July  2020, 25(7): 2453-2460. doi: 10.3934/dcdsb.2020018

Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply

1. 

Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China

2. 

School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

* Corresponding author: Junde Wu

Received  June 2019 Revised  August 2019 Published  July 2020 Early access  April 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China under grant 11301474 and the Natural Science Foundation of Guangdong Province under grant 2018A030313536

In this paper we study a nonlinear free boundary problem for the growth of radially symmetric tumor with a necrotic core. The proliferation of tumor cells depends on the concentration of nutrient which satisfies a diffusion equation within tumor and is periodically supplied by external tissues. The tumor outer surface and the inner interface of the necrotic core are both free boundaries. We give a sufficient and necessary condition for the existence and uniqueness of positive periodic solution, and show it is globally asymptotically stable under radial perturbations. Our analysis implies that tumor growth may finally synchronize the periodic external nutrient supply.

Citation: Junde Wu, Shihe Xu. Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2453-2460. doi: 10.3934/dcdsb.2020018
References:
[1]

M. Bai and S. H. Xu, Qualitative analysis of a mathematical model for tumor growth with a periodic supply of external nutrients, Pac. J. Appl. Math., 5 (2013), 217-223. 

[2]

H. BuenoG. Ercole and A. Zumpano, Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases, SIAM J. Appl. Math., 68 (2008), 1004-1025.  doi: 10.1137/060654815.

[3]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.

[4]

S. B. Cui, Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors, Interfaces and Free Boundaries, 7 (2005), 147-159.  doi: 10.4171/IFB/118.

[5]

S. B. Cui, Formation of necrotic cores in the growth of tumors: Analytic results, Acta Math. Sci. Ser. B (Engl. Ed.), 26 (2006), 781-796.  doi: 10.1016/S0252-9602(06)60104-5.

[6]

S. B. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677.  doi: 10.1006/jmaa.2000.7306.

[7]

J. Escher and A.-V. Matioc, Radially symmetric growth of nonnecrotic tumors, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 1-20.  doi: 10.1007/s00030-009-0037-6.

[8] D. B. Forger, Biological clocks, Rhythms, and Oscillations. The theory of biological timekeeping, MIT Press, Cambridge, MA, 2017. 
[9]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.  doi: 10.1007/s002850050149.

[10]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[11]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[12]

Y. D. HuangZ. C. Zhang and B. Hu, Linear stability for a free boundary tumor model with a periodic supply of external nutrients, Math. Meth. Appl. Sci., 42 (2019), 1039-1054.  doi: 10.1002/mma.5412.

[13]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.

[14]

J. D. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, J. Differential Equations, 260 (2016), 5875-5893.  doi: 10.1016/j.jde.2015.12.023.

[15]

J. D. Wu, Analysis of a mathematical model for tumor growth with Gibbs-Thomson relation, J. Math. Anal. Appl., 450 (2017), 532-543.  doi: 10.1016/j.jmaa.2017.01.051.

[16]

J. D. Wu, Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Conti. Dyn. Syst., 39 (2019), 3399-3411.  doi: 10.3934/dcds.2019140.

[17]

J. D. Wu and C. Wang, Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors, Nonlinear Anal. Real World Appl., 50 (2019), 25-33.  doi: 10.1016/j.nonrwa.2019.04.012.

[18]

J. D. Wu and F. J. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.  doi: 10.1016/j.jde.2017.01.012.

[19]

Y. H. Zhuang and S. B. 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.

show all references

References:
[1]

M. Bai and S. H. Xu, Qualitative analysis of a mathematical model for tumor growth with a periodic supply of external nutrients, Pac. J. Appl. Math., 5 (2013), 217-223. 

[2]

H. BuenoG. Ercole and A. Zumpano, Stationary solutions of a model for the growth of tumors and a connection between the nonnecrotic and necrotic phases, SIAM J. Appl. Math., 68 (2008), 1004-1025.  doi: 10.1137/060654815.

[3]

H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216.  doi: 10.1016/0025-5564(96)00023-5.

[4]

S. B. Cui, Global existence of solutions for a free boundary problem modeling the growth of necrotic tumors, Interfaces and Free Boundaries, 7 (2005), 147-159.  doi: 10.4171/IFB/118.

[5]

S. B. Cui, Formation of necrotic cores in the growth of tumors: Analytic results, Acta Math. Sci. Ser. B (Engl. Ed.), 26 (2006), 781-796.  doi: 10.1016/S0252-9602(06)60104-5.

[6]

S. B. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677.  doi: 10.1006/jmaa.2000.7306.

[7]

J. Escher and A.-V. Matioc, Radially symmetric growth of nonnecrotic tumors, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 1-20.  doi: 10.1007/s00030-009-0037-6.

[8] D. B. Forger, Biological clocks, Rhythms, and Oscillations. The theory of biological timekeeping, MIT Press, Cambridge, MA, 2017. 
[9]

A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284.  doi: 10.1007/s002850050149.

[10]

A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342.  doi: 10.1090/S0002-9947-08-04468-1.

[11]

H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.  doi: 10.1002/sapm1972514317.

[12]

Y. D. HuangZ. C. Zhang and B. Hu, Linear stability for a free boundary tumor model with a periodic supply of external nutrients, Math. Meth. Appl. Sci., 42 (2019), 1039-1054.  doi: 10.1002/mma.5412.

[13]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), R1–R91. doi: 10.1088/0951-7715/23/1/R01.

[14]

J. D. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs-Thomson relation, J. Differential Equations, 260 (2016), 5875-5893.  doi: 10.1016/j.jde.2015.12.023.

[15]

J. D. Wu, Analysis of a mathematical model for tumor growth with Gibbs-Thomson relation, J. Math. Anal. Appl., 450 (2017), 532-543.  doi: 10.1016/j.jmaa.2017.01.051.

[16]

J. D. Wu, Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors, Discrete Conti. Dyn. Syst., 39 (2019), 3399-3411.  doi: 10.3934/dcds.2019140.

[17]

J. D. Wu and C. Wang, Radially symmetric growth of necrotic tumors and connection with nonnecrotic tumors, Nonlinear Anal. Real World Appl., 50 (2019), 25-33.  doi: 10.1016/j.nonrwa.2019.04.012.

[18]

J. D. Wu and F. J. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs-Thomson relation, J. Differential Equations, 262 (2017), 4907-4930.  doi: 10.1016/j.jde.2017.01.012.

[19]

Y. H. Zhuang and S. B. 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.

[1]

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

[2]

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

[3]

Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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 and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

[11]

Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213

[12]

Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929

[13]

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

[14]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

[15]

Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13

[16]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[17]

Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092

[18]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[19]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure and Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795

[20]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (324)
  • HTML views (106)
  • Cited by (0)

Other articles
by authors

[Back to Top]