November  2020, 25(11): 4397-4410. doi: 10.3934/dcdsb.2020103

Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition

School of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China

* Corresponding author: Shangbin Cui

Received  October 2011 Published  April 2020

Fund Project: This work is supported by China National Natural Science Foundation under the grant number 11571381

In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration $ \sigma $ is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient $ \gamma $ takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.

Citation: Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103
References:
[1]

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

[2]

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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S. Cui, Analysis of a free boundary problem modelling tumor growth, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1071-1082.  doi: 10.1007/s10114-004-0483-3.  Google Scholar

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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

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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

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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

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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

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J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 1997, 1028–1047. doi: 10.1137/S0036141095291919.  Google Scholar

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

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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

[11]

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

[12]

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

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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

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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

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Y. HuangZ. 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

[16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.   Google Scholar
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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

[18]

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

[19]

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

[20]

J. Zheng and S. Cui, Analysis of a tumor-model free boundary problem with a nonliear boundary condition, J. Math. Anal. Appl., 478 (2019), 806-824.  doi: 10.1016/j.jmaa.2019.05.056.  Google Scholar

[21]

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

[22]

F. ZhouJ. 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

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

S. Cui, Analysis of a free boundary problem modelling tumor growth, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1071-1082.  doi: 10.1007/s10114-004-0483-3.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 1997, 1028–1047. doi: 10.1137/S0036141095291919.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Y. HuangZ. 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

[16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.   Google Scholar
[17]

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

[18]

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

[19]

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

[20]

J. Zheng and S. Cui, Analysis of a tumor-model free boundary problem with a nonliear boundary condition, J. Math. Anal. Appl., 478 (2019), 806-824.  doi: 10.1016/j.jmaa.2019.05.056.  Google Scholar

[21]

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

[22]

F. ZhouJ. 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

[23]

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

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