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

Jiayue Zheng; Shangbin Cui

doi:10.3934/dcdsb.2020103

Article Contents

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

This issue Previous Article Large time behavior in a predator-prey system with indirect pursuit-evasion interaction Next Article Modelling fungal competition for space:Towards prediction of community dynamics

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: September 30, 2011

Early access: April 2020

Published: November 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.