# 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  November 2018 Early access  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
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