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  December 2019

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 & Continuous Dynamical Systems - B, 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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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