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Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation
Long time dynamics of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects
1. | Dipartimento di Matematica, Universita' degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy |
2. | Dipartimento di Matematica, Universita' degli Studi di Pavia and IMATI-C.N.R., Via Ferrata 5, 27100 Pavia, Italy |
We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [
References:
[1] |
C. Cavaterra, E. Rocca and H. Wu,
Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Applied Mathematics & Optimization, 83 (2021), 739-787.
doi: 10.1007/s00245-019-09562-5. |
[2] |
P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali and E. Rocca,
Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Mathematical Models and Methods in Applied Sciences, 30 (2020), 1253-1295.
doi: 10.1142/S0218202520500220. |
[3] |
P. Colli, G. Gilardi, G. Marinoschi and E. Rocca,
Sliding mode control for a phase field system related to tumor growth, Applied Mathematics & Optimization, 79 (2019), 647-670.
doi: 10.1007/s00245-017-9451-z. |
[4] |
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, Discrete & Continuous Dynamical Systems, 10 (2017), 37-54.
doi: 10.3934/dcdss.2017002. |
[5] |
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.
doi: 10.1088/1361-6544/aa6e5f. |
[6] |
P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn-Hilliard type phase field system related to tumor growth, Discrete & Continuous Dynamical Systems, 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
[7] |
P. Colli, G. Gilardi, F. Issard-Roch and G. Schimperna,
Long time convergence for a class of variational phase-field models, Discrete & Continuous Dynamical Systems, 25 (2019), 63-81.
doi: 10.3934/dcds.2009.25.63. |
[8] |
M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. Schonbek,
Analysis of a diffuse interface model for multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.
doi: 10.1088/1361-6544/aa6063. |
[9] |
M. Ebenbeck and H. Garcke,
Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, Journal of Differential Equations, 266 (2019), 5998-6036.
doi: 10.1016/j.jde.2018.10.045. |
[10] |
M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calculus of Variations, 58 (2019), Paper No. 131, 31 pp.
doi: 10.1007/s00526-019-1579-z. |
[11] |
E. Feireisl, F. Issard-Roch and H. Petzeltova,
Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete & Continuous Dynamical Systems, 10 (2004), 239-252.
doi: 10.3934/dcds.2004.10.239. |
[12] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European Journal of Applied Mathematics, 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[13] |
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. |
[14] |
H. Garcke and K. F. Lam,
Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis, Discrete & Continuous Dynamical Systems, 37 (2017), 4277-4308.
doi: 10.3934/dcds.2017183. |
[15] |
H. Garcke, K. F. Lam and E. Rocca,
Optimal control of treatment time in a diffuse interface model of tumor growth, Applied Mathematics & Optimization, 78 (2018), 495-544.
doi: 10.1007/s00245-017-9414-4. |
[16] |
H. Garcke, K. F. Lam, E. Sitka and V. Styles,
A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1095-1148.
doi: 10.1142/S0218202516500263. |
[17] |
H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka,
A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences, 28 (2018), 525-577.
doi: 10.1142/S0218202518500148. |
[18] |
W. Hao, M. Grasselli and S. Zheng,
Convergence to equilibrium for a parabolic–hyperbolic phase-field system with Neumann boundary conditions, Mathematical Models and Methods in Applied Sciences, 17 (2007), 125-153.
doi: 10.1142/S0218202507001851. |
[19] |
J. Jiang, H. Wu and S. Zheng,
Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth, Journal of Differential Equations, 259 (2015), 3032-3077.
doi: 10.1016/j.jde.2015.04.009. |
[20] |
P. Laurençot,
Long-time behaviour for a model of phase-field type, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 126 (1996), 167-185.
doi: 10.1017/S0308210500030663. |
[21] |
G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), E7663–E7671.
doi: 10.1073/pnas.1615791113. |
[22] |
J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European Journal of Applied Mathematics, 24 (2013) 691–734.
doi: 10.1017/S0956792513000144. |
[23] |
A. Miranville, E. Rocca and G. Schimperna,
On the long time behavior of a tumor growth model, Journal of Differential Equations, 267 (2019), 2616-2642.
doi: 10.1016/j.jde.2019.03.028. |
[24] |
E. Rocca and G. Schimperna,
Universal attractor for some singular phase transition systems, Physica D: Nonlinear Phenomena, 192 (2004), 279-307.
doi: 10.1016/j.physd.2004.01.024. |
[25] |
A. Sergiu, E. Feireisl and F. Issard–Roch,
Long–time convergence of solutions to a phase–field system, Mathematical methods in the applied sciences, 24 (2001), 277-287.
doi: 10.1002/mma.215. |
[26] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[27] |
J. Xu, G. Vilanova and H. Gomez, A mathematical model coupling tumor growth and angiogenesis, PLoS ONE, 11 (2016), e0149422.
doi: 10.1371/journal.pone.0149422. |
show all references
References:
[1] |
C. Cavaterra, E. Rocca and H. Wu,
Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Applied Mathematics & Optimization, 83 (2021), 739-787.
doi: 10.1007/s00245-019-09562-5. |
[2] |
P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali and E. Rocca,
Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Mathematical Models and Methods in Applied Sciences, 30 (2020), 1253-1295.
doi: 10.1142/S0218202520500220. |
[3] |
P. Colli, G. Gilardi, G. Marinoschi and E. Rocca,
Sliding mode control for a phase field system related to tumor growth, Applied Mathematics & Optimization, 79 (2019), 647-670.
doi: 10.1007/s00245-017-9451-z. |
[4] |
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, Discrete & Continuous Dynamical Systems, 10 (2017), 37-54.
doi: 10.3934/dcdss.2017002. |
[5] |
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.
doi: 10.1088/1361-6544/aa6e5f. |
[6] |
P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn-Hilliard type phase field system related to tumor growth, Discrete & Continuous Dynamical Systems, 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
[7] |
P. Colli, G. Gilardi, F. Issard-Roch and G. Schimperna,
Long time convergence for a class of variational phase-field models, Discrete & Continuous Dynamical Systems, 25 (2019), 63-81.
doi: 10.3934/dcds.2009.25.63. |
[8] |
M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. Schonbek,
Analysis of a diffuse interface model for multispecies tumor growth, Nonlinearity, 30 (2017), 1639-1658.
doi: 10.1088/1361-6544/aa6063. |
[9] |
M. Ebenbeck and H. Garcke,
Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis, Journal of Differential Equations, 266 (2019), 5998-6036.
doi: 10.1016/j.jde.2018.10.045. |
[10] |
M. Ebenbeck and P. Knopf, Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation, Calculus of Variations, 58 (2019), Paper No. 131, 31 pp.
doi: 10.1007/s00526-019-1579-z. |
[11] |
E. Feireisl, F. Issard-Roch and H. Petzeltova,
Long-time behaviour and convergence towards equilibria for a conserved phase field model, Discrete & Continuous Dynamical Systems, 10 (2004), 239-252.
doi: 10.3934/dcds.2004.10.239. |
[12] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumour growth, European Journal of Applied Mathematics, 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[13] |
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. |
[14] |
H. Garcke and K. F. Lam,
Analysis of a Cahn–Hilliard system with non zero Dirichlet conditions modelling tumour growth with chemotaxis, Discrete & Continuous Dynamical Systems, 37 (2017), 4277-4308.
doi: 10.3934/dcds.2017183. |
[15] |
H. Garcke, K. F. Lam and E. Rocca,
Optimal control of treatment time in a diffuse interface model of tumor growth, Applied Mathematics & Optimization, 78 (2018), 495-544.
doi: 10.1007/s00245-017-9414-4. |
[16] |
H. Garcke, K. F. Lam, E. Sitka and V. Styles,
A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1095-1148.
doi: 10.1142/S0218202516500263. |
[17] |
H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka,
A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences, 28 (2018), 525-577.
doi: 10.1142/S0218202518500148. |
[18] |
W. Hao, M. Grasselli and S. Zheng,
Convergence to equilibrium for a parabolic–hyperbolic phase-field system with Neumann boundary conditions, Mathematical Models and Methods in Applied Sciences, 17 (2007), 125-153.
doi: 10.1142/S0218202507001851. |
[19] |
J. Jiang, H. Wu and S. Zheng,
Well-posedness and long-time behavior of a non-autonomous Cahn–Hilliard–Darcy system with mass source modeling tumor growth, Journal of Differential Equations, 259 (2015), 3032-3077.
doi: 10.1016/j.jde.2015.04.009. |
[20] |
P. Laurençot,
Long-time behaviour for a model of phase-field type, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 126 (1996), 167-185.
doi: 10.1017/S0308210500030663. |
[21] |
G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes, Y. J. Zhang, L. Liu, G. Vilanova and H. Gomez, Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), E7663–E7671.
doi: 10.1073/pnas.1615791113. |
[22] |
J. S. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European Journal of Applied Mathematics, 24 (2013) 691–734.
doi: 10.1017/S0956792513000144. |
[23] |
A. Miranville, E. Rocca and G. Schimperna,
On the long time behavior of a tumor growth model, Journal of Differential Equations, 267 (2019), 2616-2642.
doi: 10.1016/j.jde.2019.03.028. |
[24] |
E. Rocca and G. Schimperna,
Universal attractor for some singular phase transition systems, Physica D: Nonlinear Phenomena, 192 (2004), 279-307.
doi: 10.1016/j.physd.2004.01.024. |
[25] |
A. Sergiu, E. Feireisl and F. Issard–Roch,
Long–time convergence of solutions to a phase–field system, Mathematical methods in the applied sciences, 24 (2001), 277-287.
doi: 10.1002/mma.215. |
[26] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[27] |
J. Xu, G. Vilanova and H. Gomez, A mathematical model coupling tumor growth and angiogenesis, PLoS ONE, 11 (2016), e0149422.
doi: 10.1371/journal.pone.0149422. |
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