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Efficient and accurate sav schemes for the generalized Zakharov systems
Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core
1. | College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China |
2. | Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA |
In this paper, we present a rigorous mathematical analysis of a free boundary problem modeling the growth of a vascular solid tumor with a necrotic core. If the vascular system supplies the nutrient concentration $ \sigma $ to the tumor at a rate $ \beta $, then $ \frac{\partial\sigma}{\partial\bf n}+\beta(\sigma-\bar\sigma) = 0 $ holds on the tumor boundary, where $ \bf n $ is the unit outward normal to the boundary and $ \bar\sigma $ is the nutrient concentration outside the tumor. The living cells in the nonnecrotic region proliferate at a rate $ \mu $. We show that for any given $ \rho>0 $, there exists a unique $ R\in(\rho, \infty) $ such that the corresponding radially symmetric solution solves the steady-state necrotic tumor system with necrotic core boundary $ r = \rho $ and outer boundary $ r = R $; moreover, there exist a positive integer $ n^{**} $ and a sequence of $ \mu_n $, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solution for each $ \mu_n $ (even $ n\ge n^{**}) $.
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[2] |
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. |
[3] |
M. Bodnar and U. Foryś,
Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472.
doi: 10.3934/mbe.2005.2.461. |
[4] |
H. Bueno, G. Ercole and A. Zumpano,
Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases, SIAM J. Appl. Math., 68 (2008), 1004-1025.
doi: 10.1137/060654815. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
S. Cui,
Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors, Interfaces Free Bound., 7 (2005), 147-159.
doi: 10.4171/IFB/118. |
[11] |
S. Cui,
Formation of necrotic cores in the growth of tumors: Analytic results, Acta Math. Sci. Ser. B (Engl. Ed.), 26 (2006), 781-796.
doi: 10.1016/S0252-9602(06)60104-5. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
S. Cui, Analysis of a free boundary problem modeling the growth of necrotic tumors, arXiv: 1902.04066. Google Scholar |
[16] |
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. |
[17] |
S. Cui and Y. Zhuang,
Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.
doi: 10.1016/j.jmaa.2018.08.022. |
[18] |
J. Escher and A.-V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math. (Basel), 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
[19] |
M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
[20] |
U. Foryś and A. Mokwa-Borkowska,
Solid tumour growth analysis of necrotic core formation, Math. Comput. Modelling, 42 (2005), 593-600.
doi: 10.1016/j.mcm.2004.06.022. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
[24] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
doi: 10.3934/dcdsb.2004.4.147. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[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 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. |
[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] |
H. P. Greenspan,
Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.
doi: 10.1002/sapm1972514317. |
[33] |
H. P. Greenspan,
On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242.
doi: 10.1016/S0022-5193(76)80054-9. |
[34] |
W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. 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. |
[35] |
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. |
[36] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Contin. Dyn. Syst., 39 (2019), 2473-2510.
doi: 10.3934/dcds.2019105. |
[37] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
![]() |
[38] |
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. |
[39] |
J. D. Nagy,
The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng., 2 (2005), 381-418.
doi: 10.3934/mbe.2005.2.381. |
[40] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
![]() |
[41] |
H. Shen and X. Wei,
A qualitative analysis of a free boundary problem modeling tumor growth with angiogenesis, Nonlinear Anal. Real World Appl., 47 (2019), 106-126.
doi: 10.1016/j.nonrwa.2018.10.004. |
[42] |
H. Shen, X. Wei, C. Liu and Z. Feng, Existence and uniqueness of the stationary solution of mathematical model of necrotic tumor with the third boundary, Acta Sci. Natur. Univ. Sunyatseni, 57 (2018), 140-144. Google Scholar |
[43] |
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. |
[44] |
Z. Wang, S. Xu and H. Song,
Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2593-2605.
doi: 10.3934/dcdsb.2018129. |
[45] |
X. Wei,
Global existence for a free boundary problem modelling the growth of necrotic tumors in the presence of inhibitors, Int. J. Pure Appl. Math., 28 (2006), 321-338.
|
[46] |
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. |
[47] |
J. Wu,
Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Contin. Dyn. Syst., 39 (2019), 3399-3411.
doi: 10.3934/dcds.2019140. |
[48] |
J. Wu,
Asymptotic stability of a free boundary problem for the growth of multi-layer tumours in the necrotic phase, Nonlinearity, 32 (2019), 2955-2974.
doi: 10.1088/1361-6544/ab15a8. |
[49] |
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. |
[50] |
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. |
[51] |
S. Xu and M. Bai,
Stability of solutions to a mathematical model for necrotic tumor growth with time delays in proliferation, J. Math. Anal. Appl., 421 (2015), 955-962.
doi: 10.1016/j.jmaa.2014.07.029. |
[52] |
F. Zhou and S. Cui,
Bifurcation for a free boundary problem modeling the growth of multi-layer tumors, Nonlinear Anal., 68 (2008), 2128-2145.
doi: 10.1016/j.na.2007.01.036. |
[53] |
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. |
[54] |
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. |
[55] |
Y. Zhuang and S. Cui,
Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis, J. Differential Equations, 265 (2018), 620-644.
doi: 10.1016/j.jde.2018.03.005. |
[56] |
Y. Zhuang and S. Cui,
Analysis of a free boundary problem modeling the growth of spherically symmetric tumors with angiogenesis, Acta Appl. Math., 161 (2019), 153-169.
doi: 10.1007/s10440-018-0208-8. |
show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[2] |
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. |
[3] |
M. Bodnar and U. Foryś,
Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472.
doi: 10.3934/mbe.2005.2.461. |
[4] |
H. Bueno, G. Ercole and A. Zumpano,
Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases, SIAM J. Appl. Math., 68 (2008), 1004-1025.
doi: 10.1137/060654815. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
S. Cui,
Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors, Interfaces Free Bound., 7 (2005), 147-159.
doi: 10.4171/IFB/118. |
[11] |
S. Cui,
Formation of necrotic cores in the growth of tumors: Analytic results, Acta Math. Sci. Ser. B (Engl. Ed.), 26 (2006), 781-796.
doi: 10.1016/S0252-9602(06)60104-5. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
S. Cui, Analysis of a free boundary problem modeling the growth of necrotic tumors, arXiv: 1902.04066. Google Scholar |
[16] |
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. |
[17] |
S. Cui and Y. Zhuang,
Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis, J. Math. Anal. Appl., 468 (2018), 391-405.
doi: 10.1016/j.jmaa.2018.08.022. |
[18] |
J. Escher and A.-V. Matioc,
Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math. (Basel), 97 (2011), 79-90.
doi: 10.1007/s00013-011-0276-8. |
[19] |
M. A. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
[20] |
U. Foryś and A. Mokwa-Borkowska,
Solid tumour growth analysis of necrotic core formation, Math. Comput. Modelling, 42 (2005), 593-600.
doi: 10.1016/j.mcm.2004.06.022. |
[21] |
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. |
[22] |
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. |
[23] |
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. |
[24] |
A. Friedman,
A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.
doi: 10.3934/dcdsb.2004.4.147. |
[25] |
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. |
[26] |
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. |
[27] |
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. |
[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 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. |
[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] |
H. P. Greenspan,
Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.
doi: 10.1002/sapm1972514317. |
[33] |
H. P. Greenspan,
On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242.
doi: 10.1016/S0022-5193(76)80054-9. |
[34] |
W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. 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. |
[35] |
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. |
[36] |
Y. Huang, Z. Zhang and B. Hu,
Bifurcation from stability to instability for a free boundary tumor model with angiogenesis, Discrete Contin. Dyn. Syst., 39 (2019), 2473-2510.
doi: 10.3934/dcds.2019105. |
[37] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
![]() |
[38] |
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. |
[39] |
J. D. Nagy,
The ecology and evolutionary biology of cancer: A review of mathematical models of necrosis and tumor cell diversity, Math. Biosci. Eng., 2 (2005), 381-418.
doi: 10.3934/mbe.2005.2.381. |
[40] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
![]() |
[41] |
H. Shen and X. Wei,
A qualitative analysis of a free boundary problem modeling tumor growth with angiogenesis, Nonlinear Anal. Real World Appl., 47 (2019), 106-126.
doi: 10.1016/j.nonrwa.2018.10.004. |
[42] |
H. Shen, X. Wei, C. Liu and Z. Feng, Existence and uniqueness of the stationary solution of mathematical model of necrotic tumor with the third boundary, Acta Sci. Natur. Univ. Sunyatseni, 57 (2018), 140-144. Google Scholar |
[43] |
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. |
[44] |
Z. Wang, S. Xu and H. Song,
Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2593-2605.
doi: 10.3934/dcdsb.2018129. |
[45] |
X. Wei,
Global existence for a free boundary problem modelling the growth of necrotic tumors in the presence of inhibitors, Int. J. Pure Appl. Math., 28 (2006), 321-338.
|
[46] |
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. |
[47] |
J. Wu,
Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Contin. Dyn. Syst., 39 (2019), 3399-3411.
doi: 10.3934/dcds.2019140. |
[48] |
J. Wu,
Asymptotic stability of a free boundary problem for the growth of multi-layer tumours in the necrotic phase, Nonlinearity, 32 (2019), 2955-2974.
doi: 10.1088/1361-6544/ab15a8. |
[49] |
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. |
[50] |
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. |
[51] |
S. Xu and M. Bai,
Stability of solutions to a mathematical model for necrotic tumor growth with time delays in proliferation, J. Math. Anal. Appl., 421 (2015), 955-962.
doi: 10.1016/j.jmaa.2014.07.029. |
[52] |
F. Zhou and S. Cui,
Bifurcation for a free boundary problem modeling the growth of multi-layer tumors, Nonlinear Anal., 68 (2008), 2128-2145.
doi: 10.1016/j.na.2007.01.036. |
[53] |
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. |
[54] |
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. |
[55] |
Y. Zhuang and S. Cui,
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