标题： 多层肿瘤生长时滞自由边界问题的线性稳定性 显示英文标题

标题： The linear stability for a free boundary problem modeling multi-layer tumor growth with time delay

摘要： 我们研究一个自由边界问题，该问题模拟具有小时间延迟 $\tau$的多层肿瘤生长，表示细胞完成复制过程所需的时间。 该模型由两个椭圆方程组成，分别描述营养物质的浓度和肿瘤组织的压力，一个常微分方程描述表征时间延迟的细胞位置，以及一个偏微分方程用于自由边界。 在本文中，我们建立了该问题的适定性，即首先我们证明对于所有 $\mu>0$，存在一个唯一的平坦稳态解 $(\sigma_*, p_*, \rho_*, \xi_* )$。 该稳态解的稳定性应取决于肿瘤侵略性常数 $\mu$。 期望扰动是平坦的也是不现实的。 We show that, under non-flat perturbations, there exists a threshold $\mu_*>0$ such that $(\sigma_*, p_*, \rho_*, \xi_*)$ is linearly stable if $\mu<\mu_*$ and linearly unstable if $\mu>\mu_*$. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications. 显示英文摘要

摘要： We study a free boundary problem modeling multi-layer tumor growth with a small time delay $\tau$, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper we establish the well-posedness of the problem, namely, first we prove that there exists a unique flat stationary solution $(\sigma_*, p_*, \rho_*, \xi_* )$ for all $\mu>0$. The stability of this stationary solution should depend on the tumor aggressiveness constant $\mu$. It is also unrealistic to expect the perturbation to be flat. We show that, under non-flat perturbations, there exists a threshold $\mu_*>0$ such that $(\sigma_*, p_*, \rho_*, \xi_*)$ is linearly stable if $\mu<\mu_*$ and linearly unstable if $\mu>\mu_*$. Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.