# American Institute of Mathematical Sciences

November  2018, 23(9): 3535-3551. doi: 10.3934/dcdsb.2017213

## Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays

 1 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China 2 College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China

* Corresponding author: Shihe Xu

Received  April 2017 Revised  June 2017 Published  September 2017

Fund Project: The first two authors of this work are partially supported by NNSF of China (11301474), Foundation for Distinguish Young Teacher in Higher Education of Guangdong, China(YQ2015167) and NSF of Guangdong Province (2015A030313707), the third author is partially supported by NNSF of China(51508319,61374195,51409157)

In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case $c$ (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as $t\to ∞$. The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions.

Citation: Shihe Xu, Meng Bai, Fangwei Zhang. Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3535-3551. doi: 10.3934/dcdsb.2017213
##### References:
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##### References:
 [1] M. Bodnar, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91-95. [2] M. Bodnar and U. Foryś, Time delay in necrotic core formation, Math. Biosci. Eng., 2 (2005), 461-472. doi: 10.3934/mbe.2005.2.461. [3] H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83-117. doi: 10.1016/S0025-5564(97)00023-0. [4] H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3. [5] H. Byrne and M. 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. [6] H. M. Byrne, Modelling aspects of cancer dynamics: A review, Trans. Royal Soc. A, 364 (2006), 1563-1578. doi: 10.1098/rsta.2006.1786. [7] S. Cui, Fromation of necrotic cores in the growth of tumors: Analytic results, Aata. Math. Scientia, 26 (2006), 781-796. doi: 10.1016/S0252-9602(06)60104-5. [8] S. Cui and A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103-137. doi: 10.1016/S0025-5564(99)00063-2. [9] S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta. Math. Sinica., 21 (2005), 1071-1082. doi: 10.1007/s10114-004-0483-3. [10] S. Cui and 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. doi: 10.1016/j.jmaa.2007.02.047. [11] U. Foryś and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201-1209. doi: 10.1016/S0895-7177(03)80019-5. [12] U. Foryś and A. Mokwa-Borkowska, Solid tumour growth analysis of necrotic core formation, Math. Comput. Modelling, 42 (2005), 593-600. doi: 10.1016/j.mcm.2004.06.022. [13] U. Foryś and M. Kolev, Time delays in proliferation and apoptosis for solid avascular tumour, Mathematical Modelling of Population Dynamics, 63 (2004), 187-196. [14] 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. [15] H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317. [16] H. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9. [17] J. Hale, Theory of Functional Differential Equations, 2nd edition, Springer-Verlag, New York, 1977. doi: 10.1007/978-1-4612-9892-2. [18] J. 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. [19] J. 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. [20] J. Wu and F. 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. [21] S. Xu, Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients, Discrete and Contin. Dyn. Syst. B., 21 (2016), 997-1008. [22] S. Xu, M. Bai and X. Q. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38-47. doi: 10.1016/j.jmaa.2012.02.034.
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