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Autonomous Formation of Spatial Structures in Solutions of Parabolic Partial Differential Equations

抛物型偏微分方程解中空间结构的自主形成

基本信息

批准号：
13440050
负责人：
TAKAGI Izumi
金额：
$ 9.28万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

项目摘要

This objective of this project is to pursue the behavior of solutions of nonlinear partial differential equations of parabolic type.In collaboration with Wei-Ming Ni (University of Minnesota) and Kanako Suzuki (Tohoku University), Takagi studied the behavior of solutions of a reaction-diffusion system of activator-inhibitor type proposed by Gierer and Meinhardt and obtained the following results : (i)In the case where the initial data are constant functions, there exist solutions blowing up in finite time if the activator activates the its production stronger than that of the inhibitor. There are two types of blow-up solutions. Either (a) only the activator blows up or (b) both the activator and the inhibitor blow up. In the former case, we can choose the initial value so that the inhibitor converges to any specified positive number. (ii)In the case where the equation for the activator does not contain the source term, no solution blows up in finite time if the activator produces the inhibitor more than itself. Moreover, in this case the collapse of patterns can occur. Here, by the collapse of patterns we mean that the solution converges to the origin as the time variable tends to infinity.Nishiura studied scattering phenomena of pulse solutions and spot solutions. He found that various input/output relationships can be formed depending on the local dynamics in the neighborhood of unstable steady-states or periodic solutions and on the location of solution orbits.Yanagida considered a certain quasilinear parabolic equation and showed that the solution is either (a) globally increasing, (b) a traveling wave, or (c) extinct in finite time, depending on the initial data. The asymptotic behavior of the solution is also investigated.Kozono proved that in three dimensional exterior domains one can construct weak solutions of the Navier-Stokes equations which satisfy the strong energy inequality for all square-integrable initial data.

本计画的目标是探讨非线性抛物型偏微分方程解的性质。（明尼苏达大学）和Kanako Suzuki（东北大学），高木研究了由Gierer和Meinhardt提出的活化剂-抑制剂型反应扩散系统的解的行为，并获得了以下结果：（i）在初始数据为常数函数的情况下，如果激活剂的激活作用强于抑制剂，则存在解在有限时间内爆破。有两种类型的爆破解决方案。或者（a）仅激活剂爆破，或者（B）激活剂和抑制剂都爆破。在前一种情况下，我们可以选择初始值，使抑制收敛到任何指定的正数。（ii）在活化剂的方程不包含源项的情况下，如果活化剂产生的抑制剂多于其自身，则没有解在有限时间内爆破。此外，在这种情况下，模式的崩溃可能发生。在这里，我们的意思是崩溃的模式，解决方案收敛到原点的时间变量趋于无穷大。西浦研究了脉冲解决方案和斑点解决方案的散射现象。他发现，各种输入/输出关系可以形成取决于当地的动态在附近的不稳定的稳定状态或周期性的解决方案和对位置的解决方案orbers.Yanagida考虑了一定的准线性抛物方程，并表明，解决方案是（a）全球增加，（B）行波，或（c）灭绝在有限的时间，这取决于初始数据。Kozono证明了在三维外部区域上，对于所有平方可积的初值，可以构造出满足强能量不等式的Navier-Stokes方程的弱解。