Research on the existence and property of the solution for some partial differential equations appeared in the field of the life science and medicine

2004改进Levine和Sleeman用于Othmer和Stevens模型（1997）的方法，我们可以将Anderson-Chaplain（1998）的肿瘤血管生成的数学模型简化为一个方程。这一结果表明，尽管这些模型是独立提出的，但通过这些模型的约化方程的关系和相似性来讨论它们之间的等价性是可能的。事实上，我们可以证明Anderson-Chaplain模型（A-C）解的存在性，并通过类似的方法得到解的渐近分布。Kubo和Suzuki（2004）的Othmer-Stevens模型（O-S）。通过上述结果，我们可以研究这些模型共有的数学结构。引入O-S中的吸收情况，可以看出O-S和A-C中的吸收情况的简化方程属于同一类，O-S和A-C可以在这个意义上被识别。因此，我们找到了一个数学结构共同的，他们和有一个基本的数学结构的肿瘤血管生成在这一点上。2006我们研究上述论点更精确。事实上，我们可以直接将A-C修饰为O-S摄取的形式。因此，这些模型的共同数学结构和肿瘤血管生成的一些本质点可以在这种形式化的数学表达式中找到，这保证了我们可以从数学上研究它们之间的某些联系点。A-C和O-S分别产生于医学领域和强化随机游走理论，这一方面保证了我们可以用强化随机游走理论进行Anderson-Chaplain模型的数值计算，另一方面也保证了我们可以用强化随机游走理论进行Anderson-Chaplain模型的数值计算。

2004 Improving Levine and Sleeman's way used for Othmer and Stevens model(1997), we could reduce the mathematical model of tumour angiogenesis by Anderson-Chaplain(1998) to a single equation. The result implies some possibility to discuss the equivalence between these models by the relationship and similarity of the reduced equations of these models, though these models are proposed independently.2005 In fact, we can prove the existence of the solution of Anderson-Chaplain model (A-C) and obtain the asymptotic profile of the solution by the similar way used in. Kubo and Suzuki(2004) for Othmer-Stevens model(O-S). By the above result we may investigate the mathematical structure common to these models. Introducing the uptake case in O-S, it is seen that the reduced equations of the uptake case in O-S and A-C belong to the same class and O-S and A-C can be identified in such sense. Hence we find a mathematical structure common to them and there is an essential mathematical structure of tumour angiogenesis in this point.2006 We study the above argument more precisely. In fact, we could modify A-C to the form of the uptake in O-S directly. Therefore the mathematical structure common to these models and some essential point of tumour angiogenesis can be found in such formal mathematical expression, which assures us that we can study some contact point between them mathematically. That is, A-C can be studied from the standpoint of statistical mechanics and it also gives the medical understanding of O-S while.A-C and O-S arise from the field of medicine and the theory of reinforeced random walk respectively.On the other hand, it assures us that we can proceed the numerical computation of Anderson-Chaplain model based on the theory of reinforced random walk.