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June  2019, 39(6): 3399-3411. doi: 10.3934/dcds.2019140

Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors

Junde Wu

Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China

Received  August 2018 Published  February 2019

In this paper we study bifurcation solutions of a free boundary problem modeling the growth of necrotic multilayered tumors. The tumor model consists of two elliptic differential equations for nutrient concentration and pressure, with discontinuous terms and two free boundaries. The novelty is that different types of boundary conditions are imposed on two free boundaries. By bifurcation analysis, we show that there exist infinitely many branches of non-flat stationary solutions bifurcating from the unique flat stationary solution.

Keywords: Free boundary problem, bifurcation, stationary solutions, tumor model, necrotic phase.
Mathematics Subject Classification: Primary: 35B32, 35R35; Secondary: 35Q92.
Citation: Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140
References:
[1]

W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc., 47 (2004), 15-33.  doi: 10.1017/S0013091502000378.  Google Scholar

[2]

A. Borisovich and A. Friedman, Symmetry-breaking bifurcations for free boundary problems, Indiana. Uni. Math. J., 54 (2005), 927-947.  doi: 10.1512/iumj.2005.54.2473.  Google Scholar

[3]

H. Byrne and M. 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

[4]

L. A. Caffarelli, The regularity of free boundaries in higher dimension, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

[5]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

S. Cui, Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems, arXiv:1606.09393. Google Scholar

[7]

S. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modeling stationary growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235.  doi: 10.1137/060657509.  Google Scholar

[8]

S. Cui and J. Escher, Asymptotic behavior 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

[9]

S. Cui and J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Rational Mech. Anal., 191 (2009), 173-193.  doi: 10.1007/s00205-008-0158-9.  Google Scholar

[10]

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

[11]

A. Friedman, Cancer models and their mathematical analysis, in Tutorials in Mathematical Biosciences. III, Lecture Notes in Math., 1872, Springer, Berlin, (2006), 223–246. doi: 10.1007/11561606_6.  Google Scholar

[12]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Rational Mech. Anal., 180 (2006), 293-330.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[13]

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

[14]

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

[15]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[16]

W. Hao, J. D. Hauenstein, B. Hu and et al, 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

[17]

J. Kim, R. Stein and M. O'Haxe, Three-dimensional in vitro tissue culture models for breast cancer – a review, Breast Cancer Research and Treatment, 85 (2004), 281-291.  doi: 10.1023/B:BREA.0000025418.88785.2b.  Google Scholar

[18]

A. Kyle, C. Chan and A. Minchinton, Characterization of three-dimensional tissue cultures using electrical impedance spectroscopy, Biophysical J., 76 (1999), 2640-2648.  doi: 10.1016/S0006-3495(99)77416-3.  Google Scholar

[19]

F. Li and B. Liu, Bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate, J. Differential Equations, 263 (2017), 7627-7646.  doi: 10.1016/j.jde.2017.08.023.  Google Scholar

[20]

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

[21]

J. Wu, Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase, arXiv:1802.03112. Google Scholar

[22]

J. Wu and S. Cui, Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Meth. Appl. Sci., 38 (2015), 1813-1823.  doi: 10.1002/mma.3190.  Google Scholar

[23]

J. Wu and F. Zhou, Bifurcation analysis of a free boundary problem modelling tumour growth under the action of inhibitors, Nonlinearity, 25 (2012), 2971-2991.  doi: 10.1088/0951-7715/25/10/2971.  Google Scholar

[24]

J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.  doi: 10.1016/j.jde.2017.01.012.  Google Scholar

[25]

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

[26]

F. Zhou and J. Wu, Stability and bifurcation analysis of a free boundary problem modeling multi-layer tumors with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425.  doi: 10.1017/S0956792515000108.  Google Scholar

show all references

References:
[1]

W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc., 47 (2004), 15-33.  doi: 10.1017/S0013091502000378.  Google Scholar

[2]

A. Borisovich and A. Friedman, Symmetry-breaking bifurcations for free boundary problems, Indiana. Uni. Math. J., 54 (2005), 927-947.  doi: 10.1512/iumj.2005.54.2473.  Google Scholar

[3]

H. Byrne and M. 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

[4]

L. A. Caffarelli, The regularity of free boundaries in higher dimension, Acta Math., 139 (1977), 155-184.  doi: 10.1007/BF02392236.  Google Scholar

[5]

M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

S. Cui, Linearized stability theorem for invariant and quasi-invariant parabolic differential equations in Banach manifolds with applications to free boundary problems, arXiv:1606.09393. Google Scholar

[7]

S. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modeling stationary growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235.  doi: 10.1137/060657509.  Google Scholar

[8]

S. Cui and J. Escher, Asymptotic behavior 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

[9]

S. Cui and J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Rational Mech. Anal., 191 (2009), 173-193.  doi: 10.1007/s00205-008-0158-9.  Google Scholar

[10]

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

[11]

A. Friedman, Cancer models and their mathematical analysis, in Tutorials in Mathematical Biosciences. III, Lecture Notes in Math., 1872, Springer, Berlin, (2006), 223–246. doi: 10.1007/11561606_6.  Google Scholar

[12]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Rational Mech. Anal., 180 (2006), 293-330.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[13]

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

[14]

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

[15]

R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65-222.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[16]

W. Hao, J. D. Hauenstein, B. Hu and et al, 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

[17]

J. Kim, R. Stein and M. O'Haxe, Three-dimensional in vitro tissue culture models for breast cancer – a review, Breast Cancer Research and Treatment, 85 (2004), 281-291.  doi: 10.1023/B:BREA.0000025418.88785.2b.  Google Scholar

[18]

A. Kyle, C. Chan and A. Minchinton, Characterization of three-dimensional tissue cultures using electrical impedance spectroscopy, Biophysical J., 76 (1999), 2640-2648.  doi: 10.1016/S0006-3495(99)77416-3.  Google Scholar

[19]

F. Li and B. Liu, Bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate, J. Differential Equations, 263 (2017), 7627-7646.  doi: 10.1016/j.jde.2017.08.023.  Google Scholar

[20]

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

[21]

J. Wu, Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase, arXiv:1802.03112. Google Scholar

[22]

J. Wu and S. Cui, Bifurcation analysis of a mathematical model for the growth of solid tumors in the presence of external inhibitors, Math. Meth. Appl. Sci., 38 (2015), 1813-1823.  doi: 10.1002/mma.3190.  Google Scholar

[23]

J. Wu and F. Zhou, Bifurcation analysis of a free boundary problem modelling tumour growth under the action of inhibitors, Nonlinearity, 25 (2012), 2971-2991.  doi: 10.1088/0951-7715/25/10/2971.  Google Scholar

[24]

J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.  doi: 10.1016/j.jde.2017.01.012.  Google Scholar

[25]

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

[26]

F. Zhou and J. Wu, Stability and bifurcation analysis of a free boundary problem modeling multi-layer tumors with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425.  doi: 10.1017/S0956792515000108.  Google Scholar

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