Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: An application to a model of tumor growth
Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: An application to a model of tumor growth
复制标题
DOI：
10.1512/iumj.2003.52.2317
复制
发表时间：
2003
影响因子：
1.1
通讯作者：
A. Friedman;B. V. Bazaliy
中科院分区：
文献类型：
--
作者：
A. Friedman;B. V. Bazaliy
考虑一类椭圆型方程Ap + μ（σ -？）= 0（p > 0）和一个关于σ的抛物方程。这个问题是由肿瘤生长的模型激发的，其中p表示增殖细胞的压力，并且β是营养物的浓度。在肿瘤区域的边界上，p等于表面张力，并且p的通量等于Γ（t）的法向速度。当p = 0时，系统对于p可归结为Hele-Shaw问题，对于σ可归结为标准抛物方程。对于Hele-Shaw问题，已知存在定常的径向对称解，并且每个解在以下意义下是渐近稳定的：如果我们取一个径向对称解的小扰动作为初始值，则相应的Hele-Shaw问题有唯一的整体解，并且当t → ∞时，其自由边界收敛到球面。在本文中，我们证明了一个类似的结果耦合椭圆抛物问题提供p是小的。当p不小时，渐近稳定性的结果一般是错误的.
We consider a free boundary problem for a coupled system consisting of an elliptic equation Ap + μ(σ -?) = 0 (p > 0) for p and a parabolic equation for σ. The problem is motivated by a model of tumor growth whereby p represents the pressure of the proliferating cells and (7 is the concentration of nutrients. On the boundary Γ(t) of the tumor region, p is equal to the surface tension, and the flux of p is equal to the normal velocity of Γ(t). In the case p = 0, the system decouples into a Hele-Shaw problem for p and a standard parabolic equation for σ. For the Hele-Shaw problem it is known that there are stationary radially symmetric solutions and each one is asymptotically stable in the following sense: If we take for initial data a small perturbation of a radially symmetric solution, then the corresponding Hele-Shaw problem has a unique global solution and its free boundary converges to a sphere as t → ∞. In this paper we prove a similar result for the coupled elliptic-parabolic problem provided p is small. The asymptotic stability result is generally false if p is not small.