# American Institute of Mathematical Sciences

January  2021, 26(1): 667-691. doi: 10.3934/dcdsb.2020084

## 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

Received  September 2019 Revised  November 2019 Published  January 2021 Early access  January 2020

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 $n\ge n^{**})$.

Citation: Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084
##### 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. [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. [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.

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##### 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. [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. [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.
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