Linear stability for a free boundary problem modeling the growth of tumor cord with time delay

Haihua Zhou; Yaxin Liu; Zejia Wang; Huijuan Song; Haihua Zhou; Yaxin Liu; Zejia Wang; Huijuan Song

doi:10.3934/mbe.2024103

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2344-2365. doi: 10.3934/mbe.2024103

Previous Article Next Article

Research article Special Issues

Linear stability for a free boundary problem modeling the growth of tumor cord with time delay

1.
School of Mathematics and Statistics, Hengyang Normal University, Hengyang 421010, China
2.
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China

Academic Editor: Wenrui Hao

Received: 16 November 2023 Revised: 08 January 2024 Accepted: 09 January 2024 Published: 15 January 2024

This paper was concerned with a free boundary problem modeling the growth of tumor cord with a time delay in cell proliferation, in which the cell location was incorporated, the domain was bounded in $ \mathbb{R}^2 $, and its boundary included two disjoint closed curves, one fixed and the other moving and a priori unknown. A parameter $ \mu $ represents the aggressiveness of the tumor. We proved that there exists a unique radially symmetric stationary solution for sufficiently small time delay, and this stationary solution is linearly stable under the nonradially symmetric perturbations for any $ \mu > 0 $. Moreover, adding the time delay in the model leads to a larger stationary tumor. If the tumor aggressiveness parameter is bigger, the time delay has a greater effect on the size of the stationary tumor, but it has no effect on the stability of the stationary solution.
- free boundary problem,
- tumor cord,
- time delay,
- stability,
- stationary solution
Citation: Haihua Zhou, Yaxin Liu, Zejia Wang, Huijuan Song. Linear stability for a free boundary problem modeling the growth of tumor cord with time delay[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2344-2365. doi: 10.3934/mbe.2024103

Related Papers:

Abstract

This paper was concerned with a free boundary problem modeling the growth of tumor cord with a time delay in cell proliferation, in which the cell location was incorporated, the domain was bounded in $ \mathbb{R}^2 $, and its boundary included two disjoint closed curves, one fixed and the other moving and a priori unknown. A parameter $ \mu $ represents the aggressiveness of the tumor. We proved that there exists a unique radially symmetric stationary solution for sufficiently small time delay, and this stationary solution is linearly stable under the nonradially symmetric perturbations for any $ \mu > 0 $. Moreover, adding the time delay in the model leads to a larger stationary tumor. If the tumor aggressiveness parameter is bigger, the time delay has a greater effect on the size of the stationary tumor, but it has no effect on the stability of the stationary solution.

References

[1]	B. Bazaliy, A. Friedman, A free boundary problem for an elliptic parabolic system: application to a model of tumor growth, Commun. Partial Differ. Equations, 28 (2003), 517–560. https://doi.org/10.1081/PDE-120020486 doi: 10.1081/PDE-120020486
[2]	B. Bazaliy, A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: An application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 1265–1304.
[3]	S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol., 44 (2002), 395–426. https://doi.org/10.1007/s002850100130 doi: 10.1007/s002850100130
[4]	S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta Math. Sin. (Engl. Ser.), 21 (2005), 1071–1082. https://doi.org/10.1007/s10114-004-0483-3 doi: 10.1007/s10114-004-0483-3
[5]	S. Cui, Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors, J. Funct. Anal., 245 (2007), 1–18. https://doi.org/10.1016/j.jfa.2006.12.020 doi: 10.1016/j.jfa.2006.12.020
[6]	S. Cui, J. Escher, Asymptotic behavior of solutions of a multidimensional moving boundary problem modeling tumor growth, Commun. Partial Differ. Equations, 33 (2008), 636–655. https://doi.org/10.1080/03605300701743848 doi: 10.1080/03605300701743848
[7]	S. Cui, S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523–541. https://doi.org/10.1016/j.jmaa.2007.02.047 doi: 10.1016/j.jmaa.2007.02.047
[8]	S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal. Real World Appl., 11 (2010), 401–406. https://doi.org/10.1016/j.nonrwa.2008.11.002 doi: 10.1016/j.nonrwa.2008.11.002
[9]	S. Xu, Q. Zhou, M. Bai, Qualitative analysis of a time-delayed free boundary problem for the tumor growth under the action of external inhibitors, Math. Methods Appl. Sci., 38 (2015), 4187–4198. https://doi.org/10.1002/mma.3357 doi: 10.1002/mma.3357
[10]	J. V. Moore, P. S. Hasleton, C. H. Buckley, Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: Inferences for their cellular kinetics and radiobiology, Br. J. Cancer, 51 (1985), 407–413. https://doi.org/10.1038/bjc.1985.55 doi: 10.1038/bjc.1985.55
[11]	J. Dyson, R. Villella-Bressan, G. Webb, The steady state of a maturity structured tumor cord cell population, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 115–134. https://doi.org/10.3934/dcdsb.2004.4.115 doi: 10.3934/dcdsb.2004.4.115
[12]	J. Dyson, R. Villella-Bressan, G. Webb, The evolution of a tumor cord cell population, Commun. Pure Appl. Anal., 3 (2004), 331–352. https://doi.org/10.3934/cpaa.2004.3.331 doi: 10.3934/cpaa.2004.3.331
[13]	G. F. Webb, The steady state of a tumor cord cell population, J. Evol. Equations, 2 (2002), 425–438. https://doi.org/10.1007/PL00012598 doi: 10.1007/PL00012598
[14]	F. Zhou, S. Cui, Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord, Discrete Contin. Dyn. Syst., 21 (2008), 929–943. https://doi.org/10.3934/dcds.2008.21.929 doi: 10.3934/dcds.2008.21.929
[15]	J. Wu, F. Zhou, S. Cui, Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord, J. Math. Anal. Appl., 352 (2009), 184–205. https://doi.org/10.1016/j.jmaa.2008.06.028 doi: 10.1016/j.jmaa.2008.06.028
[16]	X. Zhao, B. Hu, The impact of time delay in a tumor model, Nonlinear Anal. Real World Appl., 51 (2020), 1–29. https://doi.org/10.1016/j.nonrwa.2019.103015 doi: 10.1016/j.nonrwa.2019.103015
[17]	A. Friedman, B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293–330. https://doi.org/10.1007/s00205-005-0408-z doi: 10.1007/s00205-005-0408-z

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)