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

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.

keywords: Tumor growth free boundary problem global existence and uniqueness asymptotic behavior stability

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