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

Jiayue Zheng; Shangbin Cui

doi:10.3934/dcdsb.2020103

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2020, Volume 25, Issue 11: 4397-4410. Doi: 10.3934/dcdsb.2020103

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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
^* Corresponding author: Shangbin Cui

Received: October 2011

Early access: April 2020

Published: November 2020

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

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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.

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Mathematics Subject Classification: 35R35; 46N60; 92B99.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptot. Anal., 35 (2003), 187-206.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.
[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.
[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.
[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.
[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.
[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.
[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.
[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.