Study on differential equations arising out of life phenomena in vivo

(1)2007年9月，我们邀请英国邓迪大学的M·查普兰教授到日本访问一周。在此情况下，我们组织了国际会议“”数学模型和分析的生物模式的形成和相关的主题研讨会“(9月25日至26日，名古屋大学)(http://www.fujita-hu.ac.jp/~akikubo/).(2)我们调查了M.Chaplan和G.Lolas的肿瘤生长和侵袭的数学模型，2006年。主要研究人员和合作研究人员需要花费大量的时间来克服这一困难。(3)我们成功地获得了Othmer和Stevens模型与Anderson和Chaplan模型之间的数学上的精确关系。我们的结果使我们能够通过形式计算和代换将它们从一个连接到另一个。(4)我们试图改进Anderson和Chaplan在一系列肿瘤血管生成和肿瘤侵袭的数学模型中使用的数学方法，以研究Gerish和Chaplan(2007)最新的非局部肿瘤侵袭模型。2009(5)本项目的主要研究员和合作研究员在更一般的框架下研究了非线性发展方程的初始0-Neumann边值问题。(6)主要研究者和他的合作者将Levine和Sleeman(1998)所使用的方法应用于发展方程的Neumann初边值问题，并成功地证明了在Dirichlet边界条件下(见研究活动4)(7)基于(2)至(6)中的数学分析，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.