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The modified Camassa-Holm equation in Lagrangian coordinates
Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor
College of Mathematical and Informational Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
We consider a free boundary problem modeling the growth of angiogenesis tumor with inhibitor, in which the tumor aggressiveness is modeled by a parameter $μ$. The existences of radially symmetric stationary solution and symmetry-breaking stationary solution are established. In addition, it is proved that there exist a positive integer $m^{**}$ and a sequence of $μ_m$, such that for each $μ_m(m > m^{**})$, the symmetry-breaking stationary solution is a bifurcation branch of the radially symmetric stationary solution.
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R. P. Araujo and D. L. S. McElwain,
A history of the study of solid tumour growth: The contribution of mathematical modeling, Bull. Math. Biol., 66 (2004), 1039-1091.
doi: 10.1016/j.bulm.2003.11.002. |
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H. M. Byrne and M. A. J. Chaplain,
Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.
doi: 10.1016/0025-5564(94)00117-3. |
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M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
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V. Cristini, J. Lowengrub and Q. Nie,
Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
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S. Cui,
Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.
doi: 10.1007/s002850100130. |
[6] |
S. Cui and J. Escher,
Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235.
doi: 10.1137/060657509. |
[7] |
S. Cui and J. Escher,
Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.
doi: 10.1080/03605300701743848. |
[8] |
S. Cui and A. Friedman,
Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103-137.
doi: 10.1016/S0025-5564(99)00063-2. |
[9] |
J. Escher and A. V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math., 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
[10] |
M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
[11] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
|
[12] |
A. Friedman, Cancer models and their mathematical analysis, in Lecture Notes in Math., Vol. 1872, Springer, (2006), 223-246. |
[13] |
A. Friedman,
Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[14] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330.
doi: 10.1007/s00205-005-0408-z. |
[15] |
A. Friedman and B. Hu,
Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equation, 227 (2006), 598-639.
doi: 10.1016/j.jde.2005.09.008. |
[16] |
A. Friedman and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
[17] |
A. Friedman and B. Hu,
Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.
doi: 10.1090/S0002-9947-08-04468-1. |
[18] |
A. Friedman and K. Y. Lam,
Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
[19] |
A. Friedman and F. Reitich,
Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.
doi: 10.1007/s002850050149. |
[20] |
A. Friedman and F. Reitich,
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
[21] |
A. Friedman and F. Reitich,
Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 341-403.
|
[22] |
H. P. Greenspan,
On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242.
doi: 10.1016/S0022-5193(76)80054-9. |
[23] |
W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. -T. Zhang,
Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core, Nonlinear Anal. Real World Appl., 13 (2012), 694-709.
doi: 10.1016/j.nonrwa.2011.08.010. |
[24] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.
doi: 10.1016/j.nonrwa.2016.12.003. |
[25] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. -L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini,
Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[26] |
Z. Wang,
Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53.
doi: 10.1016/j.nonrwa.2014.03.001. |
[27] |
J. Wu,
Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, J. Differential Equations, 260 (2016), 5875-5893.
doi: 10.1016/j.jde.2015.12.023. |
[28] |
J. Wu and S. Cui,
Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors, Nonlinearity, 20 (2007), 2389-2408.
doi: 10.1088/0951-7715/20/10/007. |
[29] |
J. Wu and S. Cui,
Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 1813-1823.
doi: 10.1002/mma.3190. |
[30] |
J. Wu and F. Zhou,
Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.
doi: 10.1016/j.jde.2017.01.012. |
[31] |
S. Xu, M. Bai and F. Zhang, Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), online. |
[32] |
C. Xue, A. Friedman and C. K. Sen,
A mathematical model of ischemic cutaneous wounds, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 16782-16787.
|
[33] |
F. Zhou, J. Escher and S. Cui,
Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors, J. Math. Anal. Appl., 337 (2008), 443-457.
doi: 10.1016/j.jmaa.2007.03.107. |
[34] |
F. Zhou and J. Wu,
Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425.
doi: 10.1017/S0956792515000108. |
show all references
References:
[1] |
R. P. Araujo and D. L. S. McElwain,
A history of the study of solid tumour growth: The contribution of mathematical modeling, Bull. Math. Biol., 66 (2004), 1039-1091.
doi: 10.1016/j.bulm.2003.11.002. |
[2] |
H. M. Byrne and M. A. J. Chaplain,
Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181.
doi: 10.1016/0025-5564(94)00117-3. |
[3] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
V. Cristini, J. Lowengrub and Q. Nie,
Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
doi: 10.1007/s00285-002-0174-6. |
[5] |
S. Cui,
Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395-426.
doi: 10.1007/s002850100130. |
[6] |
S. Cui and J. Escher,
Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235.
doi: 10.1137/060657509. |
[7] |
S. Cui and J. Escher,
Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655.
doi: 10.1080/03605300701743848. |
[8] |
S. Cui and A. Friedman,
Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103-137.
doi: 10.1016/S0025-5564(99)00063-2. |
[9] |
J. Escher and A. V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math., 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
[10] |
M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
[11] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
|
[12] |
A. Friedman, Cancer models and their mathematical analysis, in Lecture Notes in Math., Vol. 1872, Springer, (2006), 223-246. |
[13] |
A. Friedman,
Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772.
doi: 10.1142/S0218202507002467. |
[14] |
A. Friedman and B. Hu,
Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330.
doi: 10.1007/s00205-005-0408-z. |
[15] |
A. Friedman and B. Hu,
Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equation, 227 (2006), 598-639.
doi: 10.1016/j.jde.2005.09.008. |
[16] |
A. Friedman and B. Hu,
Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194.
doi: 10.1137/060656292. |
[17] |
A. Friedman and B. Hu,
Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.
doi: 10.1090/S0002-9947-08-04468-1. |
[18] |
A. Friedman and K. Y. Lam,
Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661.
doi: 10.1016/j.jde.2015.08.032. |
[19] |
A. Friedman and F. Reitich,
Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.
doi: 10.1007/s002850050149. |
[20] |
A. Friedman and F. Reitich,
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634.
doi: 10.1090/S0002-9947-00-02715-X. |
[21] |
A. Friedman and F. Reitich,
Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 341-403.
|
[22] |
H. P. Greenspan,
On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242.
doi: 10.1016/S0022-5193(76)80054-9. |
[23] |
W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. -T. Zhang,
Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core, Nonlinear Anal. Real World Appl., 13 (2012), 694-709.
doi: 10.1016/j.nonrwa.2011.08.010. |
[24] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear Anal. Real World Appl., 35 (2017), 483-502.
doi: 10.1016/j.nonrwa.2016.12.003. |
[25] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. -L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini,
Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[26] |
Z. Wang,
Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53.
doi: 10.1016/j.nonrwa.2014.03.001. |
[27] |
J. Wu,
Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, J. Differential Equations, 260 (2016), 5875-5893.
doi: 10.1016/j.jde.2015.12.023. |
[28] |
J. Wu and S. Cui,
Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors, Nonlinearity, 20 (2007), 2389-2408.
doi: 10.1088/0951-7715/20/10/007. |
[29] |
J. Wu and S. Cui,
Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 1813-1823.
doi: 10.1002/mma.3190. |
[30] |
J. Wu and F. Zhou,
Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.
doi: 10.1016/j.jde.2017.01.012. |
[31] |
S. Xu, M. Bai and F. Zhang, Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), online. |
[32] |
C. Xue, A. Friedman and C. K. Sen,
A mathematical model of ischemic cutaneous wounds, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), 16782-16787.
|
[33] |
F. Zhou, J. Escher and S. Cui,
Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors, J. Math. Anal. Appl., 337 (2008), 443-457.
doi: 10.1016/j.jmaa.2007.03.107. |
[34] |
F. Zhou and J. Wu,
Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425.
doi: 10.1017/S0956792515000108. |
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