Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 73
Number of page(s) 46
DOI https://doi.org/10.1051/cocv/2021072
Published online 13 July 2021
  1. V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces. Springer, London, New York (2010). [Google Scholar]
  2. H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Vol. 5 of North-Holland Math. Stud. North-Holland, Amsterdam (1973). [Google Scholar]
  3. H. Cartan, Calcul différentiel. Formes différentielles. Hermann, Paris (1967). [Google Scholar]
  4. E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22 (2012) 261–279. [Google Scholar]
  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. 83 (2021) 739–787. [Google Scholar]
  6. P. Colli, M.H. Farshbaf-Shaker, G. Gilardi and J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary condition. Ann. Acad. Rom. Sci. Ser. Math. Appl. 7 (2015) 41–66. [Google Scholar]
  7. P. Colli, G. Gilardi and D. Hilhorst, On a Cahn–Hilliard type phase field system related to tumor growth. Discret. Cont. Dyn. Syst. 35 (2015) 2423–2442. [Google Scholar]
  8. P. Colli, G. Gilardi, E. 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. [Google Scholar]
  9. P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth. Discret. Contin. Dyn. Syst. Ser. S 10 (2017) 37–54. [Google Scholar]
  10. 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]
  11. P. Colli, G. Gilardi and J. Sprekels, A distributed control problem for a fractional tumor growth model. Mathematics 7 (2019) 792. [Google Scholar]
  12. 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. [Google Scholar]
  13. V. Cristini, X. Li, J.S. Lowengrub and S.M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: invasionand branching. J. Math. Biol. 58 (2009) 723–763. [Google Scholar]
  14. V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach. Cambridge University Press (2010). [Google Scholar]
  15. M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M.E. Schonbek, Analysis of a diffuse interface model of multi-species tumor growth. Nonlinearity 30 (2017) 1639–1658. [Google Scholar]
  16. J. Dieudonné, Foundations of Modern Analysis. Vol. 10 of Pure and Applied Mathematics. Academic Press, New York (1960). [Google Scholar]
  17. M. Ebenbeck and P. Knopf, Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth, ESAIM: COCV 26 (2020) 1–38. [EDP Sciences] [Google Scholar]
  18. M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn–Hilliard–Brinkman equation. Calc. Var. Partial Differ. Equ. 58 (2019) 131. [Google Scholar]
  19. M. Ebenbeck and H. Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis. J. Differ. Equ. 266 (2019) 5998–6036. [Google Scholar]
  20. S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, Eur. J. Appl. Math. 26 (2015) 215–243. [Google Scholar]
  21. S. Frigeri, K.F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, edited by P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels. Vol. 22 of Springer INdAM Series. Springer, Cham (2017) 217–254. [Google Scholar]
  22. S. Frigeri, K.F. Lam, E. Rocca and G. Schimperna, On a multi-species Cahn–Hilliard–Darcy tumor growth model with singular potentials. Commun. Math Sci. 16 (2018) 821–856. [Google Scholar]
  23. S. Frigeri, K.F. Lam and A. Signori, Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities. Eur. J. Appl. Math. (2021) 1–42. [Google Scholar]
  24. H. Garcke and K.F. Lam, Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport. Eur. J. Appl. Math. 28 (2017) 284–316. [Google Scholar]
  25. H. Garcke and K.F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth. AIMS Math. 1 (2016) 318–360. [Google Scholar]
  26. H. Garcke and K.F. Lam, Analysis of a Cahn–Hilliard system with non–zero Dirichlet conditions modeling tumor growth with chemotaxis, Discr. Contin. Dyn. Syst. 37 (2017) 4277–4308. [Google Scholar]
  27. H. Garcke and K.F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, in Trends on Applications of Mathematics to Mechanics, edited by E. Rocca, U. Stefanelli, L. Truskinovski, A. Visintin. Vol. 27 of Springer INdAM Series. Springer, Cham (2018) 243–264. [Google Scholar]
  28. H. Garcke, K.F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis. Math. Models Methods Appl. Sci. 28 (2018) 525–577. [Google Scholar]
  29. H. Garcke, K.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. [Google Scholar]
  30. 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. [Google Scholar]
  31. 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]
  32. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edn. Springer-Verlag, Berlin-Heidelberg (1983). [Google Scholar]
  33. A. Hawkins-Daarud, K.G. van der Zee and J.T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Eng. 28 (2011) 3–24. [Google Scholar]
  34. D. Hilhorst, J. Kampmann, T.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. [Google Scholar]
  35. C. Kahle and K.F. Lam, Parameter identification via optimal control for a Cahn–Hilliard-chemotaxis system with a variable mobility. Appl. Math. Optim. 82 (2020) 63–104. [Google Scholar]
  36. O.A. Ladyženskaja, V.A. Solonnikov and N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. Vol. 23 of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island (1968). [Google Scholar]
  37. J.-L. Lions, Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer-Verlag, Berlin-Göttingen-Heidelberg (1961). [Google Scholar]
  38. 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. [Google Scholar]
  39. A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential. Appl. Math. Optim. 82 (2020) 517–549. [Google Scholar]
  40. A. Signori, Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach. Evol. Equ. Control Theory 9 (2020) 193–217. [Google Scholar]
  41. A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme. Math. Control Relat. Fields 10 (2020) 305–331. [Google Scholar]
  42. A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth. Asymptot. Anal. 117 (2020) 43–66. [Google Scholar]
  43. 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]
  44. J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65–96. [Google Scholar]
  45. J. Sprekels and F. Tröltzsch, Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth. ESAIM: COCV 27 (2021) 2. [EDP Sciences] [Google Scholar]
  46. J. Sprekels and H. Wu, Optimal distributed control of a Cahn–Hilliard–Darcy system with mass sources. Appl. Math. Optim. 83 (2021) 489–530. [Google Scholar]
  47. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island (2010). [Google Scholar]
  48. S.M. Wise, J.S. Lowengrub, H.B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth I: Model and numerical method. J. Theor. Biol. 253 (2008) 524–543. [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.