Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors
Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors
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DOI：
10.1090/s0002-9947-2013-05779-0
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发表时间：
2013-01
影响因子：
1.3
通讯作者：
Junde Wu;Fujun Zhou
中科院分区：
文献类型：
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作者：
Junde Wu;Fujun Zhou
在本文中，我们研究了一个自由边界问题模拟肿瘤生长与流体状组织的作用下的抑制剂。该模型包括两个椭圆方程描述的营养物和抑制剂的浓度，分别和一个斯托克斯方程的流体速度和内部压力。利用泛函方法、解析半群理论和崔氏不变性抛物型微分方程的局部相定理，证明了若径向定常解在径向扰动下渐近稳定，则存在一个非负阈值γ_∞，使得当γ > γ_∞时，该解在非径向扰动下渐近稳定.而如果0 < γ < γ *，则径向稳态解是不稳定的，特别是存在中心稳定的流形，使得如果瞬时解全局存在并且包含在径向稳态解的足够小的邻域中，那么它指数收敛于这个径向稳态解（模平移），并且它的平移位于中心稳定的流形上。结果表明一个有趣的现象，即增加抑制剂摄取对肿瘤的治疗具有积极作用，并且可以促进肿瘤的稳定性。
In this paper we study a free boundary problem modeling the growth of tumors with fluid-like tissue under the action of inhibitors. The model includes two elliptic equations describing the concentration of nutrients and inhibitors, respectively, and a Stokes equation for the fluid velocity and internal pressure. By employing the functional approach, analytic semigroup theory and Cui’s local phase theorem for parabolic differential equations with invariance, we prove that if a radial stationary solution is asymptotically stable under radial perturbations, then there exists a non-negative threshold value γ∗ such that if γ > γ∗, then it keeps asymptotically stable under non-radial perturbations. While if 0 < γ < γ∗, then the radial stationary solution is unstable and, in particular, there exists a center-stable manifold such that if the transient solution exists globally and is contained in a sufficiently small neighborhood of the radial stationary solution, then it converges exponentially to this radial stationary solution (modulo translations) and its translation lies on the center-stable manifold. The result indicates an interesting phenomenon that an increasing inhibitor uptake has a positive effect on the tumor’s treatment and can promote the tumor’s stability.