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Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors
Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China |
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.
References:
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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. |
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A. Borisovich and A. Friedman,
Symmetry-breaking bifurcations for free boundary problems, Indiana. Uni. Math. J., 54 (2005), 927-947.
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H. Byrne and M. Chaplain,
Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.
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L. A. Caffarelli,
The regularity of free boundaries in higher dimension, Acta Math., 139 (1977), 155-184.
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M. Crandall and P. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
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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. |
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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. |
| [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. |
| [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. |
| [10] |
M. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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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. |
| [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. |
| [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. |
| [21] |
J. Wu, Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase, arXiv:1802.03112. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
L. A. Caffarelli,
The regularity of free boundaries in higher dimension, Acta Math., 139 (1977), 155-184.
doi: 10.1007/BF02392236. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
M. Fontelos and A. Friedman,
Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
J. Wu, Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase, arXiv:1802.03112. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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