Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

Huijuan Song; Bei Hu; Zejia Wang

doi:10.3934/dcdsb.2020084

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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

Huijuan Song ^1, , Bei Hu ^2, and Zejia Wang ^1,

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 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^{**}) $.

Keywords: Stationary solution, free boundary problem, vascular tumor, necrotic core, bifurcation.

Mathematics Subject Classification: Primary: 35R35, 35K57, 35B32; Secondary: 35B35, 92B05.

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 & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

References:

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[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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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 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. 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]	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
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[19]	M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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 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. 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]	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
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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American Institute of Mathematical Sciences

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

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American Institute of Mathematical Sciences

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

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