DCDS-B
Analysis of a delayed free boundary problem for tumor growth
Shihe Xu
Discrete & Continuous Dynamical Systems - B 2011, 15(1): 293-308 doi: 10.3934/dcdsb.2011.15.293
In this paper we study a delayed free boundary problem for the growth of tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the process of proliferation is delayed compared to apoptosis. By $L^p$ theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as $t\rightarrow\infty.$
keywords: Asymptotic behavior. Parabolic equations Tumors Global solution
DCDS-B
Analysis of a free boundary problem for avascular tumor growth with a periodic supply of nutrients
Shihe Xu Yinhui Chen Meng Bai
Discrete & Continuous Dynamical Systems - B 2016, 21(3): 997-1008 doi: 10.3934/dcdsb.2016.21.997
In this paper we study a free boundary problem for the growth of avascular tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the supply of external nutrients is periodic. We mainly study the long time behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a positive periodic state.
keywords: global solution periodic solution free boundary problem asymptotic behavior. Tumors
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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