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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101

[2]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[3]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[4]

Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

[5]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021067

[6]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[7]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[8]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[9]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021035

[10]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109

[11]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[12]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210

[13]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021

[14]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[15]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[18]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[19]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[20]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (233)
  • HTML views (172)
  • Cited by (3)

Other articles
by authors

[Back to Top]