May 2019, 39(5): 2473-2510. doi: 10.3934/dcds.2019105

Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

2. 

Department of Applied Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

* Corresponding author: Zhengce Zhang

Received  January 2018 Revised  October 2018 Published  January 2019

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients $ \sigma $ and an elliptic equation describing the distribution of the internal pressure $ p $. The vasculature supplies nutrients to the tumor, so that $ \frac{\partial\sigma}{\partial \mathbf{n}}+\beta(\sigma-\bar{\sigma}) = 0 $ holds on the boundary, where a positive constant $ \beta $ is the rate of nutrient supply to the tumor and $ \bar{\sigma} $ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate $ \mu $. If $ 0<\widetilde{\sigma}<\overline{\sigma} $, where $ \widetilde{\sigma} $ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution $ (\sigma_S(r), p_S(r), R_S) $. In this paper, we found a function $ \mu^\ast = \mu^\ast(R_S) $ such that if $ \mu<\mu^\ast $ then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if $ \mu>\mu^\ast $ then the radially symmetric stationary solution is linearly unstable.

Citation: Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
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J. A. Adam, A simplified mathematical model of tumor growth, Math. Biosci., 81 (1986), 229-244. doi: 10.1016/0025-5564(86)90119-7.

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J. A. Adam, A mathematical model of tumor growth: Ⅲ. Comparison with experiment, Math. Biosci., 86 (1987), 213-227. doi: 10.1016/0025-5564(87)90011-3.

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J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, 1nd edition, Birkhäuser Basel, 1997. doi: 10.1007/978-0-8176-8119-7.

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B. V. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 1265-1304. doi: 10.1512/iumj.2003.52.2317.

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N. F. Britton and M. A. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci., 113 (1993), 77-89. doi: 10.1016/0025-5564(93)90009-Y.

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H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol., 14 (1997), 305-323. doi: 10.1093/imammb/14.4.305.

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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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H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216. doi: 10.1016/0025-5564(96)00023-5.

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H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1-17. doi: 10.1016/S0895-7177(96)00174-4.

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H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-658. doi: 10.1017/S0956792597003264.

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X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

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V. CristiniJ. 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, Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth, SIAM J. Math. Anal., 45 (2013), 2870-2893. doi: 10.1137/130906271.

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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 J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Ration. Mech. Anal., 191 (2009), 173-193. doi: 10.1007/s00205-008-0158-9.

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S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc., 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4.

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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.

[23]

J. Escher and A.-V. Matioc, Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 573-596. doi: 10.3934/dcdsb.2011.15.573.

[24]

J. Escher and A.-V. Matioc, Analysis of a two-phase model describing the growth of solid tumors, European J. Appl. Math., 24 (2013), 25-48. doi: 10.1017/S0956792512000290.

[25]

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

[26]

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.

[27]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.

[28]

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.

[29]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.

[30]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Holf 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.

[31]

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.

[32]

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.

[33]

A. Friedman and F. Reitich, Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Sc. Norm. Super Pisa CI. Sci., 30 (2001), 341-403.

[34]

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.

[35]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317.

[36]

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.

[37]

W. HaoJ. D. HauensteinB. HuY. LiuA. J. Sommese and Y. Zhang, Continuation along bifurcation branches for a tumor model with a necrotic core, J. Sci. Comput., 53 (2012), 395-413. doi: 10.1007/s10915-012-9575-x.

[38]

W. HaoJ. D. HauensteinB. HuT. McCoy and A. J. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, J. Comput. Appl. Math., 237 (2013), 326-334. doi: 10.1016/j.cam.2012.06.001.

[39]

W. HaoJ. D. HauensteinB. Hu and A. J. Sommese, A three-dimensional steady-state tumor system, Appl. Math. Comput., 218 (2011), 2661-2669. doi: 10.1016/j.amc.2011.08.006.

[40]

Y. D. HuangZ. C. 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.

[41]

J. S. LowengrubH. B. FrieboesF. JinY.-L. ChuangX. LiP. MacklinS. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/R01.

[42]

Z. J. 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.

[43]

J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, IMA J. Math. Appl. Med. Biol., 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39.

[44]

J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth Ⅱ: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 16 (1999), 171-211. doi: 10.1093/imammb/16.2.171.

[45]

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.

[46]

J. Wu and S. Cui, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with Stokes equations, Discrete Contin. Dyn. Syst., 24 (2009), 625-651. doi: 10.3934/dcds.2009.24.625.

[47]

J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal., 41 (2009), 391-414. doi: 10.1137/080726550.

[48]

J. Wu and S. Cui, Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth, Discrete Contin. Dyn. Syst., 26 (2010), 737-765. doi: 10.3934/dcds.2010.26.737.

[49]

F. ZhouJ. Escher and S. Cui, Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors, J. Differential Equations, 244 (2008), 2909-2933. doi: 10.1016/j.jde.2008.02.038.

[50]

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.

[51]

F. ZhouJ. Wu and S. Cui, Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors, Comm. Pure Appl. Anal., 8 (2009), 1669-1688. doi: 10.3934/cpaa.2009.8.1669.

show all references

References:
[1]

J. A. Adam, A simplified mathematical model of tumor growth, Math. Biosci., 81 (1986), 229-244. doi: 10.1016/0025-5564(86)90119-7.

[2]

J. A. Adam, A mathematical model of tumor growth: Ⅲ. Comparison with experiment, Math. Biosci., 86 (1987), 213-227. doi: 10.1016/0025-5564(87)90011-3.

[3]

J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, 1nd edition, Birkhäuser Basel, 1997. doi: 10.1007/978-0-8176-8119-7.

[4]

B. V. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 1265-1304. doi: 10.1512/iumj.2003.52.2317.

[5]

N. F. Britton and M. A. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci., 113 (1993), 77-89. doi: 10.1016/0025-5564(93)90009-Y.

[6]

H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol., 14 (1997), 305-323. doi: 10.1093/imammb/14.4.305.

[7]

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.

[8]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216. doi: 10.1016/0025-5564(96)00023-5.

[9]

H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1-17. doi: 10.1016/S0895-7177(96)00174-4.

[10]

H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-658. doi: 10.1017/S0956792597003264.

[11]

X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.

[12]

V. CristiniJ. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.

[13]

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.

[14]

S. Cui, Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth, SIAM J. Math. Anal., 45 (2013), 2870-2893. doi: 10.1137/130906271.

[15]

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.

[16]

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.

[17]

S. Cui and J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Ration. Mech. Anal., 191 (2009), 173-193. doi: 10.1007/s00205-008-0158-9.

[18]

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.

[19]

S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677. doi: 10.1006/jmaa.2000.7306.

[20]

S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc., 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4.

[21]

J. Escher and A.-V. Matioc, Radially symmetric growth of nonnecrotic tumors, Nonlinear Differ. Equ. Appl., 17 (2010), 1-20. doi: 10.1007/s00030-009-0037-6.

[22]

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.

[23]

J. Escher and A.-V. Matioc, Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 573-596. doi: 10.3934/dcdsb.2011.15.573.

[24]

J. Escher and A.-V. Matioc, Analysis of a two-phase model describing the growth of solid tumors, European J. Appl. Math., 24 (2013), 25-48. doi: 10.1017/S0956792512000290.

[25]

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

[26]

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.

[27]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.

[28]

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.

[29]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.

[30]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Holf 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.

[31]

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.

[32]

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.

[33]

A. Friedman and F. Reitich, Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Sc. Norm. Super Pisa CI. Sci., 30 (2001), 341-403.

[34]

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.

[35]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317.

[36]

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.

[37]

W. HaoJ. D. HauensteinB. HuY. LiuA. J. Sommese and Y. Zhang, Continuation along bifurcation branches for a tumor model with a necrotic core, J. Sci. Comput., 53 (2012), 395-413. doi: 10.1007/s10915-012-9575-x.

[38]

W. HaoJ. D. HauensteinB. HuT. McCoy and A. J. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, J. Comput. Appl. Math., 237 (2013), 326-334. doi: 10.1016/j.cam.2012.06.001.

[39]

W. HaoJ. D. HauensteinB. Hu and A. J. Sommese, A three-dimensional steady-state tumor system, Appl. Math. Comput., 218 (2011), 2661-2669. doi: 10.1016/j.amc.2011.08.006.

[40]

Y. D. HuangZ. C. 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.

[41]

J. S. LowengrubH. B. FrieboesF. JinY.-L. ChuangX. LiP. MacklinS. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/R01.

[42]

Z. J. 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.

[43]

J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, IMA J. Math. Appl. Med. Biol., 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39.

[44]

J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth Ⅱ: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 16 (1999), 171-211. doi: 10.1093/imammb/16.2.171.

[45]

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.

[46]

J. Wu and S. Cui, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with Stokes equations, Discrete Contin. Dyn. Syst., 24 (2009), 625-651. doi: 10.3934/dcds.2009.24.625.

[47]

J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal., 41 (2009), 391-414. doi: 10.1137/080726550.

[48]

J. Wu and S. Cui, Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth, Discrete Contin. Dyn. Syst., 26 (2010), 737-765. doi: 10.3934/dcds.2010.26.737.

[49]

F. ZhouJ. Escher and S. Cui, Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors, J. Differential Equations, 244 (2008), 2909-2933. doi: 10.1016/j.jde.2008.02.038.

[50]

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.

[51]

F. ZhouJ. Wu and S. Cui, Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors, Comm. Pure Appl. Anal., 8 (2009), 1669-1688. doi: 10.3934/cpaa.2009.8.1669.

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