American Institute of Mathematical Sciences

Advanced
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

Jiayue Zheng and  Shangbin Cui ,

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

* Corresponding author: Shangbin Cui

Received  October 2011 Published  November 2020 Early access  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.

Keywords: Free boundary problem, tumor model, nonlinear boundary condition, stationary solution, bifurcation.
Mathematics Subject Classification: 35R35; 46N60; 92B99.
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

[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. 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

[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. 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

[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. 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

[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. 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

[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

[1]

Zejia Wang, Suzhen Xu, Huijuan Song. Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2593-2605. doi: 10.3934/dcdsb.2018129
[2]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[3]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293
[4]

Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045
[5]

Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837
[6]

Shihe Xu, Yinhui Chen, Meng Bai. Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 997-1008. doi: 10.3934/dcdsb.2016.21.997
[7]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[8]

Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357
[9]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253
[10]

Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1
[11]

Meihua Wei, Yanling Li, Xi Wei. Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5203-5224. doi: 10.3934/dcdsb.2019129
[12]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[13]

Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140
[14]

Yongzhi Xu. A free boundary problem model of ductal carcinoma in situ. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 337-348. doi: 10.3934/dcdsb.2004.4.337
[15]

Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979
[16]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[17]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[18]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
[19]

Byung-Hoon Hwang, Seok-Bae Yun. Stationary solutions to the boundary value problem for the relativistic BGK model in a slab. Kinetic & Related Models, 2019, 12 (4) : 749-764. doi: 10.3934/krm.2019029
[20]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (181)
  • HTML views (255)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences