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α_(12)为零,0≤α_(21)<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)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抛物型方程组解的存在性理论,以及积分估计技术.如何建立解的积分估计是获得这个问题解的整体存在性的关键.(2)解的渐近性态.该模型有EP_1,EP_2,EP_3和EP_4四个稳态解,其中EP_1和EP_2两个平凡稳态解在任何情况下都不稳定.通过构造Lyapunov函数,我们证明了,在一定条件下EP_3全局渐近稳定,从而时变解在时间趋于无穷时将趋于EP_3,而在相反的条件下EP_4全局渐近稳定,从而时变解在时间趋于无穷时将趋于EP_4.

In this artical we study a mathematical model for cancer invasion proposed by Gatenby and Gaolinski . Tbis model is a strongly coupled degenerate reaction-diffusion system. Under the assumption that a21-0 and 0≤ α21 〈 1, we make rigorous mathematical analysis to t his 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 combined with application of H. Amann＇s exsitence thorem for general quasilinear parabolic equations and integral estimates. The key step is to establish integral estimates for solutions of this model. （2） Asymptotic behavior of solutions. This model has four stationary points EP1, EP2, EPa and EP4. The first two stationary points EPI and EP.2 are unstable in all situations. By precisely constructing Lyapunov fuetions, we prove that in certain case EP3 is globally asymptotically stable so that all solutions will converge to this stationary solution as time goes to infinity, and in the opposite ease EP4 is globally asymptotically stable so that all solutions will converge to it as time goes to infinity.