Study on differential equations arising out of life phenomena in vivo

我们邀请了英国邓迪大学的M.Chaplain教授于2007年9月到日本访问一周。借此机会，我们组织了国际会议“生物模式形成的数学建模和分析及相关主题研讨会”（25-26,9月，名古屋大学）（http://www.fujita-hu.ac.jp/~akikubo/）.(2)。我们研究了M.Chaplain和G.Lolas， 2006年肿瘤生长和侵袭的数学模型。主要研究者和共同研究者必须花费大量的时间来克服这个困难。(3)成功地获得了肿瘤血管生成的Othmer和Stevens模型与Anderson和Chaplain模型的数学精确关系。我们的结果允许我们仅通过形式计算和替换将它们彼此连接起来。(4)为了研究Gerish和Chaplain（2007）最新的非局部肿瘤侵袭模型，我们尝试改进了Anderson和Chaplain一系列肿瘤血管生成和肿瘤侵袭数学模型中的数学方法。2009(5)本项目主要研究了广义框架下非线性演化方程的初始0-Neumann边界问题。最后，为了完成这个项目，我们可以通过存在定理和非线性进化方程解的性质的框架，给出这个项目中研究的一系列肿瘤生长模型的特征（见研究活动1,2,3）。(6)主要研究者及其合作者应用了Levine和Sleeman（1998）在演化方程初始Neumann边值问题中的方法，并成功地证明了Dirichlet边界条件下爆炸解的存在性（见研究活动4）。(7)在(2)至(6)的数学分析基础上，我们对每个问题进行了计算机模拟。

(1) We invited Professor M.Chaplain from Dundee University, UK to Japan for a week, September, 2007. On this occasion we organized the international conference" "Workshop on Mathematical Modelling and Analysis of Biological Pattern Formations and the Related Topics" (25-26, September, Nagoya University) (http://www.fujita-hu.ac.jp/~akikubo/).(2) We investigated mathematical model of tumour growth and invasion by M.Chaplain and G.Lolas, 2006. The principal investigator and co-investigators have to spend much time to overcome this difficulty.(3) We succeeded to obtain the mathematically exact relation between Othmer and Stevens model and Anderson and Chaplain model of tumour induced angiogenesis. Our result allow us to connect them from one to another only by formal calculation and substitution.(4) We tried to improve our mathematical way used in a sequence of mathematical models of tumour angiogenesis and tumour invasion by Anderson and Chaplain in order to study the latest model of nonlocal tumour invasion by Gerish and Chaplain (2007).2009(5) The principal investigator and co-investigators of this project studied an initial 0-Neumann boundary problem of nonlinear evolution equations in more general frame work. Finally to complete touch to this project we could give a characterization of a sequence of tumour growth models studied in this project by this framework of existence theorem and properties of the solution of our nonlinear evolution equations(see research activities 1,2,3).(6) The main investigator and his collaborator applied the method used by Levine and Sleeman (1998) in initial Neumann boundary value problems of the evolution equation and succeeded to show the existence of blow up solution under Dirichlet boundary condition(see research activities 4)(7) Based on the mathematical analysis in (2)to (6), we made computer simulations for each problem.