一个肿瘤侵入模型的定性分析

Qualitative Analysis for a Mathematical Model of Cancer Invasion

本文研究由Gatenby和Gawlinski在CancerResearch(Vol.56,1996)上提出的一个肿瘤侵入模型。该模型是一个强耦合的退缩型反应扩散方程组.本文在为α12零,0≤α21<1的情况下,对该模型进行严格的数学分析。所获结果包括两个方面:(1)建立解的整体存在性。主要应用了逼近方法,H.Amann关于一般拟线性方程和这类方程与常微分方程耦合而成的广义的抛物型方程组的存在性理论,并且结合积分估计来证明解的存在性。如何建立解的积分估计是这个问题的关键所在。(2)研究解的渐近性态。通过构造Lyapunov函数,我们证明时变解在时间趋于无穷时将趋于稳态解。

In this article, we study a mathematical model for cancer invasion proposed by Gatenby and Gawlinski in ＂Cancer Research （Vol. 56, 1996）＂. The model is a strongly coupled degenerate reaction-diffusion system. Under the assumption that is zero and , we make rigorous mathematical analysis to this model, and obtain the following results ： （ 1 ） Global existence of solutions. We prove that this model has a unique global solution by using approximation method, H. Amann＇ s existence theorem for generalized quasilinear parabolic equations and integral estimate method. The key point is to establish integral estimate of solutions of this model. （2） Asymptotic behavior of solutions. By precisely constructing Lyapunov functions, we prove that all solutions will converge to a stationary solution as time goes to infinity.