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August  2018, 23(6): 2593-2605. doi: 10.3934/dcdsb.2018129

Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor

Zejia Wang , Suzhen Xu and  Huijuan Song ,

College of Mathematical and Informational Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author: Huijuan Song

Received  July 2017 Revised  October 2017 Published  August 2018 Early access  July 2018

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.

Keywords: Free boundary problem, tumor angiogenesis, stationary solution, bifurcation, symmetry-breaking solution.
Mathematics Subject Classification: Primary: 35B35, 35K35; Secondary: 35Q80.
Citation: Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129
References:
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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.  doi: 10.1016/0022-1236(71)90015-2.

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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.

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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.

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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.

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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.

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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.

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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.

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M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206. 

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A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159. 

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A. Friedman, Cancer models and their mathematical analysis, in Lecture Notes in Math., Vol. 1872, Springer, (2006), 223-246.

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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.

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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.

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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.

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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.

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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.

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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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