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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 |
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.
References:
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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. |
[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. 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. |
[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. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. 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. Hao, J. D. Hauenstein, B. Hu, T. 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. Hao, J. D. Hauenstein, B. 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. Huang, Z. 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. 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), 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. Zhou, J. 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. Zhou, J. 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. 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. |
[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. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. 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. Hao, J. D. Hauenstein, B. Hu, T. 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. Hao, J. D. Hauenstein, B. 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. Huang, Z. 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. 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), 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. Zhou, J. 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. Zhou, J. 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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