scholar]
  27. J.L. Lions, Equations Différentielles Opérationnelles et Problèmes aux Limites. Grundlehren, Vol. 111. Springer-Verlag, Berlin (1961). [Google Scholar]
  28. P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Well-posedness and optimal control for a Cahn-Hilliard-Oono system with control in the mass term. Discrete Contin. Dyn. Syst. Ser. S 15 (2022) 2135-2172. [Google Scholar]
  29. J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/1971) 1077-1092. [Google Scholar]
  30. J. Simon, Compact sets in the space LP(O,T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96. [Google Scholar]
  31. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, Vol. 112. American Mathematical Society (2010). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.