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

改进了Levine和Sleeman(1997)用于Othmer和Stevens模型的方法，将Anderson-Chaplan(1998)关于肿瘤血管生成的数学模型简化为一个方程。这一结果暗示了利用这些模型的约化方程的关系和相似性来讨论这些模型之间的等价性的可能性，尽管这些模型是独立提出的。实际上，我们可以用类似的方法证明Anderson-Chaplan模型(A-C)解的存在性，并得到解的渐近轮廓。Kubo和Suzuki(2004)的Othmer-Stevens模型(O-S)。根据上述结果，我们可以研究这些模型的共同数学结构。引入《O-S》中的摄取格，可以看出O-S和A-C的摄取格的约化方程属于同一类，O-S和A-C可以在这种意义上识别。因此，我们找到了它们共同的数学结构，在这一点上，存在着肿瘤血管生成的基本数学结构。2006我们更精确地研究了上述论点。事实上，我们可以直接将A-C修饰为O-S中的摄取形式。因此，在这种形式的数学表达式中可以找到这些模型的共同的数学结构和肿瘤血管生成的一些关键点，这保证了我们可以从数学上研究它们之间的一些接触点。也就是说，A-C可以从统计力学的角度来研究，它也给出了O-S的医学理解，而A-C和O-S分别产生于医学领域和增强随机游走理论，另一方面，它保证了我们可以基于增强随机游走理论来进行安德森-查普兰模型的数值计算。

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.